Direct Volume Rendering DVR Ray casting Jian Huang This set of slides references slides used by Prof Torsten Moeller Simon Fraser Prof Han Wei Shen Ohio State Papers Tuy and Tuy 1984 IEEE CG A one of the earliest volume rendering techniques Levoy 1988 IEEE CG A and later improvements Drebin Carpenter Hanrahan 1988 SIGGRAPH Direct No conversion to surface geometry Basic Idea Based on the idea of ray tracing Trace from eat each pixel as a ray into object space Compute color value along the ray Assign the value to the pixel Data Representation 3D volume data are represented by a finite number of cross sectional slices hence a 3D raster On each volume element voxel stores a data value if it uses only a single bit then it is a binary data set Normally we see a gray value of 8 to 16 bits on each voxel N x 2D arraies 3D array Data Representation 2 What is a Voxel Two definitions A voxel is a cubic cell which has a single value cover the entire cubic region A voxel is a data point at a corner of the cubic cell The value of a point inside the cell is determined by interpolation Viewing Ray Casting Where to position the volume and image plane What is a ray How to march a ray Viewing 1 1 Position the volume Assuming the volume dimensions is w x w x w We position the center of the volume at the world origin Volume center w 2 w 2 w 2 local space 0 0 0 y z x Translate T w 2 w 2 w 2 data to world matrix world to data matrix Viewing 2 2 Position the image plane Assuming the distance between the image plane and the volume center is D and initially the center of the image plane is 0 0 D Image plane 0 0 0 y z x Viewing 3 3 Rotate the image plane w position of the image plane can be defined in terms ree rotation angle with respect to x y z axes ming the original view vector is 0 0 1 then the new vector g becomes 0 sin 1 0 0 cos sin 0 0 1 0 1 0 0 cos sin sin cos sin 0 cos sin cos Viewing 4 v0 E0 u0 E v S0 u S B 0 0 0 y z x B 0 0 0 S0 0 0 D u0 1 0 0 v0 0 1 0 Now R the rotation matrix S B Dxg U 1 0 0 x R V 0 1 0 x R Viewing 5 Image Plane L x L pixels v E S u R the rotation matrix S B Dxg U 1 0 0 x R V 0 1 0 x R Then E S L 2 x u L 2 x v So Each pixel i j has coordinates P E ixu jxv We enumerate the pixels by changing i and j 0 L 1 Viewing 6 4 Cast rays Remember for each pixel on the image plane P E ixu jxv and the view vector g 0 0 1 x R So the ray has the equation Q P k d x g d the sampling distance at each step d x Q x x x p K 0 1 2 Early Methods Before 1988 Did not consider transparency did not consider sophisticated light transportation theory were concerned with quick solutions hence more or less applied to binary data non binary data require sophisticated classification compositing methods Ray Tracing Ray Casting another typical method from traditional graphics Typically we only deal with primary rays hence ray casting a natural image order technique as opposed to surface graphics how do we calculate the ray surface intersection Since we have no surfaces we need to carefully step through the volume Ray Casting Stepping through the volume a ray is cast into the volume sampling the volume at certain intervals The sampling intervals are usually equi distant but don t have to be e g importance sampling At each sampling location a sample is interpolated reconstructed from the grid voxels popular filters are nearest neighbor box trilinear tent Gaussian cubic spline Along the ray what are we looking for Example Using the nearest neighbor kernel In tuys paper Q P K x V v dxg At each step k Q is rounded off to the nearest voxel like the DDA algorithm Check if the voxel is on the boundary or not compare against a threshold If yes perform shading Basic Idea of Ray casting Pipeline Data are defined at the corners of each cell voxel The data value inside the voxel is determined using interpolation e g tri linear Composite colors and opacities along the ray path c1 c2 c3 Can use other ray traversal schemes as well Ray Traversal Schemes Intensity Max Average Accumulate First Depth Ray Traversal First Intensity First Depth First extracts iso surfaces again done by Tuy Tuy 84 Ray Traversal Average Intensity Average Depth Average produces basically an X ray picture Ray Traversal MIP Intensity Max Depth Max Maximum Intensity Projection used for Magnetic Resonance Angiogram Ray Traversal Accumulate Intensity Accumulate Depth Accumulate opacity while compositing colors make transparent layers visible Levoy 88 Raycasting volumetric compositing color opacity 1 0 object color opacity Raycasting Interpolation kernel volumetric compositing color opacity 1 0 object color opacity Raycasting Interpolation kernel volumetric compositing color c c s s 1 c opacity s 1 1 0 object color opacity Raycasting volumetric compositing color opacity 1 0 object color opacity Raycasting volumetric compositing color opacity 1 0 object color opacity Raycasting volumetric compositing color opacity 1 0 object color opacity Raycasting volumetric compositing color opacity 1 0 object color opacity Raycasting volumetric compositing color opacity object color opacity Volume Rendering Pipeline Acquired values Data preparation Prepared values shading classification Voxel colors Voxel opacities Ray tracing resampling Ray tracing resampling Sample colors Sample opacities compositing Image Pixels DCH DVR Pipeline DCH Pipeline Original data Material percentage volumes Classification Density volume Opacity volume Color volume Normals Gradient Shading Shaded volume shears Final image compositing Transformed volume Common Components of General Pipeline Interpolation reconstruction Classification or transfer function Gradient normal estimation for shading Question are normals also interpolated Levoy Interpolation eye image pixel viewing ray trilinear interpolation voxel sample point Levoy Interpolation 2 binary Closest value smooth Weighted average Levoy Gradient Normals Central difference per voxel Gx Y 1 Gy Gz y 1 z 1 X 1 vi 1 j k vi 1 j k vi j 1 k 2 vi j 1 k 2 vi j k 1 vi j k 1 2 Levoy Compositing Image order back to front using the over operator Cout Cin 1 C out in 1 Levoy Shading Phong Shading Depth Cueing C x C p k a Cp k1 k 2 d x k d N x L k s N x H n Cp color of parallel light source ka kd ks ambient diffuse specular light coefficient k1 k2 fall off constants d x distance to picture plane L normalized vector to light H …
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