![Page 1: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/1.jpg)
xi xi
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Example: Camera Image
• directions are counter-intuitive for images • Markov blanket is not just the direct neighbors
when using a directed model!1
![Page 2: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/2.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Markov Random Fields
All paths from A to B go through C, i.e. C blocks all paths.
Markov Blanket
We only need to condition on the direct neighbors of x to get c.i., because these already block every path from x to any other node.
!2
![Page 3: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/3.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Factorization of MRFs
Any two nodes xi and xj that are not connected in an MRF are conditionally independent given all other nodes:
This means: each factor contains only nodes that are connected This motivates the consideration of cliques in the graph:
• A clique is a fully connected subgraph.
• A maximal clique can not be extendedwith another node without loosing the property of full connectivity.
Clique
Maximal Clique
!3
p(xi, xj | x\{i,j}) = p(xi | x\{i,j})p(xj | x\{i,j})
![Page 4: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/4.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Factorization of MRFsIn general, a Markov Random Field is factorized as
where is the set of all (maximal) cliques and is
a positive function of a given clique xC of nodes, called the clique potential. Z is called the partition function. Theorem (Hammersley/Clifford): Any undirected model with associated clique potentials is a perfect map for the probability distribution defined by Equation (4.1). As a conclusion, all probability distributions that can be factorized as in (4.1), can be represented as an MRF.
!4
p(x) =
Qc2C c(xc)P
x0Q
c2C c(x0c)
=1
Z
Y
c2C c(xc) (4.1)
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![Page 5: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/5.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Representability• As for DAGs, we can define an I-map, a D-map
and a perfect map for MRFs.
• The set of all distributions for which a DAG exists that is a perfect map is different from that for MRFs.
Distributions with a DAG as perfect map
Distributions with an MRF as
perfect map
All distributions
!5
![Page 6: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/6.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
If a distribution p satisfies all conditional independence relationships of this graph, then we can write p as
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p(x) =1
Z 123(x1, x2, x3) 234(x2, x3, x4) 35(x3, x5)
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![Page 7: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/7.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
How to define the potentials? • Intuitively, the potential of a
clique should be high, iff the joint probability of the corres-ponding random variables is high.
!7
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p(x) =1
Z 123(x1, x2, x3) 234(x2, x3, x4) 35(x3, x5)
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![Page 8: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/8.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
How to define the potentials? • Intuitively, the potential of a
clique should be high, iff the joint probability of the corres-ponding random variables is high.
• In most cases the potential is defined using a log-linear model:
!8
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log c(xc) = �c(xc)T✓c
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Feature function Parameters
![Page 9: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/9.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
How to define the potentials? • Intuitively, the potential of a
clique should be high, iff the joint probability of the corres-ponding random variables is high.
• In most cases the potential is defined using a log-linear model
making the parameters explicit:
!9
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log c(xc) = �c(xc)T✓c
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) log p(x | ✓) =X
c2C�c(xc)
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![Page 10: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/10.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
Using numbers, e.g.: • Let all variables be binary:
• We can define features
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�ijk(xi, xj , xk) =
⇢1 if xn = 1 8n 2 {i, j, k}0 otherwise
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![Page 11: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/11.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
Using numbers, e.g.: • Let all variables be binary:
• We can define features • and determine weights
!11
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�ijk(xi, xj , xk) =
⇢1 if xn = 1 8n 2 {i, j, k}0 otherwise
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![Page 12: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/12.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
Using numbers, e.g.: • Let all variables be binary:
• We can define features • and determine weights • Then, we can compute the
(log of the ) joint probability for each realisation of the
!12
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![Page 13: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/13.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
A Simple Example
Using numbers, e.g.: • The same graph can also be
interpreted as a binary MRF • This a more specific
representation, but it is less complex (and therefore more efficient)
• In Computer Vision, we almost always use binary MRFs; they are a specific case of general MRFs:
!13
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p(x) =1
Z 12(x1, x2) 13(x1, x3) 23(x2, x3) 24(x2, x4) 34(x3, x4) 35(x3, x5)
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![Page 14: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/14.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Using Graphical Models
We can use a graphical model to do inference:
• We want to find • Some nodes in the graph are observed, for others
we want to find the posterior distribution
• Also, computing the local marginal distribution p(xn) at any node xn can be done using inference.
Question: How can inference be done with a graphical model?
We will see that, when exploiting conditional independences, we can do efficient inference.
