Compact finite-difference approximations for anisotropic image smoothing and painting

Transcription

1 CWP-593 Compact finite-difference approximation for aniotropic image moothing and painting Dave Hale Center for Wave Phenomena, Colorado School of Mine, Golden CO 80401, USA ABSTRACT Finite-difference approximation are uccinctly repreented by their tencil, a et of weight that when applied to adjacent ample of a function approximate ome differential operator. In image proceing the ample are pixel or voxel, and the differential operator mut be inverted for moothing or painting application. For efficiency in uch application requiring invere, the finitedifference tencil hould be compact, uing only a mall 3 3 et of nine pixel or a et of twenty-even voxel. From 2 2 and approximation to gradient operator I obtain 3 3 (9-point) and (27-point) tencil that approximate Laplacian operator. The latter may include tenor coefficient that make them aniotropic. By deriving finite-difference tencil for aniotropic Laplacian in thi way, that i, from approximation to gradient, we guarantee that our approximation to aniotropic Laplacian are ymmetric and poitive-emidefinite. And by chooing the gradient approximation carefully, dicretization error can be made iotropic to leading order. Key word: finite-difference numerical method 1 INTRODUCTION A Laplacian i a ymmetric differential operator that roughen. When applied to a function, it remove any contant bia or linear trend while enhancing higher frequencie. Thi behavior i apparent in the Fourier tranform of the Laplacian operator. For if F (k) denote the Fourier tranform of a function f(x) then f(x) F (k), (1) f(x) F (k). (2) The minu ign in make thi ymmetric differential operator poitive-emidefinite (SPS); it Fourier tranform = k T k i non-negative and implie amplification of higher wavenumber. An iotropic Laplacian roughen a function equally in all direction; it Fourier tranform depend on only the magnitude k of the wavenumber vector k. To roughen a function in one direction more than other, we may include a ymmetric poitiveemidefinite (SPS) tenor D in our Laplacian operator: D f(x) k T Dk F (k). (3) For thi aniotropic Laplacian operator, the direction of maximum roughening i aligned with that eigenvector of D having larget eigenvalue. In practical application involving image, the function f(x) i ampled and partial derivative mut be approximated with finite difference. If we arrange all image ample into a ingle column vector f, then an aniotropic Laplacian may be approximated by D f G T DGf, (4) where G i ome pare matrix that repreent a finitedifference approximation to a gradient. In particular, for a 2-D image with N = N 1 N 2 ample, G i a 2N N matrix G1 G =, (5) G 2 with N N pare matrix component G 1 and G 2 that

2 76 D. Hale approximate partial derivative along the firt and econd image dimenion, repectively. In the finite-difference approximation of equation 4, the tenor field D i repreented by a block-diagonal matrix D11 D 12 D =, (6) D 12 D 22 compoed of N N diagonal matrice D 11, D 12 and D 22. The 3N poibly non-zero element of thoe matrice determine the direction and extent of roughening, which may vary from ample to ample. 1.1 Image moothing If an aniotropic Laplacian roughen in certain direction, then the invere of an aniotropic Laplacian mooth in thoe ame direction, and one application of uch an invere operator i tructure-oriented image moothing (e.g., Fehmer and Höcker, 2003). Given an input image f, an aniotropically moothed output image g may be obtained a the olution to: (I + G T DG)g = f. (7) The addition of the identity matrix I to the aniotropic Laplacian G T DG ha two conequence. Firt, it make the ymmetric matrix I + G T DG poitive-definite (SPD) o that a olution g can be found efficiently uing conjugate-gradient iteration. Second, it implie that no moothing occur in eigenvector direction for which correponding eigenvalue of the tenor in D are zero. The direction and extent of