Discrete shading of three-dimensional objects from medial axis transform

Transcription

1 Pattern Recognition Letters 20 (1999) 1533± Discrete shading of three-dimensional objects from medial axis transform Jayanta Mukherjee a, *, M. Aswatha Kumar b, B.N. Chatterji c, P.P. Das a a Deartment of Comuter Science and Engineering, IIT, Kharagur , India b Deartment of Electronics and Communications Engineering, BDT College of Engineering, Davangere, Karnataka, India c Deartment of Electronics and ECE Deartment, IIT, Kharagur , India Received 6 January 1999; received in revised form 4 August 1999 Abstract In this aer, we resent a discrete shading technique using medial axis transform (MAT) of 3D binary image data based on digital generalized octagonal distances. Our method is comutationally attractive as it does not require the exlicit comutation of surface normals. We have comared our results with images rendered from voxel and octree reresentations while using analytical surface rendered images as bench marks. The quality of rendering by our method is certainly suerior to those obtained from voxel and octree reresentations. Ó 1999 Elsevier Science B.V. All rights reserved. Keywords: Medial axis transform; Medial shere reresentation; Octagonal distance; Octree; Discrete shading 1. Introduction In various alications of medical imaging, image rocessing, solid modeling, 3D simulation and scienti c visualization, 3D objects are reresented in a 3D binary matrix, where occuancy of each unit object volume cell (or voxel) is described. The shae of the voxels is taken to be a cube and thus this model is oularly known as cuberille model (Chen et al., 1985). Visualization of 3D objects reresented in this form is one of the major challenges today. For achieving this, it is not only su cient to roject the object surface oints as * Corresonding author. Tel.: ; fax: address: jay@cse.iitkg.ernet.in (J. Mukherjee) visible from the viewing direction on to a 2D screen, but shading is also required to rovide the critical deth cues. In general, shading involves comutation of the intensity (brightness) value that reaches the viewer's eye from each visible surface-oint. The intensity value may be comuted by di erent shading models taking into account of the surface normal, the characteristics of object-surfaces, the distance from the viewer's osition and the light sources (Foley and Van Dam, 1984). In this aer, we discuss shading of 3D objects reresented by voxels, which is referred to as discrete shading (Cohen et al., 1990; Tuy and Tuy, 1984; Frieder et al., 1985; Herman and Udua, 1981; Chen et al., 1985; Gordon and Reynolds, 1985; Doctor and Torborg, 1981). In conventional shading techniques, object surfaces are comosed /99/$ - see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S ( 9 9 )

2 1534 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533±1544 of olygonal meshes or mathematical surfaces which describe surface normals with high degree of accuracy at each surface oint. In contrast, in the discrete shading technique, boundary surfaces of 3D objects and their normals are unknown and must be recovered from the voxel information. In the cuberille model, a voxel has six faces with normals in the directions of the rimary axes and at most three faces are visible from a viewing direction. Based on these facts, various discrete shading techniques have been develoed. The major contentions of these work are to kee the comutation time for nding surface normals as less as ossible and to increase the accuracy of the comutation of surface normals. There lies a trade-o between these two objectives as the increase in the degree of the accuracy of extraction requires more comutation. However, the techniques are nally judged by the quality of the rendered images of known 3D geometric objects (voxel form) comared (visually) with the benchmark images (roduced by conventional rendering techniques such as Gouraud's shading (Gouraud, 1971), Phong's shading (Phong, 1975) from the mathematical descrition of these surfaces). Some of the reorted techniques of discrete shading in cuberille environment are: constant shading (Foley and Van Dam, 1985), distance-only (deth-only) shading (Tuy and Tuy, 1984; Frieder et al., 1985), image-based contextual shading (Herman and Udua, 1981), normal-based contextual shading (Chen et al., 1985), congradient shading (Gordon and Reynolds, 1985), etc. It is easy to observe that the cuberille environment suits octree encoding of the voxel data as the octree technique reresents the object volumes by a set of disjoint cubes of di erent sizes. Hence, similar methodologies have been adoted for discrete shading of octree encoded data (Doctor and Torborg, 1981). The advantage of using the octree encoded data is the reduction in the volume of the data to be rocessed (less number of cubes to be rendered). But the disadvantage lies in the extra e ort required for comuting the neighboring volume elements of a voxel that is required for more accurate comutation of surface normals. In this aer, we focus our attention to discrete shading on a non-cuberille environment, called the medial axis transform (MAT) (Rosenfeld and Pfaltz, 1968; Jain, 1989) of a 3D object. As the maximal blocks of MAT are also termed as `sheres' (on di erent metrics), the reresentation is also called as medial shere reresentation (MSR). The advantage of MSR not only lies in data reduction but also in reresenting the surface normals in more number of directions exloiting the geometry of medial sheres. 2. Medial shere reresentation using octagonal distance In 2D, binary images could be reresented by medial axis transform (MAT), which is also called as medial circle reresentation (MCR). MAT consists of a set of center and radii of maximal blocks of the image R. Here every center is located half-way between the boundary on either side. Hence the collection of all these centers forms a sort of sine for the object and is called the medial axis. The comutation of medial axis requires the distance transformation. Distance transform converts the binary image R consisting of object ixels (1) and background ixels (0), into an image where all ixels have a value corresonding to the distance of the nearest background ixel. The local maxima of the distance transformed image gives the MSR (Kumar et al., 1996). For de ning various digital distances in 2D, di erent neighborhood sequences for ath de nitions have been used (Rosenfeld and Pfaltz, 1968; Rosenfeld and Kak, 1982). The commonly used neighborhood sequences are 4-neighborhood (city block) and 8- neighborhood (chessboard). It is observed that neither the cityblock nor the chessboard distance is close to the Euclidean distance. 3D extension of MCR is known as MSR, which describes a 3D object by the set of centers and radii of medial sheres. Similar to the 2D images, the distance mas are also used in 3D images. Commonly known distance functions in 3D use 6-neighbors (face neighbor), 18-neighbors (face and edge neighbor) and 26-neighbors (face, edge and vertex neighbor). It is interesting to note that the shae of a medial shere varies if we choose di erent

