The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Distance Fields Avneesh Sud COMP 290-058, Fall 2003.

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The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Distance Fields

Avneesh Sud

COMP 290-058, Fall 2003

2The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL

Overview

WHAT are Distance Fields?HOW are they computed?WHY do we care?

Overview

Distance Functions

A scalar function f:Rn -> R representing the distance from an object O to any point P ε Rn

Object O is called a ‘site’

Distance Functions: Example

Euclidean Distance Function:f(x,y)=√x2+y2

Point in 2D

Distance Functions: Example

Euclidean Distance Function:f(x,y)=√x2+y2

Plot a graph:z=√x2+y2

Points in 2D

Distance Metrics

Lp distance metric in k dim

p = 1, “Manhattan” or City Blockp = 2, Euclideanp =∞, Max-norm

iip yxd

Distance Fields

For a set of sites, the minima of all distance functions representing the distance from a point P ε Rn to closest site“Lower Envelope” of all distance functions

Differential Geometry

S = surface moving in a normal direction with unit speedDistance of point X from surface = arrival time of the surface at X

Differential Geometry

Shortest distance from X to point p ε S is orthogonal to SLine segment Xp is a characteristicS must be smooth

What is a Voronoi Diagram?Given a collection of geometric primitives, it is a subdivision of space into cells such that all points in a cell are closer to one primitive than to any other

Voronoi Site

Voronoi Region

Ordinary Point sitesNearest Euclidean distance

GeneralizedHigher-order site geometryVarying distance metrics

Weighted Distances

Higher-orderSites

Voronoi Diagram and Distance Fields“Projection of lower envelope” of

distance functions

“Lower Envelope” of distance functions

Voronoi Diagram

Voronoi Diagram and Distance Fields

“Projection of lower envelope” of distance functionsGiven one, other can be easily computedPractical methods compute distance fields on a discrete grid

Related Terminology

Distance Transform = Distance FieldClosest Point Transform = Feature Transform = Voronoi Diagram

Why Should We Compute Them?

Useful in a wide variety of applications

Collision DetectionSurface ReconstructionRobot Motion Planning

Non-Photorealistic RenderingSurface Simplification

Mesh GenerationShape Analysis

GVD: “Clearance Information”

Points with the largest distance values to nearby obstacles

Maximally Clear Path

Density Estimation Nearest Neighbors

Distance Field: “Proximity Information”

Potential Fields – need proximity information to obstaclesDistance field of obstacles (and its gradient) used to compute repulsive forces [Khatib86]

reprep

Overview

Brute Force Algorithm (Grid)1. Initially : dp =∞, for all p ε

grid2. for each site m3. for each grid point p4. dnew = distance to m

5. if |dnew|<|dp|

6. dp = dnew

7. cpp = closest point on m8. end9. end

Brute-force Algorithm

Record ID of the closest site to each sample

Coarsepoint-sampling

result

Finerpoint-sampling

result

Brute Force Algorithm

Time Complexity?M sites, grid size N O(MN)

Space Complexity?O(N)

Immensely Parallelizable!

GPU Based Computation

Point Line Triangle

HAVOC2D, HAVOC3D [Hoff et al, 99]

Evaluate distance at each pixel for all sitesAccelerate using graphics hardwareModel is polygonal

HAVOC2D, HAVOC3DEvaluate distance at each pixel for all sitesAccelerate using graphics hardware

StepsMesh approximation of Distance FunctionsRender distance meshes using graphics hardwareReadback final buffers

GPU Based Computation

Approximating the Distance Function

Depth buffer performs minimum comparisonGraphics Hardware performs fast linear interpolation Distance Functions are non-linearPerform linear approximation to Distance Function

Approximating the Distance Function

Point Line Triangle

Avoid per-pixel distance evaluationPoint-sample the distance functionReconstruct by rendering polygonal mesh

The Error Bound

Error bound is determined by the pixel resolution farthest distance a point can be from a pixel sample point

Close-up of pixel grid

Meshing the Distance Function

Shape of distance function for a 2D point is a cone

Need a bounded-error tessellation of the cone

Rasterization to reconstruct distance valuesDepth test to perform minimum operator

Graphics Hardware Acceleration

Perspective, 3/4 view Parallel, top view

Readback Results

Distance Field

Depth Buffer

Voronoi Regions

Color Buffer

Shape of Distance Functions

Sweep apex of cone along higher-order site to obtain the shape of the distance function

Example Distance Meshes

Curves

Tessellate curve into a polylineTessellation error is added to meshing error

3D Voronoi Diagrams

Graphics hardware can generate one 2D slice at a timeSweep along 3rd dimension (Z-axis) computing 1 slice at a time

Polygonal model

Shape of 3D Distance Functions

Slices of the distance function for a 3D point site

Distance meshes used to approximate slices

Shape of 3D Distance FunctionsPoint Line segment Triangle

1 sheet of a hyperboloid

Elliptical cone Plane

Advantages

Simple idea – efficiently implemented on graphics hardwareExtensible to any type of distance function and input set

Disadvantages

Mesh Generation and TranformLarge number of triangles generated

Rasterization: Distance mesh can fill entire sliceComplexity for M sites and N slices = O(MN) Lot of Fill !

