Neighborhood Sequences on nd Hexagonal/Face-Centered-Cubic Grids

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1 Neighborhood Sequences on nd Hexagonal/Face-Centered-Cubic Grids Benedek Nagy 1 and Robin Strand 1 Department of Computer Science, Faculty of Informatics, University of Debrecen, Debrecen, Hungary nbenedek@inf.unideb.hu Centre for Image Analysis, Uppsala University, Sweden robin@cb.uu.se Abstract. The two-dimensional hexagonal grid and the three-dimensional face-centered cubic grid can be described by intersecting Z 3 and Z 4 with a (hyper)plane. Corresponding grids in higher dimensions (nd) are examined. In this paper, we define distance functions based on neighborhood sequences on these, higher dimensional generalizations of the hexagonal grid. An algorithm to produce a shortest path based on neighborhood sequences between any two gridpoints is presented. A formula to compute distance and condition of metricity are presented for neighborhood sequences using two types of neighbors. Distance transform as an application of these distances is also shown. Keywords: Digital geometry, nd grids, Neighborhood sequences. 1 Introduction It is well-known that the hexagonal grid has several advantages over the twodimensional Cartesian grid, the square grid. Most importantly, since the hexagonal grid is a densest packing, fewer samples are needed to represent a twodimensional signal (an image) when it is sampled on the hexagonal grid. Also, since there is a larger number of closest neighbors and these neighbors are such that the same adjacency relation can be used for object grid points and background grid points, and because its low rotational dependency the hexagonal grid is a good choice for two-dimensional image processing [7, 9, 10, 6, 9]. One argument for using Cartesian grids in image processing is that many functions defined on these grids are separable, which allows fast and efficient image processing. Consider, e.g., the important discrete Fourier transform. By using redundancy of the matrices used for computing the Fourier transform and the separability on the square grid, a fast Fourier transform is derived, see, e.g., [4], where the well-known Cooley-Tukey algorithm is presented. The so-obtained algorithm has computation time O (N log N), where N = k for some k. By utilizing the 6-fold symmetry of the hexagonal grid and applying a vector-radix Cooley-Tukey algorithm, the discrete Fourier transform can be computed in

2 O (N log 7 N), where N = 7 k for some k, on the hexagonal grid [10]. In fact, the number of operations needed to compute the Fourier transform on the hexagonal grid is roughly 60%(!) fewer on the hexagonal grid compared to the square grid, [10]. The densest packing in three dimensions is the face-centered cubic (fcc) grid. The fcc grid has many of the advantages over the three-dimensional Cartesian grid, the cubic grid, see [3, 6, 1, 5]. In this paper, we will consider n-dimensional generalizations of the hexagonal and fcc grids [3, 8, 1, 14, 17, 19, 4]. In [3, 6, 7], the hexagonal and fcc grids are obtained by intersecting Cartesian grids by a plane/hyperplane. For example, the hexagonal grid is obtained by intersecting Z 3 with the plane x + y + z = 0. In this paper, we consider (n 1)-dimensional grids obtained by intersecting Z n with a hyperplane. This grid is called A n 1 in [3]. We will present some digital distance functions on these grids. The distance between two points is defined as the shortest path using local steps corresponding to neighboring grid points. In digital images, distance functions defined as the minimal cost-path are often considered [0]. There are basically two approaches either weights are used between neighboring grid points [1] or a neighborhood sequence is used to define which adjacency relation is allowed in each step of the path [13, 15, 18, 7, 8]. It is also possible to combine the two approaches to get even lower rotational dependency [3, 4, 8]. Distance functions have also been defined for high dimensional and non- Cartesian grids, see e.g. [5, 11, 16, ]. To handle non-cartesian grids efficiently in a computer, one needs a good coordinate system. The points of the hexagonal grid can be addressed by two integers [9]. There is a more elegant solution using three coordinate values with zero sum reflecting the symmetry of the grid [7]. Opposed to [4], where a general theory for distance functions is presented, we will use the zero-sum coordinates in this paper. Since integer coordinates are used with this approach, clear and elegant expressions are obtained. In this paper we deal with three-dimensional grids and a family of grids in higher dimensions. Since the fcc grid is the densest packing in three dimensions, it appears frequently in nature. For example, the fcc grid is the structure of, e.g., gold (Au), silver (Ag), calcium (Ca) and aluminium (Al). In the figures, each grid point is sometimes illustrated by the corresponding picture element (pixel in D and voxel in 3D). The picture element of a given grid point is the Voronoi region of the grid point. In Figure 1, the Voronoi regions for the fcc grid is shown. Grids Described by Intersecting Cartesian Grids In [7], the hexagonal grid is described as the intersection of Z 3 and the plane x + y + z = 0. The so-obtained coordinates has some advantages. Most importantly, the points in the hexagonal grid are addressed by integer coordinates, which fits the digital geometry framework. In Section 3, we will define distance