3 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533± neighborhood sequences (Kumar et al., 1996) for forming the distance ma. In our shading algorithm we shall render each medial shere individually and the combined e ect of all the rendered sheres roduces the desired image of the 3D object considering the fact that the grahics environment rovides z-bu ering hardware feature. The comutation gets reduced as the shading of a shere does not require the comutation of surface normals by scanning the neighborhoods of any voxel. Rather, for any oint, the vector from the center of the shere to that oint, gives the desired normal direction. Hence it is imortant to know the shae of the digital medial sheres for di erent octagonal distances in 3D, which could be shown as convex olyhedrons. These have been discussed elsewhere (Kumar et al., 1994). However to make the aer self-content, a brief introduction of 3D octagonal distances (Borgefors, 1984; Das and Chatterji, 1990) and the comutation of vertices of the convex olyhedrons (reresenting digital sheres for the corresonding octagonal distance) are given below. It may be noted here that there are other tyes of distance transforms such as weighted distance transforms (Borgefors, 1986) which are also useful for obtaining MAT. However, in this work we have restricted ourselves to the class of 3D octagonal distances Octagonal distances in 3D In 2D digital lane, two tyes of motions are natural. They are the cityblock motion (movement in horizontal or vertical direction) and the chess board motion (movement in horizontal, vertical or diagonal direction). The distance d B between two oints is de ned as the length of a shortest ath between these two oints which is restricted by a articular tye of motion B. The cityblock movement is marked here as 1-neighbor or B ˆ f1g as it allows a unit change in at most one of the coordinates whereas the chessboard movement allows a unit change in both the coordinates and hence it is termed as 2-neighbor or B ˆ f2g. The distance between two oints i 1 ; j 1 and i 2 ; j 2 is given by d f1g ˆ ji 1 i 2 j jj 1 j 2 j for the city block movement and d f2g ˆ max ji 1 i 2 j; jj 1 j 2 j for the chessboard movement whereas the Euclidean q distance is given by d e ˆ i 1 i 2 2 j 1 j 2 2. In a generalization to the cityblock or the chessboard distances, generalized octagonal distances d B have been de ned in (Das and Chatterji, 1990) where the tye of motion at each ste from oint (i 1 ; j 1 ) to (i 2 ; j 2 ) is determined by a sequence of neighborhoods B ˆ fb 1 ; b 2 ;... ; b g where 8i b i ˆ 1 or 2. As it is imractical to work with an in nite sequence, one usually concerns with the neighborhood sequences which are cyclic in nature with the cycle length ˆ jbj as B ˆ fb 1 ; b 2 ;... ; b ; b 1 ; b 2 ;... ; b ;...g: It may be noted that a secial case of d B with B ˆ f1; 2g was de ned earlier in (Rosenfeld and Pfaltz, 1968) under the name d oct. It is found that the distance function de ned by neighborhood sequence B is a metric if B is sorted. These distances are referred to here as the generalized octagonal distances (Das and Chatterji, 1990). It is interesting to note that the shae of the digital circle deends uon the distance function. The di erent distance functions of neighborhood length uto 4 in 2D are {1}, {1,1,1,2}, {1,1,2}, {1,2}, {1,2,2}, {1,2,2,2} and {2}. Similarly in 3D digital sace Z 3, three tyes of motions have been identi ed. They are face neighborhood (tye 1), edge neighborhood (tye 2) and vertex neighborhood (tye 3). They are termed as B ˆ {1}, B ˆ f2g and B ˆ {3}, resectively. The notion of the neighborhood sequences also naturally extends to 3D. The neighborhood sequences of length uto 3 are {1}, {1,1,2}, {1,1,3}, {1,2}, {1,2,2}, {1,2,3}, {1,3}, {1,3,3}, {2}, {2,2,3}, {2,3}, {2,3,3}, {3} Vertex aroximation Let x r ˆ x 1 r ; x 2 r ; x 3 r be a vertex (lying in the all ositive octant) of the octagonal digital shere in 3D. The vertices of the octagonal digital shere can be comuted using the following lemma (Das and Chatterji, 1990).

