1. GPGPU: Distance Fields
Avneesh Sud and Dinesh Manocha
Feb 12, 2007

2. So Far Overview Intro to GPGPU using OpenGL
Current Architecture (Cell, G80)
Programming (CUDA, Compilers)
Applications (Vision)

3. Interesting Reading on Parallel Computing
The Landscape of Parallel Computing Research: A View from Berkeley

4. This Lecture Distance Fields and Voronoi Diagrams Hands on demo
Advanced: Optimization
Discussion: Why fast on a GPU?

5. This Lecture Distance Fields and Voronoi Diagrams
Hands on application demo
Parallel algorithm
Example code: 2D
Visual Debugging (imdebug)
Example code: 3D
Advanced: Optimization
Discussion: Why fast on a GPU?

6. Outline Distance Fields and Voronoi Diagrams Hands on application demo
Advanced: Optimization
Discussion: Why fast on a GPU?

7. Distance Field
Given a set of geometric primitives (sites), it is a scalar field representing the minimum distance from any point to the closest site
Given a set of …
Here we show an example with 3 pt sites and their 2d dist field
Sites
2D Distance field

8. Generalized Voronoi Diagram
Given a collection of sites, it is a subdivision of space into cells such that all points in a cell are closer to one site than to any other site
Voronoi Site
Voronoi cell
Distance fields are closely related to Voronoi diagrams
Given a collection of sites, a Voronoi diagram is a subdivision of space into cells, such that all points in a cell are closer to one primitive than to any other.
Make brighter
Sites
Voronoi diagram

9. Voronoi Diagram and Distance Fields
Region where distance function contributes to final distance field = Voronoi Region
… is the Voronoi region of that site
Distance field
Voronoi diagram

10. Distance Functions: 2D
A scalar function f (x) representing minimum distance from a point x to a site
A distance function is a …
The distance function can be drawn as a graph with z = value
The distance function to a pt in 2D corresponds to a cone.
graph z = f (x,y)
f (x,y)=√x2+y2

11. Distance Functions: 3D
Distance function of a site to plane is a quadric
Point Site
Circular Paraboloid
Line Site
Elliptic Cone
Plane Site
Plane

12. Why Should We Compute Them?
Collision Detection & Proximity Queries
Robot Motion Planning
Surface Reconstruction
Non-Photorealistic Rendering
Surface Simplification
Mesh Generation
Shape Analysis
Voronoi diagrams have also been useful in a wide variety of recent applications.
Here are some common examples.

13. Why Difficult? Exact Computation Analytic Boundary
Compute analytic boundaries
These algorithms compute analytic boundaries which requires representing and manipulating high-degree curves and surfaces and their intersections
This not only makes them complex and difficult to implement, but it also makes them suffer from robustness and accuracy problems.
In an attempt to overcome many of these difficulties, approximate algorithms are often used.
Analytic Boundary

14. Why Difficult? Exact Computation
Compute analytic boundaries
Boundaries composed of high-degree curves and surfaces and their intersections
Complex and difficult to implement
Robustness and accuracy problems
These algorithms compute analytic boundaries which requires representing and manipulating high-degree curves and surfaces and their intersections
This not only makes them complex and difficult to implement, but it also makes them suffer from robustness and accuracy problems.
In an attempt to overcome many of these difficulties, approximate algorithms are often used.

15. Approximate Computation
Approximate Algorithms
Approximate algorithms simplify the problem by either discretizing the sites or by discretizing the space containing the sites.
In the center image, the higher-order line site is point-sampled. The Voronoi diagram of these points forms a piecewise-linear approximation to the Voronoi boundary.
In the right image, a volumetric representation is constructed by determining which Voronoi regions contain each sample point.
We will be computing this type of approximation.
Discretize Sites
Discretize Space
GPU

16. Outline Distance Fields and Voronoi Diagrams Hands on application demo
Parallel algorithm
Example code: 2D
Visual Debugging (imdebug)
Demo: 3D
Advanced: Optimization
Discussion: Why fast on a GPU?

