Discrete Applied Mathematics. Weighted distances based on neighborhood sequences for point-lattices

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1 Discrete Applied Mathematics Contents lists available at ScienceDirect Discrete Applied Mathematics journal homepage: wwwelseviercom/locate/dam Weighted distances based on neighborhood sequences for point-lattices Robin Strand Centre for Image Analysis, Uppsala University, Box 337, SE Uppsala, Sweden a r t i c l e i n f o a b s t r a c t Article history: Received 7 February 008 Received in revised form 7 August 008 Accepted 14 August 008 Available online 5 September 008 Keywords: Digital geometry Distance functions Image processing Neighborhood sequences A path-based distance is defined as the minimal cost-path between two points One such distance function is the weighted distance based on a neighborhood sequence It can be defined using any number of neighborhood relations and weights in conjunction with a neighborhood sequence The neighborhood sequence restricts some steps in the path to a smaller neighborhood We give formulas for computing the point-to-point distance and conditions for metricity for weighted distances based on neighborhood sequences with two neighborhood relations for the general case of point-lattices 008 Elsevier BV All rights reserved 1 Introduction There are two ways to attac the problem of image processing We need to put the images in a framewor that is easy to handle a model for the images One way is to assume that the images can be modelled using the Euclidean geometry and processed in a similar way as a function defined for the continuous space Since the images are discrete by definition, a discretization of the methods using approximations, is needed at some step This approach is sufficient to solve some problems but insufficient for other problems Another point of view is to develop methods on the digital grid without resorting to Euclidean geometry This is the digital geometry approach By not maing the assumption that the image can be processed using the Euclidean geometry, the problems that arise from approximation errors are avoided As an example, consider the computation of a distance transform DT, the mapping defined by setting each object grid point to the distance to the closest bacground grid point A method that assumes that the Euclidean geometry can be used is the fast-marching method [1] With this method, a differential equation the Eional equation is approximated using finite differences The errors that are produced due to these approximations are large, see [] A distance function defined as a minimal cost-path between two points belongs to the digital geometry framewor We do not assume that the Euclidean geometry can be used, and the distance values assigned to the grid points are correct according to the definition by using, eg, the wave-front propagation technique in [3] In this paper, distance functions following the digital geometry approach will be considered The theory will be developed and the distance functions will be defined in a general setting that allows also non-standard grids The history of digital geometry can be traced bac to 1966 when the basic distance functions, the city-bloc distance and the chessboard distance, were defined for the square grid in [4] An algorithm for computing the DT was also introduced in [4] and it was proven that the algorithm produces a correct result In [5] a generalization of these distance functions, the distance based on neighborhood sequences ns-distances, was introduced With ns-distances, a sequence of neighborhood relations defines the distance function In the case of the square grid, the neighborhood sequence defines which steps are restricted to edge neighbors city-bloc distance and in which steps the vertex neighbors chessboard distance can be used address: robin@cbuuse X/$ see front matter 008 Elsevier BV All rights reserved doi:101016/jdam