moothing via equation 7 are determined entirely by the element of the block-diagonal matrix D. Thoe 3N value may be computed from the tructure of the input image f (van Vliet et al., 1995), which may vary from ample to ample. Indeed, our deire to perform patially-varying moothing i the primary reaon that we cannot imply ue fat Fourier tranform to olve equation Image painting Another application of aniotropic Laplacian i image painting, in which miing ample of a painting p are computed to be conitent with the tenor in D and known ample of p. Following Claerbout (1992), let M denote a diagonal making matrix that decribe the location of the miing ample: ( 1 ; p[i] miing M[i, i] = (8) 0 ; p[i] known, and let K be the diagonal complement of M uch that K + M = I. Then to compute the miing part Mp of p we may olve (K + MG T DGM)p = (K MG T DGK)p. (9) Figure 1. A eimic image of channel (provided by Joe Stefani of Chevron). To facilitate the olution of equation 9, I contructed the matrix on the left-hand-ide to be ymmetric and poitive-definite (SPD). Any olution p to equation 9 i alo a olution to an alternative ymmetric poitiveemidefinite (SPS) ytem analogou to that propoed by Claerbout (1992, p. 178): MG T DGMp = MG T DGKp. (10) Uing K + M = I, thi ytem i equivalent to MG T DGp = 0. (11) which imply tate that the miing part of p hould vary linearly in patially-varying direction determined by eigen-decompoition of the tenor in D. In other word, olution of equation 9 yield a tructure-oriented multi-dimenional linear interpolation of the known ample Kp of the painting p. In practice, thoe known ample could be pecified interactively or determined from other information. For eimic image of the earth uburface, the known ample could correpond to location where well log are available. Figure 1 and 2 how an example of image painting via equation 9. In thi example, I painted one pixel red near the center of the image p hown in Figure 1. The olution to equation 9 i hown in Figure 2. Although thi painting i an extreme example in which only a ingle pixel i known, it demontrate that paint flow to other ample according to the image tructure repreented by the tenor in D. The painted image in Figure 2 may vary depending on the finite-difference approximation G 1 and G 2 in equation 9. In fact, my firt attempt at image painting yielded reult more like the one hown in Figure 3, in which artifact roughly orthogonal to image feature are

3 Finite-difference approximation 77 Figure 2. The image of Figure 1 painted with a finitedifference approximation to an aniotropic Laplacian. A ingle pixel inide the white circle wa contrained to be red. All other pixel were painted uing tructure etimated from the image. Figure 4. The central portion of the image of Figure 3. Painting artifact have a checkerboard pattern correponding to high wavenumber near the patial Nyquit frequencie. aniotropic Laplacian. The purpoe of thi paper i to explain thee artifact and to derive the improved approximation that were ued to obtain the painting in Figure 2. I begin by deigning a family of 2-D finite-difference approximation to iotropic Laplacian. The deign method ued facilitate the derivation of approximation to aniotropic Laplacian. I then ue thi ame method to obtain new 3-D finite-difference approximation for both iotropic and aniotropic Laplacian. Both 2-D and 3-D approximation are compact in that they approximate derivative for any image ample uing only nearet neighbor ample: 9 ample for 2-D and 27 ample for 3-D Laplacian. Both 2-D and 3-D approximation maintain the SPS property of G T DG, which enure SPD ytem of equation 7 and 9. Compact SPD finite-difference approximation lead to the mot efficient method for olving uch equation. Figure 3. The image of Figure 1 painted with a poor finitedifference approximation to an aniotropic Laplacian. A ingle pixel inide the white circle wa contrained to be red. All other pixel were painted uing tructure etimated from the image. apparent. A cloeup view in Figure 4 how that thee artifact have a checkerboard pattern correponding to high wavenumber near the patial Nyquit frequencie. 2 ISOTROPIC LAPLACIANS Let u firt conider approximation to iotropic Laplacian in two dimenion: G T G = G T 1 G 1 + G T 2 G 2. (12) Finite-difference approximation G 1 and G 2 to partial derivative are mot uccinctly decribed by tencil that pecify the weight applied to adjacent image ample. To obtain compact tencil for G T G, I conider Artifact uch a thoe highlighted in Figure 4 are caued by poor finite-difference approximation to

