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Jagmohan presentation2008

  1. 1. IIITHyderabad Real Time Ray-Tracing Implicit Surfaces on the GPU Jag Mohan Singh IIIT, Hyderabad
  2. 2. IIITHyderabad Implicit Surfaces • Implicit Surface which can be described by an equation S(x,y,z) = 0. This can be of different kinds – Algebraic – Non- Algebraic eg. Transcedental, Irrational, Rational etc. • Implicit Surfaces are used for fluid simulation, modeling of fire, waves and natural phenomena described by equations
  3. 3. IIITHyderabad Thesis Contributions • Analytical (Exact) Root Finding at frame-rates of 1100 – 5821 for surfaces up to fourth order • Mitchell’s Interval method (first time on GPU) at frame-rates of 60 – 965 for surfaces up to fifth order • Marching Points at frame rates of 38 – 825 for arbitrary implicits • Adaptive Marching Points (a new method) for arbitrary implicits at frame rates of 60 – 920
  4. 4. IIITHyderabad Traditional Methods of Rendering • Rasterization • Ray Tracing
  5. 5. IIITHyderabad Rendering Implicit Surfaces • Polygonization using Marching Cubes – Marching Cubes gives a 3d mesh for the input implicit surface – Rasterization of this 3d mesh gives the rendering • Ray Tracing – Shoot rays towards the implicit surface and intersect them with these
  6. 6. IIITHyderabad Ray-Tracing Implicit Surfaces Can express as: f(t) = 0 Desired: smallest +ve real root t0 Normal at t0 = (Sx, Sy, Sz) at (O + D t0)‫‏‬ S(x,y,z) = 0 Ray: P = O + t D t0
  7. 7. IIITHyderabad Root Finding Methods • Analytical (Exact) exists for polynomials up to fourth order • Iterative Methods exists for arbitrary implicits but have problems related to initialization and convergence. • Searching based methods which search for the root along the ray using surface properties
  8. 8. IIITHyderabad Related Work (Exact) Loop and Blinn [ Siggraph ’06] • Piecewise algebraic surfaces up to order four. • The roots are computed by converting the polynomial to Bezier form. • Coefficients are interpolated in vertex shader. If root is inside the Bezier tetrahedron then surface normal and per-pixel lighting done. • Problems in quartic root finding due to extreme self intersections • Quadric root finding on GPU – Sigg , PBG ‘06 – Toledo, INRIA Tech Report ‘06 – Ranta , ICVGIP ‘06
  9. 9. IIITHyderabad Iterative Methods • Newton Raphson Method xn+1 = xn-f(xn)/ f’(xn) • Laguerre’s method ( Similar to Newton’s) • Newton Bisection Method Given interval [t1,t2] Choose one of the intervals [t1,tm] or [tm,t2] where tm is the midpoint
  10. 10. IIITHyderabad Interval based Iterative Methods • Newton’s Interval Method xn+1 = xn- f(xn)/ F’(xn) • Krawczyk Method xn+1 = xn-f(xn)/f’(xn) + (I- J( xn) / f’( xn)) (Xn - xn)
  11. 11. IIITHyderabad Recent Related Work (Iterative) Knoll’s Affine Arithmetic [ CGF ’08] • Compute affine extension of function as F • If 0 ε F then the interval contains the root • Compute maximum depth (dmax) of bisection based on user defined threshold • If depth is dmax then we hit the surface • Else increment depth and reduce the stepsize by half • Back recursion helps in visiting other unvisited nodes in the tree. In the worst case it can lead to visiting all the nodes of the tree.