2 64 R Strand / Discrete Applied Mathematics Fig 1 Neighbors up to order two with positive coordinate values The grid points top and the corresponding Voronoi regions bottom represent an fcc grid left, a cubic grid middle, and a bcc grid right In [6], another generalization of the chessboard and city-bloc distances, the weighted distance, was introduced With this distance function, the different local steps have different weights and the distance between two points is defined as the minimal cost-path between the points using only one neighborhood relation This approach was further developed in [7 9], where optimal weights were calculated In [10,11], some theoretical results for ns-distances were presented One of the definitions in [11] also includes the distance function obtained by using both weights and a neighborhood sequence the weighted distance based on neighborhood sequences weighted ns-distances The theory for ns-distances with periodic neighborhood sequences was further developed in [1 15], where for example formulas for point-to-point distance and conditions for metricity were given In [16 0], ns-distances for the non-periodic case were considered for standard and non-standard grids Optimal parameters, conditions for metricity, and algorithms for computing the DT for weighted ns-distances were presented in [3] for the square grid and in [1 3] for the face-centered cubic fcc and body-centered cubic bcc grids Independent of [3], some results for weighted ns-distances were derived and published in [4] In this paper, the theory for weighted ns-distances is presented for the general case of point-lattices Previous results and notation We will consider point-lattices A point-lattice is a grid that can be defined by a set of linearly independent vectors, a basis In image processing, this structure is usually called a grid Therefore, both terms are used in this paper We define the fcc and bcc grids as the point-lattices spanned by 1, 1, 0, 1, 0, 1, 0, 1, 1} and 1, 1, 1, 1, 1, 1, 1, 1, 1}, respectively In general, we can define the point-lattice or grid G spanned by the vectors v 1, v,, v n as the set } G = p R n : p = λ v, λ Z Since the grids considered here are point-lattices, it is natural to consider them as affine spaces in which the difference of two points in G is a vector We will also treat the grids G as modules ie, discrete vector spaces and write v G meaning p G for p such that v = p 0 See Fig 1 for an illustration of the fcc, cubic, and bcc grids Definition 1 R-Sector Given a set of n independent vectors of the n-dimensional grid G, v } [1n], the R-sector v 1, v,, v n is the region of R n spanned by the vectors v 1, v,, v n, ie, } v 1, v,, v n = p R n : p = λ v, λ R + Definition G-Sector The G-sector v 1, v,, v n is the set of points belonging to G which are included in the R- sector v 1, v,, v n, so v 1, v,, v n = G v 1, v,, v n Definition 3 G-Wedge Given a set of vectors v i }, a wedge W is a G-sector formed by a set v i of n vectors from }n v i } such that W does not contain any other vectors from v i }

3 R Strand / Discrete Applied Mathematics Definition 4 G-Basis-Sector A G-sector v 1, v n is a G-basis-sector if v } n is a basis of G By the definition of a basis, a G-basis-sector corresponds exactly to the set of points p such that p = n λ v, λ N A G-basis-wedge is a G-basis-sector that is also a wedge 1 Weighted distance In this section, we recall some results from [5] Definition 5 Chamfer Mas A restricted chamfer mas C is a finite set of weighted vectors v, ω [1m] G R} which satisfies the following properties: Positive weights, ω > 0 and v 0 Symmetry v, ω C v, ω C p G, W of C such that Organized in G-basis-wedges p W and W is a G-basis-wedge Convex normalized B chamfer mas polyhedron C = convb C The normalized chamfer mas polyhedron of a chamfer mas is now defined Each set of vectors v i, i = 1 n defining a G-basis-wedge also defines a simplex by the n + 1 points corresponding to these vectors normalized by the corresponding weight 0 + v i /ω i and 0 The union of the simplices is the normalized chamfer mas polyhedron When Z 3 and integer weights are considered, the normalized chamfer mas polyhedron is called the rational ball in [6] Definition 6 Normalized Chamfer Mas Polyhedron We call the polyhedron of R n that is the union of the n-dimensional simplices formed by 0 and the n vectors of each wedge of a chamfer mas C = v, ω [1m] G R} normalized by their weights, ie, ṽ i = 1 v ω i, i [1 n] for each wedge v i1,, v in of C the normalized chamfer mas polyhedron, denoted B C Given a path p = p 0, p 1,, p n = q, the differences between two consecutive points p i+1 p i are called local steps Definition 7 Path Given a chamfer mas C = v, ω [1m] G R} and two points p, q G, a path from p to q is a sequence of adjacent grid points p = p 0, p 1,, p n = q denoted P p,q = p = p 0, p 1,, p n = q Definition 8 Cost of a Path The cost C ω } Pp,q of a such a path Pp,q, where there are a local