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Distance Fields

Avneesh Sud

COMP 290-058, Fall 2003

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Overview

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

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Overview

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

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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’

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Distance Functions: Example

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

Point in 2D

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Distance Functions: Example

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

Plot a graph:z=√x2+y2

Cone

Points in 2D

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Distance Metrics

Lp distance metric in k dim

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

pk

i

p

iip yxd

1

1

),(

yx

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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

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Differential Geometry

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

S

X

d

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Differential Geometry

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

S

X

d

p

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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

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Ordinary Point sitesNearest Euclidean distance

GeneralizedHigher-order site geometryVarying distance metrics

Weighted Distances

Higher-orderSites

2.0

0.5

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

distance functions

“Lower Envelope” of distance functions

Voronoi Diagram

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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

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Related Terminology

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

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Why Should We Compute Them?

Useful in a wide variety of applications

Collision DetectionSurface ReconstructionRobot Motion Planning

Non-Photorealistic RenderingSurface Simplification

Mesh GenerationShape Analysis

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GVD: “Clearance Information”

Points with the largest distance values to nearby obstacles

Maximally Clear Path

Density Estimation Nearest Neighbors

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Distance Field: “Proximity Information”

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

0

020

reprep

if0

if111

x

kF

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Overview

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

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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

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Brute-force Algorithm

Record ID of the closest site to each sample

point

Coarsepoint-sampling

result

Finerpoint-sampling

result

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Brute Force Algorithm

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

Space Complexity?O(N)

Immensely Parallelizable!

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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

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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

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Approximating the Distance Function

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

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Approximating the Distance Function

Point Line Triangle

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

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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

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Meshing the Distance Function

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

Need a bounded-error tessellation of the cone

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Rasterization to reconstruct distance valuesDepth test to perform minimum operator

Graphics Hardware Acceleration

Perspective, 3/4 view Parallel, top view

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Readback Results

Distance Field

Depth Buffer

Voronoi Regions

Color Buffer

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Shape of Distance Functions

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

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Example Distance Meshes

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Curves

Tessellate curve into a polylineTessellation error is added to meshing error

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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

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Shape of 3D Distance Functions

Slices of the distance function for a 3D point site

Distance meshes used to approximate slices

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Shape of 3D Distance FunctionsPoint Line segment Triangle

1 sheet of a hyperboloid

Elliptical cone Plane

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Advantages

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

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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

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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

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Characteristics/Scan Conversion Algorithm

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

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Characteristics/Scan Conversion Algorithm

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

Characteristic Curve

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Characteristics/Scan Conversion Algorithm

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

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Edge Strips

Positive Distance Negative Distance

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Vertex Wedges

Positive Distance Negative Distance

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Bounds

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

Vertex Edge

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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

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3D Triangle Mesh

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

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

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3D Triangle Mesh

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

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3D Computation

Slice polyhedra into polygonsScan convert each polygon

Slices of a “cone”

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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

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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

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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 )

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Advantages

Efficiently exploits surface connectivity, minimizing fillGives signed distance fields

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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

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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

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HW-Based CSC

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

Model

Slice PolyhedraPolygons

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Reducing polygon count

Expand triangular prisms to encloses point closest to edges and faces

2D example

Bisector

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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

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Results

Stanford Bunny Model Knot Model

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Advantages

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

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DiFi

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

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Voronoi Diagram Properties

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

9 Sites, 2D

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Voronoi Diagram Properties

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

As site density increases, average spatial bounds decrease

27 Sites, 2D

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Voronoi Diagram Properties

Voronoi regions are connected

Valid for all Lp norms

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Voronoi Diagram Properties

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

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Voronoi Diagram Properties

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

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Culling Techniques

Use Voronoi region bounds for culling

Site ClassificationEstimating Z-BoundsEstimating XY-Bounds

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Site Culling: Classification

For each slice partition the set of sites S1

S3

Slicej

S4

S2

Sweep Direction

X

Z

S5

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S4

Site Culling: Classification

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

Slicej

S1

S2

S3

X

ZSweep Direction

S5

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Site Culling: Classification

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

Approaching (Aj)

Slicej

S1S3

S4

S2

X

ZSweep Direction

Aj

S5

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S4

S2

Site Culling: Classification

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

Approaching (Aj)

Intersecting (Ij)

Slicej

S1S3

X

ZSweep Direction

Aj

Ij

S5

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S4

S2

Site Culling: Classification

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

Approaching (Aj)

Intersecting (Ij)

Receding (Rj)

Slicej

S1S3

X

ZSweep Direction

Aj

Ij

Rj

S5

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S4

S2

Site Culling: Classification

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

Approaching (Aj)

Intersecting (Ij)

Receding (Rj)

Render distance functions for Intersecting sites only

Slicej

S1S3

X

ZSweep Direction

Aj

Ij

Rj

S5

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Coherence

Updating Ij

Ij+1 = Ij …

Slicej+1

S1S3

Previously Intersecting

S4

S2

X

ZSweep Direction

Ij

S5

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S4

S2

Coherence

Updating Ij

Ij+1 = Ij

+ (Aj – Aj+1) …

Slicej+1

S1S3

X

ZSweep Direction

Approaching Intersecting

Aj-Aj+1

S5

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S4

S2

Coherence

Updating Ij

Ij+1 = Ij

+ (Aj – Aj+1)

– (Rj+1 – Rj)

Slicej+1

S1S3

X

ZSweep Direction

Rj+1-Rj

Intersecting Receding

S5

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S4

S2

Coherence

Updating Ij

Ij+1 = Ij

+ (Aj – Aj+1)

– (Rj+1 – Rj)

Slicej+1

S1S3

X

ZSweep Direction

S5

Aj+1Ij+1

Rj+1

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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

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Estimating XY Bounds

Monotonic distance functions

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Estimating XY Bounds

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

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Estimating XY Bounds

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

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Estimating Depth Bound

Render distance function in layers using occlusion query

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Estimating Depth Bound

Render distance field in layers using occlusion query

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Estimating Depth Bound

Render distance field in layers using occlusion query

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Estimating Depth Bound

Render distance field in layers using occlusion query

Last visible layer bounds the max depth

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Estimating Depth Bound

Render distance field in layers using occlusion query

Last visible layer bounds the max depth

Gives max depth at end of current slice

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Updating Depth Bound

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

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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

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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!

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Overview

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

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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 ……

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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

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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

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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

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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

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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

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Voronoi Roadmap/Graph for

Dynamic Environments

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Video (3D View)

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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

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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

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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