CS 556: Computer Vision. Lecture 13
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1 CS 556: Computer Vision Lecture 1 Prof. Sinisa Todorovic sinisa@eecs.oregonstate.edu 1
2 Outline Perceptual grouping Low-level segmentation Ncuts
3 Perceptual Grouping
4 What do you see? 4
5 What do you see? Rorschach test 5
6 Gestalt Laws Arise from constraints of the real world Similarity Proximity Closure Good continuation Common fate Figure-ground Symmetry Periodicity 6
7 Gestalt Laws -- Similarity We group image parts with similar properties (color, shape, texture) Because same matter in the real world often yields same image properties 7
8 Gestalt Laws -- Proximity We tend to group nearby image parts Because matter is cohesive resulting in meaningful configurations of nearby objects 8
9 Gestalt Laws -- Closure We tend to ignore gaps and hallucinate complete, closed contours Because objects have closed surfaces, and so their image projections should have closed boundaries 9
10 Gestalt Laws -- Good Continuation We prefer to see configurations forming smooth contours Because objects have locally smooth surfaces 10
11 Gestalt Laws -- Common Fate We tend to see distinct image parts with same motion as a unit Because parts of a moving object move coherently in the same manner 11
12 Gestalt Laws -- Figure-Ground We see certain image areas as foreground or figure and the remaining areas as background or ground Because of a target oriented nature of human vision 1
13 Gestalt Laws -- Symmetry We tend to see symmetric image parts as figure Because many objects are symmetric due to functionality/growth/reproduction processes 1
14 Gestalt Laws -- Periodicity We tend to see objects in image parts that spatially repeat Because many objects are spatially periodic due to functionality/growth/reproduction processes 14
15 Issues How to formalize Gestalt laws? What Gestalt law should we apply first? Different orderings Different groupings 15
16 Top-down Interpretation from Context Context and feedback from higher levels may resolve low-level ambiguities Because of the context we may not be even aware of alternatives 16
17 Top-down Interpretation from Context Context and feedback from higher levels may resolve low-level ambiguities Because of the context we may not be even aware of alternatives 16
18 Top-down Interpretation from Context Context and feedback from higher levels may resolve low-level ambiguities Because of the context we may not be even aware of alternatives 16
19 Top-down Interpretation from Context Context and feedback from higher levels may resolve low-level ambiguities Because of the context we may not be even aware of alternatives 16
20 Top-down Interpretation from Context Context and feedback from higher levels may resolve low-level ambiguities Because of the context we may not be even aware of alternatives 16
21 Clustering as a Formalism of Perceptual Grouping Group image elements that are close/neighbors/similar Broad theory is absent at present Motivation: reducing the amount of information -- summarization more compact representation of data -- efficiency Should support application CLOSE/NEIGHBORS/SIMILAR... imply a model 17
22 Clustering as a Computational Framework of Perceptual Grouping 18
23 Example: Pixel Clustering segments or regions results of the Ncuts algorithm 19
24 Graph-based Clustering 0
25 Graph-based Clustering Image elements are represented by a graph 0
26 Graph-based Clustering Image elements are represented by a graph Clustering = Graph partitioning into subgraphs 0
27 Major Goal: Graph Partitioning
28 Major Goal: Graph Partitioning
29 Example: Pixel Clustering segments or regions results of the Ncuts algorithm
30 Graph Terminology Adjacency matrix Node degree Volume Cut Association
31 Adjacency Matrix j A = i a ij a ij =exp 1 σ SIFT i SIFT j 4
32 Adjacency Matrix j A = i a ij Affinities between pairs of nodes (i,j) in the graph a ij =exp 1 σ SIFT i SIFT j 4
33 Adjacency Matrix j A = i a ij Affinities between pairs of nodes (i,j) in the graph Example: Nodes = Pixels a ij =exp 1 σ SIFT i SIFT j 4
34 Node Degree d i = d i = j a ij 5
35 Node Degree D = 0.1 d d d N 1 1 i = i 0 d i = 1 T i D 1 i 6
36 Volume of a Subgraph G 1 G vol(g 1 )= i G 1 d i 7
37 Volume of a Subgraph D = d d d N 1 G1 = G 1 0 vol(g 1 )=1 T G 1 D 1 G1 8
38 Association within a Subgraph G 1 G assoc(g 1,G 1 )= i,j G 1 a ij 9
39 Association within a Subgraph j adjacency matrix A = i a ij assoc(g 1,G 1 )=1 T G 1 A 1 G1 0