steps v, = 1 m, is defined by: C ω } m Pp,q = a ω Since a mas C contains a basis of G, and is symmetric, such a path always exists for any pair of points p, q with positive coefficients Definition 9 Weighted Distance A weighted distance d C associated with a chamfer mas C between two points p and q in G is the minimum cost of all paths P p,q lining p to q d C p, q = } min C ω } Pp,q Given a set F = v } [1n] of n independent vectors, we define 0 F R v 1 1 v 1 v n 1 0 F = v 1 v v n detv 1, v,, v n =, v 1 n v n v n n where v = v 1, v,, v n Also [1 n], we consider the function F : G R p such that: F p F p = detv 1,, v 1, p, v +1,, v n v 1 1 v 1 1 x1 v +1 1 v n 1 v 1 v 1 x v +1 v n =, v 1 n v 1 n xn v +1 n v n n where p = x1, x,, xn

4 644 R Strand / Discrete Applied Mathematics The following Theorem 10 is proved in [5] Theorem 10 Given a chamfer mas C = v, ω [1n] G R }, defined as in Definition 5, the weighted distance of any point p lying in a wedge v i1, v in can be expressed by: d C p = 1 0 F F pω i The formula in Theorem 10 does not use explicit matrix inversions, but d C p can be expressed also using matrix inversions see also [5]: Let v 1 1 v 1 v 1 n v 1 v v n V =, v n 1 v n v n n then d C p = ω i1, ω i,, ω in V 1 p In some cases, eg, when V is the identity matrix, this formula can be more efficient 3 Distances based on neighborhood sequences Given the two sets N 1 the 1-neighborhood and N the strict -neighborhood of vectors of G, two grid points p 1, p G are r-neighbors, r 1, }, if p p 1 N r Neighbors of higher order can also be defined, but in this section, we will use only 1- and -neighbors We denote the set of -neighbors by N 1, = N 1 N The points p 1, p are adjacent if p 1 and p are r-neighbors for some r The -neighbors which are not 1-neighbors are called strict -neighbors The neighborhood relations are visualized in Fig 1 by showing the Voronoi regions the voxels corresponding to some adjacent grid points in the fcc, cubic, and bcc grids A ns B is a sequence B = bi i=1, where each bi denotes a neighborhood relation in G If B is periodic, ie, if for some fixed strictly positive l Z +, bi = bi+l is valid for all i Z +, then we write B = b1, b,, bl The concatenation of two paths P p0,p n and Q q0,q m such that p n and q 0 are adjacent is P p0,p n Q q0,q m = p 0, p 1,, p n, q 0, q 1,, q m A path is a B-path of length n if, for all i 1,,, n}, p i 1 and p i are bi-neighbors The notation 1- and strict -steps will be used for a step to a 1-neighbor and step to a strict -neighbor, respectively The numbers of 1-steps and strict -steps in a given path P are denoted 1 P and P, respectively The length of the path P is denoted L P Definition 11 Given the ns B, the ns-distance dp 0, p n ; B between the points p 0 and p n is the length of one of the shortest B-paths between the points The following notation is used: 1 = B i : bi = 1, 1 i } and B = i : bi =, 1 i } We will now restrict the discussion to point-lattices satisfying the following definition saying that N 1, is obtained by subdivision of N 1 Definition 1 Given a point-lattice G, let the set of 1-neighbors be N 1 = v i G} n i=1 and the set of strict -neighbors be N = w i G} m i=1 If N 1 and N are such that: C 1 = v i, 1} n i=1 and C = C 1 w i, 1} m i=1 are chamfer mass, i, j, : w i = v j + v, and each G-basis-wedge of N 1 is the union of some G-basis-wedges of N 1,, then G is a point-lattice wedge--generated by N 1 and N The following example is illustrated in Fig 1 Example 13 Examples of point-lattices and sets N 1 and N such that G is a point-lattice wedge--generated by N 1 and N are: The square grid Z with N 1 = ±1, 0, 0 ± 1} and N = ±1, ±1} The fcc grid F with N 1 = ±1, ±1, 0, ±1, 0 ± 1, 0, ±1, ±1} N = ±, 0, 0, 0, ±, 0, 0, 0, ±}

5 R Strand / Discrete Applied Mathematics The bcc grid B with N 1 = ±1, ±1, ±1} N = ±, 0, 0, 0, ±, 0, 0, 0, ±} An example of a point-lattice and N 1 and N such that G is not wedge--generated by N 1 and N is Z 3 with N 1 = ±1, 0, 0, 0, ±1, 0, 0, 0, ±1} N = ±1, ±1, 0, ±1, 0, ±1, 0, ±1, ±1}, since F = 1, 1, 0, 1, 0, 1, 0, 1, 1} is a wedge of N 1,, but is not a basis of Z 3, C is not a chamfer mas according to Definition 5 Definition 14 For each of the grids considered in Example 13, the neighborhoods N 1 and N defined in Example 13 are called the natural neighborhood relations In Fig 1, the natural neighborhood relations are used Lemma 15 Let G be a point-lattice wedge--generated by N 1 and N and let p, q G, let