Readback:Stalls the graphics pipelineUnsuitable for interactive applications

Using Voronoi Information

Assume point sites onlyCompute Voronoi diagram

O(M logM) in 2DO(M2) in 3D

Scan convert each Voronoi cell, computing closest distance

O(N)Edges of Voronoi polygons/polyhedra must be larger than grid spacing

Characteristics/Scan Conversion Algorithm

Use characteristics [Mauch00]2D algorithm for a piecewise linear curve

Closest points to a line segment lie within an infinite stripEach strip exactly contains the characteristic curves from that edge

Characteristic Curve

Closest points to a point lie within an infinite wedgeCharacteristic line for non-smooth points on the manifold lies “between” the normals of surrounding smooth parts

Edge Strips

Positive Distance Negative Distance

Vertex Wedges

Positive Distance Negative Distance

Bounds

Compute distance field in a band up to distance d from the curveCompute bounding polygons up to distance d

Vertex Edge

CSC algorithm – 2D

1. Initially: dij =∞, for all i,j2. for each edge e3. p = bounding polygon of e4. G = scan_convert(p)5. for each (i, j) ε G6. dnew = distance to e7. if |dnew|<|dij|8. dij = dnew

9. cpij = closest point on e10. end11. end12. for each vertex v13. …14. end

3D Triangle Mesh

Sites are triangular faces, edges or verticesCharacteristic lines have bounding polyhedra

Faces – Triangular Prims Edge - Wedge Vertices – “Cones”

3D Triangle Mesh

Sites are triangular faces, edges or verticesCharacteristic lines have bounding polyhedra

3D Computation

Slice polyhedra into polygonsScan convert each polygon

Slices of a “cone”

CSC algorithm – 3D

1. Initially: dij =∞, for all i,j2. for each face f3. p = bounding polyhedron of f4. …5. end6. for each edge e7. …8. end9. for each vertex v10. …11.end

Complexity

M sites, N grid cells within distance d of the surfacer = ratio of sum of volumes of polyhedra by volume of region within distance d

Gives amount of overlap = number of times each cell is written

Complexity = O(M +rN)

Polyhedra Construction Scan Conversion

Complexity

For small d, r ≈ 1For large d, clip bounding polyhedra to keep r small

Can increase setup cost of polyhedra construction to O(M2 )

Advantages

Efficiently exploits surface connectivity, minimizing fillGives signed distance fields

Disadvantages

Valid for manifold surfacesCannot evaluate non-linear distance functions on older generations of graphics hardwareA large number of polygons may be scan-converted for each sliceSaddle points – “Cones” invalidSetup cost

CSC on graphics hardware

Use programmable fragment programs on modern GPUs to compute distance functions at each pixel [Sigg03]

Can compute complicated functions at each pixel on current GPUs

HW-Based CSC

Compute entire slice at a timeCPU computes slicing polygonsGPU computes distance values at each pixel on slice in parallel

Slice PolyhedraPolygons

Reducing polygon count

Expand triangular prisms to encloses point closest to edges and faces

2D example

Bisector

Reducing polygon count

Expand triangular prisms to encloses point closest to edges and faces

Fragment program computes closest siteComplete Voronoi information is lost

3D example

Results

Stanford Bunny Model Knot Model

Advantages

Efficient implementation on graphics hardware – exploits parallelismReduces number of polygons drawnHandles saddle points

Fast Distance Field computation using graphics hardware [Sud03]Enhancement over HAVOCExploits Voronoi diagram properties for culling

Voronoi Diagram Properties

Within a bounded region, all voronoi regions have a bounded volume

9 Sites, 2D

Within a bounded region, all voronoi regions have a bounded volume

As site density increases, average spatial bounds decrease

27 Sites, 2D

Voronoi regions are connected

Valid for all Lp norms

High distance field coherence between adjacent slicesChange in distance function between adjacent slices is bounded Distance functions for a

point site Pi to slice Zj

High distance field coherence between adjacent slicesChange in distance function between adjacent slices is bounded Distance functions for a

point site Pi to slice Zj+1

Culling Techniques

Use Voronoi region bounds for culling

Site ClassificationEstimating Z-BoundsEstimating XY-Bounds

Site Culling: Classification

For each slice partition the set of sites S1

Slicej

Sweep Direction

For each slice partition the set of sites, using voronoi region bounds:

Slicej

ZSweep Direction

Approaching (Aj)

Slicej

ZSweep Direction

Approaching (Aj)

Intersecting (Ij)

Slicej

ZSweep Direction

Approaching (Aj)

Intersecting (Ij)

Receding (Rj)

Slicej

ZSweep Direction

Approaching (Aj)

Intersecting (Ij)