4 1536 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533±1544 Lemma 1. (a) x 1 r ˆ r. b x i r ˆ r f i f i 1 f i r mod f i 1 r mod for i ˆ 2; 3; where f i j ˆ X b i k and b i j ˆ 1 6 k 6 j b j if b j 6 i; i otherwise: (c) All other vertices of the shere are re ections and ermutations of the coordinates of this vertex. Hence to reresent a digital shere as a olytoe in 3D, we need to comute the quantities x 1 r, x 2 r and x 3 r for a given value of r. We rovide aroximate formulae below to comute x 2 r and x 3 r. We have assumed here that the neighborhood elements in B are sorted. It then guarantees that d B is always a metric and also allows the aroximation to be tight. In fact this is not a major restriction since for ˆ jbj 6 4 any metric B is sorted, and > 4 hardly has any secial use here. As B is sorted we can use an alternative reresentation for it as trilet (a 1 ; a 2 ; a 3 ) where a 1 1s are followed by a 2 2s and then by a 3 3s, i.e., B ˆ 1 a 1 2 a 23 a 3. Clearly a1 a 2 a 3 ˆ and a 1 a 2 a 3 P 0. We shall aroximate x r by x 0 r where x 0 r ˆ r; r a 2 a 3 =; ra 3 =. We rove that the di erence between x r and x 0 r is bounded by unity on any coordinate. So by this aroximation the estimated vertex does not even cross the next digital oint in any direction from its actual site. Now we are ready to resent the aroximation result. Lemma 2. For any r > 0 and for any B, we have (i) x 0 1 r x 1 r ˆ 0; (ii) 0 6 x0 2 r x 2 6 a 1 a 2 1 = x 0 3 r x 3 6 a 3 a ( 2 3 = =4 even; 6 odd: where x i 's and x 0 i 's are as defined above. For roof, lease see Aendix A. 3. Shading of 3D objects For the uroses of the comarative study on shading, we have considered three di erent reresentations, MSR, octree and voxel data for rendering Using MSR In this technique, we have considered the inut as an MSR of a 3D object. Each medial shere is rendered indeendently aided by the z-bu ering features of the grahics hardware environment. Hence the hidden surface elimination is automatically erformed by the grahics workstation. To render a medial shere, the vertex of the olyhedrons are comuted and the normal at a vertex has been taken as the direction of the vector drawn from the center of the shere to that vertex. This normal has been used to comute the intensity value at that oint by using simle shading model (Foley and Van Dam, 1984). The faces of the olyhedron are then shaded by Gouraud's interolation technique (Gouraud, 1971) using the intensity values comuted at the vertices forming a face Using voxel and octree reresentations A voxel or a leaf node of an octree is of cubic shae. Hence each face of this cube is rendered and the combined e ect roduces shaded image of the 3D object. To render a face of the cube, the intensity at each vertex of the cube is comuted by alying the same shading model followed by Gouraud's interolation technique. The normals at the vertices are drawn from the center of the cube Surface rendering In this aer, we have considered only known 3D geometric objects such as shere, cylinder and cone. They are reresented by olygonal meshes (Rogers and Adams, 1990) derived from their arametric descritions and for each olygonal face we have alied the same shading model (Foley and Van Dam, 1984) and Goraud's inter-

