kd-trees for Volume Ray-Casting

Transcription

1 kd-trees for Volume Ray-Casting Anita Schilling Special Seminar for Computer Graphics 15. January 2009 Anita Schilling kd-trees for Volume Ray-Casting 1 / 39

2 Outline 1 Introduction 2 Ray-Voxel Intersection Techniques 3 kd-trees 4 Evaluation Anita Schilling kd-trees for Volume Ray-Casting 2 / 39

3 Introduction high-performance ray-tracing scientific visualization iso-surface ray-casting Volume Raycasting with CUDA (Marsalek & Slusallek 2008) Anita Schilling kd-trees for Volume Ray-Casting 3 / 39

4 Iso-Surface Extraction Marching Cubes simple algorithm precomputed lookup table for iso-surface/voxel intersection Anita Schilling kd-trees for Volume Ray-Casting 4 / 39

5 Iso-Surface Extraction Marching Cubes holes possible only approximation of iso-surface generates too many triangles iso-surface extraction for every iso-value necessary Anita Schilling kd-trees for Volume Ray-Casting 5 / 39

6 Iso-Surface Rendering ray-casting the iso-surface exact intersection calculation of ray with volume higher image quality Volume Ray Tracing (Marmitt et. al. 2004) Anita Schilling kd-trees for Volume Ray-Casting 6 / 39

7 Direct Ray-Casting of Iso-Surface Problems efficiently finding voxels hit by ray containing iso-surface correct intersection point calculation of ray with interpolated implicit surface of voxel Anita Schilling kd-trees for Volume Ray-Casting 7 / 39

8 Ray-Voxel Intersection correct intersection point calculation is expensive Anita Schilling kd-trees for Volume Ray-Casting 8 / 39

9 Accurate Intersection Method voxel data values ρ ijk, i, j, k {0, 1} compute density at any point (u, v, w) [0, 1] 3 using trilinear interpolation ρ(u, v, w) = i,j,k {0,1} u i v j w k ρ ijk with u 0 = u, u 1 = 1 u, v 0 = v, v 1 = 1 v, w 0 = w, w 1 = 1 w Anita Schilling kd-trees for Volume Ray-Casting 9 / 39

10 Accurate Intersection Method computing density ρ(p) of any point p V spatial location of voxel cell V = [x 0... x 1 ] [y 0... y 1 ] [z 0... z 1 ] 1 transform p = (x, y, z) into voxel unit coordinate system ( p(u p 0, v p 0, w p 0 ) = x1 p x, y 1 p y, z ) 1 p z x 1 x 0 y 1 y 0 z 1 z 0 Anita Schilling kd-trees for Volume Ray-Casting 10 / 39

11 Accurate Intersection Method Ray R(T ) = O + T D O... ray origin and D... ray direction in world coordinates passing the voxel in interval [T in, T out ] 2 transform ray R(T ) to voxel unit coordinate system r(t) = a + tb p entry = r(t in = 0) p exit = r(t out = 1) Anita Schilling kd-trees for Volume Ray-Casting 11 / 39

12 Accurate Intersection Method density at every ray point within voxel ρ(t) = ρ(r(t)) = i,j,k {0,1} (u a i + tu b i )(v a j + tv b j )(w a k + tw b k ) ρ ijk Anita Schilling kd-trees for Volume Ray-Casting 12 / 39

13 Accurate Intersection Method 3 expand to ρ(t) = At 3 + Bt 2 + Ct + D with A = ijk B = ijk C = ijk D = ijk u b i v b j w b k ρ ijk (u a i v b j w b k + ub i v a j w b k + ub i v b j w a k ) ρ ijk (u b i v a j w a k + ua i v b j w a k + ua i v a j w b k ) ρ ijk u a i v a j w a k ρ ijk Anita Schilling kd-trees for Volume Ray-Casting 13 / 39

14 Approximate Method Simple Midpoint Algorithm trading quality for speed intersection set to midpoint between entry and exit point of ray blocky artifacts at size of voxels only for performance reference Anita Schilling kd-trees for Volume Ray-Casting 14 / 39

