ࡱ>  wCA Y G "bjbjَ l]$ Pp  t l8;W"WWWXvD JJR$?3vXXvWWl8WW  . HW L$[ #"Characterization of the integer-valued translation-invariant regular metrics on the discrete plane G. J. F. Banon National Institute for Space Research, INPE, Brazil Abstract We say that a metric space is regular if a straight-line (in the metric space sense) passing through the center of a sphere and any other point has at least two diametrically opposite points. Normed vector spaces have this property. Nevertheless, this property might not be satisfied in some metric spaces. In this work, we give a characterization of an integer-valued translation-invariant regular metric defined on the discrete plane, in terms of a symmetric subset B that induces through, what we call, the Minkowski product, a chain of subsets that are morphologically closed with respect to B. Keywords: Mathematical Morphology, symmetric subset, ball, regular metric space, integer-valued metric, translation-invariant metric, triangle inequality, Minkowski product, morphologial closed subset, discrete plane, computational geometry, discrete geometry, digital geometry. 1. Introduction The continuous plane or, more precisely, the two-dimensional Euclidean vector space, has good geometrical properties. For example, in such space, a closed ball is included in another one only if the radius of the latter is greater than or equal to the sum of the distance between their centers and the radius of the former. Furthermore, in such space, two closed balls intersect each other if the sum of their radii is greater than or equal to the distance between their centers. Nevertheless, not all metric spaces have these properties. In the first part of this work, we introduce the concept of regular metric space in which the above two geometrical properties are satisfied. We say that a metric space is regular if its metric satisfies three regularity axioms or equivalently if a straight-line (in the metric space sense) passing through the center of a sphere and any other point has at least two diametrically opposite points. Minkowski spaces (i.e., finite dimensional normed vector spaces) have this property. This regularity is generally lost when a metric on the continuous plane is restricted to the discrete plane, as it is the case of the Euclidean metric. In the second part of this work, we study the characterization of the integer-valued translation-invariant regular metrics on the discrete plane in terms of some appropriate symmetric subsets. We show that every such metric can be characterized in terms of a symmetric subset B that induces through, what we call, the Minkowski product, a chain of subsets that are morphologically closed with respect to B. Our characterization shows the unique way to construct integer-valued translation-invariant regular metrics on the discrete plane. This is an important issue in digital image analysis since the image domains are then discrete. In the sixties, Rosenfeld and Pfaltz  REF _Ref79383931 \h  \* MERGEFORMAT [8] have already introduced a metric property and have used it to describe algorithms for computing some distance functions by performing repeated local operations. It appears that their property is precisely a necessary condition for a metric to be regular. Actually, we came across the regularity property for a metric because we tried to find a class of metric whose metric dilations satisfy a semigroup relation (relation (9.19) of  REF _Ref41884991 \h [3]) and whose balls are morphologically closed with respect to the unit ball. This class contains, for example, the chessboard distance. In one dimension, we observed that the (discrete) convexity is not a necessary condition to have the morphological closure property, so it was useless to solve our problem. For the sake of simplicity of the presentation, in this work, we limit ourselves to the class of integer-valued metrics. This is not a serious limitation because on the discrete plane the metrics assume only a countable number of values. In Section 2, we give an axiomatic definition of regular metric space and we relate it to the Kiselman's properties of upper and lower regularity for the triangle inequality. In particular, we show that of the three axioms only two are sufficient to define the metric regularity. Independently of the definition of metric, in Section 3, we give a definition of ball based on the notions of set translation and set transposition. We introduce the Minkowski product in Section 4, and use it in Section 5 to define the notions of generated balls and radius of a ball. In Section 6, we study the properties of the balls of a regular metric space. Conversely, in Section 7, we study the properties of the metric spaces constructed from symmetric balls having a morphological closure property. Finally, in Section 8, we show the existence of a bijection between the set of integer-valued translation-invariant regular metrics defined on the discrete plane and the set of symmetric balls that satisfy the morphological closure property. 2. Regular metric space We first recall the definitions of distance, metric and metric space. Definition  SEQ Definition \* ARABIC 1 (metric space) Let E be a nonempty set. A distance d on E is a mapping from E ( E to R (the set of real numbers) satisfying, for any x and y ( E: d(x, y) ( 0, (positiveness) d(x, y) ( 0 ( x ( y, d(x, y) ( d(y, x). (symmetry) Furthermore, a distance d on E is a metric if in addtion it satisfies, for any x, y and z ( E: (iv) d(x, y) ( d(x, z) + d(z, y). (triangle inequality) A metric space (E, d) is a set E provided with a metric d on E. ( The mapping d0 from E ( E to R defined by, for any x and y ( E:  EMBED Equation.3 , is an example of metric, it is called the discrete metric. In a metric space we can define the concepts of straight-line and sphere. Furthermore, based on these concepts, we can define what we call a regular metric and a regular metric space. Let (E, d) be a metric space. For any x and y ( E, and any i ( d({x} ( E) (i.e., the image of {x} ( E through d), let us define the following subsets of E: L1(x, y) ( {z ( E: d(x, z) ( d(x, y) + d(y, z)}, L2(x, y) ( {z ( E: d(x, y) ( d(x, z) + d(z, y)}, L3(x, y) ( {z ( E: d(z, y) ( d(z, x) + d(x, y)}, and S(x, i) ( {z ( E: d(x, z) ( i}. The subsets L(x, y) ( L1(x, y) ( L2(x, y) ( L3(x, y) and S(x, i) are, respectively, the straight-line passing through the points x and y, and the sphere with center at x and of radius i.  REF _Ref79397486 \h  \* MERGEFORMAT Figure 1 and  REF _Ref79395360 \h  \* MERGEFORMAT Figure 2 in this section show examples of spheres and straight-lines. We must be aware that the above definition of straight-line is based on the concept of metric and the resulting object is generally different from the usual straight-line defined in the framework of linear vector space. Because of the symmetry property of the distances, the straight-lines have some kind of symmetry as well. Proposition  SEQ Definition \* ARABIC 2 (straight-line symmetry) Let (E, d) be a metric space. For any x and y ( E, L1(x, y) ( L3(y, x) L2(x, y) ( L2(y, x) L(x, y) ( L(y, x). ( Proof (a) Lut us prove (i). For any x, y and z ( E, z ( L1(x, y) ( d(x, z) ( d(x, y) + d(y, z) (definition of L1) ( d(z, x) ( d(y, x) + d(z, y) (symmetry of d) ( d(z, x) ( d(z, y) + d(y, x) (commutativity of +) ( z ( L3(y, x), (definition of L3) that is, L1(x, y) ( L3(y, x). (b) Let us prove (ii). For any x, y and z ( E, z ( L2(x, y) ( d(x, y) ( d(x, z) + d(z, y) (definition of L2) ( d(y, x) ( d(z, x) + d(y, z) (symmetry of d) ( d(y, x) ( d(y, z) + d(z, x) (commutativity of +) ( z ( L2(y, x), (definition of L2) that is, L2(x, y) ( L2(y, x). (c) Let us prove (iii). For any x and y ( E, L(x, y) ( L1(x, y) ( L2(x, y) ( L3(x, y) (definition of L) ( L3(y, x) ( L2(y, x) ( L1(y, x) (from (i) and (ii)) ( L1(y, x) ( L2(y, x) ( L3(y, x) (commutativity of () ( L(y, x), (definition of L) that is, L(x, y) ( L(y, x). ( We first define the regular metric space from three axioms. Definition  SEQ Definition \* ARABIC 3 (regular metric space) Let (E, d) be a metric space. The metric d on E is lower regular of type 1 if S(x, i) ( L1(x, y) ( (, for any x and y ( E, and any i ( d({x} ( E), such that d(x, y) ( i; lower regular of type 2 if S(x, i) ( L2(x, y) ( (, for any x and y ( E, and any i ( d({x} ( E), such that i ( d(x, y); upper regular if S(x, i) ( L3(x, y) ( (, for any x and y ( E, and any i ( d({x} ( E); regular if it is lower regular (of type 1 and 2) and upper regular. A metric space (E, d) is regular if its metric is regular. ( Actually, the three regularity axioms are not independent each other as we show in the next proposition. Proposition  SEQ Definition \* ARABIC 4 (axiom dependence) Let (E, d) be a metric space. The metric d on E is lower regular of type 1 if and only if (iff) it is upper regular. ( Proof (a) Let us prove that the lower regularity of type 1 implies the upper regularity. For any x and y ( E, and any i ( d({x} ( E), (d is lower regular of type 1) ( S(y, i + d(x, y)) ( L1(y, x) ( ( (definition of lower regularity of type 1) ( S(y, i + d(x, y)) ( L3(x, y) ( ( (Property (i) of  REF _Ref78357666 \h Proposition 2 - straight-line symmetry) ( ( z ( E: d(y, z) ( i + d(x, y) and d(z, y) ( d(z, x) + d(x, y) (definitions of S and L3) ( ( z ( E: d(z, y) ( i + d(x, y) and d(z, y) ( d(z, x) + d(x, y) (symmetry of d) ( ( z ( E: d(z, x) ( i and d(z, y) ( d(z, x) + d(x, y) (transitivity of ( and regularity of +) ( ( z ( E: d(x, z) ( i and d(z, y) ( d(z, x) + d(x, y) (symmetry of d) ( S(x, i) ( L3(x, y) ( (, (definitions of S and L3) that is, d is upper regular. (b) Let us prove that the upper regularity implies the lower regularity of type 1. For any x and y ( E, and any i ( d({x} ( E) such that d(x, y) ( i, (d is upper regular) ( S(y, i d(x, y)) ( L3(y, x) ( ( (definition of upper regularity) ( S(y, i d(x, y)) ( L1(x, y) ( ( (Property (ii) of  REF _Ref78357666 \h Proposition 2) (( z ( E: d(y, z) ( i d(x, y) and d(x, z) ( d(x, y) + d(y, z) (definitions of S and L1) ( ( z ( E: d(x, y) + d(y, z) ( i and d(x, z) ( d(x, y) + d(y, z) (+ versus ) ( ( z ( E: d(x, z) ( i and d(x, z) ( d(x, y) + d(y, z) (transitivity of () ( S(x, i) ( L1(x, y) ( (, (definitions of S and L1) that is, d is lower regular of type 1. ( The axiom dependence allows us to make an equivalent definition of regular metric, but simpler with only two axioms, the lower regularity of type 2 being called simply lower regularity. Corollary  SEQ Definition \* ARABIC 5 (first equivalent definition of regular metric) Let (E, d) be a metric space. The metric d on E is (i) lower regular if S(x, i) ( L2(x, y) ( (, for any x and y ( E, and any i ( d({x} ( E), such that i ( d(x, y); regular iff it is lower and upper regular. ( Proof (a) If d is regular in the sense of  REF _Ref41052844 \h Definition 3, then d is regular in the sense of the corollary statement since lower regularity means lower regularity of type 2. (b) Conversely, if d is regular in the sense of the corollary statement, then d is lower regular of type 2 and upper regular, therefore by  REF _Ref78511934 \h Proposition 4 (axiom dependence) it is also lower regular of type 1, that is, it is regular in the sense of  REF _Ref41052844 \h Definition 3. ( In order to prove another equivalent definition of regular metric, we need the following lemma. Lemma  SEQ Definition \* ARABIC 6 (straight-line and sphere intersection properties) Let d be a metric on a set E. For any x and y ( E, and any i ( d({x} ( E), (i) S(x, i) ( L1(x, y) ( ( ( d(x, y) ( i; (ii) S(x, i) ( L2(x, y) ( ( ( i ( d(x, y). ( Proof (a) Let us prove (i). For any x and y ( E, and any i ( d({x} ( E), S(x, i) ( L1(x, y) ( ( ( ( u ( E: d(x, u) ( d(x, y) + d(y, u) and d(x, u) ( i (definitions of S and L1) ( ( u ( E: i ( d(x, y) + d(y, u) (substitution) ( d(x, y) ( i. ( REF _Ref69010114 \h Definition 1 - positiveness of d) (b) Let us prove (ii). For any x and y ( E, and any i ( d({x} ( E), S(x, i) ( L2(x, y) ( ( ( ( u ( E: d(x, y) ( d(x, u) + d(u, y) and d(x, u) ( i (definitions of S and L2) ( ( u ( E: d(x, y) ( i + d(u, y) (substitution) ( i ( d(x, y). ( REF _Ref69010114 \h Definition 1 - positiveness of d) ( The next proposition allows a geometrical interpretation for the regular metrics. Proposition  SEQ Definition \* ARABIC 7 (second equivalent definition of regular metric) A metric d on E is regular iff for any x and y ( E, and any i ( d({x} ( E), the intersection between the straight-line L(x, y) and the sphere S(x, i) have at least two diametrically opposite points in the sense that it exists u and v ( S(x, i) such that u ( (L1(x, y) ( L2(x, y)) and v ( L3(x, y). ( Proof (a) Let us assume that d is a regular metric on E. Since, for any x and y ( E, and any i ( d({x} ( E), at least one of the two conditions d(x, y) ( i and i ( d(x, y) is satisfied, by  REF _Ref41052844 \h Definition 3 (regular metric space), at least one of the two properties L1(x, y) ( S(x, i) ( ( and L2(x, y) ( S(x, i) ( ( is satisfied, in other words, we have (L1(x, y) ( L2(x, y)) ( S(x, i) ( (. That is, ( u ( S(x, i): u ( (L1(x, y) ( L2(x, y)). Furthemore, by  REF _Ref41052844 \h Definition 3 again, for any x and y ( E, and any i ( d({x} ( E), L3(x, y) ( S(x, i) ( (. That is, ( v ( S(x, i): v ( L3(x, y). (b) Conversely, let us assume that for any x and y ( E, and any i ( d({x} ( E), ( u and v ( S(x, i): u ( (L1(x, y) ( L2(x, y)) and v ( L3(x, y). (b1) If i < d(x, y), true ( S(x, i) ( (L1(x, y) ( L2(x, y)) ( ( and i < d(x, y) (hypotheses) ( S(x, i) ( (L1(x, y) ( L2(x, y)) ( ( and S(x, i) ( L1(x, y) ( ( (Property (i) of  REF _Ref73272841 \h Lemma 6) ( (S(x, i) ( L1(x, y)) ( (S(x, i) ( L2(x, y)) ( ( and S(x, i) ( L1(x, y) ( ( (distributivity of ( over () ( S(x, i) ( L2(x, y) ( ( (( is unity for () ( d is lower regular. ( REF _Ref78512758 \h Corollary 5 - first equivalent definition of regular metric) If d(x, y) ( i, true ( ( u ( E: d(x, y) ( d(x, u) + d(u, y) and d(x, u) ( i (namely u ( y and Property (ii) of  REF _Ref69010114 \h Definition 1) ( S(x, i) ( L2(x, y) ( ( (definitions of S and L2) ( d is lower regular. ( REF _Ref78512758 \h Corollary 5) (b2) true ( S(x, i) ( L3(x, y) ( ( (hypothesis) ( d is upper regular. ( REF _Ref78512758 \h Corollary 5) That is, by  REF _Ref78512758 \h Corollary 5, d is regular. ( The Euclidean distance d on R2, given by, for any x ( (x1, x2) and any y ( (y1, y2) ( Z2, d(x, y) (  EMBED Equation.3 , is regular, nevertheless, when restricted to the discrete plane Z2 it is not regular. For example, the straight-line L(x, y) containing x ( (0, 0) and y ( (1, 1) consists of the points (i, i) with i ( Z, and the sphere S(x, 1) of radius 1 with center at x consists of the points (0, 1), (1, 0), (0, (1) and ((1, 0). We observe that this line and this sphere have no intersection. The left-hand side of  REF _Ref79397486 \h  \* MERGEFORMAT Figure 1 illustrates this point. Another example of non-regular metric in the discrete plane can be built from the elliptic distance d given by, for any x ( (x1, x2) and any y ( (y1, y2) ( Z2, d(x, y) (  EMBED Equation.3 , with a ( 5/3 and b ( 5/4. For example, the straight-line L(x, y) containing x ( (0, 0) and y ( (2, 1) consists of the points (2i, i) with i ( Z, and the sphere S(x, 1) of radius 1 with center at x consists of the points (1, 1), (1, ( 1), (( 1, (1) and ((1, 1). Again, both have no intersection. The right-hand side of  REF _Ref79397486 \h  \* MERGEFORMAT Figure 1 illustrates this point.  