3 (a) (b) (c) (d) Fig. 1. A unit cell of the fcc grid is shown in (a). Voronoi region (voxels) in an fcc grid (b) and a hexagonal grid (d). In (c), the fcc voxels is intersected with the plane x + y + z = 0 (a planar patch is shown). In this intersection, the hexagonal grid is embedded. based on neighborhood sequences and for this, a convenient way to describe neighboring points is needed. We will see that when describing the hexagonal grid as an intersecting plane, there is a natural way to describe the neighboring grid points. One can use a symmetric description with three values by adding a third coordinate value to have zero-sum vectors. The three values are dependent, but the description (including the neighborhood structure) is more simple and more elegant. In [6], the four-dimensional Cartesian grid Z 4 was intersected with the hyperplane x + y + z + w = 0 to obtain the fcc grid. In [3], some properties of the grid obtained by intersecting Z n with a hyperplane with zero sum for any n are presented. Here, we will consider this generalization and define distance functions on these grids. Other high-dimensional grids can be obtained by considering not only intersections of Z n with zero-sum hyperplanes, but also other hyperplanes. In [1, 14], it is noted that the two-dimensional hexagonal and triangular grids can be obtained by the union of integer points of some hyperplanes in three dimensions. In [17, 19], we showed that some well-known three-dimensional grids can be obtained in a similar manner by unions of hyperplanes intersecting Z 4. In geometry, the term (point) lattice is frequently used to describe regularly spaced arrays of points. A lattice can be defined by a finite set} of linearly inde- { n pendent vectors over Z ({v 1,..., v n }, basis) as a i v i a i Z. A lattice may be viewed as a regular tiling of a space by a primitive cell. The Bravais lattices are those kinds of lattices used in crystallography and solid state physics. Further, in this section, we recall some results from [14, 1] and introduce our definitions and notations. In Z n, we have the following natural neighborhood relations:

4 Definition 1. Let p = (p(1), p(),..., p(n)) and q = (q(1), q(),..., q(n)) be two points in Z n. For any integer k such that 0 k n, p and q are Z n k-neighbors if p(i) q(i) 1 for 1 i n and n p(i) q(i) k. When constructing the grids, we will consider intersections between Z k and hyperplanes with normal direction (1, 1,..., 1). Let Q k = {(x 1, x,..., x k ) Z k : x 1 + x + + x k = 0}. Let us start with the cubic grid Z 3. Three natural neighborhood relations on this grid are given by Definition 1. It is well-known that the grid points of Z 3 on a plane with x + y + z = 0 form a hexagonal grid [7]. The neighbor relation of the hexagonal grid has only one type of natural neighborhood, the one that corresponds to the grid points in the -neighborhood of Z 3 satisfying x + y + z = 0. Since Q 3 is the hexagonal grid and Q 4 is the face-centered-cubic grid, the grids defined by Q n are the generalizations of them. Therefore we may use the terms nd hexagonal grids and/or nd fcc grids for Q n+1. Fig.. Neighbors up to order two are shown. The grid points represent an fcc grid. The figure show eight unit-cell size of the fcc grid. In Figure size part of the fcc grid is shown, where every gridpoint has integer coordinate values. 3 Distances Based on Neighborhood Sequences on the Grids with a Projected Hyperplane in Higher Dimensions In the first part of this section we described the n-dimensional generalizations of the previously recalled hexagonal and face-centered-cubic grids. Now we will define and analyze digital distances based on some neighborhood sequences. We can define some types of neighborhood structures on these grids generalizing the concept from the previous lower dimensional grids. Let p = (p(1), p(),..., p(n)), q = (q(1), q(),..., q(n)) Q n be two points. Then their difference vector: w = p q is defined as (p(1) q(1), p() q(),..., p(n) q(n). Two points having difference vector with values only from the set { 1, 0, 1} can be

