jagmohan presentation2008
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Real Time Ray-Tracing Implicit Surfaces on the GPU
Jag Mohan SinghIIIT, Hyderabad
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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
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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
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Traditional Methods of Rendering
• Rasterization
• Ray Tracing
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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
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Ray-Tracing Implicit Surfaces
Can express as: f(t) = 0Desired: smallest +ve real root t0
Normal at t0 = (Sx, Sy, Sz) at (O + D t0)
S(x,y,z) = 0
Ray: P = O + t Dt0
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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
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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
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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
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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)
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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.
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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]
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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)
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• 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)
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Analytical Root Finding
Cubic Roots Equation (Homogenous Form) : Ax3+3Bx2w+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.
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Analytical Root FindingQuartic 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.
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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]
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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 < εMitchell, Graphics Interface 90
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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))]
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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
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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
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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.
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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)]
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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
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Results: Implicit Surfaces(Marching Points)
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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.
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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))
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Adaptive Marching Points
• Silhouette Adaptation
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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.
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Shadowing of Surfaces
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Dynamic Implicit Surfaces
• Implicit Surfaces whose equation varies with time.
Blobby Molecules and Twisted Superquadric
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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
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Mitchell’s Interval Method
96518Ding dong [3]
58027Cayley [3]
19550Tooth[4]
18652 Miter [4]
19552CrossCap[4]
17053Cushion [4]
8560 Peninsula [5]
7765Kiss [5]
6086Dervish [5]
FPSIterations Surface [Order]
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Marching Points: Results
125105150Superquadric
306260250Diamond Surface
315200250Scherk’s Surface
305160 250 Blobby 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]
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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]
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Result : ShadowsAMP (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 ShadowsSurface
[Order]Without
ShadowsWith 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
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Comparison with Knoll’s Affine Arithmetic
9416Barth Decic
17660Mitchell
170101Kleine
12088Barth Sextic
19671Tangle
178121Teardrop
21238Steiner
FPS (AMP Sign)
FPS (Knoll’s ANE)Surface
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Results: Robustness
Top row: Steiner Surface Bottom row: Cross Cap Surface(Sign Change, Taylor and Interval)
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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)
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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
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Video
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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.
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Thesis PublicationsRelated 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 EnvironmentsKedarnath 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
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Thank you!
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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 = 4 z = 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 = 4 z = 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 = 4 z = 1z = 0Position of Torus
Frame Rates for 512x512
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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.
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Results: Some More …
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Results: More Alg Surfaces