  12. 12. IIITHyderabad Related Work (Searching) LG Implicit Surfaces [ Kalra and Barr, Siggraph ’89] • Lipschitz constants (L,G) for ray tracing implicits. L is equal to maximum rate of change of f(x) over R. G is equal to maximum rate of change of g(t). • Compute Bounding Box (B) divide it into sub-bounding boxes (b) Compute L for b If |f(x0)| > Ld reject b else continue recursive subdivision. • For each ray compute bounding box extents t1,t2 and midpoint tm If |g(tm)| > Gd If F(t1) and F(t2) are of opposite signs then find the root using Newton’s method. Else there is no intersection in t1,t2 Else if |g(tm)| < Gd Call the function recursively on intervals [t1,tm] and [tm,t2]
  13. 13. IIITHyderabad Related Work (Searching) Sphere Tracing [Hart, Visual Computer ’96] • Compute while t < D d = f(r(t)) ( Geometric Distance) If d < epsilon then return t t = t + d where Geometric Distance = Signed Distance/ Lipschitz Constant (L)
  14. 14. IIITHyderabad • Roots are computed in power basis • Limitations: – Not available for polynomials of order > 4! – Difficult for non-algebraic equations Must use iterative methods for others Analytical (closed-form) Roots ( Our Work)
  15. 15. IIITHyderabad Analytical Root Finding Cubic Roots Equation (Homogenous Form) : Ax3 +3Bx2 w+3Cxw2 +Dw3 = 0 • Compute: δ1= AC-B2 , δ2 = AD-BC, δ3=BD-C2 , δ (discriminant) = 4 δ1 δ3- δ2 2 • The sign of the discriminant and the values of δis determine if it has one triple root, one double and a single real root, three distinct real roots or one real root and one complex conjugate pair as roots.
  16. 16. IIITHyderabad Analytical Root Finding Quartic Roots • The equation is first depressed by removing the cubic term t4 +pt2 +qt+r = 0 • If r is zero then the roots are the roots of cubic equation and zero. • If r is non zero then rewrite as (t2 +p)2 +qt+r = pt2 +p2 This is followed by a substitution y s.t. RHS becomes a perfect square (t2 +p+y)2 = (p+2y)t2 -qt+(y2 +2yp+p2 -r) Now, for RHS to be a perfect square its discriminant must be zero which yields a cubic equation in y. Now resubstitute to get two quadratic equations.
  17. 17. IIITHyderabad Interval Arithmetic Two Intervals a = [x, y] and b = [z, w] • Addition a + b = [x + z, y + w] • Subtraction a – b = [x – w, y – z] • Multiplication a * b = [min(xz,xw,yz,yw), max(xz,xw,yz,yw)] • Division a/b = a * (1/b) = a* [ 1/w,1/z]
  18. 18. IIITHyderabad Mitchell’s Interval-Based Method • Initialize interval to [ta, tb] = [tnear, tfar] • Compute interval extension of function f ([ta, tb]) and it’s derivative ft ([ta, tb]) • If f ([ta, tb] contains 0, root exists in it. – If ft ([ta, tb]) contains zero, multiple roots. • Divide into [ta, tm] and [tm, tb] around the midpoint • Recurse into right half only if left has no root. – Else, single root. Proceed to root finding • Continue till tb -ta < ε Mitchell, Graphics Interface 90
  19. 19. IIITHyderabad Interval Extensions • Natural: Uses end-points only. f ([ta, tb]) = [min(f(ta), f(tb)), max(f(ta), f(tb))] • Centered: f ([ta, tb]) = f (tm) + ft ([ta,tb]) * [ta - tm, tb - tm] • Exact: Use critical points ta < t1 < t2 < … < tb of f() f ([ta, tb]) = [min(f(ta), f(t1), f(t2) …, f (tb)), max (f(ta), f(t1), f(t2) …, f(tb))]
  20. 20. IIITHyderabad Mitchell’s Method: Discussion • Advantages: – Robust, based on interval arithmetic – Fast as the order is logarithmic due to bisections • Disadvantages: – Good interval extension needed • Not obvious for general functions • Not easy even for polynomials – Difficult on SIMD/GPU • Calculations in f(t), derivatives needed • Interval extension used is exact
  21. 21. IIITHyderabad Two-Step Root Finding • Bracketing the root – Find a (small) bracket/interval that contains the first positive root. – Between tnear and tfar • Find the root in the interval – Newton bisections • Always converges, no “special” situations • Best for GPU/SIMD as uniform calculations
  22. 22. IIITHyderabad Marching Points (Sign Test) • Divide the parameter domain into equal width intervals from tnear till tfar • Compute the function value at endpoints of these intervals. Return the interval with the first sign change.