5 considered as neighbors. For instance the difference vector (1, 1, 0,..., 0) corresponds to a closest neighbor. The neighbor points with four ±1 elements in their difference vector are second closest neighbors point-pair. We will define distances based on neighborhood sequences in higher dimensions using these two neighborhood relations. In this way we generalize the distances of the fcc grid to higher dimension. We may also define a more extended neighborhood structure depending on the dimension n of the space. In this way a further generalization of our distances can be obtained. The neighbor points with difference vector with k elements of ±1 are defined as strict k-neighbors. Therefore, the next formal definition of the neighborhood structure from [19] is natural. Definition. Let p = (p(1),..., p(n)) and q = (q(1),..., q(n)) two points of Q n. Then p and q are Q n k-neighbors if p(i) q(i) k and p(i) q(i) 1 for every 0 < i n. We remark that there are n types of neighbors in Q n, where the floor function is used. If there is an equality in the first condition, then the points are strict k- neighbors. In special cases n = 3, 4 the hexagonal and the fcc grids are obtained by their natural neighborhood structure (only one type of neighborhood in the hexagonal grid and two types of neighbor relations on the fcc grid). In the next subsection we define formally the distances based on neighborhood sequences on these higher dimensional grids with the general neighborhood structure. After this we analyze the distances in detail, specially the ones using only the two closest neighborhood in neighborhood sequences. 3.1 Neighborhood sequences on Q n Let p and q be two points of Q n. The sequence B = (b(i)), where the values of b(i) are possible neighborhood relations of the nd hexagonal grid (0 < b(i) n ) for all i N, is called a neighborhood sequence (on the nd hexagonal grid). A movement is called a b(i)-step when we move from a point p to a point q and they are b(i)-neighbors. Let p, q be two points and B be a neighborhood sequence. The point-sequence p = p 0, p 1,..., p m = q, in which we move from p i 1 to p i by a b(i)-step (1 i m), is called a B-path from p to q. The length of this path is the number of its steps, i.e., m. The B-distance d(p, q; B) from p to q is defined as the length of the shortest B-path(s) between them. As usual in digital geometry, the shortest path may not be unique. In the next subsection we give an algorithm that produces a shortest path between any two points p and q of Q n using a given neighborhood sequence B. Then we also give a formula to compute these path-based distances when only 1 s and s are allowed in the neighborhood sequence, furthermore some properties of these distances will also be analyzed. 3. Algorithm for producing a shortest path The next algorithm provides a shortest path from p to q using an arbitrary neighborhood sequence B.

6 Algorithm 1: Producing a shortest B-path Π in Q n and computing the B-distance. Input: p = (p(1),..., p(n)), q = (q(1),..., q(n)) and the neighborhood sequence B = (b(i)). Output: A shortest B-path from p to q and the B-distance. Initialization: Set i = 0, p 0 = p. Let Π contain the start point p. while p i q do Let i = i + 1 and p i = p i 1; Let j p and j n be the number of positive and negative values among q(h) p i(h), respectively; Let j = min{j p, j n, b(i)}; Let the values q(h) p i(h) are permuted in a monotonous order and let us modify the first and last j values: (a) Let p i(h) = p i(h) + 1 if h is a coordinate among the j largest positive values of the set of q(h) p i(h); (b) Let p i(h) = p i(h) 1 if h is a coordinate among the j smallest (negative) values of the set of q(h) p i(h); Concatenate the point p i to the path Π. end The result is Π as a shortest B-path and i is the B-distance of p and q. Algorithm 1 is a greedy algorithm working in a linear time of the sum of the coordinate differences of the points. 3.3 Formula for B-distance In this section we give a formula to compute the B-distance of any two points of the nd fcc grid, when B has elements only from the set {1, }. Theorem 1. The B-distance of the points p and q in Q n can be computed as d(p, q; B) = max n n p(i) q(i) k 1 p(i) q(i), k > b(j) j=1 with a neighborhood sequence B containing only elements of the set {1, }. Proof. Since a coordinate can be changed by at most one in a step in a B-path, d(p, q; B) max n { p(i) q(i) }. On the other side, in a step at most b(i) values { } n can be changed, therefore d(p, q; B) max k p(i) q(i) > k 1 b(j). j=1 By technical calculation it can be proven that the maximum of these two values gives the actual B-distance.

