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12 cv mil_models_for_grids
1.
Computer vision: models, learning
and inference Chapter 10 Graphical Models Please send errata to s.prince@cs.ucl.ac.uk
2.
Models for grids •
Consider models where one unknown world state at each pixel in the image – takes the form of a grid. • Loops in the graphical model so cannot use dynamic programming or belief propagation • Define probability distributions that favour certain configurations of world states – Called Markov random fields – Inference using a set of techniques called graph cuts 2 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
3.
Binary Denoising
Before After Image represented as binary discrete variables. Some proportion of pixels randomly changed polarity. 3 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
4.
Multi-label Denoising
Before After Image represented as discrete variables representing intensity. Some proportion of pixels randomly changed according to a uniform distribution. 4 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
5.
Denoising Goal Observed Data
Uncorrupted Image 5 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
6.
Denoising Goal
Observed Data Uncorrupted Image • Most of the pixels stay the same • Observed image is not as smooth as original Now consider pdf over binary images that encourages smoothness – Markov random field 6 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
7.
Markov random fields This
is just the typical property of an undirected model. We’ll continue the discussion in terms of undirected models 7 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
8.
Markov random fields Normalizing
constant (partition function) Potential function Returns positive number Subset of variables (clique) 8 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
9.
Markov random fields Normalizing
constant Cost function (partition function) Returns any number Subset of variables Relationship (clique) 9 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
10.
Smoothing example Computer vision:
models, learning and inference. ©2011 Simon J.D. Prince 10
11.
Smoothing Example Smooth solutions
(e.g. 0000,1111) have high probability Z was computed by summing the 16 un-normalized probabilities Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 11
12.
Smoothing Example
Samples from larger grid -- mostly smooth Cannot compute partition function Z here - intractable Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 12
13.
Denoising Goal Observed Data
Uncorrupted Image 13 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
14.
Denoising overview Bayes’ rule: Likelihoods:
Probability of flipping polarity Prior: Markov random field (smoothness) MAP Inference: Graph cuts 14 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
15.
Denoising with MRFs
MRF Prior (pairwise cliques) Original image, w Likelihoods Observed image, x Inference : 15 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
16.
MAP Inference
Unary terms Pairwise terms (compatability of data with label y) (compatability of neighboring labels) 16 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
17.
Graph Cuts Overview
Graph cuts used to optimise this cost function: Unary terms Pairwise terms (compatability of data with label y) (compatability of neighboring labels) Three main cases: 17 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
18.
Graph Cuts Overview Graph
cuts used to optimise this cost function: Unary terms Pairwise terms (compatability of data with label y) (compatability of neighboring labels) Approach: Convert minimization into the form of a standard CS problem, MAXIMUM FLOW or MINIMUM CUT ON A GRAPH Polynomial-time methods for solving this problem are known 18 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
19.
Max-Flow Problem Goal: To push
as much ‘flow’ as possible through the directed graph from the source to the sink. Cannot exceed the (non-negative) capacities cij associated with each edge. 19 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
20.
Saturated Edges When we
are pushing the maximum amount of flow: • There must be at least one saturated edge on any path from source to sink (otherwise we could push more flow) • The set of saturated edges hence separate the source and sink 20 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
21.
Augmenting Paths Two numbers
represent: current flow / total capacity 21 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
22.
Augmenting Paths Choose any
route from source to sink with spare capacity and push as much flow as you can. One edge (here 6-t) will saturate. 22 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
23.
Augmenting Paths Choose another
route, respecting remaining capacity. This time edge 5-6 saturates. 23 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
24.
Augmenting Paths A third
route. Edge 1-4 saturates 24 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
25.
Augmenting Paths A fourth
route. Edge 2-5 saturates 25 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
26.
Augmenting Paths A fifth
route. Edge 2-4 saturates 26 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
27.
Augmenting Paths There is
now no further route from source to sink – there is a saturated edge along every possible route (highlighted arrows) 27 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
28.