!14
argmaxx
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![Page 15: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/15.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Example Application: Denoising
Aim: Recover the noise-free image from the noisy one
We model the original image with variables and the noisy image with pixel values
!15
Noise-free image Noisy image (observation)
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![Page 16: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/16.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Example Application: Denoising
Aim: Recover the noise-free image from the noisy one
We model the original image with variables and the noisy image with pixel values
We consider the true pixel vales as hidden or latent
!16
Noise-free image Noisy image (observation)
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![Page 17: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/17.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Example Application: Denoising
!17
We define two simple edge features:
These are multiplied by parameters and :
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�(xi, yi) = xiyi<latexit sha1_base64="GPLQqt4Rn/5/6KkEgkborzGLo/E=">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</latexit><latexit sha1_base64="GPLQqt4Rn/5/6KkEgkborzGLo/E=">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</latexit><latexit sha1_base64="GPLQqt4Rn/5/6KkEgkborzGLo/E=">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</latexit><latexit sha1_base64="GPLQqt4Rn/5/6KkEgkborzGLo/E=">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</latexit>
log (xi, yi) = ⌘xiyi<latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit><latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit><latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit><latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit>
log (xi, yi) = ⌘xiyi<latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit><latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit><latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit><latexit sha1_base64="RRO0FK7ILH6GVFMPyMIbSquC0kQ=">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</latexit>
“Ising model"
![Page 18: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/18.jpg)
Example Application: Denoising
!18
With this, we can compute the joint:
and its log:
xi<latexit sha1_base64="rxBVHaA+hw0CFRz2ybgNdJiZSik=">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</latexit><latexit sha1_base64="rxBVHaA+hw0CFRz2ybgNdJiZSik=">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</latexit><latexit sha1_base64="rxBVHaA+hw0CFRz2ybgNdJiZSik=">AAACfHicZVHbihNBEO2MtzXedvXRl8YYENQwE0R9XPTFl4UVze7CJoSankrSbF+G7hpJGOYTfNVv82fEmuwgu9mCbg51OVV1Ki+NjpSmf3rJrdt37t7bu99/8PDR4yf7B09Poq+CwonyxoezHCIa7XBCmgyelQHB5gZP84vPbfz0B4aovftOmxJnFpZOL7QCYte39VzP9wfpKN2avAmyDgxEZ8fzg56aFl5VFh0pAzGeZ2lJsxoCaWWw6U+riCWoC1jiOUMHFuOs3s7ayCF7CrnwgZ8jufVerajBxrixOWdaoFXcjbXO/7Hh1SCtW8a4058WH2e1dmVF6NRl+0VlJHnZyiELHVCR2TAAFTRvINUKAihi0a4xtQoXb0Jl+FfetttHV9kcAxa8vFl6Ll/ZMTZSDiXLwgJF2U0sOxapXSQwzNGfRiTlK8eN6iNYHwEFveaDxaYep02fz5LtHuEmOBmPsnSUfX03OPzUHWhPPBcvxCuRiQ/iUHwRx2IilFiKn+KX+N37m7xMXidvL1OTXlfzTFyz5P0/K1zETQ==</latexit><latexit sha1_base64="rxBVHaA+hw0CFRz2ybgNdJiZSik=">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</latexit>
yi<latexit sha1_base64="T4bOcyBkSaDwgzBUUPvBcr4MpBQ=">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</latexit><latexit sha1_base64="T4bOcyBkSaDwgzBUUPvBcr4MpBQ=">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</latexit><latexit sha1_base64="T4bOcyBkSaDwgzBUUPvBcr4MpBQ=">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</latexit><latexit sha1_base64="T4bOcyBkSaDwgzBUUPvBcr4MpBQ=">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</latexit>
xj<latexit sha1_base64="/uiTXGk2yRxVSM8cHktxwgEDOwo=">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</latexit><latexit sha1_base64="/uiTXGk2yRxVSM8cHktxwgEDOwo=">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</latexit><latexit sha1_base64="/uiTXGk2yRxVSM8cHktxwgEDOwo=">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</latexit><latexit sha1_base64="/uiTXGk2yRxVSM8cHktxwgEDOwo=">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</latexit>
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
p(x,y | ⌘,�) = 1
Z
Y
i
exp(⌘xiyi)Y
i,j
exp(�xixj)
<latexit sha1_base64="A2q+UUgCJ+0lbdXj7CIGZFHw/uA=">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</latexit><latexit sha1_base64="A2q+UUgCJ+0lbdXj7CIGZFHw/uA=">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</latexit><latexit sha1_base64="A2q+UUgCJ+0lbdXj7CIGZFHw/uA=">AAAC73icZVJLa9tAEF6pr8R9Oe2xl6XGEIMxkim0l0JoL70EUqiT0MiI1WrlbLIPdXdUZIR+R2+l1/6k9td05KghcQa0+phvHjvzbVYq6SGK/gThvfsPHj7a2R08fvL02fPh3otjbyvHxYJbZd1pxrxQ0ogFSFDitHSC6UyJk+zyY8effBfOS2u+wLoUS81WRhaSM0BXOoRyP9EMzrOiqdvpf7huaaJlThMBbEqTDH8T+p4mhWO8idvmK/Kls3nayJZiVI1VMIbWqVyncnJNTi/ans16uk4vJjQdjqJZtDF6F8Q9GJHejtK9gCe55ZUWBrhi3p/FUQnLhjmQXIl2kFRelIxfspU4Q2iYFn7ZbNbT0jF6clpYh58BuvHezGiY9n6tM4zs5vfbXOe85sY3Sai7in6rPxTvlo00ZQXC8Kv2RaUoWNopQHPpBAe1RsC4kzgB5ecMNwuo061Knaj51FUKT251N703lc6EEzkOr1YW08/1XKAIY4prwQV52t+Y9lWoNB6YwhqDxAvgtjLYqDlk9SEDJ2t8I75t5lE7QFnibRHuguP5LI5m8ec3o4MPvUA75BV5TfZJTN6SA/KJHJEF4eRvQILdYBB+C3+EP8NfV6Fh0Oe8JLcs/P0P45Ptcg==</latexit><latexit sha1_base64="A2q+UUgCJ+0lbdXj7CIGZFHw/uA=">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</latexit>
log p(x,y | ⌘,�) = ⌘X
i
xiyi + �X
i,j
xixj � log(Z)<latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit><latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit><latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit><latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit>