Receding (Rj)

Render distance functions for Intersecting sites only

Slicej

ZSweep Direction

Coherence

Updating Ij

Ij+1 = Ij …

Slicej+1

Previously Intersecting

ZSweep Direction

Coherence

Updating Ij

Ij+1 = Ij

+ (Aj – Aj+1) …

Slicej+1

ZSweep Direction

Approaching Intersecting

Aj-Aj+1

Coherence

Updating Ij

Ij+1 = Ij

+ (Aj – Aj+1)

– (Rj+1 – Rj)

Slicej+1

ZSweep Direction

Rj+1-Rj

Intersecting Receding

Coherence

Updating Ij

Ij+1 = Ij

+ (Aj – Aj+1)

– (Rj+1 – Rj)

Slicej+1

ZSweep Direction

Aj+1Ij+1

Estimating Set Partitions

Computing exact set partition = Exact voronoi computationUse hardware based occlusion queries

Determine number of visible fragments

Compute a set of potentially intersecting sites Îjjj II ˆ

Estimating XY Bounds

Monotonic distance functions

Monotonic distance functionsVoronoi region’s XY extent bounded by depth of distance function

Monotonic distance functionsVoronoi region’s XY extent bounded by depth of distance functionEstimate max depth bounds for each site!

Estimating Depth Bound

Render distance function in layers using occlusion query

Render distance field in layers using occlusion query

Last visible layer bounds the max depth

Gives max depth at end of current slice

Updating Depth Bound

Exploit depth field coherenceGiven a depth bound for current slice, estimate a bound for next slice

Complexity

Performance depends on amount of occlusion in distance functions

Influences estimated boundsr = ratio of estimated Voronoi bounds by actual Voronoi size

Complexity O(M+rN )Worst case: r = MExpected: r = constant

Mesh Generation + Transform

Rasterization

Results

Model Polys ResolutionHAVOC

(s)DiFi(s)

Shell Charge 4460 128x126x126 31.69 3.38

Head 21764 79x106x128 52.47 13.60

Bunny 69451 128x126x100 212.71 36.21

Cassini 90879 94x128x96 1102.01 47.90

4 -20 times speedup!

Overview

Planning Based on GVD & MAT

O’Dunlaing, Sharir and Yap [1983]

Canny and Donald [1987]

Latombe [1991]

Choset, Burdick, et al. [1994-1999]

Vleugels and Overmars [1996,1997]

Guibas, Holleman and Kavraki [1999]

Wilmarth, Amato and Stiller [1999]

…… and others ……

Planning Using GVDs

Is a GVD sufficient for path planning?

GVD computed in 3D workspaceSufficient for a point robot

Use GVD along with other methods

3-DOF Potential Field Planner with GVD

Rigid robot moving on a plane in a dynamic environment [Hoff00]Combines a GVD roadmap with a local (potential field) planner

Basic Approach

Depth Buffer – providing distance function and distance gradient (finite difference)

Color Buffer – building Voronoi graphs

Combination of BothCompute weighted Voronoi graphs Voronoi vertices used for milestonesWeighted edges used for selecting pathsDistance values for quick rejection

Voronoi BoundaryFor each voxel in discrete GVD, associate a color corresponding to an object ID and a distance value to this obstacle.

To extract the boundary, use continuation method similar to iso-surface extraction -- starting from a seed point and step to next by bracketing boundary curves in a 2x2x2 region of sampled points.

Can be efficiently performed on modern GPUs using fragment programs

Constructing Voronoi Roadmap

Identify Voronoi vertices

Extract Voronoi boundary

Build Voronoi graph

Select path based on edge weights

Incrementally construct Voronoi roadmap

local planner (PFF) between milestonesquick collision rejection test & exact CD

Voronoi Roadmap/Graph for

Dynamic Environments

Video (3D View)

Other Approaches

Voronoi based sampling for PRM [Hoff, WAFR00], [Foskey01] Constraint based motion planning [Garber 02]

Planning problem formulated as a dynamic simulation problem with constraintsUses distance information for solving constraints

References (Distance Fields)

HAVOC: Hoff et al. Fast Computation of Generalized Voronoi Diagrams Using Graphics Hardware SIGGRAPH 1999 CSC: S. Mauch A fast algorithm for computing the closest point and distance transformHW-CSC: Sig et al. Signed Distance Transform Using Graphics Hardware IEEE Vis 2003DiFi: Sud and Manocha Fast Distance Field Computation Using Graphics Hardware UNC-TR03-026

References (Planning)

Hoff et al. Interactive Motion Planning Using Hardware-Accelerated Computation of Generalized Voronoi Diagrams ICRA 2000Pisula et al. Randomized Path Planning for a Rigid Body Based on Hardware Accelerated Voronoi Sampling WAFR 2000Garber and Lin Constraint-Based Motion Planning using Voronoi Diagrams WAFR 2002

The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL Distance Fields Avneesh Sud COMP 290-058, Fall 2003.

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