Augmenting Paths The saturated
edges separate the source from the sink and form the min-cut solution. Nodes either connect to the source or connect to the sink. 28 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
29.
Graph Cuts: Binary
MRF Graph cuts used to optimise this cost function: Unary terms Pairwise terms (compatability of data with label w) (compatability of neighboring labels) First work with binary case (i.e. True label w is 0 or 1) Constrain pairwise costs so that they are “zero-diagonal” 29 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
30.
Graph Construction •
One node per pixel (here a 3x3 image) • Edge from source to every pixel node • Edge from every pixel node to sink • Reciprocal edges between neighbours Note that in the minimum cut EITHER the edge connecting to the source will be cut, OR the edge connecting to the sink, but NOT BOTH (unnecessary). Which determines whether we give that pixel label 1 or label 0. Now a 1 to 1 mapping between possible labelling and possible minimum cuts 30 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
31.
Graph Construction
Now add capacities so that minimum cut, minimizes our cost function Unary costs U(0), U(1) attached to links to source and sink. • Either one or the other is paid. Pairwise costs between pixel nodes as shown. • Why? Easiest to understand with some worked examples. 31 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
32.
Example 1
32 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
33.
Example 2
33 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
34.
Example 3
34 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
35.
35 Computer vision: models,
learning and inference. ©2011 Simon J.D. Prince
36.
Graph Cuts: Binary
MRF Graph cuts used to optimise this cost function: Unary terms Pairwise terms (compatability of data with label w) (compatability of neighboring labels) Summary of approach • Associate each possible solution with a minimum cut on a graph • Set capacities on graph, so cost of cut matches the cost function • Use augmenting paths to find minimum cut • This minimizes the cost function and finds the MAP solution 36 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
37.
General Pairwise costs Modify
graph to • Add P(0,0) to edge s-b • Implies that solutions 0,0 and 1,0 also pay this cost • Subtract P(0,0) from edge b-a • Solution 1,0 has this cost removed again Similar approach for P(1,1) 37 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
38.
Reparameterization The max-flow /
min-cut algorithms require that all of the capacities are non-negative. However, because we have a subtraction on edge a-b we cannot guarantee that this will be the case, even if all the original unary and pairwise costs were positive. The solution to this problem is reparamaterization: find new graph where costs (capacities) are different but choice of minimum solution is the same (usually just by adding a constant to each solution) 38 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
39.
Reparameterization 1 The minimum
cut chooses the same links in these two graphs Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
40.
Reparameterization 2 The minimum
cut chooses the same links in these two graphs 40 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
41.
Submodularity
Subtract constant Add constant, Adding together implies 41 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
42.
Submodularity If this condition
is obeyed, it is said that the problem is “submodular” and it can be solved in polynomial time. If it is not obeyed then the problem is NP hard. Usually it is not a problem as we tend to favour smooth solutions. 42 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
43.
Denoising Results
Pairwise costs increasing Original Pairwise costs increasing 43 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
44.
Plan of Talk •
Denoising problem • Markov random fields (MRFs) • Max-flow / min-cut • Binary MRFs – submodular (exact solution) • Multi-label MRFs – submodular (exact solution) • Multi-label MRFs - non-submodular (approximate) 44 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
45.
Multiple Labels
Construction for two pixels (a and b) and four labels (1,2,3,4) There are 5 nodes for each pixel and 4 edges between them have unary costs for the 4 labels. One of these edges must be cut in the min-cut solution and the choice will determine which label we assign. 45 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
46.
Constraint Edges
The edges with infinite capacity pointing upwards are called constraint edges. They prevent solutions that cut the chain of edges associated with a pixel more than once (and hence given an ambiguous labelling) 46 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
47.
Multiple Labels
Inter-pixel edges have costs defined as: Superfluous terms : For all i,j where K is number of labels47 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
48.