  23. 23. IIITHyderabad Marching Points (Taylor Test) • Divide the parameter domain into equal width intervals from tnear till tfar • Compute the values p, q, r and s for an interval and the interval checked for sign change is [min(p,q,r,s), max(p,q,r,s)]
  24. 24. IIITHyderabad S(x, y, z) versus f(t) • S(x, y, z) is the given form. – Relatively simple with dozen or so terms – For a given t, evaluate (x, y, z) and S(.). – Good for GPU; compose shader on the fly • f(t) is different for each ray/pixel. – Evaluates to a large number of terms – About 1500 terms for a 10th order polynomial – Not suitable for GPU/SIMD
  25. 25. IIITHyderabad Results: Implicit Surfaces (Marching Points)
  26. 26. IIITHyderabad Marching Points: Discussions • Advantages: – Easy Implementation – Suited for SIMD, fast on current GPUs – No need for derivative or coefficient computation • Disadvantages: – Linear in number of intervals as all may be evaluated – Sign Test Not robust. Multiple and close roots are problems – No structured way to decide interval size.
  27. 27. IIITHyderabad Adaptive Marching Points • Algebraic distance is used as a measure for searching the root • Step-size depends on algebraic distance (S(p(t)) and silhouettes (F’(t))
  28. 28. IIITHyderabad Adaptive Marching Points • Silhouette Adaptation
  29. 29. IIITHyderabad Self Shadowing • Shoot a secondary (shadow) ray towards the light source from intersection point. • If this ray intersects the surface in between then the point is in shadow. • Only need to bracket the root; no need to find the root.
  30. 30. IIITHyderabad Shadowing of Surfaces
  31. 31. IIITHyderabad Dynamic Implicit Surfaces • Implicit Surfaces whose equation varies with time. Blobby Molecules and Twisted Superquadric
  32. 32. IIITHyderabad Analytic Roots 1200Torus Surface 1100Tooth Surface Surface Name FPS for 512x512 Sphere Quadric 5821 Cylinder Quadric 4358 Cayley Cubic 3750 Ding Dong Cubic 3400 Steiner Surface 1400 500Steiner Surface 1200Cylinder Quadric FPS for 512x512 ( Loop and Blinn) Surface Name
  33. 33. IIITHyderabad Mitchell’s Interval Method 96518Ding dong [3] 58027Cayley [3] 19550Tooth[4] 18652Miter [4] 19552CrossCap[4] 17053Cushion [4] 8560Peninsula [5] 7765Kiss [5] 6086Dervish [5] FPSIterationsSurface [Order]
  34. 34. IIITHyderabad Marching Points: Results 125105150Superquadric 306260250Diamond Surface 315200250Scherk’s Surface 305160250Blobby Surface 43041050Torus [4] 44737085Peninsula [5] 275285300Dervish [5] 260265120Heart [6] 310300125Barth[6] 225230400Hunt [6] 290285400Kleine[6] 195185250Chmutov [8] 179140300Endreass [8] 10592300Barth [10] 5360300Sarti[12] 4855400Chmutov[14] 3885400Chmutov [18] FPS (Taylor Test)FPS (Sign Test)IterationsSurface [Order]
  35. 35. IIITHyderabad Adaptive Marching Points: Results 155185100Superquadric 330360100Diamond Surface 322358100Scherk’s Surface 30032950Blobby Surface 52555524Torus [4] 43551235Peninsula [5] 28028545Dervish [5] 32042048Heart[6] 31032560Barth[6] 32524084Hunt [6] 38543548Kleine[6] 21621564Chmutov [8] 20819096Endreass [8] 115150100Barth [10] 7586100Sarti[12] 95125100Chmutov[14] 6098100Chmutov [18] FPS (Taylor Test)FPS (Sign Test)IterationsSurface [Order]
  36. 36. IIITHyderabad Result : Shadows AMP (Taylor Test)AMP (Sign Test) 155 330 300 542 435 280 310 325 310 208 115 75 95 60 Without Shadows 145 208 265 425 325 250 182 280 165 140 110 78 95 70 With Shadows Surface [Order] Without Shadows With Shadows Chmutov [18] 98 45 Chmutov [14] 125 75 Sarti [12] 86 49 Barth [10] 150 79 Endreass[8] 190 140 Labs[7] 232 155 Chmutov[6] 418 235 Hunt[6] 240 155 Dervish[5] 285 175 Kiss[5] 428 265 Tooth[4] 617 287 Blobby 329 195 Diamond 360 199 Superquadric 185 105
