Weighted Neighborhood Sequences in Non-Standard Three-Dimensional Grids Parameter Optimization

Transcription

1 Weighted Neighborhood Sequences in Non-Standard Three-Dimensional Grids Parameter Optimization Robin Strand and Benedek Nagy Centre for Image Analysis, Uppsala University, Box 337, SE-7505 Uppsala, Sweden Department of Computer Science, Faculty of Informatics, University of Debrecen, PO Box, 400, Debrecen, Hungary Abstract. Recently, a distance function was defined on the face-centered cubic and body-centered cubic grids by combining weights and neighborhood sequences. These distances share many properties with traditional path-based distance functions, such as the city-block distance, but are less rotational dependent. We introduce four different error functions which are used to find the optimal weights and neighborhood sequences that can be used to define the distance functions with low rotational dependency. Introduction When using non-standard grids such as the face-centered cubic () grid and the body-centered cubic () grid for 3D images, less samples are needed to obtain the same representation/reconstruction quality compared to the cubic grid [0]. This is one reason for the increasing interest in using these grids in, e.g., image acquisition [0], image processing [4,6,5], and visualization [,7]. Measuring distances on digital grids is of great importance both in theory and in many applications. Because of its low rotational dependency, the Euclidean distance is often used as distance function. In digital grids, however, the Euclidean distance may not be the best option [9]. Both from a theoretical point of view and for several applications path-based digital distances are better options. For example, when minimal cost-paths are computed, a distance function defined as the minimal cost path between any two points is better suited, see, e.g., [3], where the constrained distance transform is computed using the Euclidean distance resulting in a complex algorithm. The corresponding algorithm using a path-based approach is simple, fast, and easy to generalize to higher dimensions [, 8]. Examples of path-based distances are weighted distances, where weights define the cost (distance) between neighboring grid points [, 6, 4], and distances based on neighborhood sequences, where the cost is fixed but the adjacency relation is allowed to vary along the path [3,5]. These pathbased distance functions are generalizations of the well-known city-block and

2 chessboard distance function defined for the square grid in []. We will abbreviate neighborhood sequence with ns, distance based on neighborhood sequences with ns-distances, and weighted distances based on neighborhood sequences with weighted ns-distances or just wns-distances. Many approaches where the deviation from the Euclidean distance is minimized in order to find the optimal ns (ns-distances) or weights (weighted distances) have been proposed for Z. In most papers, error functions minimizing the asymptotic maximum difference of a Euclidean ball and a ball obtained by using ns-distances [3, 5, 4] or weighted distances [,, 6] are minimized. Other approaches have also been considered for ns-distances. In [7], optimal ns for the D hexagonal and triangular grids are found using a compactness ratio the ratio between the squared perimeter and the area of the convex hull of the disks obtained by using ns. In [8], the symmetric difference is used for ns in Z and in [], the following error functions are considered for ns on the and the grids: absolute error, relative error, compactness ratio, maximal inscribed ball, and minimal covering ball. In [3], a general definition allowing both weights and ns was presented. The full potential of using both weights and ns was discovered in [9], where ns and weights were together used in the sense of [3], but with the well-known natural neighborhood structure of Z. In [9], the basic theory for weighted ns-distances on the square grid is presented including a formula for the distance between two points, conditions for metricity, optimal weight calculation, and an algorithm to compute the distance transform. In [6], some results for weighted ns-distances on the and grids were presented. The theory for weighted ns-distances on the and grids was further developed in [0] by presenting sufficient conditions for metricity and algorithms that can be used to compute the distance transform and a minimal cost-path bewteen two points. The asymptotic error using the compactness ratio was used to find the optimal weights and ns for weighted ns-distances on the and grids in [6]. The analysis presented in [6] is extended in this paper by considering the relative error, the compactness ratio, the maximal inscribed ball, and the minimal covering ball. Also, we analyze the behavior when ns of finite length is used. Note that the results presented here also applies to weighted distances and ns-distances, since they are both special cases of the proposed distance function. The distance function proposed here is used to find optimal weights for the weighted distance and optimal ns for ns-distances. Basic Notions and Previous Results The following definitions of the face-centered cubic (: F) and body-centered cubic (: B) grids are used: F = {(x, y, z) : x, y, z Z and x + y + z 0 (mod )}. () B = {(x, y, z) : x, y, z Z and x y z (mod )}. ()