40 Cut between Two Graph Partitions G G G 1 G G 1 G = cut(g 1,G )= i G 1,j G a ij 1
41 Cut between Two Graph Partitions G G G 1 G G 1 G = cut(g,g 1 )= i G,j G 1 a ij
42 Cut between Two Graph Partitions G G G 1 G G 1 G = cut(g 1,G ) = cut(g,g 1 ) in general
43 Cut between Two Graph Partitions cut(g 1,G ) = vol(g 1,G ) assoc(g 1,G ) = 1 T G 1 D 1 G1 1 T G 1 A 1 G1 = 1 T G 1 (D A) 1 G1 4
44 Cut between Two Graph Partitions cut(g,g 1 )=1 T G (D A) 1 G =(1 1 G1 ) T (D A) (1 1 G1 ) 5
45 Normalized-Cut Clustering Find two partitions G1 and G of graph G So that the following criterion is minimized: min Ncut(G 1,G )= cut(g 1,G ) G 1,G vol(g 1 ) + cut(g,g 1 ) vol(g ) 6
46 Normalized-Cut Objective Function cut(g 1,G ) min G 1,G vol(g 1 ) + cut(g,g 1 ) vol(g ) 7
47 Normalized-Cut Objective Function cut(g 1,G ) min G 1,G vol(g 1 ) + cut(g,g 1 ) vol(g ) 1 T (D A)1 G min 1 G1 G 1,G 1 T + 1T (D A)1 G G D1 1 G T D1 1 G1 G G 7
48 Normalized-Cut Objective Function cut(g 1,G ) min G 1,G vol(g 1 ) + cut(g,g 1 ) vol(g ) 1 T (D A)1 G min 1 G1 G 1,G 1 T + 1T (D A)1 G G D1 1 G T D1 1 G1 G G 1 T A 1 G max 1 G1 G 1,G 1 T + 1T A 1 G G D 1 1 G T D 1 1 G1 G G 7
49 Normalized-Cut Objective Function 1 T A 1 G max 1 G1 G 1,G 1 T + 1T A 1 G G D 1 1 G T D 1 1 G1 G G 1 T D G 1 A 1 max 1 G1 G 1,G 1 T + 1T D G 1 A 1 G 1 1 G T 1 1 G1 G G 8
50 Normalized-Cut Objective Function 1 T A 1 G max 1 G1 G 1,G 1 T + 1T A 1 G G D 1 1 G T D 1 1 G1 G G normalized Laplacian 1 T D G 1 A 1 max 1 G1 G 1,G 1 T + 1T D G 1 A 1 G 1 1 G T 1 1 G1 G G 8
51 Normalized-Cut Objective Function 1 T A 1 G max 1 G1 G 1,G 1 T + 1T A 1 G G D 1 1 G T D 1 1 G1 G G 1 T D G 1 A 1 max 1 G1 G 1,G 1 T + 1T D G 1 A 1 G 1 1 G T 1 1 G1 G G normalized Laplacian 1 T L 1 G max 1 G1 G 1,G 1 T + 1T L 1 G G 1 1 G T 1 1 G1 G G 8
52 Normalized-Cut Linearization 1 T L 1 G max 1 G1 G 1,G 1 T + 1T L 1 G G 1 1 G T 1 1 G1 G G X =[ ] T indicator vector for nodes that belong to a subgraph 9
53 Normalized-Cut Linearization 1 T L 1 G max 1 G1 G 1,G 1 T + 1T L 1 G G 1 1 G T 1 1 G1 G G X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 {0, 1} n X G {0, 1} n 40
54 Normalized-Cut Linearization 1 T L 1 G max 1 G1 G 1,G 1 T + 1T L 1 G G 1 1 G T 1 1 G1 G G X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 {0, 1} n X G {0, 1} n X T G 1 X G =0 orthogonality 40
55 Normalized-Cut Relaxation X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 [0, 1] n X G [0, 1] n X T G 1 X G =0 41
56 How to Solve Normalized-Cut? X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 [0, 1] n X G [0, 1] n X T G 1 X G =0 4
57 How to Solve Normalized-Cut? X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 [0, 1] n X G [0, 1] n X T G 1 X G =0 4
58 How to Solve Normalized-Cut? X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 [0, 1] n X G [0, 1] n X = XG T X =0 G1 eigenvector of the 1 G largest eigenvalue of L 4
59 How to Solve Normalized-Cut? X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 [0, 1] n X G [0, 1] n X T G 1 X G =0 X = XG T X =0 G1 eigenvector of the 1 G largest eigenvalue of L X G =? 44
60 How to Solve Normalized-Cut? X T LX G max 1 G1 X G1,X G XG T + XT LX G G X 1 G1 X T G X G subject to: X G1 [0, 1] n X G [0, 1] n X T G 1 X G =0 X = XG T X =0 G1 eigenvector of the 1 G largest eigenvalue of L X G = eigenvector of the second largest eigenvalue of L 45
61 Ncuts: General Case cut min G 1,G,...,G K K k=1 1 T G k (D A) 1 Gk 1 T G k D 1 Gk volume 46
62 Ncuts: General Case cut min G 1,G,...,G K K k=1 1 T G k (D A) 1 Gk 1 T G k D 1 Gk volume relaxation s.t. max X 1,...,X K K k=1 X T k D 1 AX k X T k X k X k [0, 1] n k, l, X T k X l =0 46
63 Ncuts: General Case Solution: K eigenvectors of D -1 A corresponding to the K largest eigenvalues 47
64 Experiments -- Ncuts as Image Segmentation input image segments Shi&Malik
65 Experiments -- Ncuts as Image Segmentation input image segments Shi&Malik
66 Experiments -- Ncuts as Image Segmentation source: Jianbo Shi 50
67 Experiments -- Ncuts as Image Segmentation source: Jianbo Shi 51
68 Summary Gestalt laws of perceptual grouping: Similarity Proximity Closure Good continuation Common fate Figure-ground Symmetry Periodicity 5
69 Summary Ncuts: Find K partitions of graph G So as to minimize: min G 1,...,G K k cut(g k,g G k ) vol(g k ) 5
70 Summary node degree matrix D = adjacency d d 0. i d N matrix A = a ij Ncuts: K eigenvectors of D -1 A corresponding to the K largest eigenvalues 54
71 Next Class Meanshift 55
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