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1 Perceptual Organization and Linear Algebra Charless Fowlkes Computer Science Dept. University of California at Berkeley

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8 Perceptual Organization Gestalt school of psychology (1920’s) –Whole is greater than the sum of its parts Grouping Principles –Proximity, Similarity Figure/Ground Organization –Convexity, Symmetry, Familiarity

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9 Do these visual phenomenon matter for real images? Why do they exist? Can we build a computational model of this process?

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10 Q: Do these visual phenomenon matter for real images?

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12 Q: Why does the brain do this? A: Perceptual organization is a useful trick for making sense of our world. Study the statistics of natural images. If similar pixels are statistically more likely to belong to the same object/surface then this all makes sense.

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13 Q: Can we build a mathematical model of this process? A: Let’s roll up our sleeves and give it a try!

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15 Encode Pairwise Relationships as a Weighted Graph

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16 Cut the graph into two pieces

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17 Algebraic representation of a weighted undirected graph W(i,j) = similarity of vertex i and vertex j W(i,j) = W(j,i) 2 [0,1]

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18 Degree of a node: d i = j W i,j Degree matrix: D ii = j W i,j

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19 Volume of a set vol(A) = i 2 A d i

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20 Weight of a cut: cut(A,B) = i 2 A, j 2 B W i,j

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21 x x Indicator vector: x i = 1 if vertex i 2 A 0 otherwise

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22 vol(A) = i 2 A d i cut(A,A c ) = i 2 A, j 2 A c W i,j vol(A) = x T D x cut(A,A c ) = x T (D-W) x D-W is called the Graph Laplacian x x

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23 Finding a cut with small weight Minimize x T (D-W) x subject to x 2 {0,1} N Minimize x T (D-W) x subject to ||x|| 2 =1 This is an eigenvalue problem!

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24 Lets solve it. download makeW.m and showGroups.m >> [W,coords] = makeSimilarity(30,70,1); >> D = diag(sum(W,2)); >> L = D-W; >> [U,S] = eig(L); >> indicator = U(:,2) > 0; >> showGroups(coords,indicator)

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25 A problem with minimum cuts… >> [W,coords] = makeSimilarity(30,70,0.5,true);

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26 Solution: Normalized Cut Simultaneously minimize similarity between groups And maximize similarity within groups Corresponding generalized eigenvalue problem: (D-W)x = Dx

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27 Comparing Generalized Eigenvectors. >> [W,coords] = makeSimilarity(30,70,0.5,true); >> D = diag(sum(W,2)); >> L = D-W; >> [U1,S] = eig(L); >> [U2,S] = eig(L,D); >> showGroups(coords, U1(:,2) > 0 ) >> showGroups(coords, U2(:,2) > 0 )

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28 Working with real images Each pixel is a vertex of the graph Similarities between pixels correspond to weights For example, difference in grayscale value: W(i,j) = exp(-(I(i) – I(j)) 2 / 2 ) - Partitioning the graph gives a segmentation of the image

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29 Working with real images demo2.m: W = makeW2(0.01,0.3); D = diag(sum(W,1)); L = D - W; [U,S] = eig(L,D); im2 = reshape(U(:,2),size(im,1),size(im,2)); figure(2); subplot(2,1,1); imagesc(im2); axis image; subplot(2,1,2); imagesc(im2>0); axis image;

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30 Generalized eigenvalue problem has interpretation as a mass-spring system(D-W)x = Dx K is the stiffness matrix, M is the mass matrix

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32 More uses of the Laplacian For un-weighted graphs, the combinatorial Laplacian, D-A, captures connectivity of the graph –det(D-A) counts # of spanning trees –rank(D-A) = N - #components = (0 th Betti number) Spectrum of Laplacian operator on a manifold gives information about the shape of the manifold

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33 Can you hear the shape of a drum?

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34 Can you hear the shape of a drum?

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