5 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533± olation technique (Foley and Van Dam, 1984) to render the boundary surfaces of these objects. These images are taken as benchmark images and are comared with the discretely shaded images both visually as well as quantitatively by comuting a correlation measure between these images. noted that the value of r will lie between 1 and +1, with 1 indicating strong correlation. Naturally the value nearing 0 imlies oor correlation. We should also note that we are not execting any negative correlation as each reresentation has some utility in the shading comutation. 4. Measure on relative erformance of discrete shading over di erent reresentation schemes We have carried out exeriments to study the relative erformance of discrete shading techniques. For this, the same shading model has been used for roducing the images of 3D objects from di erent reresentations on the same osition of rojection screen using an identical coordinate convention. We have used di erent reresentation schemes such as voxel, octree and MSR. For MSR, we have considered di erent digital octagonal distances. The benchmark image is rovided by the surface rendered image Correlation measure Let the benchmark image be denoted as I b i; j ; i ˆ 1; 2;... ; M and j ˆ 1; 2;... ; M: Let the discretely rendered image be denoted as I d i; j ; i ˆ 1; 2;... ; M and j ˆ 1; 2;... ; M: Then we have used the simle correlation measure r as follows: r ˆ Covariance of I b and I d s:d: of I b s:d: of I d 1=N P P j i ˆ I b i;j I d i;j I b I d q 1=N P P q j i I b 2 i;j I b 2 1=N P P ; j i I2 d i;j I d 2 where I b ˆ 1=M 2 P P P P j i I b i; j and I d ˆ 1=M 2 j i I d i; j, N ˆ total number of ixels such that at least one of I b i; j or I d i; j is non-zero. We have considered background ixels as dark(0) and in the comutation of r, if we get I b i; j ˆ I d i; j ˆ 0 at any oint i; j, we have ignored them (i.e., for each occurance of such case, the value of M 2 is decreased by 1 and the total number of object ixels (N) also adjusted accordingly). This removes the biasing in the correlation measure to its higher value as both the images will have a large number of common background ixel. It may be 5. Exerimental results and discussion Three sets of shaded images obtained by rendering a shere, cylinder and cone from its MSR of di erent rersentative distances have been illustrated, resectively, in Figs. 1±3. In each case, the light direction is taken as 1; 1; 0. In each gure, the benchmark images are shown and the rendered images using MSR with some reresentative octagonal distances such as f1 1 3g; f1 2 3g and f1 3g are also resented. They are found to roduce better images (as indicated also by the correlation Table 1) than those roduced by other octagonal distances. Each gure also contains the rendered images from octree and voxel reresentations. One may observe that the quality of shaded images using MSR is certainly better than those of voxel and octree reresented ones. Table 1 describes the correlation for di erent reresentations of di erent objects for the light direction 1; 1; 0. To reduce the e ect of the light direction in the correlation measure, we have obtained correlations for di erent light directions (such as 1; 0; 0 ; 0; 1; 0 ; 0; 0; 1 ; 0; 1; 1 ; 1; 0; 1 ; 1; 1; 0 and 1; 1; 1 ) and the averages of them are shown in Table 2. From Tables 1 and 2 we observe that the quality of rendered images imroves if we use MSR rather than voxel or octree reresentation of 3D objects. In this table also we may observe that some octagonal distances such as {1 1 3}, {1 2 3}, {1 3} give better results comared to the rest. It is interesting to note here that MSR descrition of a 3D object may consist of several medial sheres of small radii 6 3 lying near the boundary of the object. Even if we ignore the shading of these sheres, the aearance of the shaded image does not degrade and in fact with our quantitative measure (described in Section 4), the quality has been shown to imrove in most of

6 1538 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533±1544 Fig. 1. Comarison of shaded images for Shere comuted from various reresentations and MSRs of di erent octagonal distances: (a) benchmark image; (b) octree reresentation; (c) voxel reresentation and MSR with distances; (d) {1 1 3}; (e) {1 2 3}; (f) {1 3}.

