Stereo & Iterative Graph-Cuts Alex Rav-Acha Vision Course Hebrew University

The stereo problem

Both images are very similar (like images that you see with your two eyes) Most of the pixels in the left image are present in the right image (except for few occlusions)

After rectification: all correspondences are along the same horizontal scan lines The stereo problem (pixels in one image simply shift horizontally in the other image)

The stereo problem The horizontal shifts between the images are sometimes called: “disparities” The Disparities relates to depth: Closer objects have larger disparities.

The stereo problem: compute the disparity map between two images

Traditional Approaches Matching rigid windows around each pixel Each window is matched independently Modern approaches Finding coherent correspondences for all pixels - “Graph cuts” - “Belief Propagation”

Window-Based Approach Compute a cost for each location Location with the lowest cost wins

General Problem : Ambiguity LeftRight scanline

Window-Based Approach Small Window Large Window noisy in low texture areas blurred boundaries

Results with best window size (still not good enough) Window-based matching (best window size) Ground truth

Graph Cuts Ground truthGraph cuts

Maximum flow problem Max flow problem: –Each edge is a “pipe” –Find the largest flow F of “water” that can be sent from the “source” to the “sink” along the pipes –Edge weights give the pipe’s capacity “source” A graph with two terminals S T “sink” a flow F

Minimum cut problem Min cut problem: –Find the cheapest way to cut the edges so that the “source” is completely separated from the “sink” –Edge weights now represent cutting “costs” a cut C “source” A graph with two terminals S T “sink”

Max flow/Min cut theorem Max Flow = Min Cut: –Maximum flow saturates the edges along the minimum cut. –Ford and Fulkerson, 1962 –Problem reduction! Ford and Fulkerson gave first polynomial time algorithm for globally optimal solution “source” A graph with two terminals S T “sink”

Min-Cut: Important Rule No subset of the cut can also be a cut This is not a minimal cut

Energy Minimization Using Iterative Graph cuts Fast Approximate Energy Minimization via Graph Cuts Yuri Boykov, Olga Veksler and Ramin Zabih Pami 2001

To do better we need a better model of images We can make reasonable assumptions about the surfaces in the world Usually assume that the surfaces are smooth Can pose the problem of finding the corresponding points as an energy (or cost) minimization: how well the pixels match up for different disparities neighboring pixels have similar disparities f-assignment

To do better we need a better model of images We can make reasonable assumptions about the surfaces in the world Usually assume that the surfaces are smooth Can pose the problem of finding the corresponding points as an energy (or cost) minimization: Data term is calculated for each pixels Smoothness is calculated on neighbor pixels f-assignment p,q-pixels

Example for Smoothness terms Quadratic L1 Truncated L1 Potts model

Constructing a Graph to Solve the Stereo Problem

The labels of each pixel are the possible disparity values

Constructing a Graph to Solve the Stereo Problem

Relation between the Energy and the Graph labeling problem Data term Smoothness term p q 1 10 {f p =10} {f q =2}

Relation between the Energy and the Graph labeling problem Data term Smoothness term p q 1 10 D p (10) V (p,q) (1, 10)

Iterative graph-cuts Use an iterative scheme to find a “good” local optimum of the energy function. In each iteration: convert the original multi-label problem to a binary one, and solve it by finding a minimal graph-cut (max-flow). The most popular scheme is the expansion move. -expansion: set the label of each pixel to be either or the current label.

Types of Moves Problem: A lot of local minima A Single Pixel Move

Types of Moves Any pixel can change its label to alpha Expansion Move

Types of Moves Claim (without proof): The difference between the optimal solution and the solution from the iterative expansion moves is bounded Expansion Move

Energy Minimization Algorithm 1.Start with arbitrary labeling f 2.Set success = 0 3.For each label –Find –If set and success =1 4.If success =1 goto (2) 5.Return f

Conditions on the Smoothness for using expansion moves: In other words: V should be a metric Note : The Quadratic smoothness is not a metric

For each pair of vertices such that we add a ‘dummy’ vertex (together with the respective edges as shown in the table).

The Relation between the cut and the Energy Given a cut C, we define a labelling f c by: The cost of a cut C is |C| = E(f C ) (plus a constant) If the cut C separates p and

The Relation between the cut and the Energy

Conditions on the Smoothness for using expansion moves: In other words: V should be a metric