3 For results that are valid for both the grid F and the grid B, the notation G is used. Two distinct grid points p = (x, y, z ),p = (x, y, z ) G are ρ-neighbors, ρ, if. x x + y y + z z 3 and. max { x x, y y, z z } ρ The points p,p are adjacent if p and p are ρ-neighbors for some ρ. The -neighbors which are not -neighbors are called strict -neighbors. The neighborhood relations are visualized in Figure by showing the Voronoi regions, i.e. the voxels, corresponding to some adjacent grid points. Fig.. The grid points corresponding to the dark and the light grey voxels are - neighbors. The grid points corresponding to the dark grey and white voxels are (strict) -neighbors. Left:, right:. A ns B is a sequence B = (b(i)) i=, where each b(i) denotes a neighborhood relation in G. If B is periodic, i.e., if for some fixed strictly positive l Z +, b(i) = b(i + l) is valid for all i Z +, then we write B = (b(), b(),...,b(l)). A path, denoted P, in a grid is a sequence p 0,p,...,p n of adjacent grid points. A path is a B-path of length n if, for all i {,,..., n}, p i and p i are b(i)-neighbors. The notation - and (strict) -steps will be used for a step to a -neighbor and step to a (strict) -neighbor, respectively. Definition. Given the ns B, the ns-distance d(p 0,p n ; B) between the points p 0 and p n is the length of (one of) the shortest B-path(s) between the points. 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 length of the (α, β)-weighted B-path P is (n l)α+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 is shorter than the length of the (α, β)-weighted B-path P. Definition. Given the ns B and the weights α, β, the weighted ns-distance d α,β (p 0,p n ; B) is the length of (one of) the minimal cost (α, β)-weighted B- path(s) between the points.

4 The following notation is used: k B = {i : b(i) =, i k} and k B = {i : b(i) =, i k}. We now recall from [6] the following two theorems giving the distance between two grid points (0, 0, 0) and (x, y, z), where x y z 0. We remark that by translation-invariance and symmetry, the distance between any two grid points is given by the formulas below. Theorem. Let the ns B, the weights α, β 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 { x+y+z d α,β (0, (x, y, z); B) = α if x y + z where k = min k (k x) α + (x k) β otherwise, ( ) x + y + z : k max, x k B. The value of k is the least integer that is not less than x+y+z such that k B +k x. Theorem. Let the ns B, the weights α, β, 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 d α,β (0, (x, y, z); B) = (k x) α + (x k) β, where ( ) x + y k = min : k max, x k B. k Here k is the least integer which is not less than x+y such that k B + k x. Not all weights and ns give metric distance functions. The following sufficient conditions for metricity were derived in [0]. Theorem 3. If N b(i) i= j+n i=j 0 < α β α b(i) j, N and then d α,β (, ; B) is a metric on the and grids. 3 Optimization of Weights and neighborhood Sequences The optimization is carried out in R 3 by finding the best shape of polyhedra corresponding to balls of constant radii using the proposed distance functions. To do this, the distance functions presented for the and grids in the