7 Remark 1. Observe that the formula holds for the hexagonal grid (n = 3 and, since there is only one type of neighborhood relation, b(i) = 1, i N). In this case d(p, q) = max { p(i) q(i) } =,,3 3P p(h) q(h) h=1 [9, 11]. The formula holds for the fcc grid, i.e., the formula presented in [, 16] is equivalent to our: Proposition 1. The formula d(p, q; B) = 3 p(i) q(i) = min k N k max, max,,3 for the fcc grid with k = {i : b(i) =, 1 i k} is equivalent to our formula with n = 4. { p(i) q(i) k } Proof. Let the point (x, y, z) be a point in the fcc grid, i.e., such that x + y + z is even. The linear transformation a = (x + y z)/ b = (x y + z)/ c = ( x + y + z)/ d = (x + y + z)/ is orthogonal, see [19]. Also, a + b + c + d = 0, so (a, b, c, d) Q 4, so it is nothing but a mapping between the different representations of the fcc grid. (i) First we note that { x + y z max { a, b, c, d } = max, = x y + z, x + y + z, } x + y + z x + y + z. (1) (ii) Assume that x y z 0 (the other cases follow by symmetry). (a) x y + z In this case, a + b + c + d = x = max{x, y, z}. We have k 1 max k a + b + c + d > b(j) j=1 = min k a + b + c + d k b(j) j=1 = min { k x k + k } = min { k k max{x, y, z} k }.

8 (b) x < y + z This case gives a + b + c + d = x y z, which implies that x + y + z > x. Now we sum up our formulas and get k 1 d(0, (a, b, c, d); B) = max a, b, c, d, k a + b + c + d > b(j) j=1 { x + y + z = max, min { k k max{x, y, z} k }} { { }} = min k x + y + z k max, max{x, y, z} k, which, with p q = (x, y, z) is the formula in the proposition. When the extended neighborhood is used in Q n (n 6) allowing not only 1 s and s in B, we have the following conjecture. Conjecture 1. The B-distance of p and q in Q n can be determined in the following way. Let W + = {i q(i) p(i) > 0} and W = {i q(i) p(i) < 0} be two sets of coordinates. Then W + and W give two subspaces of Z n. Let p + and q + the images of p and q in the subspace defined by W +, and similarly p and q in the subspace defined by W. Let d + = d(p +, q + ; B) in the subspace of Z n defined by W +, and similarly, d = d(p, q ; B) using the corresponding formula for Z n from [15, 18]. Then d(p, q; B) = max{d +, d } give the B-distance in the nd fcc grid. When only the two closest neighbor relations can be used in B the Conjecture 1 coincides with Theorem Metrical properties A distance function is called a metric, if it satisfies the following three properties: (1) positive definiteness: d(p, q) 0 for any point-pair (p, q) and d(p, q) = 0, if and only if p = q; () symmetry: d(p, q) = d(q, p) for any point-pair (p, q); (3) triangular inequality: d(p, q) + d(q, r) d(p, r) for any point-triplet (p, q, r). In this section we deal mostly with neighborhood sequences using only the closest two neighborhoods. Definition 3. For any neighborhood sequence B = (b(i)), the sequence B(j) = (b(i)) i=j is the j-shifted sequence of B. We say, that a neighborhood sequence B 1 is faster than B if the B 1 -distance is not larger than the B -distance for any point-pairs. Based on Theorem 1 one can prove the following statement.

9 Proposition. For the nd-fcc grids, a neighborhood sequence B 1 is faster than B if and only if j j b 1 (i) b (i) for all j N. This can be written in the following equivalent condition: j B 1 j B for all j N (with the notation k = {i : b(i) =, 1 i k} ). Theorem. The B-distance (using only the closest two neighborhoods) on Q n is a metric if and only if B has the following property: B(i) is faster than B for all i N. Proof. Since the first two properties of the metricity (positive definiteness and symmetry) are automatically fulfilled, the triangular inequality is crucial and can technically be proven using Theorem 1. Proposition gives a computationally efficient way of deciding if a B-distance is a metric or not. For instance, the periodic neighborhood sequence repeating (1, 1,, 1, ) generates a metric. Allowing extended neighborhood, i.e., more neighbors than the closest two in B we have the following conjecture. Conjecture. The B-distance is metrical on Q n if it is metrical on Z n. If Conjecture 1 holds, then the proof of Conjecture goes in the same manner as it works in Z n, see [13, 15]. 4 Distance Transform In this section we apply the digital distances defined above in an image processing algorithm, namely to produce distance transform. Definition 4 (Image). The image domain is a finite subset of Q n denoted I. We call the function F : I R + 0 an image. Note that real numbers are allowed in the range of F. Definition 5 (Object and background). We denote the object X and the background X. These sets have the following properties: 1. X I and X I. X X = 3. X X = I. We denote the distance transform for distances based on neighborhood sequences with DT.