Figure  SEQ Figure \* ARABIC 1 - Examples of non-regular metrics The discrete metric is not regular despite the fact that for any x and y ( E, x ( y and any i ( d0({x} ( E) ( {0, 1}, S(x, i) ( L(x, y) ( ( (since L(x, y) ( {x, y}, S(x, 0) ( {x} and S(x, 1) ( {y}). For the discrete metric, we cannot find two diametrically opposite points since L3(x, y) ( {x} and x ( S(x, 1). From now on, we restrict ourself to translation-invariant metrics on an Abelian group. Such metric property suits most image analysis problems. Definition  SEQ Definition \* ARABIC 8 (translation-invariant metric sapce) A distance or metric d on an Abelian group (E, +) is translation-invariant (t.i.) if, for any u, x and y ( E: d(x + u, y + u) ( d(x, y). A translation-invariant metric space (E, d) is a set E provided with a t.i. metric d on E. ( As it is well known, every translation-invariant metric on E can be characterized in terms of a mapping from E to R as stated in the next proposition. Let (E, +, o) be an Abelian group where o is the unit element of +. The subtraction on E, denoted , is the mapping E ( E ( (x, y)  EMBED Equation.3 x + ( y) ( E, where y is the inverse of y. For any mapping d from E ( E to R and any x ( E, let fd (x) ( d(x, o). For any mapping f from E to R and any x and y ( E let df(x, y) ( f(x y). Proposition  SEQ Definition \* ARABIC 9 (characterization of translation-invariant metric) The mapping d  EMBED Equation.3 fd from the set of translation-invariant distances on an Abelian group (E, +, o) to the set of mappings f from E to R satisfying the properties, for any x ( E: f(x) ( 0, (positiveness) f(o) ( 0,  EMBED Equation.3 , (symmetry) is a bijection and its inverse is f  EMBED Equation.3 df. Furthermore, d  EMBED Equation.3 fd is, as well, a bijection from the set of translation-invariant metrics on the Abelian group (E, +, o) to the set of mappings f from E to R satisfying the properties (i) - (iii) and the property, for any x and y ( E: (iv) f(x + y) ( f(x) + f(y) (subadditivity). ( Proof Let us divide the proof into four parts. Let d be a mapping from E ( E to R. (a1) Let us assume that d satisfies Property (i) of  REF _Ref69010114 \h Definition 1. For any x ( E, fd(x) ( d(x, o) (definition of fd) ( 0, (hypothesis) that is fd satisfies Property (i). (a2) Let us assume that d satisfies Property (ii) of  REF _Ref69010114 \h Definition 1. fd(o) ( d(o, o) (definition of fd) ( 0, (hypothesis) that is fd satisfies Property (ii). (a3) Let us assume that d satisfies Property (iii) of  REF _Ref69010114 \h Definition 1. For any x ( E,  EMBED Equation.3  (definition of fd)  EMBED Equation.3  (d is t.i.)  EMBED Equation.3  (hypothesis)  EMBED Equation.3 , (definition of fd) that is fd satisfies Property (iii). (a4) Let us assume that d satisfies Property (iv) of  REF _Ref69010114 \h Definition 1. For any x and y ( E, fd(x + y) ( d(x + y, o) (definition of fd) ( d(x + y, y) + d(y, o) (hypothesis) ( d(x, o) + d(y, o) (d is t.i.) ( fd(x) + fd(y), (definition of fd) that is fd satisfies Property (iv). In other words, from (a1) - (a3), if d is a t.i. distance then fd satisfies Properties (i) - (iii), and, from (a1) - (a4), if d is a t.i. metric then fd satisfies Properties (i) - (iv). (b) Let f be a mapping from E to R. (b1) Let us assume that f satisfies Property (i). For any x and y ( E, df(x, y) ( f(x y) (definition of df) ( 0, (hypothesis) that is df satisfies Property (i) of  REF _Ref69010114 \h Definition 1. (b2) Let us assume that f satisfies Property (ii). For any x ( E, df(x, x) ( f(x x) (definition of df) ( f(o) (group porperty) ( 0, (hypothesis) that is df satisfies Property (ii) of  REF _Ref69010114 \h Definition 1. (b3) Let us assume that f satisfies Property (iii). For any x and y ( E, df(x, y) ( f(x y) (definition of df) ( f( (x y)) (hypothesis) ( f(y x) (group property) ( df(y, x) (definition of df) that is df satisfies Property (iii) of  REF _Ref69010114 \h Definition 1. (b4) Let us assume that f satisfies Property (iv). For any x, y and z ( E, df(x, y) ( f(x y) (definition of df) ( f((x z) + (z y)) (group property) ( f(x z) + f(z y) (hypothesis) ( df(x, z) + df(z, y), (definition of df) that is df satisfies Property (iv) of  REF _Ref69010114 \h Definition 1. In other words, from (b1) - (b3), if f satisfies Properties (i) - (iii) then df is a t.i. distance and, from (b1) - (b4), if f satisfies Properties (i) - (iv) then df is a t.i. metric. (c) Let d be a t.i. distance on an Abelian group (E, +, o). For any x and y ( E,  EMBED Equation.3  (definition of df) ( d(x y, o) (definition of fd) ( d(x, y), (d is t.i.) that is, f  EMBED Equation.3 df is the left inverse of d  EMBED Equation.3 fd. (d) Let f be a mapping from E to R. For any x ( E,  EMBED Equation.3  (definition of fd) ( f(x o) (definition of df) ( f(x), (group properties) that is, f  EMBED Equation.3 df is the right inverse of d  EMBED Equation.3 fd. In other words, from (a) - (d), both mappings d  EMBED Equation.3 fd from the set of distances and the set of metrics are bijections. ( In the case of t.i. metrics, we can make an explicit relationship between the above concept of regular metric and the concepts of upper and lower regularity proposed by Kiselman  REF _Ref69021347 \h [5] in order to compare balls with different centers. Let us recall the Kiselman's definitions. Definition  SEQ Definition \* ARABIC 10 (first definition of Kilselman's regularity for a metric) Let (E, +, o) be an Abelian Group and let f a mapping from E to R satisfying Properties (i) - (iv) of  REF _Ref69177667 \h  \* MERGEFORMAT Proposition 9 (characterization of t.i. metric), then the translation-invariant metric df on (E, +, o) is lower regular (of type 1) for the triangle inequality if, for any x and y ( E such that f(y) ( f(x), there exists a point  EMBED Equation.3 ( E such that  EMBED Equation.3  and  EMBED Equation.3 ; lower regular (of type 2) for the triangle inequality if, for any x and y ( E such that f(x) ( f(y), there exists a point  EMBED Equation.3 ( E such that  EMBED Equation.3  and  EMBED Equation.3 ; upper regular for the triangle inequality if, for any x and y ( E, there exists a point  EMBED Equation.3 ( E such that  EMBED Equation.3  and  EMBED Equation.3 . ( In  REF _Ref69021347 \h [5], Kiselman has defined Axioms (i) and (iii). For the sake of compleness, we have added here Axiom (ii) and the expressions "type 1" and "type 2". Actually, we can rewrite  REF _Ref69638550 \h Definition 10 as shown below. Definition  SEQ Definition \* ARABIC 11 (second definition of Kilseman's regularity for a metric) A translation-invariant metric d on an Abelian group (E, +, o) is lower regular (of type 1) for the triangle inequality if, for any x and y ( E such that fd(y) ( fd(x), there exists  EMBED Equation.3 ( E such that  EMBED Equation.3  and  EMBED Equation.3 ; lower regular (of type 2) for the triangle inequality if, for any x and y ( E such that fd(x) ( fd(y), there exists  EMBED Equation.3 ( E such that  EMBED Equation.3  and  EMBED Equation.3 ; upper regular for the triangle inequality if, for any x and y ( E, there exists  EMBED Equation.3 ( E such that  EMBED Equation.3  and  EMBED Equation.3 . ( Because of  REF _Ref69177667 \h Proposition 9 (characterization of t.i. metric), both definitions correspond to exactly the same class of metrics. Proposition  SEQ Definition \* ARABIC 12 (definition equivalence) Let A be the subset of mappings f from E to R satisfying Properties (i) - (iv) of  REF _Ref69177667 \h  \* MERGEFORMAT Proposition 9 (characterization of t.i. metric) and let B be the subset of mappings f in A satisfying Conditions (i) - (iii) of  REF _Ref69638550 \h  \* MERGEFORMAT Definition 10, then the set (f  EMBED Equation.3 df)(B) (the image of B through the inverse of d  EMBED Equation.3 fd) is equal to the set  EMBED Equation.3 (the inverse image of B through d  EMBED Equation.3 fd). ( Proof By  REF _Ref69177667 \h  \* MERGEFORMAT Proposition 9 (characterization of t.i. metric), f  EMBED Equation.3 df is a left and right inverse for the mapping d  EMBED Equation.3 fd from the set of translation-invariant metrics to the set A, therefore, for any subset X of A, (f  EMBED Equation.3 df)(X) ( EMBED Equation.3 . So, the equality is also true for B. ( By using the second definition of Kiselman's regularity for a metric, we show now that the t.i. regular metrics satisfy the Kiselman's regularity axioms and conversely. Proposition  SEQ Definition \* ARABIC 13 (equivalent definition of translation-invariant regular metric) Let d be a translation-invariant metric on an Abelian group (E, +, o), then, (i) d is lower regular of type 1 in the sense of  REF _Ref41052844 \h  \* MERGEFORMAT Definition 3 (regular metric space) iff d is lower regular of type 1 for the triangle inequality; (ii) d is lower regular of type 2 in the sense of  REF _Ref41052844 \h  \* MERGEFORMAT Definition 3 iff d is lower regular of type 2 for the triangle inequality; (iii) d is upper regular in the sense of  REF _Ref41052844 \h  \* MERGEFORMAT Definition 3 iff d is upper regular for the triangle inequality; (iv) d is regular iff d is lower regular of type 2 and upper regular for the triangle inequality. ( Proof Let us divide the proof into three parts. (a) Let us prove (i). (a1) Let x and y ( E, then fd(x) and fd(y) ( d({o} ( E). Let us assume that fd(y) ( fd(x), (d is lower regular of type 1 in the sense of  REF _Ref41052844 \h Definition 3) ( S(o, fd(x)) ( L1(o, y) ( ( and d(o, y) ( fd(x) ( REF _Ref41052844 \h Definition 3 - regular metric space) ( (  EMBED Equation.3 ( E: d(o,  EMBED Equation.3 ) ( d(o, y) + d(y,  EMBED Equation.3 ) and d(o,  EMBED Equation.3 ) ( fd(x) (definitions of S and L1) ( (  EMBED Equation.3 ( E: d( EMBED Equation.3 , o) ( d(y, o) + d( EMBED Equation.3 , y) and d( EMBED Equation.3 , o) ( fd(x) ( REF _Ref69010114 \h Definition 1 - symmetry of d) ( (  EMBED Equation.3 ( E: d( EMBED Equation.3 , o) ( d(y, o) + d( EMBED Equation.3  y, o) and d( EMBED Equation.3 , o) ( fd(x) (d is t.i. and group properties) ( (  EMBED Equation.3  ( E: fd( EMBED Equation.3 ) ( fd(y) + fd( EMBED Equation.3  y) and fd( EMBED Equation.3 ) ( fd(x) (definition of fd) ( (d is lower regular of type 1 for the triangle inequality). ( REF _Ref73330872 \h Definition 11) (a2) Conversely, let x and y ( E, and i ( d({x} ( E) such that d(x, y) ( i, then fd(y x) ( i (since d is t.i. and symmetric); furthermore, (d is lower regular of type 1 for the triangle inequality) ( (  EMBED Equation.3 ( E: fd( EMBED Equation.3 ) ( fd(y x) + fd( EMBED Equation.3  (y x)) and fd( EMBED Equation.3 ) ( i ( REF _Ref73330872 \h Definition 11) ( (  EMBED Equation.3 ( E: d( EMBED Equation.3 , o) ( d(y x, o) + d( EMBED Equation.3  (y x), o) and d( EMBED Equation.3 , o) ( i (definition of fd) ( (  EMBED Equation.3 ( E: d( EMBED Equation.3 , o) ( d(y x, o) + d( EMBED Equation.3 + x y, o) and d( EMBED Equation.3 , o) ( i (group properties) ( (  EMBED Equation.3 ( E: d( EMBED Equation.3 + x, x) ( d(y, x) + d( EMBED Equation.3 + x, y) and d( EMBED Equation.3 + x, x) ( i (d is t.i. and group properties) ( ( u ( E: d(u, x) ( d(y, x) + d(u, y) and d(u, x) ( i (namely u ( EMBED Equation.3 + x) ( ( u ( E: d(x, u) ( d(x, y) + d(y, u) and d(x, u) ( i (d is symmetric) ( S(x, i) ( L1(x, y) ( ( (definitions of S and L1) ( (d is lower regular of type 1 in the sense of  REF _Ref41052844 \h Definition 3). ( REF _Ref41052844 \h Definition 3) (b) Let us prove (ii). (b1) Let x and y ( E, then fd(x) and fd(y) ( d({o} ( E). Let us assume that such that fd(x) ( fd(y), (d is lower regular of type 2 in the sense of  REF _Ref41052844 \h Definition 3) ( S(o, fd(x)) ( L2(o, y) ( ( and fd(x) ( d(o, y) ( REF _Ref41052844 \h Definition 3) ( (  EMBED Equation.3 ( E: d(o, y) ( d(o,  EMBED Equation.3 ) + d( EMBED Equation.3 , y) and d(o,  EMBED Equation.3 ) ( fd(x) (definitions of S and L2) ( (  EMBED Equation.3 ( E: d(o, y) ( d(o,  EMBED Equation.3 ) + d(o, y  EMBED Equation.3 ) and d(o,  EMBED Equation.3 ) ( fd(x) (d is t.i. and group properties) ( (  EMBED Equation.3 ( E: d(y, o) ( d( EMBED Equation.3 , o) + d(y  EMBED Equation.3 , o) and d( EMBED Equation.3 , o) ( fd(x) ( REF _Ref69010114 \h Definition 1 - symmetry of d) ( (  EMBED Equation.3  ( E: fd(y) ( fd( EMBED Equation.3 ) + fd(y  EMBED Equation.3 ) and fd( EMBED Equation.3 ) ( fd(x) (definition of fd) ( (d is lower regular of type 2 for the triangle inequality). ( REF _Ref73330872 \h Definition 11) (b2) Conversely, let x and y ( E, and i ( d({x} ( E) such that i ( d(x, y), then i ( fd(y x) (since d is t.i. and symmetric); furthermore, (d is lower regular of type 2 for the triangle inequality) ( (  EMBED Equation.3 ( E: fd(y x) ( fd( EMBED Equation.3 ) + fd ((y x)  EMBED Equation.3 ) and fd( EMBED Equation.3 ) ( i ( REF _Ref73330872 \h Definition 11) ( (  EMBED Equation.3 ( E: d(y x, o) ( d( EMBED Equation.3 , o) + d((y x)  EMBED Equation.3 , o) and d( EMBED Equation.3 , o) ( i (definition of fd) ( (  EMBED Equation.3 ( E: d(y x, o) ( d( EMBED Equation.3 , o) + d(y ( EMBED Equation.3 + x), o) and d( EMBED Equation.3 , o) ( i (group properties) ( (  EMBED Equation.3 ( E: d(y, x) ( d( EMBED Equation.3 + x, x) + d(y, EMBED Equation.3 + x) and d( EMBED Equation.3 + x, x) ( i (d is t.i. and group properties) ( ( u ( E: d(y, x) ( d(u, x) + d(y, u) and d(u, x) ( i (namely u ( EMBED Equation.3 + x) ( ( u ( E: d(x, y) ( d(x, u) + d(u, y) and d(x, u) ( i (d is symmetric) ( S(x, i) ( L2(x, y) ( ( (definitions of S and L2) ( (d is lower regular of type 2 in the sense of  REF _Ref41052844 \h Definition 3). ( REF _Ref41052844 \h Definition 3) (c) Let us prove (iii). (c1) Let x and y ( E, then fd(x) and fd(y) ( d({o} ( E); furthermore, (d is upper regular in the sense of  REF _Ref41052844 \h Definition 3) ( S(o, fd(y)) ( L3(o, x) ( ( ( REF _Ref41052844 \h Definition 3) ( (  EMBED Equation.3 ( E: d( EMBED Equation.3 , x) ( d( EMBED Equation.3 , o) + d(o, x) and d(o,  EMBED Equation.3 ) ( fd(y) (definitions of S and L3) ( (  EMBED Equation.3 ( E: d(o, x + EMBED Equation.3 ) ( d(o,  EMBED Equation.3 ) + d(o, x) and d(o,  EMBED Equation.3 ) ( fd(y) (d is t.i. and group properties) ( (  EMBED Equation.3 ( E: d(x + EMBED Equation.3 , o) ( d( EMBED Equation.3 , o) + d(x, o) and d( EMBED Equation.3 , o) ( fd(y) ( REF _Ref69010114 \h Definition 1 - symmetry of d) ( (  EMBED Equation.3  ( E: fd(x + EMBED Equation.3 ) ( fd(x) + fd( EMBED Equation.3 )and fd( EMBED Equation.3 ) ( fd(y) (definition of fd) ( (d is upper regular for the triangle inequality). ( REF _Ref73330872 \h Definition 11) (c2) Conversely, let x and y ( E, and i ( d({x} ( E), then, (d is upper regular for the triangle inequality) ( (  EMBED Equation.3 ( E: fd((y x) +  EMBED Equation.3 ) ( fd ( EMBED Equation.3 ) + fd(y x) and fd( EMBED Equation.3 ) ( i ( REF _Ref73330872 \h Definition 11) ( (  EMBED Equation.3 ( E: d((y x) +  EMBED Equation.3 , o) ( d( EMBED Equation.3 , o) + d(y x, o) and d( EMBED Equation.3 , o) ( i (definition of fd) ( (  EMBED Equation.3 ( E: d(y (x  EMBED Equation.3 ), o) ( d( EMBED Equation.3 , o) + d(y x, o) and d( EMBED Equation.3 , o) ( i (group properties) ( (  EMBED Equation.3 ( E: d(y, x  EMBED Equation.3 ) ( d(x, x  EMBED Equation.3 ) + d(y, x) and d(x, x  EMBED Equation.3 ) ( i (d is t.i. and group properties) ( ( v ( E: d(y, v) ( d(x, v) + d(y, x) and d(x, v) ( i (namely v ( x  EMBED Equation.3 ) ( ( v ( E: d(v, y) ( d(v, x) + d(x, y) and d(x, v) ( i (d is symmetric) ( S(x, i) ( L3(x, y) ( ( (definitions of S and L3) ( (d is upper regular in the sense of  REF _Ref41052844 \h Definition 3). ( REF _Ref41052844 \h Definition 3) (d) Property (iv) is a consequence of  REF _Ref78512758 \h Corollary 5 (first equivalent defintion of regular metric) and Properties (ii) - (iii). ( Actually, we have reused in this work the word "regular" of the Kiselman's definitions for the metric satisfying  REF _Ref41052844 \h Definition 3 because of the previous proposition. As a consequence of Property (iv), we observe that the important Kiselman's axioms are the lower regularity of type 2 and the upper regularity for the triangle inequality and not the lower regularity of type 1 for the triangle inequality.  