X Ray Rendering CSE 564 Visualization Estimate ray integral via discrete raycasting Other Rendering Techniques Points Splats Shear Warp interpolation kernel h Complete discrete ray integral pi v j h X sk X v j k Stony Brook University Computer Science Center for Visual Computing pi v j h X sk X v j sk v j h X sk X v j j 2 X Ray Point Splatting Estimate the ray integral via point projection Example projecting a volume of two points image plane 1 3 Footprint splat of point vj ext point vj k j X Ray Rendering pixel pi j Reversing the order of j and k sample sk 1 h ri h r s ds i ext ray ri Compute continuous ray integral at pi pixel pi point vj Klaus Mueller interpolation kernel h image plane 2 4 rasterize footprint pi vj h j ri j 3 rasterize footprint add footprints 4 X Ray Point Splatting Point Projection Re ordering was first recognized by Hanson and Wecksung for 2D CT Hanson 85 Each point is represented by a 3D Gaussian GV GV Later independently discovered by Westover for 3D volume rendering Westover 89 1 2 V e 0 5 0 5 x v j T V 1 x v j GV is an ellipsoid to facilitate more general grids It is a sphere for cubic grids Facilitates computation of the true ray integral not just a discrete Riemann sum raycasting A viewing matrix M transforms GV into GMVM Pre integrated footprint is stored into a table GMVM Need a kernel function for which mappings into the footprint table can be defined for any orientation The Gaussian is such a function 5 1 G T u Mv j T M 1 MVM Mvj T GMVM Heckbert 89 Zwicker 01 Point Projection 6 Blending Projection P of GMVM is screen ellipse P GMVM Note Gaussian kernels do not blend perfectly Find vj s screen projection P M vj T Find linear mapping of P GMVM into footprint table Rasterize footprint table under P GMVM at P V vj P M vj T vj GV A small ripple always remains Typical range 0 99845 1 00249 assuming a function of unity mapping ellipse p circle P GMVM The wider the Gaussians the smaller the ripple In practice a radius 2 0 in volume space works well given the appropriate Gaussian See Crawfis and Max Vis 93 for an optimized kernel Footprint table circular Gaussian 7 8 Complexity Complexity Splatting seemingly reduces the interpolation complexity by one dimension Consider magnification 1 Raycasting Raycasting interpolation of samples in 3D Splatting rasterization of footprints in 2D Commonly uses trilinear interpolation Requires 8 points to calculate one ray sample point Total complexity O 8 n3 But Splatting Uses Gaussian kernel of radius 2 Footprint rasterization touches 16 pixels Total complexity O 16 n3 9 10 Complexity Storage Complexity Does this mean that raycasting is more efficient than splatting Generally only need to store relevant points Non air points masked out points ROI points Provides easy space leaping for irregular objects Storage schemes in increasing order of spatial coherence List of points sorted by value fast iso contouring RLE list of points fast transformations and sparse Octree with hierarchical bins of points It depends Spatially intricate objects are good candidates for point based rendering splatting But the simplicity of splatting has advantages even for less favorable objects RLE 11 E Fv1v2v3v4 E Fv1v2 E Fv1v2v3 E Fv1v2v3 12 Rendering Aliasing RLE list facilitates fast incremental arithmetic for point projection in software Texture mapping hardware can also be used In perspective or at low magnifications some volume portions may be sampled below Nyquist oversampled volume grid Texture map footprint onto a square polygon Set GL blending functions etc Warp polygon according to point s screen space ellipse Align the warped polygon with the screen Project polygon to the screen undersampled volume grid screen Ray grid sampling rate volume grid sampling rate no aliasing Ray grid sampling rate volume grid sampling rate aliasing 13 14 Aliasing Anti Aliasing Effects of aliasing Adapt kernel bandwidth for proper anti aliasing Amounts to a stretch of the 3D kernel screen checkerboard tunnel constant splat kernel size terrain 15 Swan 97 Mueller 98 linearly increase splat kernel size linearly decrease splat amplitude 16 Anti Aliasing Anti Aliasing Conveniently done in perspective ray space Compute the Gaussian ellipsoid in ray space Calculate the Jacobian J of the local perspective distortion varies for each point Compute the ray space ellipsoid GJMV using J screen screen camera space camera space GMV 1 G T u Mv j T M 1 MVM ray space GJMV 1 G T x xk J 1 JMVJ ray space 17 Anti Aliasing Results Zwicker 01 generalized Gaussian ellipsoid in camera space center of Gaussian in ray space 18 Compositing Raycasting Reconstruction followed by compositing interpolation kernel h image plane pixel pi Compositing c C sk sk 1 c aliased Reconstruction of c sk sk anti aliased 19 sk 1 20 Compositing Splatting Compositing Reconstruction not separable from compositing Two strategies devised by Westover Westover 89 90 Composite every point image plane interpolation kernel h Shown in previous slide Fast and simple Leads to sparkling in animated viewing pixel pi compositing of pre integrated kernel point vj C vj vj Axis aligned sheet buffers Add splats within sheets most parallel to image plane Composite these sheets in depth order Leads to popping artifacts in animated viewing ray ri ext h ri h ri s ds ext c C v j h ri v j h ri 1 c v j h ri 1 21 22 vo lu m es lic es Axis Aligned Sheet Buffers z Image Aligned Sheet Buffers volume slices y x e at 30 Switch compositing image plan axis at 45 e l an p ge a im Eliminates popping 70 t a Popping occurs 44 7 Slicing slab cuts kernels into sections Kernel sections are added into sheet buffer Sheet buffers are composited 45 2 image plane binary cube binary cube 23 sheet buffer compositing buffer Mueller 98 24 Image Aligned Sheet Buffers Pre Classified Splatting Footprint mapping as usual Requires multiple footprint rasterizations per point axis aligned normal blurred close ups image aligned 25 Pre Classified Splatting 26 One Solution Edge Splats Edge splats Huang 98 Original edge replace normal splat by special edge splat Sampled edge Classification and shading Shortcomings pre processing required problems with discontinuities micro edges are hard to resolve Splatted with Gaussian kernel Reconstruction blurred edge image 27 28 Pre Classified Rendering Post Classified Rendering Mueller 99 Rendering Loop Classify and shade Raw density volume Splat into sheet buffer Composite sheet buffer Splat into sheet buffer Advance