CAP5415-Computer Vision Lecture 13-Image/Video Segmentation Part II. Dr. Ulas Bagci
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1 CAP-Computer Vision Lecture -Image/Video Segmentation Part II Dr. Ulas Bagci
2 Labeling & Segmentation Labeling is a common way for modeling various computer vision problems (e.g. optical flow, image segmentation, stereo matching, etc)
3 Labeling & Segmentation Labeling is a common way for modeling various computer vision problems (e.g. optical flow, image segmentation, stereo matching, etc) The set of labels can be discrete (as in image segmentation) L = {l,...,l m } with L = m
4 Labeling & Segmentation Labeling is a common way for modeling various computer vision problems (e.g. optical flow, image segmentation, stereo matching, etc) The set of labels can be discrete (as in image segmentation) L = {l,...,l m } with L = m Or continuous (as in optical flow) L R n for n
5 Labeling is a function Labels are assigned to sites (pixel locations)
6 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have = N cols.n rows
7 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have Identifying a labeling function (with segmentation) is f :! L = N cols.n rows 7
8 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have Identifying a labeling function (with segmentation) is f :! L = N cols.n rows We aim at calculating a labeling function that minimizes a given (total) error or energy 8
9 Labeling is a function Labels are assigned to sites (pixel locations) For a given image, we have Identifying a labeling function (with segmentation) is f :! L = N cols.n rows We aim at calculating a labeling function that minimizes a given (total) error or energy E(f) = X p [E data (p, f p )+ X qa(p) E smooth (f p,f q )] *A is an adjacency relation between pixel locations 9
10 Energy Function The role of an energy function in minimizationbased vision is twofold:. as the quantitative measure of the global quality of the solution and. as a guide to the search for a minimal solution. Correct solution is embedded as the minimum. 0
11 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location.
12 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location. E smooth assigns non-negative penalties by comparing the assigned labels f p and f q at adjacent positions p and q.
13 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location. E smooth assigns non-negative penalties by comparing the assigned labels f p and f q at adjacent positions p and q.
14 Segmentation as an Energy Minimization Problem E data assigns non-negative penalties to a pixel location p when assigning a label to this location. E smooth assigns non-negative penalties by comparing the assigned labels f p and f q at adjacent positions p and q. This optimization model is characterized by local interactions along edges between adjacent pixels, and often called MRF (Markov Random Field) model.
15 E(f) = X p Energy Function-Details [E data (p, f p )+ X qa(p) E smooth (f p,f q )]
16 E(f) = X p Energy Function-Details Sample Data Term: [E data (p, f p )+ X qa(p) E smooth (f p,f q )] E data (p, f p )= (p) (p = 0) = logp(p BG) (p = ) = logp(p FG)
17 E(f) = X p Energy Function-Details Sample Data Term: [E data (p, f p )+ X qa(p) E smooth (f p,f q )] E data (p, f p )= (p) (p = 0) = logp(p BG) (p = ) = logp(p FG) Sample Smoothness Term: (p, q) =K pq (p = q) where K pq = exp( (I p I q ) /( )) p, q 7
18 Energy Function-Details E(p) = X p p(p)+ X p X pq(p, q) qa(p) p = argmin pl E(p) 8
19 Energy Function-Details E(p) = X p p(p)+ X p X pq(p, q) qa(p) p = argmin pl E(p) To solve this problem, transform the energy functional into min-cut/max-flow problem and solve it! 9
20 Energy Function Unary Cost (data term) Discontinuity Cost p q 0
21 Recap: Image as a Graph p q Edge Node/vertex pixel
22 Graph Cuts for Optimal Boundary Detection (Boykov ICCV 00) F Binary label: foreground vs. background User labels some pixels B Exploit F F B Statistics of known Fg & Bg F F B Smoothness of label F B B Turn into discrete graph optimization Graph cut (min cut / max flow)
23 Graph-Cut Each pixel is connected to its neighbors in an undirected graph Goal: split nodes into two sets A and B based on pixel values, and try to classify Neighbors in the same way too!
24 Graph-Cut Each pixel is connected to its neighbors in an undirected graph Goal: split nodes into two sets A and B based on pixel values, and try to classify Neighbors in the same way too!