also P p,q be a shortest path using only local steps from C 1 Then there is a shortest path Q p,q using only local steps from C such that L P p,q = 1Pp,q = 1 Qp,q + Qp,q 1 Proof We consider the case p = 0, the general case follows by translation invariance By Definition 1, there are wedges W 1 = v 1,, v n of C 1 and W = w 1,, w n of C such that q is in W and W W 1 The point q is uniquely represented by vectors of W 1 and by vectors of W By the convexity of the normalized chamfer mas polyhedrons, the corresponding paths are shortest Since each -step in W is the sum of two 1-steps in W 1, the formula follows 31 Formulas for calculating the distance In the previous section, the B-distances were defined for G Both in theory and in applications it is useful to have formulas to compute the actual values of the distances For Z n the computation is given in [1] and in [19,7], for the periodic and not necessarily periodic cases, respectively In [0], formulas are given for the fcc and bcc grids Let d C1 be the distance function obtained by using the chamfer mas C 1 and d C be the distance function obtained by using the chamfer mas C Theorem 16 ns-distance in G Wedge--Generated by N 1 and N Let G wedge--generated by N 1 and N, the neighborhood sequence B and the points p, q G be given Then dp, q; B = min max d C p, q, d C1 p, q B}} Proof It is clear that the distance dp, q; B is such that d C p, q dp, q; B d C1 p, q If B allows, the number of -steps in Q p,q is d C1 d C and thus dp, q; B = d C p, q Since each strict -step can be exchanged for two 1-steps by Lemma 15, the distance is increased by one for each missing in B The formula follows We can now use Theorem 10 to obtain the following formula that can be used to get formulas for the distance in, eg, the square, fcc, and bcc grids Corollary 17 Let G be wedge--generated by N 1 and N, and let the ns B, and the point p G such that p v i1,, v in, v i N 1 and p w j1, w jn, w j N 1, be given Let F 1 = } v i and F = } w j Then } } d0, p; B = min max 1 l 1 m 0 F 1 p, l F 1 0 F p B F l=1 l=1 Corollary 17 is now applied to the square, fcc, and bcc grids using the natural neighborhood relations The formulas presented here are equivalent to the formulas derived in Theorem 5 in [3] and Theorems and 5 in [0] Corollary 18 B-Distance in Z Let the neighborhood sequence B and the point p = x, y such that x y 0 be given Then d0, p; B = min max x, x + y B}}

6 646 R Strand / Discrete Applied Mathematics Proof Consider F 1 = 1, 0, 0, 1} and F = 1, 0, 1, 1}, then p = x, y is in 1, 0, 0, 1 and 1, 0, 1, 1 when x y 0 and thus 1 0 F 1 p = x 0 y x 0 y = x + y F 1 and 1 0 F F p = x 1 y x 0 y = x y + y = x Corollary 19 B-Distance in F Let the neighborhood sequence B and the point p = x, y, z such that x y z 0 be given Then } } d0, p; B = min x + y + z max, x B Proof Consider first F 1 = 1, 1, 0, 1, 0, 1, 0, 1, 1} and F = F 1, then p = x, y, z is in 1, 1, 0, 1, 0, 1, 0, 1, 1 when x y z 0 and x y + z We get x x x F 1 p = y 0 1 z y 1 0 z y 0 1 z = x y z and F p = x y z Also, 0 F 1 = 0 F = Note that when x y + z, we have x+y+z x x x+y+z B for any, so the distance here is Consider F 1 = 1, 1, 0, 1, 0, 1, 1, 1, 0} and F = 1, 1, 0, 1, 0, 1,, 0, 0}, then p = x, y, z is in both the wedge 1, 1, 0, 1, 0, 1, 0, 1, 1 and the wedge 1, 1, 0, 1, 0, 1,, 0, 0 when x y z 0 and x y + z We get x x x F 1 p = y 0 1 z y 1 0 z y 0 1 z = x and x 1 1 x 1 1 x F p = y 0 0 z y 0 0 z y 0 1 z = x + y + z Now, 0 F 1 = 0 F = This gives the formula in Corollary 19 Corollary 0 B-Distance in B Let the neighborhood sequence B and the point p = x, y, z such that x y z 0 be given Then } } d0, p; B = min x + y max, x B Proof When the natural neighborhood relations are used, we can use F 1 = 1, 1, 1, 1, 1, 1, 1, 1, 1} and F = 1, 1, 1, 1, 1, 1,, 0, 0}, then p = x, y, z is in the wedge 1, 1, 1, 1, 1, 1, 1, 1, 1 and the wedge 1, 1, 1, 1, 1, 1,, 0, 0 when x y z 0 We get x x x F 1 p = y 1 1 z y 1 1 z y 1 1 z = 4x and x 1 1 x 1 1 x F p = y 1 0 z y 0 1 z y = x + y 1 1 z Also, 0 F 1 = 0 F = 4 This gives the formula in Corollary 0