10 Definition 6. The distance transform DT of an object X I is the mapping DT : I R defined by p d ( p, X; B ), where d ( p, X; B ) = min {d (p, q; B)}. q X Note that the distance transform DT holds information about the minimal distance to the background as well as information about what size of neighborhood is allowed in each step. This is used in Algorithm, where a wave-front propagation technique is used to compute the DT. Algorithm : Computing the distance transform DT for distances based on neighborhood sequences by wave-front propagation on Q n. Input: The neighborhood sequence B and an object X Q n. Output: The distance transforms DT. Initialization: Set DT (p) 0 for grid points p X and DT (p) for grid points p X. For all grid points p X adjacent to X: push (p, DT (p)) to the list L of ordered pairs sorted by increasing DT (p). while L is not empty do foreach p in L with smallest DT (p) do Pop (p, DT (p)) from L; foreach q: q, p are b (DT (p) + 1)-neighbors do if DT (q) > DT (p) + 1 then DT (q) DT (p) + 1; Push (q, DT (q)) to L; end end end end Algorithm can be used to generate balls in Q n for any n by thresholding the distance transform, where only a single grid point is in the background. 4.1 Digital balls One of the easiest and most usual ways to analyze a distance function d is to analyze the digital balls, i.e., (hyper)spheres based on d. In Figure 3 and 4, balls of radius 4 are shown. In Figure 4, balls generated by some neighborhood sequences in Q 4 (the fcc grid) are shown. Figure 3 illustrates balls in the four-dimensional grid Q 5 generated by B = (1, ). To illustrate the four-dimensional ball, the subsets {p : p(1) + p() + p(3) + p(4) + p(5) = 0, p(5) = k} for k = are shown.

11 01 elements 88 elements 390 elements k = 4 k = 3 k = 43 elements 459 elements 43 elements k = 1 k = 0 k = elements 88 elements 01 elements k = k = 3 k = 4 Fig. 3. A ball of radius 4 in the four-dimensional grid Q 5 generated by B = (1, ) is illustrated by showing the subsets {p : p(1) + p() + p(3) + p(4) + p(5) = 0, p(5) = k} for some values of k. Each obtained grid point is shown as a rhombic dodecahedron, since for each k, an fcc grid is obtained. 5 Conclusions and Future Work Non-traditional grids are used in image processing and computer graphics. Some results on the hexagonal grids can be found in [, 6, 9]. Non-standard grids are also used in higher dimensions [8, 1]. The hexagonal grid and the fcc grid are the most dense packing with the highest kissing number in D and 3D, respectively. The dual of the fcc grid gives a thinnest covering in 3D [3]. Their extensions,

12 B = (1, 1, 1, 1) B = (1, 1, 1, ) B = (1,, 1, ) B = (1,,, ) B = (,,, ) 309 elements 405 elements 459 elements 483 elements 489 elements Fig. 4. Balls of radius 4 in Q 3 (the fcc grid) obtained by some neighborhood sequences. that were analysed in this paper, are also important grids. The dual of the nd fcc grids is a thinnest covering in 4D, 5D, 6D, 7D, 8D, 1D and 16D [3], therefore our work is not only for theoretic interest, but it is worth to consider some of these grids in applications. Their advantage and the easy-to-handle property of the classical integer lattice Z n can be used in a common frame by our method. Since there are various neighbors in the grid, one may use distances varying them along a path. In this way a large class of flexible digital distances are defined and used, namely distances based on neighborhood sequences. Some results as an algorithm to generate a shortest path between two points, and hence, compute their distance were presented for the general case. In a restricted case, using only the two closest neighborhood relations, further properties were presented, such as formula for the distance and conditions of metrical distances. Some problems left open using more extended neighborhood structure, we addressed some of them as conjectures, their proofs are topics of future work. Our result can also be applied in digital geometry and in several applications when (digital) distances can/should be used. As an example the distance transform is presented. References 1. G. Borgefors, Distance transformations in digital images, Computer Vision, Graphics, and Image Processing 34(3) (1986) V. E. Brimkov and R. Barneva, Honeycomb vs square and cubic models, Electronic Notes in Theoretical Computer Science 46 (001). 3. J. H. Conway, N. J. A. Sloane and E. Bannai, Sphere-packings, lattices, and groups, (Springer-Verlag, New York, NY, USA 1988). 4. J. W. Cooley, P. Lewis and P. Welch, The fast Fourier transform and its applications, IEEE Trans. on Education 1(1) (1969) C. Fouard, R. Strand and G. Borgefors, Weighted distance transforms generalized to modules and their computation on point lattices, Pattern Recognition 40(9) (007) I. Her, Description of the F.C.C. lattice geometry through a four-dimensional hypercube. Acta Cryst. A 51 (1995)

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