4 78 D. Hale approximation G 1 and G 2 with the following tencil: r G 1 =, G 2 =, r r r r 0, r + = 1. (13) The condition r + = 1 enure that G T 1 G 1 and G T 2 G 2 are 2nd-order approximation, auming unit image ampling interval. Gathering image ample with weight provided by the tencil in equation 13 i equivalent to multiplication by the matrice G 1 and G 2. Scattering with the ame weight i equivalent to multiplication by their tranpoe G T 1 and G T 2. Here i a mall fragment of a computer program that both gather and catter in thi way to compute g = G T Gf: for (int i2=1; i2<n2; ++i2) { for (int i1=1; i1<n1; ++i1) { float f1r = f[i2 ][i1 ]-f[i2 ][i1-1]; float f1 = f[i2-1][i1 ]-f[i2-1][i1-1]; float f2r = f[i2 ][i1 ]-f[i2-1][i1 ]; float f2 = f[i2 ][i1-1]-f[i2-1][i1-1]; float g1 = r*f1r+*f1; float g2 = r*f2r+*f2; // gather end float g1r = g1*r; // catter begin float g1 = g1*; float g2r = g2*r; float g2 = g2*; g[i2 ][i1 ] = g1r+g2r; g[i2 ][i1-1] -= g1r-g2; g[i2-1][i1 ] += g1-g2r; g[i2-1][i1-1] -= g1+g2; Although thi program fragment ignore boundarie, it can be eaily modified to implement zero-value or zerolope boundary condition. Gathering and cattering in thi way uing the 2 2 tencil of equation 13 i equivalent to imply gathering with the following 3 3 tencil: G T G = r (14) Compact finite-difference approximation to Laplacian are often pecified in thi way, a a 3 3 tencil for G T G (e.g., Patra and Karttunen, 2005). I focu intead on the 2 2 tencil for G 1 and G 2, the component of G, becaue thoe will facilitate compact SPS finite-difference approximation G T DG to aniotropic Laplacian with patially-varying tenor in D. The condition r 0 and r + = 1 imply a family of finite-difference approximation for which 0 r 1/4. Figure 5 how three different member of thi family, correponding to r = 0, 1/12, and 1/4, with their Fourier tranform /6-2/3-1/6-2/3 10/3-2/3-1/6-2/3-1/6-1/2 0-1/ /2 0-1/2 (a) r = 0 (b) r = 1/12 (c) r = 1/4 Figure 5. Stencil and Fourier tranform for three different finite-difference approximation to Laplacian. Fourier amplitude diplayed are clipped between zero (dark blue) and four (dark red). Wavenumber and are in the range [ π, π] radian per ample. The three approximation diplayed in Figure 5 are comparable near the origin, for low wavenumber and, where contour are nearly circular with amplitude that well approximate the ideal Fourier tranform = Thi iotropy can be important in application uch a image moothing and painting, where reult hould not depend on ampling direction. For higher wavenumber, error are viible in all three approximation, but contour for r = 1/12 appear mot nearly circular. In fact, error for thi middle approximation are well-known to be iotropic to 2nd order, and among all compact 9-point tencil thi approximation to the iotropic Laplacian operator i optimal in that ene (e.g., Patra and Karttunen, 2005). Let u define dicretization error a a function E(k) of wavenumber k by the Fourier tranform pair G T G [1 E(k)]. (15)