7 R Strand / Discrete Applied Mathematics Theorem 16, but not Corollary 17 is valid also for the cubic grid Z 3, since the chamfer mas C 1 with N 1 as defined in Example 13 satisfies all conditions for a chamfer mas Therefore, the distance in the region spanned by a wedge W 1 of C 1 is defined by only W 1 and W such that W W 1 We get the following corollary of Theorem 16 Corollary 1 B-Distance in Z 3 Let the neighborhood sequence B and the point p = x, y, z such that x y z 0 be given Then }} x + y + z min max, x + y + z if x < y + z B and x, y, z F }} d0, p; B = x + y + z 1 min 1 + max, x + y + z 1 if x < y + z B and x, y, z F }} min max x, x + y + z B if x y + z Proof The natural neighborhood relations are used, so we consider F 1 = 1, 0, 0, 0, 1, 0, 0, 0, 1} and F = 1, 0, 0, 1, 1, 0, 1, 0, 1}, then p = x, y, z is in 1, 0, 0, 0, 1, 0, 0, 0, 1 and 1, 0, 0, 1, 1, 0, 1, 0, 1 when x y + z We get x x x F 1 p = y 1 0 z y 0 0 z y 0 0 z = x + y + z and 0 F 1 p = 1 Also, x x x F p = y 0 1 z y 1 0 z y 0 1 z = x and 0 F p = 1 When x < y + z, the point x, y, z is in the wedge spanned by the vectors 1, 1, 0, 1, 0, 1, 0, 1, 1 This is not a basis for Z 3, but it is a basis for F If x, y, z F, then x, y, z = a 1 1, 1, 0 + a 1, 0, 1 + a 3 0, 1, 1 for some non-negative integers a 1, a, a 3, so the distance is given by using 1, 0, 0, 0, 1, 0, 0, 0, 1} for d C1 and 1, 1, 0, 1, 0, 1, 0, 1, 1} for d C calculated in Corollary 19 If x, y, z F, a 1-step is needed to reach x, y, z, so x, y, z = 1, 0, 0 + a 1 1, 1, 0 + a 1, 0, 1 + a 3 0, 1, 1 for some non-negative integers a 1, a, a 3 Therefore, the shortest path has length as given in the corollary 3 Metricity of the distance function Definition A function d : G G R is a metric on G if it satisfies the following conditions: 1 p, q G : dp, q 0 and dp, q = 0 iff p = q; p, q G : dp, q = dq, p; 3 p, q, r G : dp, q + dq, r dp, r For the distance function generated by a neighborhood sequence B to be a metric, some conditions on B must be fulfilled Necessary and sufficient conditions for periodic neighborhood sequences on Z n to generate a metric is presented in [1], for non-periodic neighborhood sequences in Z n in [16], and the fcc and bcc grids in [0] In this section, conditions for metricity for distances based on neighborhood sequences on point-lattices wedge--generated by some N 1 and N are derived Inspired by the wor of Nagy in [16], we give the following definition Definition 3 Let B 1 and B be two neighborhood sequences The relation B 1 B B 1 is faster than B is defined as dp, q; B 1 dp, q; B p, q G Observe that the distance function is used in the definition of the relation There is a natural question: how one can decide whether a neighborhood sequence B 1 is faster than B without calculating the actual distances of all possible point-pairs It is easy to chec, that the relation does not depend on the distances of points, but only on the neighborhood sequences Theorem 4 For any point-lattice wedge--generated by N 1 and N, a neighborhood sequence B 1 is faster than a neighborhood sequence B if and only if j b 1 i i=1 j b i for all j N i=1

8 648 R Strand / Discrete Applied Mathematics It can be written in the following equivalent condition: j B 1 j B for all j N Proof This theorem is a consequence of Theorem 16 Definition 5 For any neighborhood sequence B = bi i=1, the sequence Bj = bi i=j is the j-shifted sequence of B Theorem 6 The distance function based on a neighborhood sequence B is a metric on G if and only if Bi B for all i N Proof Property 1 in Definition is trivially fulfilled Property : Let p, p G, B be a neighborhood sequence, and p = p + q 0, p + q 1, p + q,, p + q n = p be a shortest B-path between p and p By the lattice-structure of the grids, if p + q i, p + q i+1 are r-neighbors, then so are p q i, p q i+1 It follows that p = p q 0, p q 1, p q,, p q n is a shortest B-path between p and p q n Obviously, for any p, q G : dp, p q; B = dp, p + q; B The distance functions are also translation-invariant, so dp + q, p; B = dp, p q; B The result follows The triangular inequality property 3 is now considered Assume, for some fixed j Z, Bj is not faster than B Then there are points p, q, r G such that p = p 0, p 1,, p j, r = r 0, r 1, r,, r = q is a shortest B-path between p and q and such that by assumption dr, q; B < dr, q; Bj Now, dp, q; B = dp, r; B + dr, q; Bj > dp, r; B + dr, q; B Assume now that Bi B for all i N Let p, q, r G and the B-path P p,r P r,q = p = p 0, p 1,, p i, r = r 0, r 1, r,, r = q be such that P p,r is a shortest B-path and P r,q is a shortest Bj-path By Definition 3, dr, q; B dr, q, Bi Now, dp, q; B = dp, r; B + dr, q; Bi dp, r; B + dr, q; B 4 Weighted distances based on neighborhood sequences We now generalize the distance functions defined in Sections 1 and 3 by combining the two concepts For weighted distances based on