  37. 37. IIITHyderabad Comparison with Knoll’s Affine Arithmetic 9416Barth Decic 17660Mitchell 170101Kleine 12088Barth Sextic 19671Tangle 178121Teardrop 21238Steiner FPS (AMP Sign)FPS (Knoll’s ANE)Surface
  38. 38. IIITHyderabad Results: Robustness Top row: Steiner Surface Bottom row: Cross Cap Surface (Sign Change, Taylor and Interval)
  39. 39. IIITHyderabad Limitations Chmutov 20 and 30 (Exterior, Interior)‫‏‬ • Numerical precision is a issue large number of roots are present in the exterior of Chmutov Surface [0.99,1.0] • Taylor test produces false roots for extreme self intersections (Cushion and Piriform)‫‏‬
  40. 40. IIITHyderabad What do we need on the GPU? • Number format: – Exact implementation of IEEE 754 – (Limited) Double precision support • Beam-Tracing: – Transfer roots from one pixel to neighbour • Recursive ray-tracing – Fixed stack on GPU
  41. 41. IIITHyderabad Video
  42. 42. IIITHyderabad Conclusions • MP and AMP methods are widely applicable in terms of Implicit Surfaces and are also SIMD amenable as cost per root finding is low • Analytical Method has limited applicability However it is SIMD amenable • Mitchell’s method has limited applicability and is not SIMD amenable.
  43. 43. IIITHyderabad Thesis Publications Related Publications • GPU Objects Sunil Mohan Ranta , Jag Mohan Singh and P.J. Narayanan Proc. Fifth Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP), LNCS Volume 4338, Pages 352-363, 2006, Madurai, India • Real time Ray tracing of Implicit Surfaces on the GPU Jag Mohan Singh and P. J. Narayanan IEEE Transactions on Visualization and Computer Graphics, 2008 (Under Revision) Other Publications • Progressive Decomposition of Point Clouds without Local Planes Jag Mohan Singh and P. J. Narayanan LNCS Volume 4338, Pages 364-375, Proc. of Indian Conference on Computer Vision, Graphics and Image Processing (ICVGIP), 2006 • Point Based Representations for Hierarchical Environments Kedarnath Thangudu , Lakshmi Gade,Jag Mohan Singh, and P J Narayanan. Pages 574-578, IEEE Computer Society Press, Proc. of International Conference on Computing: Theory and Applications(ICCTA),2007
  44. 44. IIITHyderabad Thank you!
  45. 45. IIITHyderabad CPU and GPU versions 2.1132.1132.1132.113CPU Mitchell 0.71960.73100.79390.8124CPU Point Sampling 1662.01662.01662.01665.0GPU Mitchell 7207909221000GPU Point Sampling z = 5z = 4z = 1z = 0Position of Sphere 1.1081.1081.1081.109CPU Mitchell 0.6560.6640.72580.7626CPU Point Sampling 952.2952.2953.4955.4GPU Mitchell 367.5447.5724.7825.3GPU Point Sampling z = 5z = 4z = 1z = 0Position of Cubic Sphere (Quadratic)‫‏‬ Ding Dong (Cubic)‫‏‬ Torus (Quartic)‫‏‬ 0.1850.1850.1850.186CPU Mitchell 0.17590.17910.20740.2202CPU Point Sampling 379.23379.23380.16381.06GPU Mitchell 230.06273.50383.77410.96GPU Point Sampling z = 5z = 4z = 1z = 0Position of Torus Frame Rates for 512x512
  46. 46. IIITHyderabad Discussion CPU vs GPU • SIMD amenable AMP method GPU is able to achieve higher speedups than for Mitchell’s method. • Interval method is faster for lower order surfaces than AMP. This advantage is nullified for higher order surfaces.
  47. 47. IIITHyderabad Results: Some More …
  48. 48. IIITHyderabad Results: More Alg Surfaces