5 previous section are stated in a form that is valid for all points (x, y, z) R 3, where x y z 0. Note that this gives the asymptotic shape of the balls. The following distance functions are considered: { d x+y+z α,β (0, (x, y, z); γ) = α if x y + z (k x) α + (x k) β otherwise, and d where k = min k ( x + y + z : k max, x γ α,β (0, (x, y, z); γ) = (k x) α + (x k) β, where ( ) x + y x k = min : k max,, k γ where k R and γ R, 0 γ is the fraction of the steps where -steps are not allowed (so k B and k B corresponds to γk and ( γ)k, respectively). Note that k x/( γ) if and only if ( γ)k + k x, which is analogous to the condition k B +k x of Theorems and. In this way we obtain a generalization of the distance functions in discrete space G valid for all points (x, y, z) where x y z 0 in continuous space R 3. By considering d α,β (0, (x, y, z); γ) = r and d α,β (0, (x, y, z); γ) = r, (3) for some radius r, the points on a sphere of constant radius r are found. When γ R (0 < γ < ) the functions d and d can be understood as asymptotic approximations to the distance functions of Theorems and for large values of x. For any triplet α, β, γ (α, β > 0 and 0 γ ), (3) defines polyhedra P in R 3. The vertices of the polyhedra are derived in [6]. Based on this, up to permutation of the coordinates and change of signs, the vertices of polyhedra with radius r are ( r ( r γ γα + β βγ, γ γα + β βγ, γ γα + β βγ, ) γ γα + β βγ, 0 ) γ and r γα + β βγ ( and r ) α, ) α, 0 for d and ( α, α, ) for d α The shape of the polyhedra obtained for some values of α, β, γ are shown in Figure for the and grids, respectively. In approximations the ratio of α and β matters. Let A P be the surface area and V P the volume of the (region enclosed by the) polyhedron P. The values of A P and V P are determined by the vertices of the polyhedra. Let B r be a Euclidean ball of radius r. The following error functions are considered E = max p,q P ( ) p q (relative error) (4)

6 Fig.. Shapes of balls for d α,β (, ; γ) (left 5 5 block) and d α,β(, ; γ) (right 5 5 block) for a fixed radius r, α =, and (left to right) γ = 0,0.5, 0.5, 0.75, and (top to bottom) β =,.5,.5,.75,. A 3 P VP E = 36π (compactness ratio) (5) E 3 = min P/V Br ) (maximal inscribed ball) r:b r P (6) E 4 = min (V Br /V P ) (minimal covering ball) r:p B r (7) These error functions attain their minimum value when A is the surface area and V is the volume of a Euclidean ball. The values of α, β, and γ that minimize the error functions are computed numerically. All error functions attain a minimum value within the domain 0 < α β α, 0 γ, so the computation is straight-forward. For the relative error on the grid, the optimum is obtained on a region as shown in Figure 3. The optimal values are found in Table and visualized by the shape of the corresponding polyhedra in Figure 4. In Table, the asymptotic behavior is shown by letting α = and β be constant and for each k, k 000 using a ns B of length k that approximates the optimal fraction γ. The values of the error functions and k are plotted in the figures. In Table, C can be any value in the range C 5/3. (8)

7 Moreover, C 3 can be any value in the range 0 C 3 ( ) and C any value satisfying the following inequalities (see also Figure 3): C 3 + C 3 C 3 C 3 C min ( 3 C3 ( 3 + ) C 3, 5 3 ). (9) Table. Performance of wns-, weighted- (w), and ns-distances using the error functions E E 4 defined in the text. The optima of the error functions are attained whenever t is a strictly positive real number. The values shown in bold are fixed in the optimization. The parameters C, C, and C 3 are related as is shown in (8) and (9). Relative error, E Name α β γ E α β γ E w t C t 0.47 t.547t ns [/3, ].47 wns t C t C 3.47 t t [/3, ].47 Compactness ratio, E Name α β γ E α β γ E w t.530t t.808t 0.85 ns wns t.486t t.99t Maximal inscribed ball, E 3 Name α β γ E 3 α β γ E 3 w t (5/3)t t.808t 0.85 ns wns t (5/3)t t.99t Minimal covering ball, E 4 Name α β γ E 4 α β γ E 4 w t t t.547t ns / wns t.79t t t / The plots in Table show how the error functions perform for ns of finite lengths. neighborhood sequences obtained by the following recursive formula are used { if k b(k + ) = B < γk, otherwise.