neighborhood sequences, both weights and a neighborhood sequences are used Let the real numbers α and β the weights and a path P of length n, where exactly l l n adjacent grid points in the path are strict -neighbors, be given The cost of the α, β-weighted B-path P is n lα + lβ The B-path P between the points p 0 and p n is a minimal cost α, β-weighted B-path between the points p 0 and p n if no other α, β-weighted B-path between the points has lower cost than the cost of the α, β-weighted B-path P Definition 7 Given the ns B and the weights α, β, the weighted ns-distance d α,β p 0, p n ; B is the cost of one of the minimal cost α, β-weighted B-paths between the points Remar 8 Hereafter, we will mainly consider two weights α and β as real numbers α and β such that 0 < α β α This is natural since: a -step should be more expensive than a 1-step since strict -neighbors are intuitively at a larger distance than 1- neighbors projection property and two 1-steps should be more expensive than a -step otherwise no -steps would be used in a minimal cost-path 41 Formulas for calculating the distance Lemma 9 states that for any p, q there are shortest B-paths that are also minimal cost B-paths when α β α The numbers of 1-steps and -steps in such a path P, ie, 1 P and P, respectively are given in Lemma 30 Lemma 9 Let G wedge--generated by N 1 and N, the ns B, the weights α, β such that 0 < α β α, and the point p G be given There is a shortest B-path P such that P is also a minimal cost α, β-weighted B-path Proof Let P be a shortest B-path such that 1 from Lemma 15 is fulfilled Assume that there is a B-path P such that the cost of the α, β-weighted B-path P is lower than the cost of the α, β-weighted B-path P Since P is a shortest B-path, we have L P L P The cost of the α, β-weighted B-path P is 1 P α + P β By assumption, 1 P α + P β < 1 P α + P β We get the following cases: i P > P Since the length of a shortest path using steps from C 1 is 1 P + P by Lemma 15, we have 1 P + P 1 P + P Thus, 3

9 1 P 1 P P P 1 P 1 P α Rewriting assumption 3 gives P P ii P < P since L R Strand / Discrete Applied Mathematics P P α 1 P 1 P α < β, which implies α < β 1 P α + P β < 1 P α + P β by 3 and L P P P α + P β < L P P α + P β L P and L P = 1P + P We have P P β < P P α β < α since P < P P P β Thus P P α 1 P 1 P α < This proves that assumption 3 implies P = P Rewriting 3 with P = P P gives L < L P, which contradicts the fact that P is a shortest B-path ie, L P L P Therefore 3 is false, so n l α + l β n lα + lβ, which means that P is a minimal cost α, β-weighted B-path Lemma 30 Let G wedge--generated by N 1 and N, the ns B, the weights α, β such that 0 < α β α, and the point p G be given There is a minimal cost α, β-weighted B-path with the following property: 1 Pp,q = dp, q; B d C1 p, q and Pp,q = d C1 p, q dp, q; B 4 5 Proof Let Q p,q be a shortest path using only 1-steps a shortest 1-path and let P p,q be shortest B-path that fulfills 1 in Lemma 15 We have d C1 p, q = 1 Qp,q = 1 Pp,q + Pp,q and dp, q; B = 1 Pp,q + Pp,q The formulas 4 and 5 follow This path is of minimal cost by Lemma 9 We get the following theorem by summing up the results from Lemmas 9 and 30 Theorem 31 Weighted ns-distance in G Wedge--Generated by N 1 and N Let G wedge--generated by some N 1 and N, the ns B, the weights α, β such that 0 < α β α, and the points p, q G be given Then d α,β p, q; B = dp, q; B d C1 p, q α + d C1 p, q dp, q; B β The following Corollaries follow from Theorem 31 using the results from Corollaries 18 0 The formulas presented here are equivalent to the formulas presented in Theorem 8 in [3] and Theorems 1 and in [1] Corollary 3 Let the ns B, the weights α, β st 0 < α β α, and the point x, y Z, where x y 0, be given The weighted ns-distance between 0 and x, y is given by d α,β 0, x, y; B = x y α + x + y β where = min : max x, x + y B Corollary 33 Let the ns B, the weights α, β st 0 < α β α, and the point x, y, z F, where x y z 0, be given The weighted ns-distance between 0 and x, y, z is given by α if x y + z d α,β 0, x, y, z; B = x α + x β otherwise, x + y + z where = min : max, x B Corollary 34 Let the ns B, the weights α, β st 0 < α β α, and the point x, y, z B, where x y z 0, be given The weighted ns-distance between 0 and x, y, z is given by