5 Finite-difference approximation 79 The Fourier tranform of our finite-difference tencil G T G (equation 14) i the product of the ideal factor and a econd factor [1 E(k)]. The econd factor include the error E(k) that we would like to be zero or, failing that, at leat iotropic to O( ). From the erie expanion of the Fourier tranform of G T G we find that for r = 1/12 the dicretization error E(k) = k O(k4 ) (16) i indeed iotropic to O( ). For r 1/12, the firt 2nd-order term of thi error i aniotropic; it depend on the wavenumber vector k and not jut it magnitude k. The coefficient r and of G 1 and G 2 for the optimal approximation are eaily found from the condition r 0, r + = 1 and r = 1/12. They are r = 1/2 + 1/ 6, = 1/2 1/ 6. The coefficient for r = 1/4 are imply r = = 1/2. Thee coefficient imply a imple averaging of finite difference in G 1 and G 2 defined by equation 13. Oberve however that the Fourier tranform diplayed in Figure 5c implie that thi approximation attenuate high wavenumber near Nyquit, in addition to low wavenumber near the origin. Thi obervation can help u undertand the image painting artifact in Figure 4. When the finite-difference operator of Figure 5c i applied to an input contant or checkerboard image f, the output image g = G T Gf i zero. The invere of uch an operator will tend to amplify the ame contant or checkerboard feature. 3 ANISOTROPIC LAPLACIANS From finite-difference approximation G 1 and G 2 we may alo contruct approximation D G T DG = ˆG»D T 1 G T 11 D 12 G1 2 D 12 D 22 G 2 (17) to aniotropic Laplacian. We imply gather with G, multiply by D, and catter with G T. The correponding computer program might look like thi: for (int i2=1; i2<n2; ++i2) { for (int i1=1; i1<n1; ++i1) { float f1r = f[i2 ][i1 ]-f[i2 ][i1-1]; float f1 = f[i2-1][i1 ]-f[i2-1][i1-1]; float f2r = f[i2 ][i1 ]-f[i2-1][i1 ]; float f2 = f[i2 ][i1-1]-f[i2-1][i1-1]; float f1 = r*f1r+*f1; float f2 = r*f2r+*f2; float g1 = f1*d11[i2][i1]+f2*d12[i2][i1]; float g2 = f1*d12[i2][i1]+f2*d22[i2][i1]; float g1r = g1*r; float g1 = g1*; float g2r = g2*r; float g2 = g2*; g[i2 ][i1 ] = g1r+g2r; g[i2 ][i1-1] -= g1r-g2; g[i2-1][i1 ] += g1-g2r; g[i2-1][i1-1] -= g1+g2; The gather-catter ymmetry in thi program enure an SPS implementation of G T DG. A above, thi implementation i equivalent to one that gather with a 9-point tencil. However, the latter implementation would require a more complex evaluation of tencil coefficient that vary from ample to ample inide the innermot loop. 3.1 The problem Although the approximation implemented by thi computer program i SPS, it i a rather poor approximation except for the pecial cae r = 1/4. To ee the problem, let u firt conider the approximation with r = 0, for three different et of contant coefficient d 11, d 12 and d 22 of the tenor D. Figure 6a how the 9-point tencil for the cae d 11 = 1, d 12 = d 22 = 0. A expected, thi tencil approximate a econd derivative in the direction of the vertical axi x 1. The problem i apparent in the other two tencil hown in Figure 6b and 6c; thee tencil approximate econd derivative in the direction x 1 = x 2 and x 1 = x 2, repectively. Both tencil approximate the appropriate derivative; but they do o with very different dicretization error. The tencil in Figure 6c hould ideally be a rotated verion of the tencil in Figure 6b, but thi clearly i not the cae. Image moothing or painting with uch tencil will yield urpriing difference for feature oriented at angle of 45 degree and +45 degree with repect to the ampling grid. Stencil for r = 1/12 how the ame problem, a indicated in Figure 7. Although r = 1/12 i optimal for the iotropic cae d 11 = d 22 = 1, d 12 = 0 hown in Figure 5, thi value i clearly not optimal for the aniotropic cae hown in Figure 7. Stencil for r = 1/4, hown in Figure 8, do how the expected (±45-degree) ymmetry and in thi ene are better approximation to aniotropic Laplacian. However, they unfortunately alo exhibit the ame attenuation at high wavenumber near Nyquit that we have already een in Figure 5. Thi attenuation of high wavenumber in approximation to aniotropic Laplacian will caue amplification of thoe ame wavenumber in image moothing and painting application. Thee wavenumber near the patial Nyquit frequencie are apparent in Figure 4, where they appear a a checkboard pattern of painting artifact. Thee high wavenumber are enhanced when we invert approximation like thoe hown in Figure 8.

