comp 775: graph cuts and continuous maximal flows

Comp 775: Graph Cuts and Continuous Maximal Flows

Marc Niethammer, Stephen PizerDepartment of Computer Science

University of North Carolina, Chapel Hill

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Representations

BackgroundGraph Cuts and Continuous Maximal Flows

Classifying individual pixels versus finding an optimal separatingcurve/surface between object and background.

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From curve evolution to pixel/voxel labeling


Example: Chan-Vese model Indicator function

Putting it all together: Relaxed indicator function

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Convex vs. Non-Convex


Non-convex

Initial contour

Initial contour Final contour

Final contour

Images: Bresson et al.

Convex

Convex

Prone to locallyoptimal solutions.

Independent ofinitial condition,optimal solution isguaranteed.

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From Curve Evolution to Pixel/Voxel Labeling


Convex, continuous, constrained optimization problems.

... can also include region-based terms, appearance information, orientation-dependency

Advantages:- Convex -> Globally optimal- No metrication artifacts- Straightforward parallel

implementations

convex

non-convex

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Continuous Maximal Flow [Appleton]

Continuous Maximal FlowGraph Cuts and Continuous Maximal Flows

Continuous version of maximal flow [Appleton].Results in a PDE and can be solved as such.

Energy to be minimized:

More general form (can easily include local classifiers):

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Segmentation with Continuous Max-Flow


Images: Unger

Thresholding

Continuous max-flowwith seeds.

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Continuous Maximal Flow


Energy to be minimized:

Introduce the auxiliary variable p, which we maximize for:

Now solve this by gradient descent.

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Continuous Maximal Flow


The variation is

The gradient descent scheme becomes

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Chan-Vese (=Otsu-Thresholding w/ spatial regularity)

Continuous Max-FlowGraph Cuts and Continuous Maximal Flows

Iterative solution method is related to solving a wave equation.

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Ex.: Segmentation with Continuous Max-Flow


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3D Example


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Segmentation by Graph-Cut

Graph-CutGraph Cuts and Continuous Maximal Flows

Example: Binary Segmentation

Images from ECCV Tutorial, Kumar/Kohli

Need to partition the picture into foreground and background.

Alternative approach through graph construction (vs. PDE)

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Graph G=(V,E)

Images from ECCV Tutorial, Kumar/Kohli

Approach: Interpret the image as a graph,where pixels are connected to its neighbors.

Goal: Cut the graph into pieces to obtain the desired image partition. Assign a label to every pixel.

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Optimization Problem: Minimize

Looking at it on a pixel-by-pixel basis (where f is the labeling):

Problem: We cannot try all possible pairings for the labels f. Need an efficient algorithm to solve this problem.

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Maximum Flow and Minimum Cut


Solution: 1) Transform problem to maximizing nework flow 2) Use algorithms for network flows on images.

Preview: Graph structure for binary labeling

Pixels are connected to neighbors (pairwise interaction cost) and to source and target vertices (data cost).

Images: Boykov

Cut separating source (s) from sink (t) gives the segmentation.

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Max-Flow/Min-Cut Theorem


For any network having a single origin and a single destination node, the maximum possible flow from origin to destination equals the minimum cut value for all the cuts in the network.

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Example: Ford & Fulkerson

Segmentation by Graph CutGraph Cuts and Continuous Maximal Flows

Pixel weights: s to node flow: high if near source (bkg) class, low if notPixel weights: Node to t flow: high if near target class, low if notNode to neighbor flow: high in both directions if near intensities

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One way to compute maximal flow: 1) Pick any viable path (no zero flow)2) Subtract minimum flow from each segment on path3) Add minimum flow on reverse path

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1) Pick path (no zero flow)

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2) Subtract minimum flow from each segment on path3) Add minimum flow on reverse path

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- Cut at zeros. - Cost of min-cut is 4. - Divides the nodes (pixels) into

two groups.- Sum of original values of zeros equals the maximum flow.

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Look at the difference between initial and final capacities.Ignore negative capacities.This is where everything flows. Flow is conserved at nodes.

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Why update the backward flow?


Image: John Chinneck

Can undo flows. Final flow does not include the central edge.

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Image Segmentation


Allows (amongst many things) to compute binary segmentations.

Image: Boykov

Original Noisy Reconstructed

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Multi-Label Case


Image: Boykov

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Multi-Label Case


Image: Boykov

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Influence of Neighborhood Choice


Minimum cost cut (standard 4-neighborhoods)

Minimum length geodesic contour (image-based Riemannian metric)

Images: Boykov

Can choose different weighted neighborhood to reduce metrication errors. (Or use continuous maximal flow.)

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Interactive Segmentations


Images: Boykov