8 The value of γ is shown in Table. For the cases when γ is not uniquely defined, we use constant values within the allowed interval. The same thing applies to β C C Fig.3. The domain for C and C 3 in Table 4 Conclusions By introducing a number of error functions that all favor round balls (in the Euclidean sense), the weighted ns-distance is analyzed for the and grids. It turns out that the optimal parameters for the special cases of weighted distances (γ = 0 or B = ()) and ns-distances (α = β = ) are also found from this procedure by keeping one of the parameters fixed in the optimization. The same weights and neighborhood sequences as were derived for weighted distances [4,6] and ns-distances [5] are found in this paper. Figure gives an overview of this fact weighted distances are shown in the left columns (γ = 0) and ns-distances are shown in the top rows (β = ). We also note that, as expected, the value of the error functions for the wns distance function are lower than (or, in some cases, equal to) the weighted distance and ns-distance. In the optimization, we let γ represent the fraction of :s in the ns. We note that when γ is fixed to 0 (for weighted distances), this corresponds to a ns with only :s. This can be attained for a neighborhood sequence of any length. Therefore, in this case the optimum is not asymptotic and thus, the error is valid also for short distances (between e.g. neighboring grid points). When γ is also subject to optimization (i.e. when the neighborhood sequence is used to define the distance function), the error functions have an asymptotic behavior. However, some of the optima for the relative error (E ) are located on regions where E is constant. For example, E for the weighted ns is optimal when γ = 0, i.e. for the weighted distance, and therefore the optimum is attained for any neighborhood sequence consisting of only :s. See Table and Figure 3. This indicates that this error function, which has been widely used in the literature, is not well-suited

9 for finding the optimal weights and ns here. The reason that E is minimal on a region (and not a point) is that there are two vertices and two surfaces that have points that can be at minimal distance (up to symmetry). Thus, there are more degrees of freedom than the restrictions in the optimization process. For the other error functions (E E 4 ), differentiable functions are defined and they have all a single minimum, see Table. We note that the error functions E (compactness ratio) and E 3 (maximal inscribed ball) give the same asymptotic optimal result for the grid and the same for ns-distances on the grid, see Table. However, as is seen in Table, the error functions perform differently for finite neighborhood sequences. This illustrates that the different error functions are different, even though they all are used to approximate the Euclidean distance. Different applications require different aspects of the roundness of the balls. Analysing the plots in Table, we see that the error converges quite fast and that a ns of length (period) 0 is sufficient in general. The application in which the distance function will be applied should be used to select which error function that should be considered. Also, by using Theorem 3, it is easy to find neighborhood sequences such that the resulting distance function is a metric, which is preferable in many applications. Intuitively, the polyhedra that best approximate the Euclidean ball are given by a distance function where both γ and β are non-trivial, see Figure. From the optimal shapes in Figure 4, we see that this is what, e.g., the compactness ratio E favors. Thus, without any specific application in mind, we suggest the parameters B = (, ), β =.486α for the grid and β =.99α for the grid. This gives E =.76 for the grid (the optimum is.757 and the approximated value is.760) and E =.59 (the optimum is.578) for the grid. We conclude that we get a good approximation of the optimal values also with a short neighborhood sequence. Acknowledgements The authors thanks for the reviewers for their useful comments. The research is supported by Digital geometry with applications in image processing in three dimensions (DIGAVIP) at Center for Image Analysis, Uppsala University. References. Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics, and Image Processing 34 (986) Carr, H., Theussl, T., Möller, T.: Isosurfaces on optimal regular samples. In G.- P. Bonneau, S. Hahmann, C.D.H., ed.: Proceedings of the symposium on Data visualisation 003, Eurographics Association (003) Coeurjolly, D., Miguet, S., Tougne, L.: D and 3D visibility in discrete geometry: an application to discrete geodesic paths. Pattern Recognition Letters 5(5) (004)