25 Each pixel = node Add two nodes F & B Graph-Cut Labeling: link each pixel to either F or B F Desired result F F F F F B B B B B
26 Cost Function: Data term Put one edge between each pixel and both F & G Weight of edge = minus data term F B
27 Cost Function: Smoothness term Add an edge between each neighbor pair Weight = smoothness term F B 7
28 Min-Cut Energy optimization equivalent to graph min cut Cut: remove edges to disconnect F from B Minimum: minimize sum of cut edge weight F cut B 8
29 Min-Cut Source 9 v v Graph (V, E, C) Vertices V = {v, v... v n } Edges E = {(v, v )...} Costs C = {c (, )...} Sink 9
30 Min-Cut What is a st-cut? Source An st-cut (S,T) divides the nodes between source and sink. 9 v v What is the cost of a st-cut? Sum of cost of all edges going from S to T Sink = 0
31 Min-Cut What is a st-cut? Source An st-cut (S,T) divides the nodes between source and sink. v v 9 Sink + + = 7 What is the cost of a st-cut? Sum of cost of all edges going from S to T What is the st-mincut? st-cut with the minimum cost
32 How to compute min-cut? Source 9 v v Sink Solve the dual maximum flow problem Compute the maximum flow between Source and Sink Constraints Edges: Flow < Capacity Nodes: Flow in & Flow out Min-cut\Max-flow Theorem In every network, the maximum flow equals the cost of the st-mincut
33 Max-Flow Algorithms Source v v Flow = 0 9 Sink Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
34 Max-Flow Algorithms Source v v Flow = 0 9 Sink Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
35 Max-Flow Algorithms Source - 9 v v - Flow = 0 + Sink Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
36 Max-Flow Algorithms Source v v Flow = 0 9 Sink Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
37 Max-Flow Algorithms Source v v Flow = 0 9 Sink Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity 7
38 Max-Flow Algorithms Source v v Flow = 0 9 Sink Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity 8
39 Max-Flow Algorithms Source v v Flow = + 0 Sink 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity 9
40 Max-Flow Algorithms Source v v Flow = 0 Sink 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity 0
41 Max-Flow Algorithms Source v v Flow = 0 Sink 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
42 Max-Flow Algorithms Source v v Flow = Sink 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
43 Max-Flow Algorithms Source v v Flow = Sink 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
44 Max-Flow Algorithms Source v v Flow = Sink 0 Augmenting Path Based Algorithms. Find path from source to sink with positive capacity. Push maximum possible flow through this path. Repeat until no path can be found Algorithms assume non-negative capacity
45 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 8 Find the path in the residual graph sink End
46 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 8 Find the path in the residual graph sink End
47 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 8 Find the path in the residual graph sink End flow = 7
48 Another Example-Max Flow - source 9 Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) 8- Find the path in the residual graph sink End flow = 8
49 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow = 9
50 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow = 0
51 Another Example-Max Flow source 9 Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
52 Another Example-Max Flow source 9- Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
53 Another Example-Max Flow source Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
54 Another Example-Max Flow source Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
55 Another Example-Max Flow source Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
56 Another Example-Max Flow source - Ford & Fulkerson algorithm (9) Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) - Find the path in the residual graph sink End flow =
57 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow = 7
58 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow = 8
59 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow = 9
60 Another Example-Max Flow - source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) - Find the path in the residual graph sink End flow = 0
61 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
62 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
63 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
64 Another Example-Max Flow source Ford & Fulkerson algorithm (9) - 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) - Find the path in the residual graph sink End flow =