17. Brute-force Algorithm
Record ID of the closest site to each sample point
The brute-force approach is very simple:
We begin with a bounded region of space containing the Voronoi sites.
We then point-sample the space and for each sample point, we compute distances to all sites and record the ID and distance of the closest site.
This forms the volumetric approximations shown in the right two images.
This approach is simple and very easy to generalize since it only requires the distance between a site and a sample point.
In addition, the most dominant error results from the point-sampling density which is simple to bound.
Coarse point-sampling result
Finer point-sampling result

18. Slight Variation…   = 
The brute-force approach is very simple:
We begin with a bounded region of space containing the Voronoi sites.
We then point-sample the space and for each sample point, we compute distances to all sites and record the ID and distance of the closest site.
This forms the volumetric approximations shown in the right two images.
This approach is simple and very easy to generalize since it only requires the distance between a site and a sample point.
In addition, the most dominant error results from the point-sampling density which is simple to bound.
For each site, compute distances to all sample pts
Given sites and uniform sampling
Composite through minimum operator
Record IDs of closest sites

19. GPU Algorithm…   =  For each site, compute distances to all pixels
The brute-force approach is very simple:
We begin with a bounded region of space containing the Voronoi sites.
We then point-sample the space and for each sample point, we compute distances to all sites and record the ID and distance of the closest site.
This forms the volumetric approximations shown in the right two images.
This approach is simple and very easy to generalize since it only requires the distance between a site and a sample point.
In addition, the most dominant error results from the point-sampling density which is simple to bound.
For each site, compute distances to all pixels
Given sites and frame buffer
Composite through depth test
Read-back IDs of closest sites

20. GPU Algorithm: 2D Demo: Point site Point coord Pixel coord
(uniform parameter)
Pixel coord

21. GPU Algorithm: 2D Source
Initialization
Setup GL State (Depth, Render Target)
Setup fragment program
Fragment program
Computation: For each point site
Set program parameters
Execute fragment program
Display
Display results

22. GPU Algorithm: 2D Source
Show source …
Compile cg source and show assembly

23. GPU Algorithm: Debugging
Visual debugging with imdebug (by Bill Baxter)
Steps
Modify fragment program
Readback and display buffer contents

24. GPU Algorithm: Debugging
Example

25. GPU Algorithm: 2D Demo: Line site
End-Point coords
(uniform parameters)
Pixel coord
Careful: Equation is to an infinite line

26. GPU Algorithm: 2D
Line segment: Region closer to interior of line segment
In remaining region?

27. GPU Algorithm: 3D Graphics hardware computes one 2D slice
Sweep along 3rd dimension (Z-axis) computing 1 slice at a time
In order to use the graphics hardware for constructing 3D Voronoi diagrams, we compute the volume one slice at a time.
This animation shows the Voronoi diagram computed over a teapot in 3D.
Subsequently everything is in 3D
- Computing distance at each voxel to all sites is slow for large # of sites and…
3D Voronoi Diagram

28. Outline Distance Fields and Voronoi Diagrams Hands on application demo
Advanced: Optimization
Discussion: Why fast on a GPU?

29. GPU Optimizations Where to optimize? Make fragment program run faster
GPU / Application dependent optimizations
Reduce memory bandwidth
Reduce number of invocations of fragment program
Geometric culling

30. GPU Optimizations: Recommended Reading
Practical Performance Analysis and Tuning
GPU Programming Guide
GPU Gems 2
GPU Computation Strategies and Tips (Ian Buck)
GPU Program Optimization (Cliff Woolley)

31. GPU Optimizations Where to optimize? Make fragment program run faster
GPU / Application dependent optimizations
Reduce memory bandwidth
Reduce number of invocations of fragment program
Geometric culling

32. Optimization: Fragment Program
Reduce number of instructions!
Do we need dist(x, p) or dist2(x, p)?
Advantage: dist() requires an additional reciprocal sqrt
Show code + demo