10 4. Das, P.P., Chatterji, B.N.: Octagonal distances for digital pictures. Information Sciences 50 (990) Das, P.P.: Best simple octagonal distances in digital geometry. Journal of Approximation Theory 68 (99) Fouard, C., Strand, R., Borgefors, G.: Weighted distance transforms generalized to modules and their computation on point lattices. Pattern Recognition 40(9) (007) Hajdu, A., Nagy, B.: Approximating the Euclidean circle using neighbourhood sequences. In: Proceedings of 3 rd Hungarian Conference on Image Processing, Domaszék. (00) Hajdu, A., Hajdu, L.: Approximating the Euclidean distance using non-periodic neighbourhood sequences. Discrete Mathematics 83 (004) Klette, R., Rosenfeld, A.: Digital Geometry: Geometric Methods for Digital Image Analysis (The Morgan Kaufmann Series in Computer Graphics). Morgan Kaufmann (004). 0. Matej, S., Lewitt, R.M.: Efficient 3D grids for image reconstruction using spherically-symmetric volume elements. IEEE Transactions on Nuclear Science 4(4) (995) Nagy, B., Strand, R.: Approximating Euclidean distance using distances based on neighbourhood sequences in non-standard three-dimensional grids. In: IWCIA. (006) Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. Journal of the ACM 3(4) (966) Rosenfeld, A., Pfaltz, J.L.: Distance functions on digital pictures. Pattern Recognition (968) Strand, R., Borgefors, G.: Distance transforms for three-dimensional grids with non-cubic voxels. Computer Vision and Image Understanding 00(3) (005) Strand, R., Nagy, B.: Distances based on neighbourhood sequences in non-standard three-dimensional grids. Discrete Applied Mathematics 55(4) (007) Strand, R.: Weighted distances based on neighbourhood sequences in non-standard three-dimensional grids. In Ersbøll, B.K., Pedersen, K.S., eds.: SCIA. Volume 45 of Lecture Notes in Computer Science., Springer (007) Strand, R., Stelldinger, P.: Topology preserving marching cubes-like algorithms on the face-centered cubic grid. In: Proceedings of 4th International Conference on Image Analysis and Processing (ICIAP 07), Modena, Italy. (007) Strand, R., Malmberg, F., Svensson, S.: Minimal cost-path for path-based distances. In: Proceedings of 5 th International Symposium on Image and Signal Processing and Analysis (ISPA 007), Istanbul, Turkey. (007) Strand, R.: Weighted distances based on neighbourhood sequences. Pattern Recognition Letters 8(5) (007) Nagy, B., Strand, R.: Weighted neighbourhood sequences in non-standard threedimensional grids metricity and algorithms. Submitted for publication (007).. Verwer, B.J.H., Verbeek, P.W., Dekker, S.T.: An efficient uniform cost algorithm applied to distance transforms. IEEE Transactions on Pattern Analysis and Machine Intelligence (4) (989) Verwer, B.J.H.: Local distances for distance transformations in two and three dimensions. Pattern Recognition Letters () (99) Yamashita, M., Ibaraki, T.: Distances defined by neighborhood sequences. Pattern Recognition 9(3) (986)

11 Table. Optimal values of E E 4 (vertical axis) on the and grid for neighbourhood sequences of length k (0 < k 000, horizontal axis showing log k) with α =. See Table for asymptotic optima. E ns weighted ns ns weighted ns β = β =.5 β = β = γ = γ = 0 γ = 0.5 γ = 0.5 E ns weighted ns ns weighted ns β = β =.486 β = β =.99 γ = γ = γ = γ = E 3 ns weighted ns ns weighted ns β = β = 5/3 β = β =.99 γ = γ = γ = γ = E 4 ns weighted ns ns weighted ns β = β =.79 β = β = γ = γ = γ = /3 γ = /3

12 E β = β = β =.5 β =.547 β = β = γ = 0 γ = γ = 0.3 γ = 0 γ = γ = E β =.530 β = β =.486 β =.808 β = β =.99 γ = 0 γ = γ = γ = 0 γ = γ = E 3 β = 5/3 β = β = 5/3 β =.808 β = β =.99 γ = 0 γ = γ = γ = 0 γ = γ = E 4 β = β = β =.79 β =.547 β = β = γ = 0 γ = γ = γ = 0 γ = /3 γ = /3 Fig.4. Shapes of balls using α = and values of β and γ that minimize E E 4, see Table.