7 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533± Fig. 2. Comarison of shaded images for Cylinder comuted from various reresentations and MSRs of di erent octagonal distances: (a) benchmark image; (b) octree reresentation; (c) voxel reresentation and MSR with distances; (d) {1 1 3}; (e) {1 2 3}; (f) {1 3}.

8 1540 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533±1544 Fig. 3. Comarison of shaded images for Cone comuted from various reresentations and MSRs of di erent octagonal distances: (a) benchmark image; (b) octree reresentation; (c) voxel reresentation and MSR with distances; (d) {1 1 3}; (e) {1 2 3}; (f) {1 3}.

9 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533± Table 1 Correlation between the benchmark images and the rendered images obtained from MSR, octree and voxel data for light direction 1; 1; 0 B Shere Cylinder Cone Octree Voxel Table 2 Average correlation between the benchmark images and the rendered images obtained from MSR, octree and voxel data averaged for di erent light directions ( 1; 0; 0 ; 0; 1; 0 ; 0; 0; 1 ; 0; 1; 1 ; 1; 0; 1 ; 1; 1; 0 and 1; 1; 1 ) B Shere Cylinder Cone Octree Voxel the cases. Hence in our study we have also considered shading of truncated MSR (TMSR) of a 3D object. In TMSR, sheres of smaller sizes are ignored. These small sheres are situated at the boundary oints. Elimination of such small sheres results in reduction in the storage requirement as well as in smoothing the object. Due to the smoothing, the correlation is imroved (shown in Table 3). It could be noted that with the TMSRs for sheres, the correlations are almost similar to those of the MSR cases. As all the medial sheres in the MSR are of radii greater than the threshold value, there is no imrovement in correlation measures in such cases. However, for the cylinder, the correlation is imroved from 0.49 to by using TMSR for the distance function {1} and from to for the distance function {3}. Similarly for cone, the correlation is imroved from to for the distance function {1} (see Tables 2 and 3). It may, however, be noted that truncation has hardly any e ect for distances that roduce good shading e ect. Besides quality, another advantage of this scheme is that it does not require any exlicit surface-normal comutation. Even it does not require any extra storage for coding the normals at the boundary oints. The size of the data is also less. The major bottleneck of the system is the use of the z-bu ering which slows the seed. MSR is not a resorted data structure. In the voxel and the octree reresentations many rendering techniques have e ciently used these features for seeding u the comutation and avoiding the z-bu ering comutation. But one can reartition medial sheres into several grous so that arallel rendering tasks could be emloyed indeendently on those grous which will seed u the oeration. Table 3 Correlation between the benchmark images and the rendered images obtained from TMSR for di erent light directions and comared with ordinary MSR Cylinder Cone B MSR TMSR MSR TMSR

10 1542 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533± Conclusion In this aer, we have resented a shading technique using MSR, which does not require exlicit comutation of surface normals. We have also studied the suitability of a set of generalized octagonal distance for this urose and found that {1 1 3}, {1 3}, {1 2 3} distances erform well in this regard. In comarison with the shaded images obtained by the octree and the voxel reresentations under similar conditions, it has been observed that MSR erforms better in most of the cases (irresective of the use of any 3D octagonal distance). Another interesting observation is that the use of TMSR imroves the quality of shading in some cases as it smoothens the aearance of the shaded surface. It is interesting to note also that TMSR occuies less storage sace as it runes many sheres of smaller radii lying on the boundary. Acknowledgements The comments of the anonymous reviewers for imroving the aer in its nal form are gratefully acknowledged. Aendix A. Proof of Lemma 2 Lemma 2. For any r > 0 and for any B, we have (i) x 0 1 r x 1 r ˆ 0; (ii) 0 6 x0 2 r x 2 6 a 1 a 2 1 = x 0 3 r x 3 6 a 3 a 2 3 ( = =4 even; 6 odd; where x i 's and x 0 i 's are as defined in the text. Proof. From the de nition of B and f i 's we have B 1 ˆ 1 a 1 a 2 a 3 ; B 2 ˆ 1 a 1 2 a 2 a 3 ; B 3 ˆ 1 a 1 2 a 2 3 a 3 ; f 1 ˆ a 1 a 2 a 3 ˆ ; f 2 ˆ a 1 2 a 2 a 3 ; f 3 ˆ a 1 2a 2 3a 3 : Let r ˆ s t; 0 6 t 6 1, br=c ˆ s and r mod ˆ t: Putting the exressions of f i 's in x j 's, we get x 2 ˆ s f 2 f 1 f 2 t t; x 0 2 ˆ s t a 2 a 3 by definition ; x 0 2 x 2 ˆ s t a 2 a 3 s a 2 a 3 f 2 t t ˆ t= a 2 a 3 t f 2 t : We now have three cases. Case 1. t ˆ 0. Clearly, x 0 2 x 2 ˆ 0: Case t 6 a 1 : As f 2 t ˆ t, we get x 0 2 x 2 ˆ t= a 2 a 3 ˆ t= a 1 ˆ t ta 1 = : Case 3. a t 6 a 1 a 2 a 3. Again, f 2 t ˆ a 1 2 t a 1 ˆ 2t a 1 imlies x 0 2 x 2 ˆ t a 2 a 3 a 1 t ˆ t a 1 a 1 t ˆ a 1 t a 1 ˆ 1 t a 1 : Therefore, x 0 2 x 2 P 0 always, i.e., x 0 2 P x 2: We get the bound on the other side by combining the cases as x 0 2 x a max a 1 ; 1 a 1 1 a 1 6 max a 1 a2 1 ; a 1 a2 1 a 1 6 a 1 a2 1 : We maximize h a ˆ a a 2 =; for 0 6 a 6 ; dh 2a ˆ 1 da ˆ 0 ) a ˆ 2 and d2 h da ˆ 2 2 < 0: Therefore, h a j aˆ=2 ˆ ˆ 2 4 ˆ 4 :