33. Optimization: Fragment Program
Do we need to evaluate (x – p) in fragment program?

34. Optimization: Fragment Program
Do we need to evaluate (x – p) in fragment program?
Rasterization/G80 lectures: GPUs have VERY FAST dedicated hardware for linear interpolation (lerp)
Lerp color, textures, normals across triangle vertices

35. GPU: Linear Interpolation
Color example

36. Optimization: Fragment Program
Evaluate (x – p) at polygon vertices and use dedicated hardware to lerp at each pixel !
What about line / triangle sites?
Can be linearly interpolated too !
More details later

37. GPU Optimizations Where to optimize? Make fragment program run faster
GPU / Application dependent optimizations
Reduce memory bandwidth
Reduce number of invocations of fragment program
Geometric culling

38. Optimization: Memory Bandwidth
Reduce number of texture lookups, framebuffer writes
Pack data into fewer channels
Is bandwidth limited?

39. Optimization: Memory Bandwidth
Reduce number of texture lookups, framebuffer writes
Pack data into fewer channels
How?

40. Optimization: Memory Bandwidth
Pack data into fewer channels
Using fp32 render target
32 bit = 4 billion site ids
We can use only 1 channel (red) for writing site id instead of 4 channels (RGBA)

41. GPU Optimizations Where to optimize? Make fragment program run faster
GPU / Application dependent optimizations
Reduce memory bandwidth
Reduce number of invocations of fragment program
Geometric culling

42. Linear Factorization
Distance vector field: Gives vector from a point in 3D to closest point on a site
Line Site
Distance Vectors

43. Linear Factorization Distance functions are non-linear (quadric)
Distance Vectors can be factored into linear terms
Linearly interpolated along each axis

44. Linear Factorization: 2D
Distance vectors are linearly interpolated
Line Segment e f

45. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated f e p

46. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated f e b p a

47. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated f e b p a

48. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated f e b p a

49. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated f e b p a

50. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated f e e b p a

51. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated f b e p a c

52. Linear Factorization: 3D
Distance vectors are bi-linearly interpolated
Distance vector in interior of a convex polygon = convex combination of distance vectors at vertices
Convex polygon: from geometry of site

53. Domain Computation Compute a convex polytope bounding Voronoi region
Non-manifold sites
Manifold sites
Intersect polytope with each slice to get convex polygonal domain
‘Clamping’ computation – reduce fill

54. Domain Computation: Non-Manifold Point
Polygon
Slice

55. Domain Computation: Line Segment
Polygon
Slice

56. Domain Computation: Triangle
Prism
Polygon
Slice

57. Domain Computation: Manifold Edge
Triangular Prisms given by incident triangles
Prism
Polygon
Slice

58. Domain Computation: Manifold Vertex
Slice
“Cone” given by half-plane intersections
Compute a bounding right circular cone
Valid for hyperbolic points

59. GPU Based Algorithm Vertex Processor Fragment Processor Raster Ops
Pentium IV 3.4Ghz
NVIDIA GeForce 7800 GTX
Compute bounding polygon
Compute distance vectors
Bi-linear interpolation
Compute Norm
Compute Min
280 GFLOPS
Vertex Processor
Fragment Processor
Raster Ops
Rasterizer
CPU
25.6 GFLOPS
35 GFLOPS
Texture
1.3 TFLOPS
GPU Memory

60. Outline Distance Fields and Voronoi Diagrams Hands on application demo
Advanced: Optimization
Discussion: Why fast on a GPU?

61. Distance Field Timings
CSC (CPU)
HAVOC
DiFi: GPU
240x
313x
100 10 1
0.1
Disclaimer: Non optimized CPU code (No SSE), includes geometric culling

62. Efficiency: Parallelism
Brute-force: Insanely parallelizable
All fragment processors (cores) are being utilized

63. Efficiency: Dedicated H/W
Graphics hardware efficiently performs bilinear interpolation of vertex attributes
Texture coordinates
Color
Normals (Phong shading)
Fast depth test = Atomic compare and set
(32bit FP)

64. Efficiency: Bandwidth
GPU: High bandwidth to framebuffer
CPU: Set of sites + grid does NOT fit in L2 cache