15 Linear Interpolation Method linearly interpolate intersection point on the ray t hit = t in + (t out t in ) ρ iso ρ in ρ out ρ in significantly more costly two tri- or bilinear interpolations fails in more complex cases if function has 2 roots such that entry and exit densities are both larger or smaller than ρ iso Anita Schilling kd-trees for Volume Ray-Casting 15 / 39

16 Neubauer s Method repeated linear interpolation t = t 0 + (t 1 t 0 ) ρ iso ρ 0 ρ 1 ρ 0 if sign(ρ(r(t)) ρ iso ) = sign(ρ 0 ρ iso ) then t 0 = t, ρ 0 = ρ(r(t)) else t 1 = t, ρ 1 = ρ(r(t)) typically 2 or 3 times fails in more complex cases falsely returns last intersection point if 3 intersections are within a voxel but 2 of them are in the first ray segment Anita Schilling kd-trees for Volume Ray-Casting 16 / 39

17 Accurate Intersection Method slow and correct or fast and sometimes incorrect methods Key Observations only need first intersection with implicit surface repeated linear interpolation does find the correct root, if the start interval contains exactly one root Anita Schilling kd-trees for Volume Ray-Casting 17 / 39

18 Accurate Intersection Method find ray parameter t where ρ(t) = ρ iso f (t) = ρ(t) ρ iso = 0 with t [t in = 0, t out = 1], smallest root of f (t) = At 3 + Bt 2 + Ct + D ρ iso in interval [0, 1] Anita Schilling kd-trees for Volume Ray-Casting 18 / 39

19 Accurate Root Finding Method 4 compute extrema e 0, e 1 of f (t) = At 3 + Bt 2 + Ct + D ρ iso with f (t) = 3At 2 + 2Bt + C = 0 5 compute density value at start point f (e 0 ) and end point f (e 1 ) of interval Anita Schilling kd-trees for Volume Ray-Casting 19 / 39

20 Accurate Root Finding Method 6 if sign(f (t in )) sign(f (e 0 )) then interval contains exactly one root else advance ray to next segment Anita Schilling kd-trees for Volume Ray-Casting 20 / 39

21 Accurate Root Finding Method 7 apply repeated linear interpolation (2-3 times) efficient computation of any density value along the ray with ρ(t) 8 transform voxel unit ray parameter t hit back to world coordinate ray parameter T hit 9 calculate intersection point p intersect = R(T hit ) Anita Schilling kd-trees for Volume Ray-Casting 21 / 39

22 Find the right voxel for intersection test every voxel whether voxel contains iso-value and ray hits voxel Anita Schilling kd-trees for Volume Ray-Casting 22 / 39

23 Uniform Grid Traversal split voxel grid into uniform macro-cells store min/max iso-values for macro-cells test whether macro-cell contains iso-value test whether ray hits macro-cell test every single voxel of macro-cell Anita Schilling kd-trees for Volume Ray-Casting 23 / 39

24 kd-tree empty space skipping in polygon ray-tracing scenes with varying primitive density kd-trees often outperform other datastructures Anita Schilling kd-trees for Volume Ray-Casting 24 / 39

25 kd-tree apply to volume data split node at center of largest dimension parent node stores min/max iso-range of children easily determined recursively test for iso-value before following down branch Anita Schilling kd-trees for Volume Ray-Casting 25 / 39

26 kd-tree middle split of scene cuts through objects better split into separate objects and empty space build kd-tree with surface area heuristic Anita Schilling kd-trees for Volume Ray-Casting 26 / 39

27 kd-tree with Surface Area Heuristic apply to volume data assumption there is always some empty space around only optimized for one iso-value for initial evaluation precompute surface area with summed area table evaluate all possible split locations on building Anita Schilling kd-trees for Volume Ray-Casting 27 / 39

28 Surface Area of Iso-Surface summed area tables for one iso-value sat(x, y) = i(x, y ) = sat(x 1, y) + i(x, y ) x x,y y y y Rectangle area SA rectangle = B x,d y x =A x,y =A y i(x, y ) SA rectangle = sat(a) + sat(c) sat(b) sat(d) Anita Schilling kd-trees for Volume Ray-Casting 28 / 39

