Corp. Research Princeton, NJ Cut Metrics and Geometry of Grid Graphs Yuri Boykov, Siemens Research,...

Corp. ResearchPrinceton, NJ

Cut Metrics and

Geometry of Grid Graphs

Yuri Boykov, Siemens Research, Princeton, NJjoint work with

Vladimir Kolmogorov, Cornell University, Ithaca, NY

Corp. ResearchPrinceton, NJOutline

I: “Cut Metrics” vs. “Path Metrics” on Graphs

II: Integral Geometry and Graph Cuts (Euclidean case) • Cauchy-Crofton formula for curve length and surface area• Euclidean Metric and Graph Cuts

III: Differential Geometry and Graph Cuts• Approximating continuous Riemannian metrics• Geodesic contours and minimal surfaces via Graph Cuts• Graph Cuts vs. Level-Sets

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Part I:

“Cut Metrics” vs. “Path Metrics” on Graphs

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Path metrics are relevant for graph applications based on Dijkstra style optimization.(e.g. Intelligent Scissors method in vision)

“Length” is naturally defined for any “path” connecting two nodes along graph edges.

Standard “Path Metrics” on graphs

A

B

ABe

eAB ||||||

The properties of path metrics are relatively straightforward and were studied in the past

Corp. ResearchPrinceton, NJ“Distance Maps” for Path Metrics

We assume here that each edge cost equals its Euclidean (L2) length

Consider all graph nodes equidistant (for a given path metric) from a given node.

4 neighborhoodsystem

8 neighborhoodsystem

256 neighborhoodsystem

Corp. ResearchPrinceton, NJCut Metrics on graphs

Cut metrics are relevant for graph applications based on Min-Cut style optimization.

(e.g. Interactive Graph Cuts and Normalized Cuts in vision) “Length” is naturally defined for any cut (closed

contour or surface) that separates graph nodes.

Ce

eC ||||||C

Corp. ResearchPrinceton, NJCut Metrics vs. Path Metrics

Both cut and path metrics are determined by the graph topology (t.e. neighborhood system and edge weights)

In both cases “length” is defined as a sum of edge costs for a set of edges. It is either a cut-set that separates nodes or a path-set connecting nodes. (Duality?)

Cuts naturally define surface “area” on 3D grids. Path metric is limited to curve “length” and can not define “area” in 3D.

Cut-based notion of “length” (“area”) can be extended to open curves (surfaces) on the imbedding space (or ).2R 3R

C = cost of edges that cross C

odd number of times||||C

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Cut metric “distance” for graphs with homogeneous topology

1e2e

3e4e

5e

6e7e

8e

Consider all edges on a gridke

ke

2

|||sin|||}{#

k

kk

ea

e

aecrossa

a

a k

kkgc eawa ||1

||||2

k-th edge cost

||

2

kk e

e

arbitrary fixed homogeneous neighborhood system

Corp. ResearchPrinceton, NJ“Distance Maps” for Cut Metrics

Consider all graph nodes equidistant (for a given cut metric) from a given node.

Here we took inversely proportional to Euclidean length .

kw || ke

4 neighborhoodsystem

8 neighborhoodsystem

256 neighborhoodsystem

Looks just like Path Metrics, does not it?

Corp. ResearchPrinceton, NJMotivation

Cut Metrics are “trickier” than Path Metrics. Why care about Cut Metrics?

Relevant for a large number of cut-based methods currently used (in vision). Inappropriate cut metric results in significant geometric artifacts.

The domain of cut-based methods is significantly more interesting than that of path-based techniques. (E.g., optimizations of hyper-surfaces on N-D grids.)

New theoretically interesting connections between graph theory and several branches of geometry.

New applications for graph based methods.

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Part II:

Integral Geometry and Graph Cuts (Euclidean case)

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Integral Geometry andCauchy-Crofton formula

CL

Any line L is determined by two parameters

2

0

space of all lines

...... dddL

Lebesgue measure

||||2 CdLnL

L Euclidean length

of contour C

a number of timesline L intersects C

A measure of all lines that cross C ?

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Example of an application for Cauchy-Crofton formula

dLnC L 21||||

nC42

||||

L

Ln 21 4

4 families of parallel lines { , , , }

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||2

2

k

kk e

w

2

1

3

4

Cut Metric approximatingEuclidean Metric

Edge weights are positive!

|| 1

2

1 e

|| 2

2

2 e

k

kkkn 21dLnC L 2

1||||

k k

kk e

n||2

2

1e

2e3e4e

arbitrary fixed homogeneous neighborhood system

C

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Part III:

Differential Geometry and Graph Cuts

Corp. ResearchPrinceton, NJNon-Euclidean Metric

constaAaa TA ||||

a

uAua

ag TA

||

||||)(

Consider normalized lengthof a vector with angle

under metric A

)()(

g

constr

Corp. ResearchPrinceton, NJ

dwg |)sin(|~)(0

Cut Metric approximatingNon-Euclidean Metric

a k

kkgc eawa ||1

||||2

d

ew

a

a gc

|)sin(|||

||

||||

02

kk we ,

positive edge weights!

Substitute and consider infinitesimally small wwk

2

)(")(~

ggw

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“Distance Maps” for Cut Metricsin Non-Euclidean case

Consider all graph nodes equidistant (for a given cut metric) from a given node.

4 neighborhoodsystem

8 neighborhoodsystem

256 neighborhoodsystem

2

)(")(

||

2

gg

ew

Corp. ResearchPrinceton, NJGeneral Riemannian Metric on R

n

C

Metric varies continuously over points in Rnx)(xg

C

dsgC )(|||| xx

x

xxg

yyg

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Cauchy-Crofton formulain case of Riemannian metric on R

dLnC L 21||||

Euclidean Case)( 21

CLx

General Riemannian CasedLgg

CCL

)2

)(")((||||

x

xx

CL x

C

n

L

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||||||||0,0

CC gc

Cut Metric approximatingRiemannian Space

2

)(")(

||

2

sss gg

ew

Theorem: if

then 0|| e

e

s

sw

C

Corp. ResearchPrinceton, NJ“Geo-Cuts” algorithm

||ˆ||||ˆ|| CC gc

Build a graph with a Cut Metric

approximating givenRiemannian metric

Besides length, certain additional contour properties can be added to the energy!

Minimum s-t cut generates Geodesic (minimum length) contour C for a given Cut Metric under fixed boundary conditions

C

Corp. ResearchPrinceton, NJGeo-Cuts vs. Level-Sets

Level-Sets generate a local minimum geodesic contour (minimal surface) but can be applied to almost any contour energy

Geo-Cuts find a global minimum but can be applied to a restricted class of contour energies

Gradient descent method VS. Global minimization method

Corp. ResearchPrinceton, NJConclusions

Introduced a notion of “Cut Metrics” on graphs• compared with previously known “path metrics”

Established connections between geometry of graph cuts and concepts of integral and differential geometry • Graph cuts work as a partial sum for an integral in Cauchy-Crofton formula

for contour length and surface area • Any non-Euclidean metric space can be approximated by graphs with

appropriate topology

Proposed “Geo-Cuts” algorithm for globally optimal geodesic contours (in 2D) and minimal surfaces (in 3D)• alternative to Level-Sets approach