Download presentation

Presentation is loading. Please wait.

Published byQuinn Willie Modified over 3 years ago

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

2
2

3
3

4
4

5
5

6
6

7
7

8
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

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

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

11
11

12
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.

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

14
14

15
15 Encode Pairwise Relationships as a Weighted Graph

16
16 Cut the graph into two pieces

17
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]

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

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

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

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

22
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

23
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!

24
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)

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

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

27
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 )

28
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

29
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;

30
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

31
31 Videos

32
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

33
33 Can you hear the shape of a drum?

34
34 Can you hear the shape of a drum?

Similar presentations

OK

1 Stella X. Yu 1,2 Jianbo Shi 1 Robotics Institute 1 Carnegie Mellon University Center for the Neural Basis of Cognition 2 Understanding Popout through.

1 Stella X. Yu 1,2 Jianbo Shi 1 Robotics Institute 1 Carnegie Mellon University Center for the Neural Basis of Cognition 2 Understanding Popout through.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on water softening techniques of neutralization Ppt on pre ignition problems Jit ppt on manufacturing engineering Ppt on bank concurrent audit Ppt on obesity management association Ppt on nature and human soul Ppt on jpeg image compression Ppt on bluetooth applications for pc Ppt on limits and derivatives definition Ppt on polarisation of light