6 80 D. Hale /12-5/6-1/ /6 5/3 1/ (a) d 11 = 1, d 12 = d 22 = 0-1/12-5/6-1/12 (a) d 11 = 1, d 12 = d 22 = /2-1/6-2/3 1/ /3 7/3-2/3 1/2-1 0 (b) d 11 = d 12 = d 22 = 1/2 1/3-2/3-1/6 (b) d 11 = d 12 = d 22 = 1/ / / /2 0 0 (c) d 11 = d 12 = d 22 = 1/2-1/2 0 0 (c) d 11 = d 12 = d 22 = 1/2 Figure 6. Finite-difference approximation G T DG for r = 0. Figure 7. Finite-difference approximation G T DG for r = 1/ A olution Recall that the approximation for r = 1/12 i in one ene optimal; it i the mot iotropic approximation of the iotropic Laplacian. For aniotropic Laplacian we can improve thi approximation by conidering two additional approximation G 1 and G 2 to the gradient operator. G 1a = G 1b = r r r r, G 2a = r, G 2b = r r r,, (18) In hindight, we had no reaon above to prefer one or the other of the approximation G 1a or G 1b ; likewie for G 2a and G 2b. In equation 13 above, I choe the tencil G 1a and G 2a, but I might jut a well have choen G 1b or G 2b. With two choice for each of the finite-difference approximation G 1 and G 2, we can form a total of four different G T DG. Let u average all four to obtain D 1»D ˆGT 1a G T 11 D 12 G1a 2a 4 D 12 D 22 G 2a + ˆG»D T 1a G T 11 D 12 G1a 2b D 12 D 22 G 2b + ˆG»D T 1b G T 11 D 12 G1b 2a D 12 D 22 G 2a + ˆG»D «T 1b G T 11 D 12 G1b 2b. (19) D 12 D 22 G 2b Thi average of four SPS matrice i clearly SPS. For r = 1/12 and contant tenor coefficient, thi average yield the 9-point tencil hown in Figure 9. Thee tencil exhibit deired ymmetrie the tencil in Figure 9c i a rotated verion of that in Figure 9b and they do not attenuate wavenumber near the patial Nyquit frequencie. The finite-difference approx-

7 Finite-difference approximation 81-1/4-1/2-1/4-1/12-5/6-1/12 1/2 1 1/2 1/6 5/3 1/6-1/4-1/2-1/4 (a) d 11 = 1, d 12 = d 22 = 0-1/12-5/6-1/12 (a) d 11 = 1, d 12 = d 22 = 0-1/ /3-1/3 1/ /3 5/3-1/ /2 (b) d 11 = d 12 = d 22 = 1/2 1/6-1/3-1/3 (b) d 11 = d 12 = d 22 = 1/ /2 1/6-1/3-1/ /3 5/3-1/3-1/2 0 0 (c) d 11 = d 12 = d 22 = 1/2-1/3-1/3 1/6 (c) d 11 = d 12 = d 22 = 1/2 Figure 8. Finite-difference approximation G T DG for r = 1/4. Figure 9. Finite-difference approximation obtained by averaging four different G T DG for r = 1/12. imation in equation 19 i the one that I ued in the image painting example of Figure 2. The ame average of four approximation for r = 0 yield a lightly different finite-difference approximation that wa derived from a mixed finite-element method by Arbogat et al. (1997). I favor the approximation hown here with r = 1/12 becaue a dicued above it error are more iotropic. Implementation of the approximation in equation 19 need not require a factor of four increae in computational cot, becaue the averaging can be performed analytically. The derivation i tediou but the reult i imple, even for the cae of patially varying tenor coefficient. For that cae, define ymmetric tenor D at the four corner of an image ample with indice (i 1, i 2) a follow: D(i 1 1, 2 i2 1 ) a00 b 00 2 b 00 c 00 D(i 1 1, 2 i2 + 1 ) a10 b 10 2 b 10 c 10 D(i 1 + 1, 2 i2 1 ) a01 b 01 2 b 01 c 01 D(i 1 + 1, 2 i2 + 1 ) a11 b 11 2 b 11 c 11 Then, uing the finite-difference approximation of equation 19, the 9-point tencil centered on the image ample

