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1、Normalized Cuts and Image Segmentation,Jianbo Shi and Jitendra Malik, Presented by: Alireza Tavakkoli,Image Segmentation,Image segmentation,How do you pick the right segmentation?,Bottom up segmentation: - Tokens belong together because they are locally coherent. Top down segmentation: - Tokens grou
2、ped because they lie on the same object.,“Correct” segmentation,There may not be a single correct answer. Partitioning is inherently hierarchical. One approach we will use in this presentation: “Use the low-level coherence of brightness, color, texture or motion attributes to come up with partitions
3、”,Outline,Introduction Graph terminology and representation. “Min cuts” and “Normalized cuts”. Other segmentation methods using eigenvectors. Conclusions.,Outline,Introduction Graph terminology and representation. “Min cuts” and “Normalized cuts”. Other segmentation methods using eigenvectors. Concl
4、usions.,Graph-based Image Segmentation,Image (I),Graph Affinities (W),Intensity Color Edges Texture,Slide from Timothee Cour (http:/www.seas.upenn.edu/timothee),Graph-based Image Segmentation,Image (I),Slide from Timothee Cour (http:/www.seas.upenn.edu/timothee),Graph Affinities (W),Intensity Color
5、Edges Texture,Graph-based Image Segmentation,Image (I),Eigenvector X(W),Slide from Timothee Cour (http:/www.seas.upenn.edu/timothee),Graph Affinities (W),Intensity Color Edges Texture,Graph-based Image Segmentation,Image (I),Eigenvector X(W),Discretization,Slide from Timothee Cour (http:/www.seas.up
6、enn.edu/timothee),Graph Affinities (W),Intensity Color Edges Texture,Outline,Introduction Graph terminology and representation. “Min cuts” and “Normalized cuts”. Other segmentation methods using eigenvectors. Conclusions.,Graph-based Image Segmentation,V: graph nodes E: edges connection nodes,G = V,
7、E,Pixels Pixel similarity,Slides from Jianbo Shi,Graph terminology,Similarity matrix:,Slides from Jianbo Shi,Affinity matrix,Similarity of image pixels to selected pixel Brighter means more similar,Reshape,N*M pixels,N*M pixels,M pixels,N pixels,Warning the size of W is quadratic with the number of
8、parameters!,Graph terminology,Degree of node:,Slides from Jianbo Shi,Graph terminology,Volume of set:,Slides from Jianbo Shi,Graph terminology,Slides from Jianbo Shi,Cuts in a graph:,Representation,Partition matrix X: Pair-wise similarity matrix W: Degree matrix D: Laplacian matrix L:,Pixel similari
9、ty functions,Intensity,Texture,Distance,Pixel similarity functions,Intensity,Texture,Distance,here c(x) is a vector of filter outputs. A natural thing to do is to square the outputs of a range of different filters at different scales and orientations, smooth the result, and rack these into a vector.
10、,Definitions,Methods that use the spectrum of the affinity matrix to cluster are known as spectral clustering. Normalized cuts, Average cuts, Average association make use of the eigenvectors of the affinity matrix. Why these methods work?,Spectral Clustering,Data,Similarities,* Slides from Dan Klein
11、, Sep Kamvar, Chris Manning, Natural Language Group Stanford University,Eigenvectors and blocks,Block matrices have block eigenvectors: Near-block matrices have near-block eigenvectors:,eigensolver,1= 2,2= 2,3= 0,4= 0,eigensolver,1= 2.02,2= 2.02,3= -0.02,4= -0.02,* Slides from Dan Klein, Sep Kamvar,
12、 Chris Manning, Natural Language Group Stanford University,Spectral Space,Can put items into blocks by eigenvectors: Clusters clear regardless of row ordering:,e1,e2,e1,e2,e1,e2,e1,e2,* Slides from Dan Klein, Sep Kamvar, Chris Manning, Natural Language Group Stanford University,Outline,Introduction
13、Graph terminology and representation. “Min cuts” and “Normalized cuts”. Other segmentation methods using eigenvectors. Conclusions.,How do we extract a good cluster?,Simplest idea: we want a vector x giving the association between each element and a cluster We want elements within this cluster to, o
14、n the whole, have strong affinity with one another We could maximize But need the constraint This is an eigenvalue problem - choose the eigenvector of W with largest eigenvalue.,Criterion for partition:,Minimum cut,First proposed by Wu and Leahy,A,B,Problem! Weight of cut is directly proportional to