REF _Ref73414444 \h Proposition 13 may be a short cut for the proof that some t.i. metrics are regular as we show now for the chessboard distance which is of particular interest in image processing. The set Z2 equipped with the addition ((x1, x2), (y1, y2))  EMBED Equation.3  (x1 + y1, x2 + y2), denoted +, forms an Abelian group (Z2, +, o) with o ( (0, 0) as unitary element. Definition  SEQ Definition \* ARABIC 14 (chessboard distance) The chessboard distance d8(x, y) between x ( (x1, x2) and y ( (y1, y2) in Z2 is the natural number defined by d8(x, y) ( max{| x1 y1|, | x2 y2|}. The mapping d8: (x, y)  EMBED Equation.3  d8(x, y) from Z2 ( Z2 to R, is called the chessboard distance. ( Actually, the chessboard distance d8 is a metric. Furthermore, with respect to the Abelian group (Z2, +, o), d8 is by construction tanslation-invariant. Let f8 ( EMBED Equation.3 , and let z ( (z1, z2) be a point in Z2 then, f8(z) ( d8(z, o) (definition of fd) ( max{| z1 0 |, | z2 0 |} ( REF _Ref69099153 \h Definition 14- def. of d8) ( max{| z1 |, | z2 |}. (properties of + on Z) In other words, f8 is given by, for any z ( (z1, z2) ( Z2, f8(z) ( max{| z1 |, | z2 |}. Let us show that d8 is regular. Proposition  SEQ Definition \* ARABIC 15 (example of regular metric space) The metric space (Z2, d8) is regular. ( Proof (a) Let us show that d8 is lower regular (of type 2) for the triangle inequality. Let x ( (x1, x2) and y ( (y1, y2) ( Z2 such that f8(x) ( f8(y), let i ( f8(x) and let  EMBED Equation.3 ( (sign(y1).min{| y1|, i}, sign(y2).min{| y2 |, i}), then f8( EMBED Equation.3 ) ( max{| sign(y1).min{| y1|, i} |, | sign(y2).min{| y2 |, i} |} (expressions of f8 and  EMBED Equation.3 ) ( max{min{| y1|, i}, min{| y2 |, i}} (i ( 0) ( i (by hypothesis, i ( max{| y1|, | y2 |}) ( f8(x). (hypothesis) Furthermore, for k ( 1, 2, | yk  EMBED Equation.3 | ( | yk sign(yk).min{| yk |, i} | (expression of  EMBED Equation.3 ) ( | sign(yk).yk min{| yk |, i} | (properties of sign and  EMBED Equation.3 ) ( | | yk | min{| yk |, i} |. (definition of  EMBED Equation.3 ) If | y1| ( | y2 |, then | y1  EMBED Equation.3 | ( | | y1| min{| y1|, i} | (see above, k ( 1) (  EMBED Equation.3  (definition of min) and | y2  EMBED Equation.3 | ( | | y2 | min{| y2|, i} | (see above, k ( 2) ( | y2 | i (by hypothesis, i ( | y2 |) therefore, | y1  EMBED Equation.3 | ( | y2  EMBED Equation.3 |. In other words, if | y1| ( | y2 |, then max{| y1  EMBED Equation.3 |, | y2  EMBED Equation.3 |} ( | y2  EMBED Equation.3 | (see above) ( | y2 | i (see above) ( max{| y1 |, | y2 |} i. (hypothesis) If | y2 | ( | y1|, then, under the same arguments, max{| y1  EMBED Equation.3 |, | y2  EMBED Equation.3 |} ( max{| y1 |, | y2 |} i. Hence, in any case, max{| y1  EMBED Equation.3 |, | y2  EMBED Equation.3 |} ( max{| y1 |, | y2 |} i. Finally, f8( EMBED Equation.3 ) + f8(y  EMBED Equation.3 ) ( i + f8(y  EMBED Equation.3 ) (see above) ( i + max{| y1  EMBED Equation.3 |, | y2  EMBED Equation.3 |} (expression of f8) ( i + max{| y1 |, | y2 |} i (see above) ( max{| y1 |, | y2 |} (cancelation) ( f8(y) (expression of f8) That is, for any x and y ( Z2 such that f8(x) ( f8(y), there exists a point  EMBED Equation.3 ( Z2 such that  EMBED Equation.3  and  EMBED Equation.3 . In other words, d8 is lower regular (of type 2) for the triangle inequality. (b) Let us show that d8 is upper regular for the triangle inequality. Let x ( (x1, x2) and y ( (y1, y2) ( Z2, let i ( f8(y) and let  EMBED Equation.3 ( (sign(x1).i, sign(x2).i), then f8( EMBED Equation.3 ) ( max{| sign(x1).i |, | sign(x2).i |} (expression of  EMBED Equation.3 ) ( max{i, i} (i ( 0) ( i (max is idempotent) ( f8(y). (hypothesis) Furthermore, f8(x +  EMBED Equation.3 ) ( max{| x1 +  EMBED Equation.3 |, | x2 +  EMBED Equation.3 |} (expression of f8) ( max{| x1 + sign(x1).i |, | x2 + sign(x2).i |} (expression of  EMBED Equation.3 ) ( max{| sign(x1).x1 + i |, | sign(x2).x2 + i |} (properties of sign and  EMBED Equation.3 ) ( max{| | x1| + i |, | | x2 | + i |} (definition of  EMBED Equation.3 ) ( max{| x1| + i, | x2 | + i} (i ( 0) ( max{| x1 |, | x2|} + i (distribuitvity of + over max) ( f8(x) + i (expression of f8) ( f8(x) + f8( EMBED Equation.3 ). (see above) That is, for any x and y ( Z2, there exists a point  EMBED Equation.3 ( Z2 such that  EMBED Equation.3  and  EMBED Equation.3 . In other words, d8 is upper regular for the triangle inequality. Therefore, by applying Property (iv) of  REF _Ref73414444 \h Proposition 13 (equivalent definition of translation-invariant regular metric), we may conclude that d8 is regular. ( The upper part of  REF _Ref79395360 \h  \* MERGEFORMAT Figure 2 shows the "straight-line" L(x, y) relative to the chessboard distance d8 passing through the point x and y and the "sphere" S(x, 2) of center x and radius 2. As it was expected from  REF _Ref79397692 \h Proposition 7, we observe that they have a nonempty intersection (see the lower part of the figure). Furthermore, this intersection have at least two diametrically opposite points, for instance, the points u and v. In this figure, u ( L2(x, y) and v ( L3(x, y). Rosenfeld and Pfaltz  REF _Ref79383931 \h  \* MERGEFORMAT [8] have shown in their Proposition 6 that d8 satisfies the following property: for any y ( Z2 such that 1 ( f8(y), there exists a point  EMBED Equation.3 ( Z2 such that  EMBED Equation.3  and  EMBED Equation.3 . This result is a consequence of the lower regularity (of type 2) of d8. More interesting for us is their counter example showing that the octogonal distance doct ( sup{d8, g} where g is the distance given by, for any x ( (x1, x2) and y ( (y1, y2) in Z2, g(x, y) ( 2(| x1 y1| + | x2 y2| + 1)/3, doesn't satisfy this property. In other words, such octagonal distance is an example of non-regular metric.  EMBED Word.Picture.8  Figure  SEQ Figure \* ARABIC 2 - Intersection of a diameter and a sphere. Before ending this section, we show that the good geometrical properties of the Euclidean vector space mencioned at the beginning of the introduction are satisfied in a regular metric space. First, we show that the lower regularity property for a distance is a necessary and sufficient condition to have the usual ball intersection property of the Euclidean vector space. Proposition  SEQ Definition \* ARABIC 16 (ball intersection in a lower regular metric space) Let (E, d) be a metric space then, (i) for any x and y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), ( z ( E: d(x, z) ( i and d(z, y) ( j ( d(x, y) ( i + j, (ii) (E, d) is lower regular iff, for any x and y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j. ( Proof Let us prove Property (i). For any x and y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), ( z ( E: d(x, z) ( i and d(z, y) ( j ( ( z ( E: d(x, z) + d(z, y) ( i + d(z, y) and i + d(z, y) ( i + j (addition is increasing) ( ( z ( E: d(x, z) + d(z, y) ( i + j (transitivity of () ( d(x, y) ( i + j. ( REF _Ref69010114 \h Definition 1 - triangle inequality and transitivity of () (b) Let us prove the if part of Property (ii). For any x and y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j) (particular case) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, z) + d(z, y) ( i + j) (+ is increasing) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, y) ( d(x, z) + d(z, y) ( i + j) (triangle inequality) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, y) ( d(x, z) + d(z, y) ( i + j ( d(x, y)) (logical derivation) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, z) + d(z, y) ( i + j) (anti-symmetry of () ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, z) + d(z, y) ( i + j and d(x, z) + d(z, y) ( i + d(z, y and d(x, z) + d(z, y) ( d(x, z) + j) (+ is increasing) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, z) + d(z, y) ( i + j and i + j ( i + d(z, y) and i + j ( d(x, z) + j) (substitution) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, z) + d(z, y) ( i + j) (anti-symmetry of () ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j and d(x, z) + d(z, y) ( d(x, y)) (logical derivation) ( (d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(x, z) + d(z, y) ( d(x, y)). (logical derivation) That is, under the hypothesis, for any x and y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}) such that d(x, y) ( i + j, we have ( z ( E: d(x, z) ( i and d(x, z) + d(z, y) ( d(x, y), but this implies that, for any x and y ( E, and any i ( d({x} ( E), i ( d(x, y), we have ( z ( E: d(x, z) ( i and d(x, z) + d(z, y) ( d(x, y), or equivalently by definitions of S and L2, for any x and y ( E, and any i ( d({x} ( E), i ( d(x, y), we have S(x, i) ( L2(x, y) ( ( which proves, by  REF _Ref78512758 \h  \* MERGEFORMAT Corollary 5 (first equivalent definition of regular metric), that d is lower regular. (c) Let us prove the ony-if part Property (ii). For any x and y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), such that d(x, y) ( i + j, (b1) if i ( j ( 0, then y ( x, therefore ( z ( E: d(x, z) ( i and d(z, y) ( j, namely z ( y ( x; (b2) if i ( 0 and j ( 0, then d(x, y) ( j, therefore ( z ( E: d(x, z) ( i and d(z, y) ( j, namely z ( x; (b3) if i ( 0 and j ( 0, then d(x, y) ( i, therefore ( z ( E: d(x, z) ( i and d(z, y) ( j, namely z ( y; (b4) if i ( 0 and j ( 0, and if d(x, y) < i or d(x, y) < j then ( z ( E: d(x, z) ( i and d(z, y) < j, or d(x, z) < i and d(z, y) ( j, namely z ( y or z ( x; (b5) if i ( d(x, y) and j ( d(x, y), d is lower regular ( S(x, i) ( L2(x, y) ( ( ( REF _Ref78512758 \h  \* MERGEFORMAT Corollary 5) ( ( z ( E: d(x, z) + d(z, y) ( d(x, y) and d(x, z) ( i (definitions of S and L2) ( ( z ( E: i + d(z, y) ( d(x, y) and d(x, z) ( i; (substitution) furthermore, d(x, y) ( i + j ( ( z ( E: d(x, z) ( i and d(z, y) ( j (lower regularity of d (see above) and + is double-side increasing) ( ( z ( E: d(x, z) ( i and d(z, y) ( j (reflexivity of () ( In order to illustrate better the previous proposition, we recall the definition of ball derived from a metric and we give a corollary. We call ball of center x ( E and radius i ( d({x} ( E), derived from a metric d, the subset Bd(x, i) ( {z ( E: d(x, z) ( i}; in the next section we will give another definition of ball. Corollary  SEQ Definition \* ARABIC 17 (ball intersection in a lower regular metric space) Let (E, d) be a metric space then, (i) for any x, y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), Bd(x, i) ( Bd(y, j) ( ( ( d(x, y) ( i + j, (ii) (E, d) is lower regular iff, for any x, y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), d(x, y) ( i + j ( Bd(x, i) ( Bd(y, j) ( (. ( Proof Properties (i) and (ii) are, respectively, equivalent to Properties (i) and (ii) of  REF _Ref26778152 \h Proposition 16 (ball intersection in a lower regular metric space) since (( z ( E: d(x, z) ( i and d(z, y) ( j) is equivalent to (Bd(x, i) ( Bd(y, j) ( () by definition of Bd(x, i). (  REF _Ref79462067 \h  \* MERGEFORMAT Figure 3 illustrates a counter example showing that the elliptic metric restricted to the discrete plane depicted in  REF _Ref79397486 \h  \* MERGEFORMAT Figure 1 doesn't satisfy Property (ii) of  REF _Ref79462313 \h Corollary 17. This is another way to conclude that this elliptic metric is not regular. In  REF _Ref79462067 \h  \* MERGEFORMAT Figure 3 the ellipses are the locus of points at distance 1 to their centers of symmetry x and y and the elliptic distance between these centers is d(x, y) ( 9/5 which is less than 2. In other words, the condition d(x, y) ( i + j (with i ( j ( 1) is satisfied. Nevertheless, as we can see on this figure, the ball intersection Bd(x, 1) ( Bd(y, 1) is empty.  Figure  SEQ Figure \* ARABIC 3 - Lack of intersection (discrete elliptic distance).  REF _Ref80096547 \h  \* MERGEFORMAT Figure 4 shows two examples of lack of intersection in the octagonal metric space (Z2, doct). On the left hand-side the two balls are of radius i ( j ( 1 and their centers x and y, 2 units apart. On the right-hand-side the two balls are of radius i ( 1 and j ( 3, and their centers x and y, 4 units apart. On both cases the two balls have no intersections despite the fact that doct(x, y) ( i + j. The right-hand side example is due to Rosenfeld and Pfaltz  REF _Ref79383931 \h  \* MERGEFORMAT [8].  REF _Ref26778152 \h Proposition 16 will be a key result to prove  REF _Ref26778315 \h Proposition 53 in Section 6 and consequently  REF _Ref26781288 \h Proposition 62 of Section 8 showing that the lower regular metrics can be reconstructed from their unit balls using the Minkowski product of Section 4.  