11 J. Mukherjee et al. / Pattern Recognition Letters 20 (1999) 1533± a 1, however, is an integer. Hence if is odd, the maxima will occur either at 1 =2 or at 1 =2. We check as follows: Case 1. even. a ˆ 2 and h a ˆ 4 : Case 2. odd. Take a ˆ b=2c ˆ 1 =2, h a ˆ ˆ 2 1 : 4 Again take a ˆ d=2e ˆ 1 =2; h a ˆ We get h a ˆ h a ˆ 2 1 : 4 Therefore, ˆ 2 1 : x 0 2 x 2 6 a 1 a2 1 ( even; a ˆ ; 2 1 odd; a ˆ b c or 4 2 de: 2 Next we consider the aroximation on x 3. x 3 ˆ s f 3 f 2 f 3 t f 2 t So, ˆ sa 3 f 3 t f 2 t and x 0 3 ˆ ra 3 ˆ s t a 3 by definition : x 0 3 x 3 ˆ ta 3 f 2 t f 3 t : Again, we have four cases. Case 1. t ˆ 0; clearly, x 0 3 x 3 ˆ 0: Case t 6 a 1, x 0 3 x 3 ˆ ta 3 : Case 3. a t 6 a 1 a 2 f 2 t ˆ 2t a 1 and f 3 t ˆ 2t a 1 imlies x 0 3 x 3 ˆ ta 3 : Case 4. a 1 a 2 1 < a 1 a 2 a 3 for a 3 P 1 So, f 2 t ˆ 2t a 1 ; f 3 t ˆ a 1 2a 2 3 t a 1 a 2 ˆ 3t 2a 1 a 2 imly x 0 3 x 3 ˆ ta 3 2t a 1 3t 2a 1 a 2 ˆ ta 3 a 1 a 2 t ˆ a 1 a 2 1 t : Combining the cases, 0 6 x 0 3 x 3 6 max a 1 a 2 a 3 ; a 1 a max a 1 a 2 a 3 ; a 1 a 2 a a 1 a 2 a 3 References ˆ a 3 a 3 ˆ a 3 a2 3 a 1 a 2 1 ( 6 odd: =4 even; Borgefors, G., Distance transformations in arbitrary dimensions. Comuter Vision Grahics and Image Processing 27, 321±345. Borgefors, G., Distance transformations in digital images. Comuter Vision Grahics and Image Processing 34, 344± 371. Chen, L.S., Herman, G.T., Reynolds, A., Udua, J.K., Surface shading in the cuberille environment. IEEE Comuter Grahics and Alications 5 (12), 33±43. Cohen, D., Kaufman, A., Bakalash, R., Bergman, S., Real time discrete shading. Visual Comuter 6, 16±27. Das, P.P., Chatterji, B.N., Octagonal distances for digital ictures. Information Sciences 50, 123±150. Doctor, L.J., Torborg, J.G., Dislay techniques for octree, encoded objects. IEEE Comuter Grahics and Alications 1 (7), 29±38.

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