Example Cuts Must cut
links from before cut on pixel a to after cut on pixel b. 48 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
49.
Pairwise Costs
Must cut links from before cut on pixel a to after cut on pixel b. Costs were carefully chosen so that sum of these links gives appropriate pairwise term. If pixel a takes label I and pixel b takes label J 49 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
50.
Reparameterization
50 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
51.
Submodularity We require the
remaining inter-pixel links to be positive so that or By mathematical induction we can get the more general result 51 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
52.
Submodularity If not submodular
then the problem is NP hard. 52 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
53.
Convex vs. non-convex
costs Quadratic Truncated Quadratic Potts Model • Convex • Not Convex • Not Convex • Submodular • Not Submodular • Not Submodular 53 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
54.
What is wrong
with convex costs? Observed noisy image Denoised result • Pay lower price for many small changes than one large one • Result: blurring at large changes in intensity 54 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
55.
Plan of Talk •
Denoising problem • Markov random fields (MRFs) • Max-flow / min-cut • Binary MRFs - submodular (exact solution) • Multi-label MRFs – submodular (exact solution) • Multi-label MRFs - non-submodular (approximate) 55 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
56.
Alpha Expansion Algorithm •
break multilabel problem into a series of binary problems • at each iteration, pick label and expand (retain original or change to ) Initial Iteration 1 labelling (orange) Iteration 2 Iteration 3 (yellow) (red) 56 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
57.
Alpha Expansion Ideas •
For every iteration – For every label – Expand label using optimal graph cut solution Co-ordinate descent in label space. Each step optimal, but overall global maximum not guaranteed Proved to be within a factor of 2 of global optimum. Requires that pairwise costs form a metric: 57 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
58.
Alpha Expansion Construction
Binary graph cut – either cut link to source (assigned to ) or to sink (retain current label) Unary costs attached to links between 58 source, sink andSimon J.D.nodes appropriately. Computer vision: models, learning and inference. ©2011 pixel Prince
59.
Alpha Expansion Construction
Graph is dynamic. Structure of inter-pixel links depends on and the choice of labels. There are four cases. 59 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
60.
Alpha Expansion Construction
Case 1: Adjacent pixels both have label already. Pairwise cost is zero – no need for extra edges. 60 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
61.
Alpha Expansion Construction
Case 2: Adjacent pixels are . Result either • (no cost and no new edge). • (P( ), add new edge). 61 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
62.
Alpha Expansion Construction
Case 3: Adjacent pixels are . Result either • (no cost and no new edge). • (P( ), add new edge). • (P( ), add new edge). 62 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
63.
Alpha Expansion Construction
Case 4: Adjacent pixels are . Result either • (P( ), add new edge). • (P( ), add new edge). • (P( ), add new edge). 63 • (no cost Prince no new edge). Computer vision: models, learning and inference. ©2011 Simon J.D. and
64.
Example Cut 1
64 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
65.
Example Cut 1
Important! 65 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
66.
Example Cut 2
66 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
67.
Example Cut 3
67 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
68.
Denoising Results
68 Computer vision: models, learning and inference. ©2011 Simon J.D. Prince
69.
Conditional Random Fields
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 69
70.
Directed model for
grids Cannot use graph cuts as three-wise term. Easy to draw samples. Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 70
71.
Applications • Background subtraction
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 71
72.
Applications • Grab cut
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 72
73.
Applications • Stereo vision
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 73
74.
Applications • Shift-map image
editing Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 74
75.
Computer vision: models,
learning and inference. ©2011 Simon J.D. Prince 75
76.
Applications • Shift-map image
editing Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 76
77.
Applications • Super-resolution
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 77
78.
Applications • Texture synthesis
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 78
79.
Image Quilting Computer vision:
models, learning and inference. ©2011 Simon J.D. Prince 79
80.
Applications • Synthesizing faces
Computer vision: models, learning and inference. ©2011 Simon J.D. Prince 80
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