Figure  SEQ Figure \* ARABIC 4 - Lack of intersection (octagonal distance). Finally, we show that the upper regularity property for a distance is a sufficient condition to have the usual ball inclusion property of the Euclidean vector space, see also  REF _Ref69021347 \h [5, Theorem 5.19]. Proposition  SEQ Definition \* ARABIC 18 (ball inclusion in an upper regular metric space) Let (E, d) be a metric space then, for any x, y ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), (i) i + d(x, y) ( j ( Bd(x, i) ( Bd(y, j), (ii) if (E, d) is upper regular then Bd(x, i) ( Bd(y, j) ( i + d(x, y) ( j. ( Proof (a) Let us prove Property (i). For any x, y and z ( E, any i ( d({x} ( E) and any j ( d(E ( {y}), z ( Bd(x, i) ( d(x, z) ( i (def. of Bd) ( d(x, y) + d(x, z) ( d(x, y) + i (positivness of d and + is increasing) ( d(y, x) + d(x, z) ( d(x, y) + i (symmetry of d) ( d(y, z) ( d(y, x) + d(x, z) ( d(x, y) + i (triangle inequality) ( d(y, z) ( d(x, y) + i (transitivity of () ( d(y, z) ( i + d(x, y) (commutativity of +) ( d(y, z) ( d(x, y) + i ( j (hypothesis) ( d(y, z) ( j (transitivity of () ( z ( Bd(y, j), (def. of Bd) that is, by inclusion definition, under the hypothesis, Bd(x, i) ( Bd(y, j). (b) Let us prove Property (ii). Let d be an upper regular metric, let x and y ( E, let i ( d({x} ( E) and let j ( d(E ( {y}). We divide the proof in five parts. (b1) Let us prove that Bd(x, i) ( Bd(y, j) ( d(y, x) ( j: Bd(x, i) ( Bd(y, j) ( x ( Bd(y, j) (d(x, x) ( 0 ( x ( Bd(x, i)) ( d(y, x) ( j. (def. of Bd) (b2) By  REF _Ref78511934 \h Proposition 4 (axiom dependence), if d is upper regular, then d is lower regular of type 1. Furthermore, true ( S(y, j) ( L1(y, x) ( ( and S(x, i) ( L3(x, y) ( ( (d is lower regular of type 1 and by (b1) d(y, x) ( j, and d is upper regular) ( ( u ( E: d(y, u) ( j and d(y, u) ( d(y, x) + d(x, u) and ( v ( E: d(x, v) ( i and d(v, y) ( d(v, x) + d(x, y) (definitions of S, L1 and L3) (b3) For any v ( E, d(x, v) ( i ( v ( Bd(x, i) (def. of Bd) ( v ( Bd(y, j) (Bd(x, i) ( Bd(y, j)) ( d(y, v) ( j. (def. of Bd) (b4) For some u and v ( E, d(x, v) + d(x, y) ( d(v, x) + d(x, y) (symmetry of d) ( d(v, y) ((b2)) ( d(y, v) (symmetry of d) ( j ((b3)) ( d(y, u) ((b2)) ( d(y, x) + d(x, u) ((b2)) ( d(x, y) + d(x, u), (symmetry of d) that is, for some u and v ( E, by regularity of +, d(x, v) ( d(x, u). (b5) For some u and v ( E, i + d(x, y) ( d(x, v) + d(x, y) ((b2)) ( d(x, u) + d(x, y) ((b4)) ( d(x, y) + d(x, u) (commutativity of +) ( d(y, x) + d(x, u) (symmetry of d) ( d(y, u) ((b2)) ( j, ((b2)) that is, if d is upper regular, then Bd(x, i) ( Bd(y, j) ( i + d(x, y) ( j. ( 3. Balls The ball definition in this section is independent of the concept of distance seen in the previous section. Instead, it is based on the concepts of set translation and transposition. Because of our interest in digital image processing, the balls will be considered as subsets of the discrete plane Z2 (the Cartesian product of the set of integers by itself). We denote by P(Z2) the collection of all the subsets of Z2. Let N be the set of natural numbers: 0, 1, 2, A subset X of Z2 is finite if there exists a natural number n and a bijection between X and the subset of natural numbers {1, 2, , n}. We denote by #X the natural number n, and by #X < ( the finiteness of X. The Abelian group (Z2, +, o) equipped with the scalar multiplication (j, (x1, x2))  EMBED Equation.3  j(x1, x2) ( (jx1, jx2) with operand j in Z forms a module over the ring Z  REF _Ref69202843 \h [6, p. 166]. We begin recalling the definition of translated version of a subset. Definition  SEQ Definition \* ARABIC 19 (translated version) The translated version of a subset X of Z2 by a point u in Z2 is the subset Xu ( {y ( Z2: y u ( X}. ( The next proposition shows a regularity property of the translated version: if Xu ( Xv then u ( v. This property will be important to prove the uniqueness of the concept of radius of a ball in Section 5. The proof of this property relies on a basic poset theorem. Let X and Y be two subsets of Z2. The complement of X in Z2 is denoted Xc, that is, for any x ( Z2, x ( Xc ( x ( X, and the difference between X and Y is denoted X Y, that is, X Y ( X ( Yc. Proposition  SEQ Definition \* ARABIC 20 (translated version property) If the subset X of Z2 is finite, then the mapping u  EMBED Equation.3 Xu from Z2 to P(Z2) is injective. Furthermore, for any u and v in Z2 such that u ( v,  EMBED Equation.3 ( and  EMBED Equation.3 ( or, equivalently, Xu ( Xv ( Xv and Xu ( Xv ( Xu. ( Proof Let u be a point in Z2 and let X be a subset of Z2, by the regularity of the addition on Z2, the mapping x  EMBED Equation.3 x + u from X to Xu is a bijection, therefore, if X is finite then Xu is finite as well. Let u ( Z2 and let Ru be the binary relation on Z2 given by, for any x and y ( Z2, x Ru y ( ( j ( N: y ( x + ju. Let us first prove that Ru is a partial order. For any x ( Z2, x Ru x is true since for j ( 0, x ( x + ju by group property, that is Ru is reflexive. If u ( o, for any x, y ( Z2, x Ru y and y Ru x imply x ( y (and y ( x). If u ( o, for any x and y ( Z2, x Ru y and y Ru x ( ( j, k ( N: y ( x + ju and x ( y + ku (definition of Ru) ( ( j, k ( N: y ( x + ju and x ( y + ku and y ( y + (j + k)u (by substitution and module property) ( ( j, k ( N: y ( x + ju and x ( y + ku and (j + k)u ( o (o is unitary element of +) ( ( j, k ( N: y ( x + ju and x ( y + ku and j + k ( 0 (u ( o and Z has no zero divisor) ( y ( x + ju and x ( y + ku and j ( k ( 0 (natural number property) ( y ( x, (natural number property) that is, Ru is antisymmetric. For any x, y and z ( Z2, x Ru y and y Ru z ( ( j, k ( N: y ( x + ju and z ( y + ku (definition of Ru) ( ( j, k ( N: z ( x + (j + k)u (by substitution) ( ( i ( N: z ( x + iu (namely, i ( j + k) ( x Ru z, (definition of Ru) that is, Ru is transitive. Let u and v be two points in Z2, u ( v and let X be a finite subset of Z2, then Xu is finite as well. Therefore, by Theorem 3 of Birkhoff  REF _Ref41885811 \h [4] or Theorem 3 of Szsz  REF _Ref41885836 \h [7], ( x ( Xu such that x is maximal with respect to Rv u. Let y be the point in Z2 such that y ( x + v u, then x Rv u y (doing j ( 1 in the definition of Rv u), y ( Xv (by translated version definition) and y ( x (by regularity of the addition on Z2). Since u ( v and x is maximal, x Rv u z for no z in Xu {x}, consequently y doesn't belong to Xu. In other words, there exists a point in Xv (namely y) which doesn't belong to Xu, that is,  EMBED Equation.3 ( and Xu ( Xv. Furthermore, for any u, v ( Z2,  EMBED Equation.3 ( ( Xv ( Xu (B A) ( ( ( B ( A) ( Xu ( Xv ( Xv. (B ( A ( (A ( B) ( B) Finally, by changing the role of u and v, for any u and v ( Z2, u ( v,  EMBED Equation.3 ( and Xv ( Xu ( Xu. ( We go on recalling the definition of transpose of a subset. Definition  SEQ Definition \* ARABIC 21 (transpose) The transpose of a subset X of Z2 is the subset Xt ( {x ( Z2 : x ( X}. ( A subset X of Z2 is symmetric (with respect to the origin o) iff it is equal to its transpose, that is X ( Xt or equivalently, iff x ( X ( x ( X. Z2 is an example of symmetric subset. We denote by S the sub collection of all the finite symmetric subsets of Z2, i.e., S ( {X ( P(Z2): #X ( ( and X ( Xt} and by S+ the sub collection S + {Z2}. From the operations of translation and transposition, we can build the sub collection of symmetric finite subsets and their translated versions, that we call balls. The symmetry assumption was made in order to establish, in Sections 6 and 7, the relationship with distance (which is a symmetric mapping). Definition  SEQ Definition \* ARABIC 22 (ball) A subset X of Z2 is a ball iff ( u ( Z2 such that Xu ( S+. ( By definition of S+, the set Z2 is a ball. Symmetric subsets are balls (doing u ( o). The 3 by 3 discrete square, denoted by B8, consisting of the points (0, 0), (0, 1), (1, 1) (1, 0), (1,  EMBED Equation.3 ), (0,  EMBED Equation.3 ), ( EMBED Equation.3 , EMBED Equation.3 ), ( EMBED Equation.3 , 0) ( EMBED Equation.3 , 1) is ball since it is a symmetric subset. Another important examples of balls (important from the theoretical point of view) are the singletons (the subsets containing just one point). Proposition  SEQ Definition \* ARABIC 23 (ball example) Any singleton of Z2 is a ball. ( Proof For any x ( Z2, ({x}-x)t ( {x x}t ( REF _Ref26767636 \h Definition 19 - translated version) ( {o}t (group property) ( { o} ( REF _Ref26766383 \h Definition 21 - transpose) ( {o} (group property) ( {x x} (group property) ( {x}-x , ( REF _Ref26767636 \h Definition 19) in other words, for any x ( Z2, ( u ( Z2 (namely u ( x) such ({x}u)t ( {x}u , that is, by  REF _Ref26766674 \h Definition 22, {x} is a ball of Z2. ( We denote by B+ the sub collection of all the balls and by B the sub collection of finite balls, i.e., B ( B+ ( {Z2}. As a consequence of these definitions, S ( B ( B+ ( P(Z2). The opposite u of the point u appearing in the ball definition is a center of symmetry for the ball, it is called its center. Definition  SEQ Definition \* ARABIC 24 (ball center) Let X be a ball (X ( B+), if X is finite, u ( Z2 is the center of X iff Xu ( S, and if X is Z2, its center is the origin o. ( Hence, by  REF _Ref26766674 \h Definition 22 (ball), a ball of Z2 has always a center, furthermore, if the ball is finite, this center is unique as shown in the next proposition. If the ball is Z2, by convention, we assume that its center is the origin o. Proposition  SEQ Definition \* ARABIC 25 (ball center construction) Let X be a finite ball (i.e., X ( B), u is the center of X iff u (  EMBED Equation.3 . ( Proof Let R be a relation on Z2 given by x R y ( y ( x or y ( x. By group properties, R is an equivalent relation. Let u1 be a center of X and let u2 (  EMBED Equation.3 . Let us prove that u1 ( u2. u2 - u1 (  EMBED Equation.3  u1 (def. of u2) (  EMBED Equation.3  (group property) (  EMBED Equation.3  (v ( x u1) (  EMBED Equation.3  (u1 center of X (  EMBED Equation.3 ( S) ( o, (group properties) that is, u1 ( u2. ( We denote by center(X) the center of X ( B+. If the center of X is u, then we say that X is with center at u. The center of a ball is not sufficient to characterize a ball, we need one more parameter that we call here the ball matrix. Definition  SEQ Definition \* ARABIC 26 (ball matrix) Let X be a ball (X ( B+), the matrix of X, denoted by matrix(X), is the ball  EMBED Equation.3 , that is, matrix(X) (  EMBED Equation.3 . ( The ball matrix is a symmetric ball. Proposition  SEQ Definition \* ARABIC 27 (ball matrix property) Let X be a ball (i.e., X ( B+), then matrix(X) ( S+. ( Proof For any X ( B+, (matrix(X))t (  EMBED Equation.3  (ball matrix definition) (  EMBED Equation.3  (center definition) ( matrix(X), (ball matrix definition) that is, matrix(X) ( S+. ( The balls with center at origin o are symmetric and every finite symmetric subset is a ball with center at origin. This, and other properties of the balls with center at origin are shown in the next proposition. Proposition  SEQ Definition \* ARABIC 28 (properties of a ball with center at origin) Let B be a ball (i.e., B ( B+), then the following statements are equivalent: (i) for any x, y ( Z2, x ( By ( y ( Bx, (ii) B ( S+, (iii) center(B) ( o, (iv) matrix(B) ( B. ( Proof (a) Let us prove (i) ( (ii). For any finite ball B (i.e., B ( B) and any x and y ( Z2, x ( By ( y ( Bx (hypothesis (i)) ( x ( (Bt)y, (Property (2) of Prop. 4.10 of  REF _Ref41884181 \h [2]) that is, for y ( o, Bt ( B, in other words, B ( S. Furthermore, for B ( Z2, B ( S+ as a consequence of the definition of S+. (b) Let us prove (i) ( (ii). For any symmetric ball B and any x and y ( Z2, x ( By ( x y ( B ( REF _Ref26767636 \h Definition 19 - translated version) ( y x ( B (hypothesis (ii)) ( y ( Bx . ( REF _Ref26767636 \h Definition 19) (c) Let us prove (ii) ( (iii). For any finite ball B, B ( S ( B ( Bt (definition of S) ( Bo ( (Bo)t (Property (1) of Prop. 4.6 of  REF _Ref41884181 \h [2]) ( center(B) ( o. ( REF _Ref34232659 \h Definition 24 - ball center) For B ( Z2, B ( S+ and center(B) ( o are both true as a consequence of the definitions of S+ and center of a ball. (d) Let us prove (iii) ( (iv). For any ball B, matrix(B) (  EMBED Equation.3  ( REF _Ref36376336 \h Definition 26 - ball matrix) (  EMBED Equation.3  (hypothesis (iii)) (  EMBED Equation.3  (group property) ( B (Property (1) of Prop. 4.6 of  REF _Ref41884181 \h [2]) (e) Let us prove (iii) ( (iv). For any ball B,  EMBED Equation.3 ( matrix(B) ( REF _Ref36376336 \h Definition 26) ( B (hypothesis (iv)) ( Bo, (Property (1) of Prop. 4.6 of  REF _Ref41884181 \h [2]) (  EMBED Equation.3  (group property) that is, if B is finite, by  REF _Ref32469616 \h Proposition 20 (translated version property), center(B) ( o, if B ( Z2, by convention, center(B) ( o as well. ( The sub collection of balls is closed under translation as we show in the next proposition. Proposition  SEQ Definition \* ARABIC 29 (ball translation) Let X be a finite ball (i.e., X ( B) and let v ( Z2, then Xv is a finite ball (i.e., Xv ( B). Furthermore, center(Xv) ( center(X) + v and matrix(Xv) ( matrix(X). ( Proof Let X be a finite ball and let u ( center(X). For any v ( Z2, ((Xv) (u + v))t ( (X u)t (Prop. 4.6 of  REF _Ref41884181 \h [2]) ( X u (X is a ball and u is its center) ( (Xv) (u + v), (Prop. 4.6 of  REF _Ref41884181 \h [2]) in other words, for any X ( B and any v ( Z2, ( u ( Z2 : ((Xv)u)t ( (Xv)u, namely u ( (center(X) + v), that is, Xv ( B and by  REF _Ref34232659 \h Definition 24 (ball center), center(Xv) ( center(X) + v. Let us prove the last statement. For any X ( B and any v ( Z2, matrix(Xv) ( EMBED Equation.3  (ball matrix definition) (  EMBED Equation.3  (Property (2) of Prop. 4.6 of  REF _Ref41884181 \h [2]) (  EMBED Equation.3  (first statement) (  EMBED Equation.3  (group property) ( matrix(X). (ball matrix definition) ( Their center and matrix characterize the finite balls. Proposition  SEQ Definition \* ARABIC 30 (ball characterization) The mapping from B to Z2 ( S, X  EMBED Equation.3 (center(X), matrix(X)) is a bijection, its inverse is (u, B)  EMBED Equation.3 Bu. ( Proof Let us divide the proof in four parts. (a) By definition of center, for any X ( B, center(X) ( Z2. Furthermore, by  REF _Ref35501202 \h Proposition 27 (ball matrix property), for any X ( B, matrix(X) ( S. (b) Let us prove that for any u ( Z2 and any B ( S, Bu ( B. For any u ( Z2 and any B ( S, let v ( u, then, ((Bu)v)t ( (Bu + v)t (Property (2) of Prop. 4.6 of  REF _Ref41884181 \h [2]) ( (Bo)t (v ( u and inverse definition) ( Bt (Property (1) of Prop. 4.6 of  REF _Ref41884181 \h [2]) ( B (B ( S) ( (Bu)v , (using the same arguments as above) that is, there exists a point v in Z2 such that ((Bu)v)t ( (Bu)v. (c) Let us prove that (u, B)  EMBED Equation.3 Bu is a left inverse for X  EMBED Equation.3 (center(X), matrix(X)). For any X ( B,  EMBED Equation.3  (  EMBED Equation.3  ( REF _Ref36376336 \h Definition 26 - ball matrix) (  EMBED Equation.3  (Property (2) of Prop. 4.6 of  REF _Ref41884181 \h [2]) ( Xo (inverse definition) ( X. (Property (1) of Prop. 4.6 of  REF _Ref41884181 \h [2]) (d) Let us prove that (u, B)  EMBED Equation.3 Bu is a right inverse for X  EMBED Equation.3 (center(X), matrix(X)). For any u ( Z2 and any B ( S, (center(Bu), matrix(Bu)) ( (center(B) + u, matrix(B)) ( REF _Ref26775027 \h Proposition 29 - ball translation) ( (o + u, B) (B ( S implies (iii) and (iv) of  REF _Ref26776185 \h Proposition 28 - properties of a ball with center at origin) ( (u, B). (group property) Therefore, from (a) - (d), X  EMBED Equation.3 (center(X), matrix(X)) has an inverse (which is (u, B)  EMBED Equation.3 Bu), consequently it is a bijection. ( In order to combine geometrically the balls, we recall the Minkowski addition and subtraction definitions  REF _Ref41905163 \h [2],  REF _Ref41884991 \h [3],  REF _Ref69202843 \h [6]. Definition  SEQ Definition \* ARABIC 31 (Minkowski addition and subtraction) Let A and B be two subsets of Z2, their Minkowski sum is the subset A ( B ( {a + b: a ( A, b ( B} and their Minkowski difference is the subset A ( B ( {x ( Z2: (b ( B, ( a ( A: x ( a b}. (A, B) EMBED Equation.3 A ( B is the Minkowski addition and (A, B)  EMBED Equation.3 A ( B is the Minkowski subtraction from P(Z2) ( P(Z2) to P(Z2). ( The sub collection of balls with center at origin is closed under Minkowski's addition as we show in the next proposition. Proposition  SEQ Definition \* ARABIC 32 (symmetric ball addition) Let X and Y be two symmetric balls (i.e., X, Y( S+), then X ( Y is a symmetric ball (i.e., X ( Y ( S+). ( Proof For any X and Y( S+, (X ( Y)t ( Xt ( Yt (Minkowski's addition property) ( X ( Y, (X and Y( S) that is, X ( Y ( S+. ( As a consequence of the above symmetric ball addition result, in the next proposition we show that the sub collection of balls is closed under Minkowski's addition as well. Proposition  SEQ Definition \* ARABIC 33 (ball addition) Let X and Y be two finite balls (i.e., X and Y ( B), then X ( Y is a ball (i.e., X ( Y ( B), furthermore, center(X ( Y) ( center(X) + center(Y) and matrix(X ( Y) ( matrix(X) ( matrix(Y). ( Proof Let divide the proof into four parts. (a) For any X and Y ( B, X ( Y (  EMBED Equation.3  (  EMBED Equation.3  ( REF _Ref35253779 \h Proposition 30 - ball characterization) ( EMBED Equation.3  (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( EMBED Equation.3  (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( EMBED Equation.3 . (Property (2) of Prop. 4.6 of  REF _Ref41884181 \h [2]) (b) For any X, Y ( B, by  REF _Ref35501202 \h Proposition 27 (ball matrix property) and by  REF _Ref35501499 \h Proposition 32 (symmetric ball addition) matrix(X) ( matrix(Y) ( S, therefore, by Part (a) of the proof, it exists u ( Z2 (namely, u ( (center(X) + center(Y)) such that (X ( Y)u ( S. That is, by  REF _Ref26766674 \h Definition 22 (ball definition), X ( Y ( B. (c) For any X, Y ( B, center(X ( Y) (  EMBED Equation.3  (Part (a) of the proof) (  EMBED Equation.3  ( REF _Ref26775027 \h Proposition 29 - ball translation) (  EMBED Equation.3  (matrix(X) ( matrix(Y) ( S, therefore, by  REF _Ref26776185 \h Proposition 28 (properties of a ball with center at origin),  EMBED Equation.3 ) (  EMBED Equation.3 . (group property) (d) For any X and Y ( B, matrix(X ( Y) (  EMBED Equation.3  (Part (a) of the proof) (  EMBED Equation.3  ( REF _Ref26775027 \h Proposition 29 - ball translation) (  EMBED Equation.3 . (as in (c) above, by  REF _Ref26776185 \h Proposition 28) ( Before continuing our study about the ball collection, we need to introduce the Minkowski product. 