Corp. Research Princeton, NJ Cut Metrics and Geometry of Grid Graphs Yuri Boykov, Siemens Research, Princeton, NJ joint work with Vladimir Kolmogorov, Cornell University, Ithaca, NY

Corp. Research Princeton, NJ Outline 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. Research Princeton, NJ Part I: “Cut Metrics” vs. “Path Metrics” on Graphs

Corp. Research Princeton, NJ n Path metrics are relevant for graph applications based on Dijkstra style optimization. (e.g. Intelligent Scissors method in vision) n “Length” is naturally defined for any “path” connecting two nodes along graph edges. Standard “Path Metrics” on graphs n The properties of path metrics are relatively straightforward and were studied in the past

Corp. Research Princeton, 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 neighborhood system 8 neighborhood system 256 neighborhood system

Corp. Research Princeton, NJ Cut Metrics on graphs n Cut metrics are relevant for graph applications based on Min-Cut style optimization. (e.g. Interactive Graph Cuts and Normalized Cuts in vision) n “Length” is naturally defined for any cut (closed contour or surface) that separates graph nodes. C

Corp. Research Princeton, NJ Cut Metrics vs. Path Metrics n Both cut and path metrics are determined by the graph topology (t.e. neighborhood system and edge weights) n 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?) n Cuts naturally define surface “area” on 3D grids. Path metric is limited to curve “length” and can not define “area” in 3D. n Cut-based notion of “length” (“area”) can be extended to open curves (surfaces) on the imbedding space (or ). C = cost of edges that cross C odd number of times

Corp. Research Princeton, NJ Cut metric “distance” for graphs with homogeneous topology Consider all edges on a grid a a k-th edge cost arbitrary fixed homogeneous neighborhood system

Corp. Research Princeton, 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. 4 neighborhood system 8 neighborhood system 256 neighborhood system Looks just like Path Metrics, does not it?

Corp. Research Princeton, NJ Motivation n Cut Metrics are “trickier” than Path Metrics. n Why care about Cut Metrics? n Relevant for a large number of cut-based methods currently used (in vision). Inappropriate cut metric results in significant geometric artifacts. n 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.) n New theoretically interesting connections between graph theory and several branches of geometry. n New applications for graph based methods.

Corp. Research Princeton, NJ Part II: Integral Geometry and Graph Cuts (Euclidean case)

Corp. Research Princeton, NJ Integral Geometry and Cauchy-Crofton formula C L Any line L is determined by two parameters space of all lines Lebesgue measure Euclidean length of contour C a number of times line L intersects C A measure of all lines that cross C ?

Corp. Research Princeton, NJ Example of an application for Cauchy-Crofton formula 4 families of parallel lines {,,, }

Corp. Research Princeton, NJ Cut Metric approximating Euclidean Metric Edge weights are positive! arbitrary fixed homogeneous neighborhood system C

Corp. Research Princeton, NJ Part III: Differential Geometry and Graph Cuts

Corp. Research Princeton, NJ Non-Euclidean Metric a Consider normalized length of a vector with angle under metric A

Corp. Research Princeton, NJ Cut Metric approximating Non-Euclidean Metric a positive edge weights! Substitute and consider infinitesimally small

Corp. Research Princeton, NJ “Distance Maps” for Cut Metrics in Non-Euclidean case Consider all graph nodes equidistant (for a given cut metric) from a given node. 4 neighborhood system 8 neighborhood system 256 neighborhood system

Corp. Research Princeton, NJ General Riemannian Metric on R n C Metric varies continuously over points in R n

Corp. Research Princeton, NJ Cauchy-Crofton formula in case of Riemannian metric on R Euclidean Case General Riemannian Case C n L

Corp. Research Princeton, NJ Cut Metric approximating Riemannian Space Theorem: if then C

Corp. Research Princeton, NJ “Geo-Cuts” algorithm Build a graph with a Cut Metric approximating given Riemannian 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

Corp. Research Princeton, NJ Geo-Cuts vs. Level-Sets n Level-Sets generate a local minimum geodesic contour (minimal surface) but can be applied to almost any contour energy n 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. Research Princeton, NJ Conclusions n Introduced a notion of “Cut Metrics” on graphs compared with previously known “path metrics” n 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 n Proposed “Geo-Cuts” algorithm for globally optimal geodesic contours (in 2D) and minimal surfaces (in 3D) alternative to Level-Sets approach