10 650 R Strand / Discrete Applied Mathematics Fig Balls of radius 0 in the fcc grid with α = 1, β = 15, and B = 1, left and the bcc grid with α = 1, β = 1, and B = 1, right d α,β 0, x, y, z; B = x α + x β x + y where = min : max, x B Remar 35 It is straightforward to show that Z is isomorphic with F z=0 = F x, y, z : z = 0} We shall now see that the weighted ns-distance is equivalent for these two grids The formula in Corollary 33 for weighted ns-distance on F α if x y + z d α,β 0, x, y, z; B = x α + x β otherwise, x + y + z where = min : max, x B reduces to d α,β 0, x, y, 0; B = x α + x β x + y + z where = min : max, x B on F z=0 By using the one-to-one mapping M : Z F z=0 this is a similarity mapping since it can be constructed by rotation, uniform scaling, and reflection defined by x, y x + y, x y, 0, we get d α,β 0, x, y ; B = x y α + x + y β where = min : max x, x + y B This is exactly the formula for weighted ns-distances on Z given in Corollary 3 Also, the formula is valid for x y z = 0 for x, y, z F z=0, which is x + y x y 0 for x, y Z Rewriting this gives x y 0, which agrees with Corollary 3 To illustrate the discrete distance functions, balls of radius 0 in the fcc grid with α = 1, β = 15, and B = 1, and the bcc grid with α = 1, β = 1, and B = 1, are shown in Fig These weights minimize the compactness ratio for the asymptotic shape of the balls, see [1,3] Each grid point is represented by its Voronoi region 4 Metricity of the distance function Example 36 Let B =,, 1, α =, and β = 3 For the points p 1 = 0, 0, 0, p = 4, 0, 0, and p 3 = 6, 0, 0 in fcc or bcc, we have d α,β p 1, p 3 ; B = 10 > 9 = = d α,β p 1, p ; B + d α,β p, p 3 ; B, so the triangular inequality is violated and therefore, d,3, ;,, 1 is a metric on neither F nor B

11 R Strand / Discrete Applied Mathematics Example 37 Let B =, 1, 1, α = 3, and β = ie, weights that do not satisfy the inequalities in Remar 8 Consider the points p 1 = 0, 0, 0, p =, 0, 0, and p 3 =,, 0 in fcc or bcc We have d α,β p 1, p 3 ; B = = 6 > 4 = + = d α,β p 1, p ; B + d α,β p, p 3 ; B in both the fcc and bcc grids Again, the triangular inequality is violated and therefore, d 3,, ;, 1, 1 is a metric on neither F nor B It is obvious that the distance functions are positive definite for positive weights They are also symmetric since C 1 and C formed by N 1 and N are chamfer mass according to Definition 5 To establish the condition of metricity, only the triangular inequality needs to be be proved As Examples 36 and 37 show, the triangular inequality is related to the weights and the number of occurrences of 1 and and their positions in the ns We eep the weights within the interval 0 < α β α To prove the result about metricity, Theorem 39, we need some conditions also on the ns, namely the faster than relation in Lemma 38 Let G wedge--generated by N 1 and N, the weights α, β such that 0 < α β α and the points p, q G be given If B 1 B B 1 is faster than B, then d α,β p, q; B 1 d α,β p, q; B Proof By Definition 3 it follows that dp, q; B 1 = dp, q; B L for some non-negative integer L Using the formula in Theorem 31, we get d α,β p, q; B 1 = dp, q; B 1 ˆ α + ˆ dp, q; B1 β = dp, q; B ˆ α + ˆ dp, q; B β + β αl dp, q; B ˆ α + ˆ dp, q; B β = d α,β p, q; B, where ˆ = dc1 p, q Theorem 39 If Bi B i N and 0 < α β α, then d α,β, ; B is a metric on a point-lattice G wedge--generated by N 1 and N Proof The positive definiteness and symmetry are trivial We prove the triangular inequality Let p, r, q G be given Let the B-path P p,q = P p,r P r,q be such that P p,r is a minimal cost α, β-weighted B-path and P r,q is a minimal cost α, β-weighted BL P p,r + 1-path Let also Qr,q be a minimal cost α, β-weighted B-path between r and q By Lemma 38, C α,β Pr,q Cα,β Qr,q Now, d α,β p, q; B C α,β Pp,q = Cα,β Pp,r + Cα,β Pr,q C α,β Pp,r + Cα,β Qr,q = d α,β p, r; B + d α,β r, q; B The proof presented here is shorter and more intuitive than the proof valid only for the fcc and bcc grids presented in [] Note that using the natural conditions for the weights α and β, the condition of metricity is exactly the same as for the nonweighted case see Theorem 6 Thus, to chec that a weighted ns-distance function is a metric can be done with the same efficiency as for the non-weighted case 5 Conclusions and future wor The power of defining weighted ns-distances in a general setting has been illustrated by presenting results for the special cases of the square, the fcc, and the bcc grid and also the cubic grid By using the general case, we give here tools for applying the weighted ns-distances to other point-lattices directly One lin between the theoretical results presented