4. Minkowski product Within the collection of balls we can identify sub collections in which the balls are mutually related through an external binary operation between a natural number and a subset, that we call the Minkowski product. We denote by N+ the set of extended natural numbers (i.e., the natural numbers plus an element denoted () with the usual addition extended in such a way that, for any j ( N+, j + ( ( ( + j ( ( and with the usual order extended in such a way that, for any j ( N+, j ( (. Definition  SEQ Definition \* ARABIC 34 (Minkowski product) Let B ( P(Z2) such that B ( (, and let j ( N+, the Minkowski product of B by j is the subset jB of Z2 given by jB (  EMBED Equation.3 . ( The Minkowski product jB should be distinguished from the result of the usual scaling operation of B by j used in the continuous case. The scaling transform of a subset B by a real number j is the subset denoted scaling(j, B) and given by scaling(j, B) ( {jx: x ( B}. In the discrete case, both results, jB and scaling(j, B), may be different as it can be verified when B is the 3 by 3 discrete square B8. The difference goes on even if we try to preserve the "interior" of the square using a slightly modified version of the scaling transform given by scaling2(j, B) ( {ix: x ( B and i ( [0, j]}.  REF _Ref79921109 \h Figure 5 shows (from left to right and top to bottom) the B8 square, its transformation when applying the above two scaling transformations and at last the Minkowski product, with j ( 2. We observe that these expansion definitions lead to different results. In the discrete domain Z2, the Minkowski product is more appropriate for our purpose than the scaling operation (see Section 7). We now verify the mix distributivity of this product and then some other elementary properties. Based on the mix distributivity, we could establish in Section 5  REF _Ref33112159 \h Proposition 44 and  REF _Ref30654522 \h Proposition 48 about the intersection and inclusion of generated balls. Proposition  SEQ Definition \* ARABIC 35 (mix distributivity of the Minkowski product) Let B ( P(Z2) such that B ( (, for any i and j ( N+, (i + j)B ( (iB) ( (jB). (  EMBED Word.Picture.8  Figure  SEQ Figure \* ARABIC 5 - Minkowski product versus scaling Proof Let us divide the proof in three parts. (a) Let us prove that for any i ( N, (i + 1)B ( (iB) ( B. If i ( 0, (i + 1)B ( 1B (hypothesis) ( B ( REF _Ref74134284 \h Definition 34 - Minkowski's) ( {o}( B (Property (4) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( (0B) ( B ( REF _Ref74134284 \h Definition 34) ( (iB) ( B (hypothesis) If i EMBED Equation.3 0, (i + 1)B ( (((i + 1) 1)B) ( B (product definition) ( (iB) ( B. (natural number property) (b) Let us prove the proposition statement for any i and j ( N. If j ( 0, (i + j)B ( iB (hypothesis) ( (iB) ( {o} (Property (4) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( (iB) ( (jB); (hypothesis) if j ( 1, (i + j)B ( (i + 1)B (hypothesis) ( (iB) ( B (Part (a) of the proof) ( (iB) ( (1B) ( REF _Ref74134284 \h Definition 34) ( (iB) ( (jB); (hypothesis) if j EMBED Equation.3 1, (i + j)B ( ((i + j 1) + 1)B (natural number property) ( ((i + j 1)B) ( B (Part (a) of the proof) ( ((iB) ( ((j 1)B)) ( B (is true for j 1) ( (iB) ( (((j 1)B) ( B) (associativity of () ( (iB) ( (jB). (Part (a) of the proof) (c) Let us prove the proposition statement for any i ( N+ and j ( (: (i + j)B ( (i + ()B (hypothesis) ( (B (assumption on the extended addition) ( Z2 ( REF _Ref74134284 \h Definition 34) ( iB ( Z2 (iB ( (,) ( (iB) ( ((B) ( REF _Ref74134284 \h Definition 34) ( (iB) ( (jB). (hypothesis) The same result can be obtained by changing the hypotheses on i and j. That is, from (b) and (c), the product is mix distributive. ( For convenience, let us assume that # Z2 ( (. Proposition  SEQ Definition \* ARABIC 36 (elementary properties of the Minkowski product) Let B ( P(Z2) such that B ( (, for any i ( N and any j ( N+, (i) if o ( B then o ( jB; (ii) if o ( B then the mapping j EMBED Equation.3 jB from N+ to P(Z2) is increasing; (iii) if j ( 0 and #B > 1 then #(jB) > 1; (iv) #(iB) ( (#B)i; (v) if B ( S then jB ( S+. ( Proof (a) Let us prove Property (i). By  REF _Ref74134284 \h Definition 34 (Minkowski product), o ( jB is true for j ( 0, 1 and (. Let us assume that it was true for j 1 (1< j < (), o ( (j 1)B (hypothesis) ( ((j 1)B) ( B (o ( B and Property (6) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( jB, ( REF _Ref74134284 \h Definition 34) that is, by inclusion definition, o ( jB. (b) Let us prove Property (ii). Let i and j ( N, such that i ( j, iB ( iB ( (j i)B (o ( (j i)B (Property. (i)) and Property (6) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( jB. (j ( i + (j i) and  REF _Ref32385489 \h Proposition 35) Let i ( N+ and let j ( (, then i EMBED Equation.3  j, furthermore, iB ( Z2 ( REF _Ref74134284 \h Definition 34) ( (B ( REF _Ref74134284 \h Definition 34) ( jB. (hypothesis) That is the mapping j EMBED Equation.3 jB from N+ to P(Z2) is increasing. (c) Let us prove Property (iii). By hypothesis, #(jB) > 1 is true for j ( 1 and it is always true for j ( (. Let us assume that it was true for j 1 (1 < j < (), and let x ( (j 1)B, #(jB) ( #(((j 1)B) ( B) ( REF _Ref74134284 \h Definition 34)  EMBED Equation.3  #({x} ( B) (Minkowski's addition is increasing (Property (6) of Exerc. 4.9 of  REF _Ref41884181 \h [2]) and cardinality property) ( #(Bx) (Property (1) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( #B (X EMBED Equation.3 Xx is a bijection) > 1. (hypothesis) (d) Let us prove Property (iv). #(iB)  EMBED Equation.3  (#B)i is true for i ( 1. Let us assume that it was true for i 1 (1 < i), #(iB) ( #(((i 1)B) ( B) ( REF _Ref74134284 \h Definition 34) ( #EMBED Equation.3 (equivalent definition of ( (Property (1) of Prop. 4.12 of  REF _Ref41884181 \h [2])) ( (#B)(#( i 1)B) (union property) ( (#B)(#(B))i - 1 (hypothesis) ( (#(B))i. (arithmetic property) (e) Let us prove Property (v). Let B ( Bt, by Minkowski product definition, jB ( (jB)t is true for j ( 0 (since {o} is symmetric), 1 and ( (since Z2 is symmetric as well). Let assume that it was true for j 1 (1 < j < (), (jB)t ( (((j 1)B) ( B)t ( REF _Ref74134284 \h Definition 34) ( ((j 1)B) ( B (hypotheses and  REF _Ref35501499 \h Proposition 32 (symmetric ball addition)) ( jB. ( REF _Ref74134284 \h Definition 34) ( Property (v) says that if a subset is symmetric then its product by any natural number is still symmetric. Next proposition about the regularity of the Minkowski product will be useful for the definition of radius of a ball in the next section. Proposition  SEQ Definition \* ARABIC 37 (regularity of the Minkowski product) Let B ( P(Z2), such that B is finite and #B > 1, then the mapping j EMBED Equation.3 #(jB) from N+ to N+, is injective and increasing. Furthermore, under the above assumption on B, the mapping j EMBED Equation.3 jB from N+ to P(Z2) is injective (i.e. iB ( jB ( i ( j). ( Proof Let i and j ( N, such that i EMBED Equation.3  j, jB ( iB ( (j i)B (j ( i + (j i) and  REF _Ref32385489 \h Proposition 35 - mix distributivity of the Minkowski product) (EMBED Equation.3. (equivalent definition of ( (Property (1) of Prop. 4.12 of  REF _Ref41884181 \h [2])) By Property (iii) of  REF _Ref32405633 \h Proposition 36, if i EMBED Equation.3  j, #((j i)B) > 1. Let u1 and u2 ( (j i)B,  EMBED Equation.3 +  EMBED Equation.3 (  EMBED Equation.3 (  EMBED Equation.3  (subset addition definition) ( jB . (see above equality) Hence, if i EMBED Equation.3  j,  EMBED Equation.3  <  EMBED Equation.3 +  EMBED Equation.3   EMBED Equation.3 ( ( by  REF _Ref32469616 \h Proposition 20 - translated version property) (  EMBED Equation.3 +  EMBED Equation.3  (X  EMBED Equation.3 Xu is a bijection) (  EMBED Equation.3  (property of the union of disjoint subsets)  EMBED Equation.3 . (see above inclusion) Let i ( N and let j ( (, then i < j, furthermore, #(iB) ( (#B)i (by Property (iv) of  REF _Ref32405633 \h Proposition 36) < ( (B is finite) ( # Z2 (by convention) ( #((B) ( REF _Ref74134284 \h Definition 34 - Minkowski product) ( #(jB). (hypothesis) Furthermore, for any i and j ( N+ i ( j ( i EMBED Equation.3  j or j EMBED Equation.3  i (N+ is a chain) (  EMBED Equation.3 <  EMBED Equation.3  or  EMBED Equation.3  EMBED Equation.3   EMBED Equation.3  (the previous conclusion) (  EMBED Equation.3 (  EMBED Equation.3  (N+ is a chain) ( iB ( jB. (iB and jB have not the same elements) That is, the mapping j EMBED Equation.3 #(jB) from N+ to N+, is injective and increasing, and the mapping j EMBED Equation.3 jB from N+ to P(Z2), is injective. ( Based on the Minkowski product, we define, in the next section, the notions of generated balls and of ball radius. In Section 7, the ball radius will be used to derive a metric from a symmetric ball. 5. Generated balls Let A and B be two subsets of Z2. If #B > 1 and B is finite, then we say that A is a multiple of B iff there exists an element j ( N+ such that A ( jB. In this case, we say that the set B divide A and we call the natural number j (which is unique under the restrictions on B) the quotient of A by B and we denote it  EMBED Equation.3 . The uniqueness of quotient is a consequence of  REF _Ref32548310 \h Proposition 37 (regularity of the Minkowski product): let i and j be two quotients of A by B, by definition of the quotient, iB ( A and jB ( A, that is, iB ( jB, therefore, since j EMBED Equation.3 jB is injective ( REF _Ref32548310 \h Proposition 37), i ( j. With a finite symmetric ball B we can associate, through the Minkowski product, a sub collection of balls. We say that B induces a sub collection or that the sub collection is generated by B. Definition  SEQ Definition \* ARABIC 38 (sub collection generated by a ball) Let B be a finite symmetric ball (i.e., B ( S), such that B ( {o}. The sub collection generated by B, denoted by BB, is the set of all the balls whose matrices are multiple of B, that is, BB ( {X ( B+: ( j ( N+, matrix(X) ( jB}. ( For any j ( N and any B ( S, by Property (v) of  REF _Ref32405633 \h Proposition 36 (elementary properties of the Minkowski product) and by  REF _Ref26776185 \h Proposition 28 (property of a ball with center at origin), matrix(jB) ( jB, that is jB ( BB, in other words, BB is never empty, moreover, it always contains Z2. We call the elements of BB generated balls and we say that they are generated from the prototype ball B. The assumption that Z2 was a ball is convenient in the sense that, in this way, every point in Z2 is contained in at least one generated ball. Proposition  SEQ Definition \* ARABIC 39 (example of generated balls) Let B be a finite symmetric ball (i.e., B ( S), such that B ( {o}. Any singleton of Z2 belongs to BB (the sub collection generated by B). ( Proof For any x ( Z2, by  REF _Ref39656736 \h Proposition 23 (ball example) {x} ( B+. Furthermore, for any x ( Z2, matrix({x}) ( matrix({o}x) ( REF _Ref26767636 \h Definition 19 - translated version) ( matrix({o}) ( REF _Ref26775027 \h Proposition 29 - ball translation) ( {o} ( REF _Ref26776185 \h Proposition 28 (property of a ball with center at origin), since {o} ( S+) ( 0B, ( REF _Ref74134284 \h Definition 34 - Minkowski product) that is, ( j ( N+, matrix({x}) ( jB (namely, j ( 0). Consequently, by  REF _Ref35667733 \h  \* MERGEFORMAT Definition 38 (sub collection generated by a ball), for any x ( Z2, {x} ( BB. ( As a direct consequence of the definition of a generated ball, we can parameterize its matrix by a natural number that we call its radius. Definition  SEQ Definition \* ARABIC 40 (radius of a generated ball) Let B be a finite symmetric ball (i.e., B ( S), such that B ( {o}. The radius of X ( BB, denoted radiusB(X) is the quotient of matrix(X) by B, that is, radiusB(X) (  EMBED Equation.3 . ( The uniqueness of the quotient guarantees the uniqueness of the radius. We observe that the same ball generated from different ball prototypes may have different radius. A ball prototype plays the role of a unit ball. We are now ready to show that the generated balls can be completely characterized in terms of their centers and their radius. Proposition  SEQ Definition \* ARABIC 41 (characterization of the generated balls) Let B be a finite symmetric ball (i.e., B ( S), such that B ( {o}. The mapping from BB ( {Z2} to Z2 ( N X  EMBED Equation.3 (center(X), radiusB(X)) is a bijection and its inverse is (x, j)  EMBED Equation.3 (jB)x. Furthermore, for any X ( BB, X ( EMBED Equation.3 , and for any (x, j) ( Z2 ( N, (x, j) ( (center((jB)x), radiusB((jB)x)). ( Proof Let us divide the proof in four parts. (a) For any X ( BB ( {Z2}, by  REF _Ref34232659 \h Definition 24 (ball center), center(X) ( Z2 and by  REF _Ref35529502 \h Definition 40 (radius of generated ball), radiusB(X) ( N. (b) For any x ( Z2 and any j ( N, matrix((jB)x) ( matrix(jB) ( REF _Ref26775027 \h Proposition 29 - ball translation) ( jB, (Property (v) of  REF _Ref32405633 \h Proposition 36 (elementary properties of the Minkowski product) and  REF _Ref26776185 \h Proposition 28 (property of a ball with center at origin)) that is, by  REF _Ref32548310 \h Proposition 37 (regularity of the Minkowski product) ( i ( N, matrix((jB)x) ( iB, namely i ( j, and by  REF _Ref35667733 \h Definition 38 (sub collection generated by a ball) (jB)x ( BB ( {Z2}. (c) Let us prove that (x, j)  EMBED Equation.3 (jB)x is a left inverse for X  EMBED Equation.3 (center(X), radiusB(X)). For any X ( BB and any x ( Z2, x (  EMBED Equation.3  ( x (  EMBED Equation.3  ( REF _Ref35529502 \h Definition 40) ( x (  EMBED Equation.3  ( REF _Ref36376336 \h Definition 26 - ball matrix) ( x (  EMBED Equation.3  (quotient property) ( x (  EMBED Equation.3  (translated version property) ( x ( X, (group property) that is, (x, j)  EMBED Equation.3 (jB)x is a left inverse. (d) Let us prove that the (x, j)  EMBED Equation.3 (jB)x is a right inverse for X  EMBED Equation.3 (center(X), radiusB(X)). For any x ( Z2 and any j ( N, center((jB)x) ( center(jB) + x ( REF _Ref26775027 \h Proposition 29 - ball translation) ( o + x (Property (v) of  REF _Ref32405633 \h Proposition 36 and  REF _Ref26776185 \h Proposition 28 (properties of a ball with center at origin)) ( x. (group property) For any x ( Z2 and any j ( N, radiusB((jB)x) (  EMBED Equation.3  (( REF _Ref35529502 \h Definition 40) (  EMBED Equation.3  (above result) (  EMBED Equation.3  (translated version property) (  EMBED Equation.3 , (group property) in other