Volume Representation by Gradient and Distance Fields - Spring ...

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CSG Operations with Voxelized Solids3.3 Advanced CSG OperationPV1V2QRAP1P2BFigure 3.7: Classification of the region T A in subregions P, Q and RIn the second step we add the missing information about the surface B to voxel V . Wedistinguish two cases:• If there are at least in one direction two nearest voxels to V in T B , we use information fromthese voxels. For instance, if we want to evaluate density and gradient in voxel V i, j,k andwe know density and gradient in voxels V i+1, j,k , V i+2, j,k , we set:d i, j,k = 2d i+1, j,k − d i+2, j,k⃗n i, j,k = 2⃗n i+1, j,k − ⃗n i+2, j,k . (3.11)If we can do this estimation for pairs of voxels in more than one direction, the final valueswill be calculated as an average of these estimations.• If there is no such pair of voxels in the neighbourhood of V , we use the information determinedin the first step for point P. The gradient of the point P is calculated in the sameway as its density. We evaluate the difference x of the densities in points V , P from theknown vectors ⃗n, ⃗u = −→ PV . Then voxel V has values:d V = d P + x⃗n V = ⃗n P . (3.12)As we can see in Figures 3.1, 3.2 third column, sharp edges are now without artifacts. Theyare visibly smoothed, but it is the only way to represent them in a grid with the given resolutionif the representability criterion is to be fulfilled. Similarly, in the union of two almost touchingspheres, the thin gap between them is not representable, and therefore is filled.We have not used any antialiasing method, so some artifacts can occur in pictures aroundobject contours. It is not a problem of the representation but the rendering technique and can beeasy solved by exploiting common routines.We successfully solved the problem of representability of CSG solids by rounding sharpedges and small details, which would otherwise result in reconstruction artifacts. Such rounding38
CSG Operations with Voxelized Solids3.3 Advanced CSG OperationV(a)nnuxVP(b)Figure 3.8: Addition of missing information for voxel in the critical area(a) using neighbouring voxels, (b) using point P calculated previously.39
Page 1 and 2: FACULTY OF MATHEMATICS, PHYSICS AND
Page 3 and 4: Statutory Declaration:I do solemnly
Page 6 and 7: CONTENTSCONTENTS4.4.2 Rectification
Page 8 and 9: LIST OF FIGURESLIST OF FIGURES3.10
Page 10 and 11: Introduction1.1 Volume Graphics and
Page 12 and 13: Introduction1.3 Surface Normal Esti
Page 14 and 15: Introduction1.4 Object Representabi
Page 16 and 17: Introduction1.4 Object Representabi
Page 18 and 19: Representation of Volume Data Sets2
Page 34 and 35: Chapter 3CSG Operations with Voxeli
Page 36 and 37: CSG Operations with Voxelized Solid
Page 40 and 41: gCSG Operations with Voxelized Soli
Page 48 and 49: Sharp Details Correction Method4.1
Page 68 and 69: Implementation5.2 The vxtRL Library
Page 70 and 71: Implementation5.2 The vxtRL Library
Page 72 and 73: Bibliography[1] A. Kaufman, D. Cohe

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