here and image processing applications is the distance transform, ie, assigning to each object grid point the distance to the closest bacground grid point In an ongoing project, we have shown that the distance transform can be computed efficiently on any grid that is wedge--generated by some N 1 and N Also, we have derived some results about finding the set of centers of maximal balls in this general setting These results are still unpublished In Corollary 1, we saw that for some cases, Theorem 16 is valid also with weaer conditions on the neighborhoods N 1 and N The definition used here allows only two neighborhood relations with a number of restrictions, for example the chamfer mass should be organized in basis-wedges and be convex This is somewhat limiting and it is our intention to derive results for these distances also in a more general setting

12 65 R Strand / Discrete Applied Mathematics Acnowledgements The author is grateful to Prof Gunilla Borgefors for her valuable help during the preparation of this manuscript Many thans also to the anonymous referees for their insightful comments that improved the quality of the manuscript References [1] JA Sethian, Level Set Methods and Fast Marching Methods, Cambridge University Press, 1999 [] JA Bærentzen, On the implementation of fast marching methods for 3D lattices, Tech Rep, Informatics and Mathematical Modelling, Technical University of Denmar, DTU, Richard Petersens Plads, Building 31, DK-800 Kgs Lyngby, [3] R Strand, Weighted distances based on neighbourhood sequences, Pattern Recognition Letters [4] A Rosenfeld, JL Pfaltz, Sequential operations in digital picture processing, Journal of the ACM [5] A Rosenfeld, JL Pfaltz, Distance functions on digital pictures, Pattern Recognition [6] U Montanari, A method for obtaining seletons using a quasi-euclidean distance, Journal of the ACM [7] G Borgefors, Distance transformations in arbitrary dimensions, Computer Vision, Graphics, and Image Processing [8] G Borgefors, Distance transformations in digital images, Computer Vision, Graphics, and Image Processing [9] BJH Verwer, Local distances for distance transformations in two and three dimensions, Pattern Recognition Letters [10] M Yamashita, N Honda, Distance functions defined by variable neighbourhood sequences, Pattern Recognition [11] M Yamashita, T Ibarai, Distances defined by neighbourhood sequences, Pattern Recognition [1] PP Das, PP Charabarti, Distance functions in digital geometry, Information Sciences [13] PP Das, PP Charabarti, BN Chatterji, Generalized distances in digital geometry, Information Sciences [14] PP Das, BN Chatterji, Octagonal distances for digital pictures, Information Sciences [15] PP Das, BN Chatterji, Hyperspheres for digital geometry, Information Sciences [16] B Nagy, Distance functions based on neighbourhood sequences, Publicationes Mathematicae Debrecen [17] A Fazeas, A Hajdu, L Hajdu, Lattice of generalized neighbourhood sequences in nd and D, Publicationes Mathematicae Debrecen [18] A Fazeas, A Hajdu, L Hajdu, Metrical neighborhood sequences in Z n, Pattern Recognition Letters [19] B Nagy, Distances with neighbourhood sequences in cubic and triangular grids, Pattern Recognition Letters [0] R Strand, B Nagy, Distances based on neighbourhood sequences in non-standard three-dimensional grids, Discrete Applied Mathematics [1] R Strand, Weighted distances based on neighbourhood sequences in non-standard three-dimensional grids, in: Proceedings of SCIA 007, in: LNCS, 45, Springer, Aalborg, Denmar, 007, pp [] R Strand, B Nagy, Weighted neighbourhood sequences in non-standard three-dimensional grids metricity and algorithms, in: Proceedings of DGCI 008, in: LNCS, 499, Springer, Lyon, France, 008, pp 01 1 [3] R Strand, B Nagy, Weighted neighbourhood sequences in non-standard three-dimensional grids parameter optimization, in: Proceedings of IWCIA 008, in: LNCS, 4958, Springer, Buffalo, NY, 008, pp 51 6 [4] A Hajdu, L Hajdu, R Tijdeman, General neighborhood sequences in Z n, Discrete Applied Mathematics [5] C Fouard, R Strand, G Borgefors, Weighted distance transforms generalized to modules and their computation on point lattices, Pattern Recognition [6] E Remy, E Thiel, Optimizing 3D chamfer mass with norm constraints, in: Proceedings of IWCIA 000, Caen, France, 000 pp [7] B Nagy, Metric and non-metric distances on Z n by generalized neighbourhood sequences, in: Proceedings of ISPA 005, Zagreb, Croatia, 005 pp 15 0