65 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
66 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow =
67 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Find the path from source to sink While (path exists) flow += maximum capacity in the path Build the residual graph ( subtract the flow) Find the path in the residual graph sink End flow = 7
68 Another Example-Max Flow source Ford & Fulkerson algorithm (9) 7 Why is the solution globally optimal? sink flow = 8
69 Another Example-Max Flow S source Ford & Fulkerson algorithm (9) 7 Why is the solution globally optimal?. Let S be the set of reachable nodes in the residual graph sink flow = 9
70 Another Example-Max Flow S source 9 8 sink Ford & Fulkerson algorithm (9) Why is the solution globally optimal?. Let S be the set of reachable nodes in the residual graph. The flow from S to V - S equals to the sum of capacities from S to V S 70
71 Another Example-Max Flow / / / 0/ / 0/ 0/ source / 8/8 sink / 0/ flow = 8/9 / 0/ / / / / Individual flows obtained by summing up all paths Ford & Fulkerson algorithm (9) Why is the solution globally optimal?. Let S be the set of reachable nodes in the residual graph. The flow from S to V - S equals to the sum of capacities from S to V S. The flow from any A to V - A is upper bounded by the sum of capacities from A to V A. The solution is globally optimal 7
72 Another Example-Max Flow 7 source sink 9 8 S T cost = 8 source sink 9 8 S T
73 Another Example-Max Flow S source 9 cost = 8 sink T 7
74 Another Example-Max Flow source 9 x x x x x x C(x) = x + 9x + x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + (-x ) + 8(-x ) + (-x ) 8 sink 7
75 Another Example-Max Flow source 9 x x x x x x C(x) = x + 9x + x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + (-x ) + (-x ) + (-x ) + x + x (-x ) + x (-x ) + (-x ) 8 sink 7
76 Another Example-Max Flow source 9 x x x x x x 8 sink C(x) = x + 9x + x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + (-x ) + (-x ) + (-x ) + x + x (-x ) + x (-x ) + (-x ) x + x (-x ) + x (-x ) + (-x ) = + x (-x ) + x (-x ) 7
77 Another Example-Max Flow source 9 x x x x x x sink C(x) = + x + x + x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + (-x ) + (-x ) + (-x ) + x (-x ) + x + x (-x ) + x (-x ) + (-x ) x + x (-x ) + x (-x ) + (-x ) = + x (-x ) + x (-x ) 77
78 Another Example-Max Flow source x x x x x x C(x) = + x + x + x + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) + (-x ) + x (-x ) + x (-x ) + (-x ) + (-x ) + x (-x ) sink 78
79 Another Example-Max Flow source C(x) = + x + x + x (-x ) + x (-x ) + 7x (-x ) + x (-x ) + x (-x ) + x (-x ) + x (-x ) x x 7 + x (-x ) + (-x ) + x (-x ) + x (-x ) x x cost = 0 x cost = 0 x S sink T S set of reachable nodes from s min cut All coefficients positive Must be global minimum 79
80 History of Max-Flow Algorithms Augmenting Path and Push-Relabel n: #nodes m: #edges U: maximum edge weight Algorithms assume nonnegative edge weights 80
81 Ford-Fulkerson Alg. In each iteration of the Ford-Fulkerson method, we find some augmenting path p and increase the flow f on each edge of p by the residual capacity cf(p). If u and v are not connected by an edge in either direction, we assume implicitly that f[u, v] = 0. The capacities c(u, v) are assumed to be given along with the graph, and c(u, v) = 0 if (u, v) E. 8
82 Example Segmentation in a Video n-links s = 0 st-cut E(x) x * w pq t = Image Flow Global Optimum 8
83 Recap: Data Term Without spatial relationship, data term can be the same for regions that Belong to different objects. Image patches show their similarities although they pertain to different objects 8
84 Segmentation Example 8
85 Segmentation Example 8
86 Segmentation Example 8
87 Optimality? Seeds GrabCut (iterative GC With box prior) 87
88 Vascular Surface Segmentation via GC 0// 88
89 GC Refinement Kohli and Torr 89
90 GC Segmentation in Videos Segmentation of human lame walk in a video sequences Kohli and Torr 90
91 Slide Credits and References FredoDurand of MIT M. Tappen of Amazon R. Szelisky, Univ. of Washington/Seatle J.Malcolm, Graph Cut in Tensor Scale Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D images. Yuri Boykov and Marie-Pierre Jolly. In International Conference on Computer Vision, (ICCV), vol. I, A Comparative Study of Energy Minimization Methods for Markov Random Fields. Rick Szeliski, Ramin Zabih, Daniel Scharstein, Olga Veksler, Vladimir Kolmogorov, Aseem Agarwala, Marshall Tappen, Carsten Rother. ECCV 00 P. Kumar, Oxford University. 9
Dr. Ulas Bagci
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