words, by quotient definition, radiusB((jB)x)B ( jB, therefore, by uniqueness of the quotient (consequence of  REF _Ref32548310 \h Proposition 37), radiusB((jB)x) ( j. That is, (x, j)  EMBED Equation.3 (jB)x is a right inverse. Therefore, from (a) - (d), the mapping X  EMBED Equation.3 (center(X), radiusB(X)) has an inverse (which is (x, j)  EMBED Equation.3 (jB)x) and consequently is a bijection. Furthermore, from (c), for any X ( BB, X ( EMBED Equation.3 , and from (d), for any (x, j) ( Z2 ( N, (x, j) ( (center((jB)x), radiusB((jB)x)). ( The radius inherits some properties of the Minkowski's addition. Proposition  SEQ Definition \* ARABIC 42 (radius properties) Let B be a finite symmetric ball (i.e., B ( S), such that B ( {o}, then (i) radiusB({o}) ( 0 (ii) radiusB (B) ( 1 (iii) for any X, Y ( BB ( {Z2}, radiusB(X ( Y) ( radiusB(X) + radiusB(Y). ( Proof (a) Let us prove (i). radiusB({o}) ( radiusB(0B) ( REF _Ref74134284 \h Definition 34 - Minkowski product) ( 0. ( REF _Ref30655022 \h Proposition 41 - characterization of the generated balls) (b) Let us prove (ii). radiusB(B) ( radiusB(1B) ( REF _Ref74134284 \h Definition 34) ( 1. ( REF _Ref30655022 \h Proposition 41) (c) Let us prove (iii). For any X, Y ( BB ( {Z2}, radiusB (X ( Y) (  EMBED Equation.3  ( REF _Ref35529502 \h Definition 40) (  EMBED Equation.3  ( REF _Ref30655022 \h Proposition 41) ( EMBED Equation.3  (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2] and center(X ( Y) ( center(X) + center(Y) by  REF _Ref26775032 \h Proposition 33 - ball addition) ( EMBED Equation.3 , ( REF _Ref32385489 \h Proposition 35 - mix distributivity of the Minkowski product) in other words, by quotient definition, radiusB(X ( Y)B ( (radiusB(X) + radiusB(Y))B, therefore, by uniqueness of the quotient (consequence of  REF _Ref32548310 \h Proposition 37), radiusB(X ( Y) ( radiusB(X) + radiusB(Y). ( Within the poset (BB, () generated by a ball B, we can identify some useful chains: the chains of generated balls with center at given points. Let B be a finite symmetric ball (i.e., B ( S), such that B ( {o}, and let BB(x) be the sub collection of BB consisting of all the balls with center at a given point x of Z2, that is, BB(x) ( {X ( BB : center(X) ( x}. In particular, BB(o) ( {jB: j ( N+} (by Property (iv) of  REF _Ref26776185 \h Proposition 28 - properties of the generated balls with center at origin). Proposition  SEQ Definition \* ARABIC 43 (chain of generated balls with center at a given point) Let B be a finite symmetric ball (i.e., B ( S), such that o ( B and B ( {o}. Then the mapping j  EMBED Equation.3 (jB)x from (N+, () to (BB(o), () when x ( o, and from (N, () to (BB(x), () when x ( o, is a poset isomorphism and the sub collection (BB(x), () is a chain. Its inverse is X  EMBED Equation.3  radiusB(X). The sub collection BB(x) has a smaller element which is {x} and a greater one which is Z2 when x ( o. ( Proof Let B be a finite symmetric ball (i.e., B ( S), such that o ( B and B ( {o}, and let x ( Z2. We observe that o ( B and B ( {o} ( #B > 1. For any j1 and j2 ( N+, j1 ( j2 ( j1B ( j2B ( REF _Ref32548310 \h Proposition 37 regularity of the Minkowski product) ( (j1B)x ( (j2B)x , (translation is injective) that is, the mapping j  EMBED Equation.3 (jB)x from N+ to BB(o) when x ( o, and from N to BB(x) when x ( o, is injective. By definition of BB(x), it is surjective, that is it a bijection. From  REF _Ref30655022 \h Proposition 41 (characterization of the generated balls), its inverse is X  EMBED Equation.3  radiusB(X). Furthermore, for any j1, j2 ( N+, j1 ( j2 ( j1B ( j2B (j  EMBED Equation.3 jB is increasing (Property (ii) of  REF _Ref32405633 \h Proposition 36)) ( (j1B)x ( (j2B)x , (translation is increasing) that is, the mapping j  EMBED Equation.3 (jB)x from N+ to BB(o) when x ( o, and from N to BB(x) when x ( o, is increasing. Conversely, let B1 and B2 be two balls in BB with center at x, and let j1 and j2 their respective radius, j2 ( j1 and j1 ( j2 ( B2 ( B1 and B1 ( B2 (j  EMBED Equation.3 (jB)x is injective and increasing) ( B2 ( B1 and (B1 ( B2 or B2 ( B1) (anti-symmetry of inclusion) ( B2 ( B1 and B1 ( B2 (logical derivation) ( B1 ( B2 , (logical derivation) in other words, B1 ( B2 ( j1 ( j2 or j1 ( j2 (above result) ( j1 ( j2 , ((N+, () is a chain) this proves that the mapping j  EMBED Equation.3 (jB)x is two-sided increasing. Consequently, the mapping j  EMBED Equation.3 (jB)x is a poset isomorphism, and the sub collection (BB(x), () is isomorphic to the chain (N+, (). That is, this sub collection is a chain as well. The smaller element (0) of N+ maps to {x} which is therefore the smaller element of BB(x), and the greater element (() of N+ maps to Z2 which is therefore the greater element of BB(x). ( Based on the mixt distributivity of the Minkowski product, we can establish the following geometrical property. This proposition shows that the intersection between generated balls occurs under the same condition as for the balls derived from an Euclidean distance on the continuous plane for example. Proposition  SEQ Definition \* ARABIC 44 (generated balls versus intersection) Let B be a finite symmetric ball (i.e., B ( S), for any points x and y in Z2 and any numbers i and j in N+, y ( ((i + j)B)x ( (iB)x ( (jB)y ( (. ( Proof Let x and y ( Z2 and let i and j ( N+, y ( ((i + j)B)x ( y ( ((iB) ( (jB))x ( REF _Ref32385489 \h Proposition 35 - mix distributivity of the product) ( y ( (iB)x ( (jB) (translation invariance of ( (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2])) ( y ( EMBED Equation.3 (equivalent definition of ( (Property (1) of Prop. 4.12 of  REF _Ref41884181 \h [2])) ( ( z ( (iB)x : y ( (jB)z (union definition) ( ( z ( (iB)x : z ( (jB)y ( REF _Ref26776185 \h Proposition 28 - properties of a ball with center at origin) ( (iB)x ( (jB)y ( (. (definition of empty set) ( We now recall the definitions of erosion by a structuring element and of B-border. We will use the latter definition in the study of the generated ball border. Definition  SEQ Definition \* ARABIC 45 (erosion by a structuring element) Let B be a subset of Z2. The erosion by B is the mappings from P(Z2) to P(Z2) (B : X  EMBED Equation.3 X ( B. The subset B is the structuring element of (B. ( Definition  SEQ Definition \* ARABIC 46 (B-border) Let B be a subset of Z2. The B-border of a subset X of Z2 is the subset (B(X) ( X (B (X). ( We observe that the B-border is an inner border: (B(X) ( X. Proposition  SEQ Definition \* ARABIC 47 (B-border properties) Let B and X be two subsets of Z2 and let x be a point in Z2, then if #B > 1 then ((B({o}) ( {o} ({o} is an invariant) ((B(X))x ( (B(Xx) (translation-invariance) X and B ( S ( (B(X) ( S. (symmetry preservation) ( Proof (a) Let us prove (i). (B({o}) ( {o} (B ({o}) ( REF _Ref26776107 \h Definition 46 - B-border) ( {o} {u ( Z2: (b ( B, ( x ( {o}: u ( x b} ( REF _Ref26774894 \h Definition 31- Minkowski's addition and subtraction) ( {o} {u ( Z2: (b ( B, u ( o b} (logical derivation) ( {o} {u ( Z2: (b ( B, u ( b} (o is unity of +) ( {o} ( (#B > 1) ( {o} (definition of and () (b) Let us prove (ii). For any x ( Z2, ((B(X))x ( (X (B (X))x ( REF _Ref26776107 \h Definition 46) ( Xx ((B (X))x (translation-invariance of ) ( Xx (X ( B)x ( REF _Ref26776140 \h Definition 45 - erosion by a subset) ( Xx Xx ( B (translation invariance of ( (Property (4) of Prop. 4.13 of  REF _Ref41884181 \h [2])) ( Xx (B (Xx) ( REF _Ref26776140 \h Definition 45) ( (B(Xx). ( REF _Ref26776107 \h Definition 46) (c) Let us prove (iii). For any u ( Z2, u ( (B(X) ( u ( X (B (X) ( REF _Ref26776107 \h Definition 46) ( u ( X (X ( B) ( REF _Ref26776140 \h Definition 45) ( u ( X and EMBED Equation.3  u ( (X ( B) (def. of ) ( u ( X and EMBED Equation.3 (b ( B, ( x ( X: u ( x b ( REF _Ref26774894 \h Definition 31 - Minkowski's addition and subtraction) ( u ( X and EMBED Equation.3 ( EMBED Equation.3 ( B, ( EMBED Equation.3 ( X: u (  EMBED Equation.3  EMBED Equation.3  (writing  EMBED Equation.3 ( x and  EMBED Equation.3 ( b, and hypothesis X and B ( S) ( u ( X and EMBED Equation.3  u ( (X ( B) ( REF _Ref26774894 \h Definition 31) ( u ( X and EMBED Equation.3  u ( (B (X) ( REF _Ref26776140 \h Definition 45) ( u ( (B(X). ( REF _Ref26776107 \h Definition 46) ( In order to be able to characterize the integer-valued t.i. regular metrics later on, we now introduce the concept of "closed" sub collection of balls. A sub collection of balls BB is B-closed if any members of BB is morphologically B-closed, that is, for any X ( BB, X satisfies the equation X ( (X ( B) ( B. If BB is B-closed then we say for convenience that B has the closure property. The balls multiple of the 3 by 3 discrete square B8 of Section 3, are examples of morphologically closed balls with respect to B8 (see Section 6). But, for example, the ball B ( (2B8 1B8) + 0B8 is not morphologically closed with respect to itself since, in this case, (B ( B) ( B ( 2B8 and consequently (B ( B) ( B ( B. Under the closure property the sufficient condition to have nested balls becomes necessary. Proposition  SEQ Definition \* ARABIC 48 (generated balls versus inclusion) Let B be a finite symmetric ball (i.e., B ( S). For any points x and y in Z2 and any numbers i in N and j in N+, (i) x ( (jB)y ( (iB)x ( ((i + j)B)y; (ii) if B has the closure property, then (iB)x ( ((i + j)B)y ( x ( (jB)y. ( Proof Let B be a finite symmetric ball (i.e., B ( S). (a) Let us prove Property (i). For any x and y ( Z2, any i ( N and j ( N+, (iB)x ( EMBED Equation.3  (x ( (jB)y) ( (jB)y ( iB (equivalent definition of ( (Property (1) of Prop. 4.12 of  REF _Ref41884181 \h [2])) ( (jB ( iB)y (translation invariance of ( (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2])) ( ((i + j)B)y , ( REF _Ref32385489 \h Proposition 35- mix distributivity of the Minkowski product) that is, x ( (jB)y ( (iB)x ( ((i + j)B)y. (b) Let us prove Property (ii). Let x and y ( Z2, let i ( N and let j ( N+. If i ( 0, x ( {x} (singleton definition) ( {o}x ( REF _Ref26767636 \h Definition 19 - translated version) ( (0B)x ( REF _Ref74134284 \h Definition 34 - Minkowski product) ( ((i + 0)B)y (hypothesis) ( (iB)y, (natural number property) that is, (0B)x ( ((0 + j)B)y ( x ( (jB)y . If i ( 0, x( {x} (singleton definition) ( (iB)x ( iB (( property) ( ((i + j)B)y ( iB (hypothesis and ( is increasing) ( ((((i 1) + j) + 1)B)y ( iB (properties of + on N+) ( ((((i 1) + j)B) ( B)y ( iB ( REF _Ref32385489 \h Proposition 35) ( ((((i 1) + j)B)y ( B) ( iB (Property (5) of Prop. 4.12 of  REF _Ref41905163 \h [2]) ( ((((i 1) + j)B)y ( B) ( (B ( (i 1)B) ( REF _Ref32385489 \h Proposition 35) ( (((((i 1) + j)B)y ( B) ( B) ( (i 1)B (Property (2) of Prop. 4.13 of  REF _Ref41884181 \h [2]) ( (((i 1) + j)B)y ( (i 1)B (B has the closure property and  REF _Ref30655022 \h Proposition 41 - characterization of the generated balls) ( (jB)y , (repeating the four previous steps (i 1) times) that is, (iB)x ( ((i + j)B)y ( x ( (jB)y . ( We observe that the last implication in Property (ii) of  REF _Ref30654522 \h Proposition 48 may be false when B has not the closure property. We can construct the following counter example: let B ( (2B8 1B8) + 0B8, let x ( 1B8 0B8 and let y ( o, then (1B)x ( (2B)y but x ( (1B)y.  REF _Ref80438122 \h  \* MERGEFORMAT Figure 6 illustrates this counter example. On the left-hand side we see that the dark gray ball (1B)x is included in the light gray ball (2B)y (the 9 by 9 square), nevertheless we see on the right-hand side that x doesn't belong to (1B)y.  EMBED Word.Picture.8  Figure  SEQ Figure \* ARABIC 6 - Counter example for  REF _Ref30654522 \h  \* MERGEFORMAT Proposition 48. The closure property for a propotype ball allows us to "go backward" along a chain of generated balls. Proposition  SEQ Definition \* ARABIC 49 (erosion of a generated ball) Let B be a subset of Z2. For any x ( Z2 and j ( N {0}, (B((jB)x) ( ((j 1)B)x if B has the closure property, then (B((jB)x) ( ((j 1)B)x. ( Proof For any subset B of Z2 having the closure property, any x ( Z2 and any j ( N {0}, (B ((jB)x) ( (jB)x ( B ( REF _Ref26776140 \h Definition 45 - erosion by a subset) ( (jB ( B)x (translation invariance of ( (Property (4) of Prop. 4.13 of  REF _Ref41884181 \h [2])) ( (((j 1)B) ( B)x ( B ( REF _Ref74134284 \h Definition 34 - Minkowski product) ( (((j 1)B)x ( B) ( B (translation invariance of ( (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2]))  EMBED Equation.3  ((j 1)B)x (( since B has the closure property and  REF _Ref30655022 \h Proposition 41 (characterization of the generated balls); ( since closing is extensive) ( Next proposition will be useful in Section 7 to show that a metric induced by a ball having the closure property is regular ( REF _Ref74295620 \h Proposition 61). Proposition  SEQ Definition \* ARABIC 50 (generated balls versus B-border) Let B be a finite symmetric ball (i.e., B ( S) 1 having the closure property and such that #B > 1. For any points x and y in Z2 and any numbers i in N and j in N+, (i) x ( (B((jB)y) ( (B(((i + j)B)y) ( (B((iB)x) ( (; (ii) x ( (B(((i + j)B)y) ( (B((iB)x) ( (B((jB)y) ( (. ( Proof (a) Let us prove Property (i). Let x and y ( Z2, let i ( N and let j ( N+. If j ( 0, x ( (B((0B)y) ( x ( (B({o}y) ( REF _Ref74134284 \h Definition 34 - Minkowski product) ( x ( ((B({o}))y (Property (ii) of  REF _Ref74042999 \h Proposition 47 - B-border properties) ( x ( {o}y (#B > 1 and Property (i) of  REF _Ref74042999 \h Proposition 47) ( x ( {y} ( REF _Ref26767636 \h Definition 19 - translated version) ( x ( y. (singleton definition) Furthermore, if x ( (B((0B)y) (B(((i + 0)B)y) ( (B((iB)x) ( (B((iB)y) ( (B((iB)x) (natural number property) ( (B((iB)x) ( (B((iB)x) (by hypothesis, as shown above x ( y) ( (B((iB)x) (( is idempotent) ( (iB)x (B((iB)x) ( REF _Ref26776107 \h Definition 46 - B-border) ( (iB)x ((i 1)B)x (B has the closure porperty and Property (ii) of  REF _Ref74193094 \h Proposition 49 - erosion of a generated ball) ( (, ( REF _Ref32548310 \h Proposition 37 - regularity of the Minkowski product) that is, x ( (B((0B)y) ( (B(((i + 0)B)y) ( (B((iB)x) ( (. If j ( 0, x ( (B((jB)y) ( x ( (jB)y (B((jB)y) ( REF _Ref26776107 \h Definition 46) ( x ( (B((jB)y) (def. of ) ( x ( ((j 1) B)y (Property (i) of  REF _Ref74193094 \h Proposition 49) ( (iB)x ( ((i + j 1)B)y (B has the closure property and Property (ii) of  REF _Ref30654522 \h Proposition 48 - generated balls versus inclusion) ( (iB)x ( (B(((i + j)B)y) (B has the closure property and Property (ii) of  REF _Ref74193094 \h  \* MERGEFORMAT Proposition 49) ( (iB)x ( ((B(((i + j)B)y))c ( (; (inclusion property) that is, x ( (B((jB)y) ( X ( ((B(Y))c ( (, where X ( (iB)x and Y ( ((i + j)B)y. Furthermore, for any j ( N+, x ( (B((jB)y) ( x ( (jB)y (B((jB)y) ( REF _Ref26776107 \h Definition 46) ( x ( (jB)y (def. of ) ( (iB)x ( ((i + j)B)y (Property (i) of  REF _Ref30654522 \h Proposition 48) ( X ( Y (definitions of X and Y) ( (B(X) ( (B(Y) (erosion is increasing (Proposition. 