Real time Ray-Casting of Algebraic Surfaces

Transcription

1 Real time Ray-Casting of Algebraic Surfaces Martin Reimers Johan Seland Center of Mathematics for Applications University of Oslo Workshop on Computational Method for Algebraic Spline Surfaces Thursday 13. September

2 Introduction Raycasting An algebraic surface is the zero set of a polynomial f (x, y, z) = f ijk x i y j z k = 0. 0 i+j+k d Raycasting amounts to shooting rays inside a view frustum (VF) and determine if they intersect the surface. Can miss thin features Conceptually easy Embarrassingly parallel

3 Introduction For a ray r pq (t) raycasting amounts to finding the smallest t [0, 1] s. t. f ((1 t)n pq + tf pq ) = f (r pq (t)) = 0. We would like to work on a univariate polynomial in Bernstein form, d f (r pq (t)) = c pqk Bk d (t) = 0. k=0 Assume a screen resolution of (m + 1) (n + 1) pixels. Pixel (p, q) corresponds to a ray through p and the pixel with coordinates (p/m, q/n). far plane r pq (t) near plane p

4 Introduction Overview Our algorithm thus consist of the following steps: 1. Compute ray coefficients C = (c pqk )

5 Introduction Overview Our algorithm thus consist of the following steps: 1. Compute ray coefficients C = (c pqk ) 2. For each pixel (p, q) 2.1 Seek the smallest root t [0, 1] of f (r pq (t)). 2.2 Compute a color for pixel (p, q).

6 Introduction Overview Our algorithm thus consist of the following steps: 1. Compute ray coefficients C = (c pqk ) 2. For each pixel (p, q) 2.1 Seek the smallest root t [0, 1] of f (r pq (t)). 2.2 Compute a color for pixel (p, q). 3. Optionally, perform postprocessing 3.1 Detect singularities 3.2 Antialias

7 Ray coefficient computation The View Frustum Form Idea: Parameterize the view frustum over the unit cube, s. t. (u, v, 0) and (u, v, 1) maps to points on the near and far plane. v L far plane near plane u p w A ray in the view frustum is given by: r pq (w) = L(p/m, q/n, w). We define the View Frustum Form to be: g = f L : [0, 1] 3 R.

8 Ray coefficient computation Using the composition g = f L, f (L( p m, q n, w)) = g( p m, q d,d,d n, w) = = i,j,k=0 g ijk B d i ( p m )Bd j ( q n )Bd k (w) d d,d g ijk Bi d ( p m )Bd j ( q n ) B k d (w). k=0 i,j=0 Yielding univariate ray equations of degree d, f (r pq (t)) = } {{ } c pqk d c pqk Bk d (t). k=0

9 Ray coefficient computation Computing VFF Coefficients The VFF coefficients G = (g ijk ) can be found in a number of ways: Blossoming [DeRose et.al. 1993]. Recursion [Sederberg/Zundel 1989]. Interpolation (our preferred approach). Choose (d + 1) 3 distinct interpolation points (u p, v q, w r ) on a grid. Solve d,d,d i,j,k=0 Ω q {}}{ g ijk Bi d (u p ) Bj d (u q )Bj d (u r ) = f (L(u p, v q, w r )) }{{}}{{} Ω p Ω r Needs inverse of Bernstein collocation matrices Ωp = (Bi d (u p )). Use Chebyshev interpolation points for stability. Not dependent on the representation of f.

10 Ray coefficient computation Benefits of the View Frustum Form For fixed m, n, d precompute basis functions Ck = MG k N T, M = (Bi d (p/m)), N = (Bj d(q/n)). Can pre-evaluate inverse of collocation matrix Ω 1. Reduce algorithmic complexity Evaluation of c pqk requires (d + 1) 2 muls/adds. Evaluation of f requires (d + 1)(d + 2)(d + 3)/6 muls/adds. Univariate ray equations for root finding. f (r pq (t)) = d c pqk Bk d (t) k=0

11 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

12 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

13 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

14 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

15 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

16 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

17 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

18 Root finding B-Spline Based Root Finding The univariate ray equations can be expressed as B-Splines. Idea: Insert knot(s) at the first intersection of the control polygon. Repeat Second order convergence for simple roots [Mørken and Reimers, 2006].

19 Root finding Properites of rootfinder Stable computations (convex combinations) Similar to Newtons, but unconditionally convergent to smallest zero (no guessing) Quadratic convergence rate to simple zeros Recent extension [Mørken/Reimers] converge quadratically to multiple zeros Similar method [Reimers] for computing e.g. max f (x) or min f (x)

20 Root finding Root finding variations The knot insertion framework is very flexible, and allow for variations: Can emulate Bézier subdivision by inserting d knots at a time. (Lane/Risenfeld, Rockwood, Schneider). Preconditioning, insert knots from neighboring rays. Estimate root multiplicity and detect roots of n th derivative. (Strictly alternating control polygon.) Detect critical points, use as start value to search for singularities.

21 Postprocessing Singularity detection 1. For misses, find smallest absolute value along ray, w Flag as singularity if: g(p/m, q/n, w 0 ) + g(p/m, q/n, w 0 ) < ɛ. How to determine ɛ? Vulnerable to scaling. x 2 y 3 = 0.

22 Postprocessing Antialiasing Due to discrete sampling, aliasing effects will occur. Suppose neighboring pixels p 1, p 2 differ. I.e. p 1 p 2 < ɛ. We seek a point s on the separating curve between p 1 and p 2. color(p 1 ) := (1.0 α) color(p 1 ) + α color(p 2 ) v silhouette n 2 α p 2 s = (u, v) p 1 u

23 Postprocessing Antialiasing II At silhouettes g(s) = 0 and g w (s) = 0. Use Newton methods on h(v, w) := (g(s), g w (s)) = (0, 0). Restrict to plane between p 1, p 2. If leaving domain, search for g(s) = 0.

24 Gallery and performance Postprocessing

25 Postprocessing Future work Interval spline methods: Topological correctness. Empty-space skipping. Bounding box calculations Efficient data structures for splines

26 Postprocessing Thank you for listening Questions? Contact info Johan S. Seland Tel:

27 Postprocessing Mørken and Reimers An unconditionally convergent method for computing zeros of splines and polynomials Math. of Comp. 76, 2006

28 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e-1

29 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e-2

30 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e-2

31 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e-3

32 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e e-4

33 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e e e-6

34 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e e e e-8

35 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e e e e e-12

36 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e e e e e e-16

37 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e e e e e e e-24

38 Postprocessing Zero Algorithm [MrkenReimers 2007] Idea: Repeated knot insertion at zeros of F t Repeat for j = 0, 1, until convergence or F t j has no zeros 1. Find the smallest value x j+1 such that F t j (x j+1 ) = 0 or stop 2. Let t j+1 = t j {x j+1 } and form F t j Error x j z : 1.41e e e e e e e e e e-24