8 82 D. Hale with indice (i 1, i 2) i r b 00 a 00 a 10 b 10 a 00 + a 01 + a 10 + a 11 c 00 c 01 +b 00 b 01 b 10 + b 11 c 10 c 11 +c 00 + c 01 + c 10 + c 11 b 01 a 01 a 11 b 11 a 00 + a 10 a 00 c 00 +c 00 + c 10 a 10 c 10 a 00 + a 01 a 00 a 01 a 10 a 11 a 10 + a 11 +c 00 + c 01 c 00 c 01 c 10 c 11 +c 10 + c 11 a 01 c 01 a 01 + a 11 a 11 c 11 +c 01 + c 11 For r = 0 only the firt part of thi tencil i ignificant, and thi part i the finite-difference approximation of Arbogat et al. (1997). Addition of the econd part with r = 1/12 yield a finite-difference approximation with error that are more iotropic. For contant tenor coefficient a = d 11, b = d 12 and c = d 22 thi tencil i conitent with the three example in Figure 9. For patially varying tenor coefficient, the variou um and difference in thi 9-point tencil maintain the important SPS property of our finite-difference approximation. A hown above, a impler way to maintain thi SPS property i to gather and catter conitently, a illutrated by the following program fragment: for (int i2=1; i2<n2; ++i2) { for (int i1=1; i1<n1; ++i1) { float a = 0.5f*d11[i2][i1]; float b = 0.5f*d12[i2][i1]; float c = 0.5f*d22[i2][i1]; float t = 2.0f*r*(a+c); float fpp = f[i2 ][i1 ]; float fpm = f[i2 ][i1-1]; float fmp = f[i2-1][i1 ]; float fmm = f[i2-1][i1-1]; float apppm = (a-t)*(fpp-fpm); float ampmm = (a-t)*(fmp-fmm); float bppmm = (b+t)*(fpp-fmm); float bpmmp = (b-t)*(fpm-fmp); float cppmp = (c-t)*(fpp-fmp); float cpmmm = (c-t)*(fpm-fmm); g[i2 ][i1 ] = apppm+bppmm+cppmp; g[i2 ][i1-1] -= apppm+bpmmp-cpmmm; g[i2-1][i1 ] += ampmm+bpmmp-cppmp; g[i2-1][i1-1] -= ampmm+bppmm+cpmmm; Note that thi implementation require for each image ample only one evaluation of tenor element D LAPLACIANS The methodology ued above to derive a finite-difference approximation to a 2-D aniotropic Laplacian extend naturally to three dimenion. The two important tep are (i) Find approximation G 1, G 2, and G 3 to component of the gradient uch that G T G = G T 1 G 1 + G T 2 G 2 + G T 3 G 3 ha (to econd order) iotropic dicretization error. (ii) Average all poible combination of thee component in approximation G T DG to the aniotropic Laplacian. The combination G T DG will have the form G T DG = ˆG D 11 D 12 D 13 G 1 T 1 G T 2 G T 4 3 D 12 D 22 D G 2 5. (20) D 13 D 23 D 33 G 3 Compact tencil for G 1, G 2, and G 3 are rotated verion of each other: G 1 = -t - t G 2 = -t t G 3 = - -t - - -r - -r r - -r r r t 0, rt = 2, r t = 1. (21) In thi notation, the back part of each tencil i pecified left of the front part. The condition rt = 2 follow from deired ymmetrie in G T 1 G 1, G T 2 G 2 and G T 3 G 3. The condition r t = 1 enure that G T 1 G 1, G T 2 G 2 and G T 3 G 3 are 2nd-order approximation. Thee tencil for G 1, G 2, and G 3 lead to compact point tencil for G T DG. In the iotropic cae where D = I, we have t G T G = G T 1 G 1 + G T 2 G 2 + G T 3 G 3 (22) For thi cae Patra and Karttunen (2005) give condition for which compact 3-D tencil have (to 2nd order) iotropic dicretization error, and they lit example of tencil that meet thoe condition. In three dimenion, unlike two dimenion, more than one uch tencil i poible. However, none of the iotropic tencil they cite can be expreed in the form of equation 22, in term of approximation to component of gradient. In other word, they do not correpond to approximation G 1, G 2, and G 3 that we could ue in the aniotropic form of equation 20. By carefully chooing the coefficient r, and t ubject to the condition cited above, I found a new finite- r,,,

9 Finite-difference approximation 83 difference approximation with the deired form. The coefficient are t = 5/12 1/ 6, r = (1 t) 2, = rt. Thee coefficient lead to the following 27-point tencil: G T G = c 1 c 2 c 1 c 2 c 3 c 2 c 1 c 2 c 1 c 2 c 3 c 2 c 3 c 4 c 3 c 2 c 3 c 2 c 1 c 2 c 1 c 2 c 3 c 2, c 1 c 2 c 1 c 1 = 1/48, c 2 = 1/8, c 3 = 5/12, c 4 = 25/6. (23) Thi tencil meet the condition for iotropic dicretization error pecified by Patra and Karttunen (2005), who cite a different 27-point tencil propoed by Spotz and Carey (1995): c 1 = 1/30, c 2 = 1/10, c 3 = 