GPGPU: Distance Fields

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

2D Distance field
Sites
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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
Voronoi diagram
Sites
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Voronoi Diagram and Distance Fields

• Region where distance function contributes to
  final distance field Voronoi Region

Distance field
Voronoi diagram
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Distance Functions 2D

• A scalar function f (x) representing minimum
  distance from a point x to a site

graph z f (x,y)
f (x,y)vx2y2
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Distance Functions 3D

• Distance function of a site to plane is a quadric

Point Site Circular Paraboloid
Line Site Elliptic Cone
Plane Site Plane
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Why Should We Compute Them?

• Collision Detection Proximity Queries
• Robot Motion Planning
• Surface Reconstruction
• Non-Photorealistic Rendering
• Surface Simplification
• Mesh Generation
• Shape Analysis

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Why Difficult?

• Exact Computation
• Compute analytic boundaries

Analytic Boundary
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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

15
Approximate Computation
Approximate Algorithms
Discretize Sites
Discretize Space
GPU
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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?

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Brute-force Algorithm
Record ID of the closest site to each sample point
Coarsepoint-samplingresult
Finerpoint-samplingresult
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Slight Variation
?
?

?
For each site, compute distances to all sample pts
Given sites and uniform sampling
Composite through minimum operator
Record IDs of closest sites
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GPU Algorithm
?
?

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For each site, compute distances to all pixels
Given sites and frame buffer
Composite through depth test
Read-back IDs of closest sites
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GPU Algorithm 2D

• Demo Point site

Point coord (uniform parameter)
Pixel coord
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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

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GPU Algorithm 2D Source

• Show source
• Compile cg source and show assembly

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GPU Algorithm Debugging

• Visual debugging with imdebug (by Bill Baxter)
• http//www.billbaxter.com/projects/imdebug/index.h
  tml
• Steps
• Modify fragment program
• Readback and display buffer contents

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GPU Algorithm Debugging

• Example

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GPU Algorithm 2D

• Demo Line site

End-Point coords (uniform parameters)
Pixel coord
Careful Equation is to an infinite line
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GPU Algorithm 2D

• Line segment Region closer to interior of line
  segment

In remaining region?
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GPU Algorithm 2D Source

• Show source

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GPU Algorithm 3D

• Graphics hardware computes one 2D slice
• Sweep along 3rd dimension (Z-axis) computing 1
  slice at a time

3D Voronoi Diagram
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Outline

• Distance Fields and Voronoi Diagrams
• Hands on application demo
• Advanced Optimization
• Discussion Why fast on a GPU?

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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

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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)

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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

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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

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Optimization Fragment Program

• Do we need to evaluate (x p) in fragment
  program?

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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

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GPU Linear Interpolation

• Color example

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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

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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

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Optimization Memory Bandwidth

• Reduce number of texture lookups, framebuffer
  writes
• Pack data into fewer channels
• Is bandwidth limited?

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Optimization Memory Bandwidth

• Reduce number of texture lookups, framebuffer
  writes
• Pack data into fewer channels
• How?

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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)

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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

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Linear Factorization

• Distance vector field Gives vector from a point
  in 3D to closest point on a site

Line Site
Distance Vectors
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Linear Factorization

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

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Linear Factorization 2D

• Distance vectors are linearly interpolated

Line Segment
e
f
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Linear Factorization 3D

• Distance vectors are bi-linearly interpolated

f
e
p
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Linear Factorization 3D

• Distance vectors are bi-linearly interpolated

f
e
b
p
a
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Linear Factorization 3D

• Distance vectors are bi-linearly interpolated

f
e
b
p
a
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Linear Factorization 3D

• Distance vectors are bi-linearly interpolated

f
e
b
p
a
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Linear Factorization 3D

• Distance vectors are bi-linearly interpolated

f
e
b
p
a
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Linear Factorization 3D

• Distance vectors are bi-linearly interpolated

f
e
e
b
p
a
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Linear Factorization 3D

• Distance vectors are bi-linearly interpolated

f
b
e
p
a
c
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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

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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

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Domain Computation Non-Manifold Point
Point
Polygon
Slice
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Domain Computation Line Segment
Line Segment
Polygon
Slice
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Domain Computation Triangle
Triangle
Prism
Polygon
Slice
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Domain Computation Manifold Edge
Edge

• Triangular Prisms given by incident triangles

Prism
Polygon
Slice
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Domain Computation Manifold Vertex
Slice

• Cone given by half-plane intersections
• Compute a bounding right circular cone
• Valid for hyperbolic points

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GPU Based Algorithm
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
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Outline

• Distance Fields and Voronoi Diagrams
• Hands on application demo
• Advanced Optimization
• Discussion Why fast on a GPU?

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

• Brute-force Insanely parallelizable
• All fragment processors (cores) are being utilized

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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)
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Efficiency Bandwidth

• GPU High bandwidth to framebuffer
• CPU Set of sites grid does NOT fit in L2 cache