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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
Corp. ResearchPrinceton, NJ
Part I:
“Cut Metrics” vs. “Path Metrics” on Graphs
Corp. ResearchPrinceton, NJ
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
Corp. ResearchPrinceton, NJ
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.
Corp. ResearchPrinceton, NJ
Part II:
Integral Geometry and Graph Cuts (Euclidean case)
Corp. ResearchPrinceton, NJ
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 ?
Corp. ResearchPrinceton, NJ
Example of an application for Cauchy-Crofton formula
dLnC L 21||||
nC42
||||
L
Ln 21 4
4 families of parallel lines { , , , }
Corp. ResearchPrinceton, NJ
||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
Corp. ResearchPrinceton, NJ
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
Corp. ResearchPrinceton, NJ
“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
Corp. ResearchPrinceton, NJ
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
Corp. ResearchPrinceton, NJ
||||||||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