7/15, c 4 = 64/15. A noted above, thi different tencil doe not have correponding approximation G 1, G 2 and G 3 that we can ue in equation 20. Before inerting the tencil G 1, G 2 and G 3 of equation 21 into equation 20, we firt recognize that each of thee 3-D tencil can be written in four different way. Thoe four way are analogou to the two different way we expreed the 2-D tencil G 1 and G 2 in equation 18. Since each of the 3-D tencil G 1, G 2 and G 3 can be written four different way, we have a total of 64 = combination of the form of equation 20. By averaging all 64 combination, we obtain a compoite 3-D finite-difference approximation for an aniotropic Laplacian. Thi large number of combination to average can be reduced if we conider only pairwie combination uch a G T 1 D 11G 1 (four way) or G T 1 D 12G 2 (ixteen way). In any cae, the averaging can again be performed analytically and, although too lengthy to be included in thi paper, the reult i analogou to that obtained above by averaging 2-D approximation. 5 CONCLUSION The method decribed here for finite-difference approximation of aniotropic Laplacian i traightforward. We begin with compact finite-difference approximation to component of the gradient. We then average ymmetric poitive-emidefinite combination of thoe gradient approximation to obtain the deired approximation to aniotropic Laplacian. Finite-difference approximation obtained in thi way are guaranteed to be ymmetric and poitive-emidefinite. Thi deign method yield 9-point tencil for 2-D approximation and 27-point tencil for 3-D approximation of aniotropic Laplacian. Our 2-D approximation i a generalization of that propoed by Arbogat et al. (1997), in that for one particular gradient approximation we obtain the coefficient of their 9-point tencil. However, an alternative gradient approximation in our method lead to a 9-point tencil with iotropic dicretization error. Arbogat et al. (1997) alo propoed a 19-point tencil for a 3-D approximation, and our 27-point tencil i not a generalization of their. Indeed, I have not found a way to modify the deign propoed here to obtain uch a 19-point tencil. A 19-point tencil i attractive becaue it implie lower computational cot. In both 2-D and 3-D approximation, required ymmetry in the reulting 9-point or 27-point finitedifference tencil implie only one free parameter in the correponding gradient approximation. I chooe thi parameter to obtain tencil with iotropic dicretization error. All finite-difference method exhibit dicretization error. By chooing method for which thoe error are iotropic to leading order, we may reduce artifact aociated with ampling grid in application uch a image moothing and image painting. REFERENCES Arbogat, T., M.F. Wheeler and I. Yotov, 1997, Mixed finite element for elliptic problem with tenor coefficient a cell-centered finite-difference: SIAM Journal of Numerical Analyi, 34, Claerbout, J.F., 1992, Earth ounding analyi proceing veru inverion: Blackwell Scientific Publication. Fehmer, G.C., C.F.W. Höcker, 2003, Fat tructural interpretation with tructure-oriented filtering: Geophyic, 68, Patra, M. and M. Karttunen, 2005, Stencil with iotropic dicretization error for differential operator: Numerical Method for Partial Differential Equation, 22, Spotz, W.F. and G.F. Carey, 1996, A high-order compact formulation for the 3D Poion equation: Numerical Method for Partial Differential Equation, 12, van Vliet, L.J., and P.W. Verbeek, 1995, Etimator for orientation and aniotropy in digitized image: Proceeding of the firt annual conference of the Advanced School for Computing and Imaging ASCI 95, Heijen (The Netherland),

10 84 D. Hale