3.3 of  REF _Ref41884181 \h [2])) ( ((B(Y))c ( ((B(X))c. (complementation is decreasing) Hence, x ( (B((jB)y) (  EMBED Equation.3  (see above) (  EMBED Equation.3  (intersection property) ( (X ( Y) ( (((B(Y))c ( ((B(X))c) ( ( (substitution) ( (X ( ((B(X))c) ( (Y ( ((B(Y))c) ( ( (associativity and commutativity of intersection) ( (B(X ) ( (B(Y) ( (. ( REF _Ref26776107 \h Definition 46) That is, for any j ( N+, x ( (B((jB)y) ( (B(((i + j)B)y) ( (B((iB)x) ( (. (b) Let us prove Property (ii). For any x and y ( Z2, any i ( N and j ( N+. If j ( 0, x ( (B(((i + 0)B)y) ( x ( (B((iB)y) (natural number property) ( x ( ((B((iB))y (Property (ii) of  REF _Ref74042999 \h Proposition 47) ( y ( ((B((iB))x ( REF _Ref26776185 \h Proposition 28 (properties of a ball with center at origin) and Property (iii) of  REF _Ref74042999 \h Proposition 47) ( y ( ((B((iB))x and y ( {y} (singleton definition) ( y ( ((B((iB))x and y ( {o}y ( REF _Ref26767636 \h Definition 19) ( y ( ((B((iB))x and y ( ((B({o}))y (#B > 1 and Property (i) of  REF _Ref74042999 \h Proposition 47) ( y ( ((B((iB))x and y ( (B({o}y) (Property (ii) of  REF _Ref74042999 \h Proposition 47 - B-border properties) ( y ( ((B((iB))x and y ( (B((0B)y) ( REF _Ref74134284 \h Definition 34) ( ((B((iB))x ( (B((0B)y) ( (. (intersection and empty set definitions) If j ( 0, x ( (B(((i + j)B)y) ( x ( ((i + j)B)y (B(((i + j)B)y) ( REF _Ref26776107 \h Definition 46) ( x ( (B(((i + j)B)y) (def. of ) ( x ( ((i + j 1) B)y (Proprety (i) of  REF _Ref74193094 \h  \* MERGEFORMAT Proposition 49) ( (iB)x ( ((j 1) B)y ( (, ( REF _Ref33112159 \h Proposition 44 - generated balls versus intersection) ( (iB)x ( (((j 1) B)y)c (property of () ( (iB)x ( ((B((jB)y))c, (B has the closure porperty and Property (ii) of  REF _Ref74193094 \h  \* MERGEFORMAT Proposition 49) that is, x ( (B(((i + j)B)y) ( X ( ((B(Y))c, where X ( (iB)x and Y ( (jB)y. Exploring the symmetry role between X and Y, x ( (B(((i + j)B)y) ( x ( ((B((i + j)B))y (Property (ii) of  REF _Ref74042999 \h Proposition 47) ( y ( ((B((i + j)B))x ( REF _Ref26776185 \h Proposition 28 (properties of a ball with center at origin) and Property (iii) of  REF _Ref74042999 \h Proposition 47) ( (jB)y ( ((B((iB)x))c, (as above) that is, x ( (B(((i + j)B)y) ( Y ( ((B(X))c. Furthermore, x ( (B(((i + j)B)y) ( x ( ((i + j)B)y (B(((i + j)B)y) ( REF _Ref26776107 \h Definition 46) ( x ( ((i + j)B)y (def. of ) ( (iB)x ( (jB)y ( (. ( REF _Ref33112159 \h Proposition 44 - generated balls versus intersection) ( X ( Y ( (. (definitions of X and Y) Hence, x ( (B(((i + j)B)y) (  EMBED Equation.3  (see above) (  EMBED Equation.3  (intersection property) ( (X ( ((B(Y))c) ( (Y ( ((B(X))c) ( ( (substitution) ( (X ( ((B(X))c) ( (Y ( ((B(Y))c) ( ( (associativity and commutativity of intersection) ( (B(X ) ( (B(Y) ( (. ( REF _Ref26776107 \h Definition 46) That is, for any j ( N+, x ( (B(((i + j)B)y) ( (B((iB)x) ( (B((jB)y) ( (. ( We observe that the implication in Property (i) of  REF _Ref74289529 \h Proposition 50 may be false when BB is not B-closed, for example, let B ( (2B8 1B8) + 0B8, let x ( 1B8 0B8 and let y ( o, then x ( (B((2B)y) but (B((3B)y) ( (B((1B)x) ( (.  REF _Ref80445076 \h  \* MERGEFORMAT Figure 7 illustrates this counter example. On this figure we see that the point x belongs to the dark gray ball border (B((2B)y), nevertheless we see that the light gray ball border (B((3B)y) has no intersection with the ball border (B((1B)x) (depicted as a set of squares).  EMBED Word.Picture.8  Figure  SEQ Figure \* ARABIC 7 - Counter example for  REF _Ref74289529 \h  \* MERGEFORMAT Proposition 50. In the last proposition of this section we present some properties of the radii of generated balls. Part of these properties will be used in the proof of  REF _Ref73786079 \h Proposition 59. Properties (ii) and (iii) will be used in future work. Proposition  SEQ Definition \* ARABIC 51 (radius properties of nested generated balls) Let B be a finite symmetric ball (i.e., B ( S), such that o ( B and B ( {o}. For any balls X and Y in BB, (i) X ( Y ( radiusB(X) ( radiusB(Y); (ii) X ( Y and X ( Y ( radiusB(X) < radiusB(Y); (iii) (center(Y) ( Bcenter(X) and radiusB(X) < radiusB(Y)) ( X ( Y. (center(Y) ( center(X) and radiusB(X) ( radiusB(Y)) ( X ( Y. ( Proof For any balls X and Y in BB, let x ( center(X), let y ( center(Y), let i ( radiusB(X) and let j ( radiusB(Y). (a) Let us prove (i): X ( Y ( (radiusB(X)B)center(X) ( (radiusB(Y)B)center(Y) ( REF _Ref30655022 \h Proposition 41 - characterization of the generated balls) ( (iB)x ( (jB)y (definitions of x, y, i and j) ( #(iB)x ( #(jB)y (property of the cardinality of nested subsets) ( #(iB) ( #(jB) (X  EMBED Equation.3 Xu is a bijection) ( iB ( jB or jB ( iB (property of the cardinality of nested subsets) ( iB ( jB ( REF _Ref26778740 \h  \* MERGEFORMAT Proposition 43 - chain of generated balls with center at a given point (here the point is the origin o)) ( i ( j (j  EMBED Equation.3 jB is a poset isomorphism ( REF _Ref26778740 \h Proposition 43)) ( radiusB(X) ( radiusB(Y) (definitions of i and j) (b) Let us prove (ii): X ( Y and X ( Y ( (iB)x ( (jB)y and (iB)x ( (jB)y ( REF _Ref30655022 \h  \* MERGEFORMAT Proposition 41 and definitions of x, y, i and j) ( #(iB)x < #(jB)y (property of the cardinality of nested subsets) ( #(iB) < #(jB) (X  EMBED Equation.3 Xu is a bijection) ( jB ( iB (property of the cardinality of nested subsets) ( iB ( jB and iB ( jB ( REF _Ref26778740 \h  \* MERGEFORMAT Proposition 43) ( i < j (j  EMBED Equation.3 jB is a poset isomorphism ( REF _Ref26778740 \h Proposition 43)) ( radiusB(X) < radiusB(Y) (definitions of i and j) (c) Let us prove (iii): center(Y) ( Bcenter(X) ( y ( Bx (definitions of x and y) ( x ( By ( REF _Ref26776185 \h Proposition 28 - properties of a ball with center at origin) ( (iB)x ( ((i + 1)B)y (Property (i) of  REF _Ref30654522 \h Proposition 48 (generated balls versus inclusion) with j ( 1) ( (radiusB(X)B)center(X) ( ((i + 1)B)y (definitions of x and i) ( X ( ((i + 1)B)y ( REF _Ref30655022 \h Proposition 41) ( X ( ((radiusB(X) + 1)B)y (definitions of i) ( X ( ((radiusB(Y))B)y (radiusB(X) + 1 ( radiusB(Y) and the mapping j  EMBED Equation.3 (jB)y is a poset isomorphism ( REF _Ref26778740 \h  \* MERGEFORMAT Proposition 43)) ( X ( ((radiusB(Y))B)center(Y) (definitions of y) ( X ( Y. ( REF _Ref30655022 \h Proposition 41) (d) Let us prove (iv): true ( center(Y) ( center(X) and radiusB(X) ( radiusB(Y) (hypothesis) ( x ( y and i ( j (def. of x, y, i and j) ( x ( y and (iB)x ( (jB)y (the mapping j  EMBED Equation.3 (jB)x is a poset isomorphism ( REF _Ref26778740 \h  \* MERGEFORMAT Proposition 43)) ( (iB)x ( (jB)y (logical derivation) ( (radiusB(X)B)center(X) ( (radiusB(Y)B)center(Y) (def. of x, y, i and j) ( X ( Y. ( REF _Ref30655022 \h Proposition 41) ( We are now ready to begin the study of the relationship between metrics and symmetric balls. 6. From metric to symmetric ball With a t.i. metric, we can associate a ball with center at origin. Let (Z2, d) be a t.i. metric space, the unit ball of (Z2, d), denoted by Bd, is the set of all the points at a distance less than or equal to one from the origin, that is, Bd ( {u ( Z2: fd(u) ( 1}. An exhaustive inspection of the nine points of B8 shows that  EMBED Equation.3 , in other words, the 3 by 3 discrete square is the unit ball of the chessboard metric space (Z2, d8). In the next proposition, we show three properties of the unit ball. Proposition  SEQ Definition \* ARABIC 52 (properties of the unit ball) For any metric d on Z2, Bd ( S, o ( Bd and (iii) Bd ( {o}. ( Proof (a) Let us prove (i). For any u ( Z2, u ( Bd ( d(u, o) ( 1 (def. of Bd) ( d( u, o) ( 1 (symmetry of d) ( u ( Bd, (def. of Bd) that is, Bd ( S. (b) Let us prove (ii): true ( d(o, o) ( 0 (( of Properties (i) of  REF _Ref69010114 \h Definition 1 - metric space) ( d(o, o) ( 1, (natural number property) that is, by definition of Bd, o ( Bd. (c) Let us prove (iii). For any u ( Z2, d(u, o) ( 1 ( u ( o, (( of Properties (i) of  REF _Ref69010114 \h Definition 1 - metric space) that is, Bd ( {o}. ( Property (i) of the previous proposition shows that the unit ball is really a ball in the sense of  REF _Ref26766674 \h Definition 22 and that its center is the origin. In the next proposition, we show a relationship between a t.i. lower regular metric and the Minkowski product of its unit ball. Proposition  SEQ Definition \* ARABIC 53 (property of the generated balls in a lower regular metric space) Let (Z2, d) be a t.i. metric space, for any x and y ( Z2 and any j ( N, (i) x y ( jBd ( d(x, y) ( j (ii) if d is lower regular, then d(x, y) ( j ( x y ( jBd. ( Proof Let us make a recursive proof. For any x and y ( Z2, x y ( 1Bd ( x y ( Bd ( REF _Ref74134284 \h Definition 34 - Minkowski product) ( fd(x y) ( 1 (unit ball def.) ( d(x y, o) ( 1 (fd def.) ( d(x, y) ( 1, (translation-invariance of d) That is, x y ( jBd ( d(x, y) ( j is true for j ( 1. Let us assume it was true for j 1 (j > 1), for any x and y ( Z2, x y ( jBd ( x y ( ((j 1)Bd) ( Bd ( REF _Ref74134284 \h Definition 34) ( (u ( Bd : x y ( ((j 1)Bd)u (equivalent definition of ( (Property (1) of Prop. 4.12 of  REF _Ref41884181 \h [2])) ( (u ( Bd : x (y + u) ( (j 1)Bd ( REF _Ref26767636 \h Definition 19 - translated version) ( (u ( Z2 : u ( Bd and x (y + u) ( (j 1)Bd (logical derivation) ( (v ( Z2 : x v ( Bd and v y ( (j 1)Bd (v ( x u) ( (v ( Z2 : x v ( 1Bd and v y ( (j 1)Bd ( REF _Ref74134284 \h Definition 34) ( (v ( Z2 : d(x, v) ( 1 and d(v, y) ( j 1 (hypothesis and first part of the proof) ( d(x, y) ( j. (by substitution of i and j of  REF _Ref26778152 \h Proposition 16 (ball intersection in a lower regular metric space), by j 1 and 1, respectively; hence, ( is true under the hypothesis that d is lower regular) ( The next corollary illustrates better the previous proposition. We recall that Bd(y, j) is the ball of center y and radius j, derived from the distance d (see Section 3). Comparing the notations, we have Bd(o, 1) ( Bd. Corollary  SEQ Definition \* ARABIC 54 (property of the generated balls in a lower regular metric space) Let (Z2, d) be a t.i. metric space, for any y ( Z2 and any j ( N, (i) (jBd(o, 1))y ( Bd(y, j) (ii) if d is lower regular, then Bd(y, j) ( (jBd(o, 1))y. ( Proof For any x and y ( Z2, and any j ( N, x ( (jBd(o, 1))y ( x ( (j{z ( Z2: d(o, z) ( 1})y (def. of Bd(y, j)) ( x y ( j{z ( Z2: d(o, z) ( 1}  REF _Ref26767636 \h Definition 19 - translated version) ( x y ( j{z ( Z2: d(z, o) ( 1} (symmetry of d) ( x y ( jBd (def. of Bd) ( d(x, y) ( j (( by Property (i) of  REF _Ref26778315 \h Proposition 53 (property of the generated balls in a lower regular metric space); ( by Property (ii) of  REF _Ref26778315 \h Proposition 53 and hypothesis) ( d(y, x) ( j (symmetry of d) ( x ( Bd(y, j). (def. of Bd(y, j)) ( We can apply  REF _Ref26778315 \h Proposition 53 to the chessboard distance. In this way, we see how to obtain the chessboard balls by using the Minkowski product. Corollary  SEQ Definition \* ARABIC 55 (property of the Minkowski product of the unit ball of the chessboard metric space) If d8 is the chessboard distance, then for any x and y ( Z2, and any j ( N, x y ( jB8 ( d8(x, y) ( j. ( Proof The result is a consequence of  REF _Ref26778315 \h Proposition 53 (property of the generated balls in a lower regular metric space) since B8 is the unit ball of the chessboard metric space and d8 is regular by  REF _Ref26778358 \h Proposition 15 (example of regular metric space). ( Next proposition, which is consequence  REF _Ref26778315 \h Proposition 53, will be used in Section 8 to characterize the regular metric. Proposition  SEQ Definition \* ARABIC 56 (closure property of the unit ball of a regular metric space) Let (Z2, d) be a t.i. metric space. If d is regular, then Bd has the closure property. ( Proof Let assume that d is regular. We divide the proof in four parts. (a) Let us prove that for any j ( N and any x ( Z2, (Bd)x ( (j + 1)Bd ( d(x, o) ( j, (Bd)x ( (j + 1)Bd ( (1Bd)x ( (j + 1)Bd ( REF _Ref74134284 \h Definition 34 - Minkowski product) ( (1Bd(o, 1))x ( (j + 1)Bd(o, 1) (definitions of Bd(y, j) and Bd) ( (1Bd(o, 1))x ( ((j + 1)Bd(o, 1))o (o is unit element of +) ( Bd(x, 1) ( Bd(o, j + 1) ( REF _Ref82089127 \h Corollary 54 (property of the generated balls in a lower regular metric space) , ( is true under the hypothesis that d is lower regular) ( 1 + d(x, o) ( j + 1. (d is upper regular and Property (ii) of  REF _Ref82088452 \h Proposition 18 - ball inclusion in an upper regular metric space) ( d(x, o) ( j. (+ is double-side increasing) (b) Let us prove that, for any j ( N, (((jBd) ( Bd) ( Bd) ( jBd. For any j ( N and any x ( Z2, x ( ((jBd) ( Bd) ( Bd ( x ( ((j +1)Bd) ( Bd (product def.) ( (Bd)x ( (j + 1)Bd (Minkowski's subtraction property) ( d(x, o) ( j (Part (a)) ( x ( jBd, (hypothesis (d is lower regular) and Property (ii) of  REF _Ref26778315 \h Proposition 53) that is, for any j ( N, (((jBd) ( Bd) ( Bd) ( jBd . (c) For any j ( N, ((jBd) ( Bd) ( Bd is the closing of jBd by Bd, therefore by closing extensivity (Property (3) of Prop. 6.21 of  REF _Ref41884181 \h [2]) jBd ( (((jBd) ( Bd) ( Bd). Hence, from Parts (b) and (c), if d is regular, then, for any j ( N, (((jBd) ( Bd) ( Bd) ( jBd, in other words, jBd is Bd-closed. (d) For any X ( BB, let x ( center(X) and let i ( radiusB(X), (X ( Bd) ( Bd) ( ((radiusB(X)Bd)center(X) ( Bd) ( Bd ( REF _Ref30655022 \h Proposition 41 - characterization of the generated balls) ( ((iBd)x ( Bd) ( Bd (def. of x and i) ( ((iBd) ( Bd)x ( Bd (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( (((iBd) ( Bd) ( Bd)x (Property (5) of Prop. 4.12 of  REF _Ref41884181 \h [2]) ( (iBd)x (Part (c)) ( (radiusB(X)Bd)center(X) ( X, ( REF _Ref30655022 \h Proposition 41) that is, if d is regular, then EMBED Equation.3 is Bd-closed. ( Based on the regularity of the Chessboard distance, we have the following result. Corollary  SEQ Definition \* ARABIC 57 (closure property of the Minkowski product of the 3 by 3 discrete square) Let B8 be the 3 by 3 discrete square with center at origin, for any j ( N, jB8 is B8-closed. ( Proof By  REF _Ref26778358 \h Proposition 15 (example of geometrical metric space) (Z2, d8) is regular. Furthermore, B8 is the unit ball of (Z2, d8):  EMBED Equation.3 . Therefore, by  REF _Ref41817758 \h Proposition 56 (closure property of the unit ball of a regular metric space), for any j ( N, jB8 is B8-closed. ( Let Bc be the collection of finite symmetric balls (i.e., B ( S) having the closure property and such that o ( B and B ( {o}, that is, Bc ( {B ( S: BB is B-closed, o ( B and B ( {o}}. Let Mr be the set of integer-valued t.i. regular metrics, that is, Mr ( {d (  EMBED Equation.3 : d is t.i. and regular}. Before closing this section, we verify that the image of Mr through d  EMBED Equation.3 Bd is contained in Bc. In other words, the unit ball of a metric space which metric is integer-valued t.i. regular and defined on the discrete plane has the closure property. Proposition  SEQ Definition \* ARABIC 58 (properties of the Minkowski product of the unit ball of a regular metric space) (d  EMBED Equation.3 Bd)(Mr) ( Bc. ( Proof By  REF _Ref80160426 \h Proposition 52 (properties of the unit ball) Bd ( S, o ( B and B ( {o}, and by  REF _Ref41817758 \h Proposition 56 (closure property of the unit ball of a regular metric space) d being regular, BB is Bd-closed. Therefore, for any d ( Mr, Bd ( Bc. ( 7. From symmetric ball to metric By using the Minkowski product, with a symmetric ball containing the origin, we can associate a t.i. metric. Let B be a finite symmetric ball (i.e., B ( S), such that o ( B and B ( {o}. For any x ( Z2, let Mx (  EMBED Equation.3 . By  REF _Ref26778740 \h Proposition 43 (chain of generated balls with center at a given point), Mx ( BB(o) and it is the smallest ball with center at origin that contains the point x. Let fB be the mapping from Z2 to N+ given by, for any x ( Z2, fB(x) ( radiusB(Mx). In the next