15、 the number of edges in the cut.,Normalized Cut,Normalized cut or balanced cut:,Finds better cut,Normalized Cut,Volume of set (or association):,A,B,Normalized Cut,Volume of set (or association): Define normalized cut: “a fraction of the total edge connections to all the nodes in the graph”:,A,B,A,B,
16、Define normalized association: “how tightly on average nodes within the cluster are connected to each other”,A,B,Subject to:,Observations(I),Maximizing Nassoc is the same as minimizing Ncut, since they are related: How to minimize Ncut? Transform Ncut equation to a matricial form. After simplifying:
17、,Rayleigh quotient,NP-Hard! ys values are quantized,Instead, relax into the continuous domain by solving generalized eigenvalue system: Which gives: Note that so, the first eigenvector is y0=1 with eigenvalue 0. The second smallest eigenvector is the real valued solution to this problem!,Observation
18、s(II),min,Algorithm,Define a similarity function between 2 nodes. i.e.: Compute affinity matrix (W) and degree matrix (D). Solve Use the eigenvector with the second smallest eigenvalue to bipartition the graph. Decide if re-partition current partitions. Note: since precision requirements are low, W
19、is very sparse and only few eigenvectors are required, the eigenvectors can be extracted very fast using Lanczos algorithm.,Discretization,Sometimes there is not a clear threshold to binarize since eigenvectors take on continuous values. How to choose the splitting point? Pick a constant value (0, o
20、r 0.5). Pick the median value as splitting point. Look for the splitting point that has the minimum Ncut value: Choose n possible splitting points. Compute Ncut value. Pick minimum.,Use k-eigenvectors,Recursive 2-way Ncut is slow. We can use more eigenvectors to re-partition the graph, however: Not
21、all eigenvectors are useful for partition (degree of smoothness). Procedure: compute k-means with a high k. Then follow one of these procedures: Merge segments that minimize k-way Ncut criterion. Use the k segments and find the partitions there using exhaustive search. Compute Q (next slides).,e1,e2
22、,e1,e2,Example,Eigenvectors,Segments,Experiments,Define similarity: for point sets. for brightness images. for HSV images. in case of texture.,Experiments (I),Point set segmentation: (a) Pointset generated by Poisson process. (b) Segmentation results.,Experiments (II),Synthetic images:,Experiments (
23、III),Weather radar:,Experiments (IV),Motion segmentation,Outline,Introduction Graph terminology and representation. “Min cuts” and “Normalized cuts”. Other segmentation methods using eigenvectors. Conclusions.,Other methods,Average association Use the eigenvector of W associated to the biggest eigen
24、value for partitioning. Tries to maximize: Has a bias to find tight clusters. Useful for Gaussian distributions.,A,B,Other methods,Average cut Tries to minimize: Very similar to normalized cuts. We cannot ensure that partitions will have a a tight within-group similarity since this equation does not
25、 have the nice properties of the equation of normalized cuts.,Other methods,Other methods,20 points are randomly distributed from 0.0 to 0.5 12 points are randomly distributed from 0.65 to 1.0,Normalized cut,Average cut,Average association,Other methods,20 points are randomly distributed from 0.0 to
26、 0.5 12 points are randomly distributed from 0.65 to 1.0,Normalized cut,Average cut,Average association,Other methods,20 points are randomly distributed from 0.0 to 0.5 12 points are randomly distributed from 0.65 to 1.0,Normalized cut,Average cut,Average association,Outline,Introduction Graph termi
27、nology and representation. “Min cuts” and “Normalized cuts”. Other segmentation methods using eigenvectors. Conclusions.,Conclusions,Good news: Simple and powerful methods to segment images. Flexible and easy to apply to other clustering problems. Bad news: High memory requirements (use sparse matri
28、ces). Very dependant on the scale factor for a specific problem.,Thank you!,The End!,Examples,Spectral Clutering,Images from Matthew Brand (TR-2002-42),Spectral clustering,Makes use of the spectrum of the similarity matrix of the data to cluster the points.,Solve clustering for affinity matrix,w(i,j) distance node i to node j,Graph terminology,Similarity matrix:,Degree of node:,Volume of set:,Graph cuts:,