29 Surface Area of Iso-Surface no directly computed surface area 3d summed area table for volume data SA = sat(g) + sat(e) + sat(b) + sat(d) sat(a) sat(f) sat(h) sat(c) Anita Schilling kd-trees for Volume Ray-Casting 29 / 39

30 kdtree with Surface Area Heuristic splitcost = max SA left SA left nocells left + SA right SA parent SA right nocells right splitcost = max splitcost = min SA left SA left nocells left + SA right nocells right nocells parent nocells left nocells parent + SA right SA parent nocells right nocells parent Anita Schilling kd-trees for Volume Ray-Casting 30 / 39

31 kd-tree with Surface Area Heuristic 3 possibilities 1 SA parent = 0 no further split 2 SA parent = nocells parent split in center of largest dimension 3 0 < SA parent < nocells parent evaluate every possible split location if maximum node dimension treshold then no split Anita Schilling kd-trees for Volume Ray-Casting 31 / 39

32 Evaluation Datasets Dataset Dimensions Resolution Inner Ear bit Mouse Skeleton bit Teddy Bear bit Tooth bit Engine bit Head bit Neghip bit Anita Schilling kd-trees for Volume Ray-Casting 32 / 39

33 Evaluation Results Threshold Tree No. of Nodes Max. Depth Build Time Render Time Engine with ρ iso = 0 4 kd ,31 15,27 4 SAH ,34 86,52 8 kd ,34 34,16 8 SAH ,17 85,95 Engine with ρ iso = kd ,21 2,49 4 SAH ,79 4,16 8 kd ,34 5,93 8 SAH ,23 4,23 Anita Schilling kd-trees for Volume Ray-Casting 33 / 39

34 Evaluation Engine Dataset Anita Schilling kd-trees for Volume Ray-Casting 34 / 39

35 Evaluation Results (2) Threshold Tree No. of Nodes Max. Depth Build Time Render Time Neghip with ρ iso = kd ,004 2,97 4 SAH ,4 4,12 8 kd ,01 10,18 8 SAH ,38 4,47 Neghip with ρ iso = kd ,04 1,81 4 SAH ,34 2,87 8 kd ,02 4,83 8 SAH ,34 3,1 Anita Schilling kd-trees for Volume Ray-Casting 35 / 39

36 Evaluation Neghip Dataset Anita Schilling kd-trees for Volume Ray-Casting 36 / 39

37 Evaluation Results (3) Dataset Theshold Tree Intersect Calls Ray-Box Tests Volume Access Engine 4 kd ρ iso = 0 4 SAH kd SAH Engine 4 kd ρ iso = SAH kd SAH Neghip 4 kd ρ iso = SAH kd SAH Neghip 4 kd ρ iso = SAH kd SAH Anita Schilling kd-trees for Volume Ray-Casting 37 / 39

38 In Progress Conclusion normal kd-tree performs better heuristic kd-tree needs less intersection calls hybrid kd-tree empty space around center of volume data set cut away empty space cells with heuristic if ratio of SA to number of cells gets large enough use normal kd-tree Anita Schilling kd-trees for Volume Ray-Casting 38 / 39

39 The End. Thank you for your attention. Anita Schilling kd-trees for Volume Ray-Casting 39 / 39

40 Literature G. Marmitt, A. Kleer, I. Wald, H. Friedrich, P. Slusallek: Fast and Accurate Ray Voxel Intersection Techniques for Iso-Surface Ray Tracing, Vision, Modeling and Visualization (VMV), I. Wald, H. Friedrich, G. Marmitt, P. Slusallek, H-P. Seidel: Faster Isosurface Ray Tracing using Implicit KD-Trees, IEEE Transactions on Visualization and Computer Graphics (IEEE VIS), V. Havran: Heuristic Ray Shooting Algorithms, PhD dissertation, CVUT, Prague Anita Schilling kd-trees for Volume Ray-Casting 40 / 39