proposition, we give the relationship between the ball border and its radius. Property (i) will be useful to prove the subadditivity of fB, and Property (iii) to prove its regularity. Proposition  SEQ Definition \* ARABIC 59 (ball border versus ball radius) Let B be a finite symmetric ball (i.e., B ( S), such that o ( B and B ( {o}. For any x ( Z2 and any finite X ( BB(o), (i) x ( X ( fB(x) ( radiusB(X), (ii) x ( (BX ( fB(x) ( radiusB(X), (iii) if B has the closure property, then fB(x) ( radiusB(X) ( x ( (BX. ( Proof Let x ( Z2, let Mx ( EMBED Equation.3  and let X ( BB(o). (a) Let us prove ( of (i). First, x ( X ( X ( {U ( BB(o): x ( U} (set definition) ( Mx ( X. (definition of Mx and property of  EMBED Equation.3 ) Second, fB(x) ( radiusB(Mx) (definitions of Mx and fB(x)) ( radiusB(X). (Mx ( X and Property (i) of  REF _Ref26780560 \h Proposition 51 - radius properties of nested generated balls) (b) Let us prove ( of (i). First, radiusB(Mx) ( fB(x) (definitions of Mx and fB(x)) ( radiusB(X), (hypothesis) that is, under the hypothesis, radiusB(Mx) ( radiusB(X). Second, x ( M (property of  EMBED Equation.3 ) ( X, (radiusB(Mx) ( radiusB(X) and Property (iv) of  REF _Ref26780560 \h Proposition 51) that is, under the hypothesis x ( X. (c) Let us prove Properties (ii) and (iii). Let us assume that B has the closure property and let j ( radiusB(X), x ( (B(X) ( x ( X  EMBED Equation.3  ( REF _Ref26776107 \h Definition 46 - B-border) ( x ( jB  EMBED Equation.3  (X ( jB by  REF _Ref30655022 \h Proposition 41 - characterization of the generated balls) ( x ( jB ((j 1)B) (( by Property (i) of  REF _Ref73980900 \h Proposition 49 (erosion of a generated ball) and ( by Property (ii) of  REF _Ref73980900 \h Proposition 49 since B has the closure property) ( x ( jB and x ( (j 1)B (def. of the set difference) ( fB(x) ( radiusB(jB) and radiusB((j 1)B) < fB(x) (Property (i)) ( fB(x) ( j and (j 1) < fB(x) ( REF _Ref30655022 \h Proposition 41) ( j ( fB(x) ( j (property of < on N) ( fB(x) ( j (anti-symmetry of () ( fB(x) ( radiusB(X). (definition of j) ( The mapping fB has almost all the properties of a norm as shown in the next proposition. Nevertheless, it doesn't verify the property: fB(jx) (  EMBED Equation.3 fB(x) for any j ( Z (for example, if B is a five by five square and x ( (1, 1), then fB(2x) ( 1 and 2fB(x) ( 2), instead it has a weaker property: the symmetry. Proposition  SEQ Definition \* ARABIC 60 (properties of fB) Let B be a finite symmetric ball (i.e., B ( S), such that o ( B and B ( {o}. For any x and y ( Z2, (i) fB(x) ( 0, (positiveness) (ii) x ( o ( fB(x) ( 0, (iii) fB(( x) ( fB(x), (symmetry) (iv) fB(x + y) ( fB(x) + fB(y). (subadditivity) ( Proof Let us prove (i). Since the radius of a ball is a natural number (see  REF _Ref35529502 \h Definition 40 - radius of a generated ball), for any x ( Z2, fB(x) ( 0. Let us prove (ii).For any x ( Z2, x ( o ( fB(x) ( radiusB( EMBED Equation.3 ) (def. of fB(x)) ( fB(x) ( radiusB({o}) ( REF _Ref26778740 \h Proposition 43 - chain of generated balls with center at a given point) ( fB(x) ( 0. (Property (i) of  REF _Ref34278866 \h Proposition 42 - radius property) Let us prove (iii). For any x ( Z2, fB(( x) ( radiusB( EMBED Equation.3 ) (def. of fB(x)) ( radiusB( EMBED Equation.3 ) (BB(o) ( S+) ( fB(x). (def. of fB(x)) Let us prove (iv). Let us divide the proof in three parts. (a) For any x ( Z2, let us prove that x ( fB(x)B, true ( fB(x) ( fB(x) (reflexivity of () ( fB(x) ( radiusB(fB(x)B) (fB(x)B ( BB(o) (definition of BB(o)) and  REF _Ref30655022 \h Proposition 41 - characterization of the generated balls) ( x ( fB(x)B. (Property (i) of  REF _Ref73786079 \h Proposition 59 - ball border versus ball radius) (b) For any x, y ( Z2, let prove that x + y ( (fB(x) + fB(x))B: true ( x ( fB(x)B and y ( fB(y)B (Part (a)) ( x + y ( (fB(x)B) ( (fB(y)B) ( REF _Ref26774894 \h Definition 31 - Minkowski's addition) ( x + y ( (fB(x) + fB(y))B ( REF _Ref32385489 \h Proposition 35 - mix distributivity of the product) (c) For any x, y ( Z2, fB(x + y) ( radiusB((fB(x) + fB(y))B) (Part (b) and Property (i) of  REF _Ref73786079 \h Proposition 59) ( fB(x) + fB(y). ( REF _Ref30655022 \h Proposition 41) ( With a mapping fB we can associate the mapping dB (  EMBED Equation.3 . By  REF _Ref69177667 \h Proposition 9 (characterization of translation-invariant metric) and  REF _Ref37084802 \h Proposition 60 (properties of fB) dB is a t.i. metric; we call it the metric induced by B or simply induced metric. Let x and y ( Z2, from the definition of df in Section 2, we have dB(x, y) ( fB(x y). It is interesting to note that by substituing the Minkowski product by the scaling operations: scaling or scaling2, of Section 4, the above construction doesn't lead to a metric. As we can see in  REF _Ref80075783 \h  \* MERGEFORMAT Figure 8, the triangle inequality is not satisfied when the induced distance is obtained by using the scaling transformations. Before closing this section, we verify that the image of Bc through B  EMBED Equation.3 dB is contained in Mr.  EMBED Word.Picture.8  Figure  SEQ Figure \* ARABIC 8 - Induced distances using Minkowski product and scaling. Proposition  SEQ Definition \* ARABIC 61 (properties of the metric induced by a ball having the closure property) (B  EMBED Equation.3 dB)(Bc) ( Mr. ( Proof We must verify four properties. (a) By  REF _Ref69177667 \h Proposition 9 (characterization of translation-invariant metric) and  REF _Ref37084802 \h Proposition 60 (properties of fB) dB is a t.i. metric. (b) For any B in Bc, by construction, fB is a mapping from Z2 to N+, therefore, by definition of induced metric, dB(Z2 ( Z2) ( N+. (c) Let us prove the lower regularity. Let x and y ( Z2, let i ( N, and let j ( dB(x, y) such that i ( j, true ( dB(x, y) ( j (definition of j) ( fB(x y) ( j (definition of induced metric) ( fB(x y) ( radiusB(jB) ( REF _Ref30655022 \h Proposition 41 - characterization of the generated balls) ( x y ( (B(jB) (B has the closure property and Property (iii) of  REF _Ref73786079 \h Proposition 59 - ball border versus ball radius) ( x ( ((B((jB))y ( REF _Ref26767636 \h Definition 19 - translated version) ( x ( (B((jB)y) (Property (ii) of  REF _Ref74042999 \h Proposition 47 - B-border properties) ( (B((iB)x) ( (B(((j i)B)y) ( ( (Property (ii) of  REF _Ref76551731 \h Proposition 50 - generated balls versus B-border) ( ( u ( Z2: u ( (B((iB)x) and u ( (B(((j i)B)y) (intersection and empty set definitions) ( ( u ( Z2: u ( ((B(iB))x and u ( ((B((j i)B))y (Property (ii) of  REF _Ref74042999 \h Proposition 47) ( ( u ( Z2: u x ( (B(iB) and u y ( (B((j i)B) ( REF _Ref26767636 \h Definition 19) ( ( u ( Z2: fB(u x) ( radiusB(iB) and fB(u y) ( radiusB((j i)B) (Property (ii) of  REF _Ref73786079 \h Proposition 59) ( ( u ( Z2: fB(u x) ( i and fB(u y) ( j i ( REF _Ref30655022 \h Proposition 41) ( ( u ( Z2: dB(u, x) ( i and dB(u, y) ( j i (definition of induced metric) ( ( u ( Z2: dB(x, u) ( i and dB(u, y) ( j i (symmetry of dB) ( ( u ( Z2: dB(x, y) ( dB(x, u) + dB(u, y) and dB(x, u) ( i (definition of j and substitution) ( S(x, i) ( L2(x, y) ( ( (definitions of S and L2 with respect to dB) ( dB is lower regular. ( REF _Ref78512758 \h Corollary 5 - first equivalent definition of regular metric) (d) Let us prove the upper regularity. Let x and y ( Z2, let i ( N, and let j ( dB(x, y), true ( dB(x, y) ( j (definition of j) ( x ( (B((jB)y) (as in (b)) ( (B(((i + j)B)y) ( (B((iB)x) ( ( (Property (i) of  REF _Ref76551731 \h Proposition 50) ( ( u ( Z2: dB(u, y) ( i + j and dB(u, x) ( i (as in (b)) ( ( u ( Z2: dB(u, y) ( dB(u, x) + dB(x, y) and dB(u, x) ( i (definition of j and substitution) ( S(x, i) ( L3(x, y) ( ( (definitions of S and L3 with respect to dB) ( dB is upper regular. ( REF _Ref41052844 \h Definition 3) Hence, from (a) - (d) and by  REF _Ref78512758 \h Corollary 5, for any B ( Bc, dB ( Mr. ( 8. Relationship between regular metrics and symmetric balls having the closure property We now establish two propositions showing that the mapping B  EMBED Equation.3 dB of the Section 7 is a left and right inverse of the mapping d  EMBED Equation.3 Bd of Section 6. Let M2 be the set of integer-valued t.i. lower regular metrics, that is, M2 ( {d (  EMBED Equation.3 : d is t.i. and lower regular}. Proposition  SEQ Definition \* ARABIC 62 (existence of a left inverse) The mapping B  EMBED Equation.3 dB is a left inverse for the mapping d  EMBED Equation.3 Bd from M2 to the collection of finite symmetric balls B (i.e., B ( S), such that o ( B and B ( {o}. ( Proof (a) From  REF _Ref80160426 \h Proposition 52 (properties of the unit ball), for any metric d, Bd ( S, o ( Bd and Bd ( {o}. (b) For any d ( M2, any x and y ( Z2 and any j ( N, d(x, y) ( j ( x y ( jBd (d is lower regular and  REF _Ref26778315 \h Proposition 53 - property of the generated balls in a lower regular metric space) (  EMBED Equation.3  (Property (i) of  REF _Ref73786079 \h Proposition 59 - ball border versus ball radius) (  EMBED Equation.3  ( REF _Ref30655022 \h \* MERGEFORMAT Proposition 41 - characterization of the generated balls) (  EMBED Equation.3 . (definition of induced metric) That is, by anti-symmetry of (, for any x and y ( Z2, d(x, y) (  EMBED Equation.3 . In other words, by mapping equality definition, d (  EMBED Equation.3 . (  REF _Ref26781288 \h Proposition 62 shows that every integer-valued t.i. lower regular metric can be reconstructed from its unit ball using the Minkowski product. Since Mr ( M2, this is also true for the regular metric. In other words, its unit ball uniquely defines any (lower) regular metric space. Corollary  SEQ Definition \* ARABIC 63 (chessboard distance as a derived distance) If d8 is the chessboard distance, then d8 ( EMBED Equation.3 . ( Proof Since B8 is the unit ball of the chessboard metric space,  EMBED Equation.3 . By  REF _Ref26778358 \h Proposition 15 (example of regular metric space) (d8, Z2) is a regular metric space, therefore by  REF _Ref26781288 \h Proposition 62 (existence of a left inverse) d8 ( EMBED Equation.3 . ( Proposition  SEQ Definition \* ARABIC 64 (existence of a right inverse) The mapping B  EMBED Equation.3 dB is a right inverse for the mapping d  EMBED Equation.3 Bd from Mr to Bc. ( Proof (a) By  REF _Ref74295620 \h Proposition 61 (properties of the metric induced by a ball having the closure property) for any B ( Bc, dB ( Mr. (b) For any B ( Bc,  EMBED Equation.3 ( {u ( Z2: dB(u, o) ( 1} (definition of Bd) ( {u ( Z2: fB(u) ( 1} (definition of induced metric) ( {u ( Z2: radiusB( EMBED Equation.3 ) ( 1} (definition of fB) ( {u ( Z2: u ( {o} or u ( B} (by  REF _Ref26778740 \h Proposition 43 (chain of generated balls with center at a given point) BB(o) is a chain and {o} and B are the only two generated balls with radius less than or equal to 1) ( {u ( Z2: u ( B} (o ( B) ( B. (set definition) That is,  EMBED Equation.3 ( B. ( We are now ready to state our characterization theorem. Theorem  SEQ Definition \* ARABIC 65 (characterization of integer-valued translation-invariant regular metrics) The mapping d  EMBED Equation.3 Bd from Mr to Bc is a bijection. Its inverse is the mapping B  EMBED Equation.3 dB. ( Proof Let us divide the proof into two parts. (a) By  REF _Ref74322234 \h Proposition 58 (properties of the Minkowski product of the unit ball of a regular metric space) for any d ( Mr, Bd ( Bc. By  REF _Ref26781288 \h Proposition 62 (existence of a left inverse), d  EMBED Equation.3 Bd from Mr ( M2 to Bc is one-to-one. (b) By  REF _Ref74322953 \h Proposition 64 (existence of a right inverse), d  EMBED Equation.3 Bd from Mr to Bc is onto. Hence, from (a) and (b), d  EMBED Equation.3 Bd from Mr to Bc is a bijection and its inverse is B  EMBED Equation.3 dB. ( 9. Conclusion In the first part of this work we have introduced a definition of regular metric space and showed its relation to the Kiselman's regularity axioms for translation-invariant metrics. We have shown, in particular, that the lower regularity of type 1 is a redundant axiom in the definition of regular metrics. In the second part, we have established a one-to-one relationship between the set of integer-valued and translation-invariant regular metrics defined on the discrete plane, and the set of symmetric balls satisfying a special closure property. From this result we now know how to construct a regular metric on the discrete plane. To this end, we choose in the discrete plane a symmetric ball B that has the closure property, i.e., that induces, through the Minkowski product, a chain of generated balls that are morphologically closed with respect to B. Then the distance of a point x to the origin is given by the radius (in the sense of the Minkowski product) of the smallest ball of the chain, that contains x. This construction shows that to preserve in the discrete plane the regularity property of the Euclidean metric on the continuous plane, we have to reach a compromise between a good approximation of a continuous ball and thin contours. "Closer" B from an Euclidean continuous ball, bigger B and thicker the borders in the discrete plane. Actually it will be interesting in a future work to give a proof that if B is the intersection of an Euclidean continuous ball with the discrete plane then the generated balls are morphologically closed with respect to B. The proof should be based on a closure property of the convex subsets of the continuous plane  REF _Ref41884991 \h [3, Proposition 9.8]. The regular metric characterization we have proved will also be very useful to derive in future work some important geometrical properties of the skeletons of "expanded" subsets of the discrete plane  REF _Ref81734440 \h [1]. Acknowledgments We wish to thank Arley F. Souza for the compelling discussions about isotropic skeletons. This work was partially supported by CNPq (Conselho Nacional de Desenvolvimento Cientfico e Tecnolgico) under contracts 300966/903. References [ SEQ Reference \* ARABIC 1] Banon, G. J. F. New insight on digital topology. In: International Symposium on Mathematical Morphology and its Applications to Image and Signal Processing, 5., 26-28 June 2000, Palo Alto, USA. Proceedings... 2000. p. 138-148. Published as: INPE-7884-PRE/3724. [ SEQ Reference \* ARABIC 2] Banon, G. J. F.; Barrera, J. Bases da morfologia matemtica para a anlise de imagens binrias. 2. ed. So Jos dos Campos: INPE, 1998. Posted at: <HYPERLINK "http://150.163.8.23:1905/rep/dpi.inpe.br/banon/1998/06.30.17.56?mirror=dpi.inpe.br/banon/2001/01.11.16.21.34&metadatarepository=dpi.inpe.br/banon/2001/01.11.16.06.25"http://iris.sid.inpe.br:1912/rep/dpi.inpe.br/banon/1998/06.30.17.56>. Access in: 2003, Apr.17. [ SEQ Reference \* ARABIC 3] Heijmans, H. J. A. M. Morphological image operators. Boston: Academic, 1994. [ SEQ Reference \* ARABIC 4] Birkhoff, G. Lattice theory. 3. ed. Providence, Rhode Island: American Mathematical Society, 1967. [ SEQ Reference \* ARABIC 5] Kiselman, C. O. Digital geometry and mathematical morphology. S.n.t. Lecture Notes, Spring Semester, 2002. Posted at: . Access in 2002, Dec. 18. [ SEQ Reference \* ARABIC 6] Serra, J. Image analysis and mathematical morphology. London: Academic. 1982. [ SEQ Reference \* ARABIC 7] Szsz, G. Thorie des treillis. Paris: Dunod, 1971. 227p. [ SEQ Reference \* ARABIC 8] Rosenfeld, A.; Pfaltz, J. L. Digital functions on digital pictures. Pattern Recognition, v.1, n. 1, p. 33-61, 1968. 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