![Page 19: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/19.jpg)
Example Application: Denoising
!19
Our aim now is to find the hidden states such that this log of the joint is maximal (or at least very high). Simple approach is Iterated Conditional Modes (ICM): 1. Initialize all by corresponding 2. For all nodes : 1. set to +1 and to -1 and evaluate 2. keep the value that gives higher log joint
This will keep or increase the joint in every step The nodes can be visited in order or randomly
log p(x,y | ⌘,�) = ⌘X
i
xiyi + �X
i,j
xixj � log(Z)<latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit><latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit><latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit><latexit sha1_base64="x8vhrlNeaXVkx9xAOXPlE9E93ZY=">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</latexit>
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
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![Page 20: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/20.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Result of ICM
!20
Noise-free image Noisy image (observation)
Result of ICM
![Page 21: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/21.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
General Inference in MRFs
• In general, we do not have such an easy model • Therefore, we need more general inference
methods for MRF • The major aim is to exploit sparsity in the
graphical model to make inference efficient
!21
![Page 22: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/22.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Inference on a Chain
The joint probability is given by
The marginal at x3 is
In the general case with N nodes we have
and
!22
![Page 23: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/23.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Inference on a Chain
• This would mean KN computations! A more efficient way is obtained by rearranging:
Vectors of size K
!23
![Page 24: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/24.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Inference on a Chain
In general, we have
!24
![Page 25: Example: Camera Image€¦ · x i x i PD Dr. Rudolph Triebel Computer Vision Group Machine Learning for Computer Vision Example: Camera Image •directions are counter-intuitive for](https://reader034.vdocuments.us/reader034/viewer/2022051604/5ff998c9a62be6468e3e85b1/html5/thumbnails/25.jpg)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Inference on a Chain
The messages µα and µβ can be computed recursively:
Computation of µα starts at the first node and computation of µβ starts at the last node.
!25
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Inference on a Chain
• The first values of µα and µβ are:
• The partition function can be computed at any node:
• Overall, we have O(NK2) operations to compute the marginal
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Inference on a Chain
To compute local marginals: •Compute and store all forward messages, .
•Compute and store all backward messages,
•Compute Z once at a node xm:
•Computefor all variables required.
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
More General Graphs
The message-passing algorithm can be extended to more general graphs:
Directed Tree PolytreeUndirected
Tree
It is then known as the sum-product algorithm. A special case of this is belief propagation.
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f(x1, x2, x3) = p(x1)p(x2)p(x3 | x1, x2)
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Factor Graphs
• The Sum-product algorithm can be used to do inference on undirected and directed graphs. • A representation that generalizes directed and
undirected models is the factor graph.
Directed graph Factor graph
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Factor Graphs
• The Sum-product algorithm can be used to do inference on undirected and directed graphs. • A representation that generalizes directed and
undirected models is the factor graph.
Undirected graph Factor graph
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fa
fb
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Factor Graphs
Factor graphs • can contain multiple factors
for the same nodes • are more general than
undirected graphs • are bipartite, i.e. they consist
of two kinds of nodes and all edges connect nodes of different kind
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x1 x3
x4
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fa
PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Factor Graphs
• Directed trees convert to tree-structured factor graphs
• The same holds for undirected trees
• Also: directed polytrees convert to tree-structured factor graphs
• And: Local cycles in a directed graph can be removed by converting to a factor graph
!32
x1 x3
x4
x1 x3
x4
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
The Sum-Product Algorithm
Assumptions: • all variables are discrete • the factor graph has a tree structure The factor graph represents the joint distribution as a product of factor nodes:
The marginal distribution at a given node x is
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p(x) =Y
s
fs(xs)
p(x) =X
x\x
p(x)
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
p(x) =X
x\x
Y
s2ne(x)Fs(x,Xs)
p(x) =Y
s2ne(x)
X
Xs
Fs(x,Xs) =Y
s2ne(x)µfs!x(x)
The Sum-Product Algorithm
For a given node x the joint can be written as
!34
Product of all factors associated with fs
p(x) =Y
s2ne(x)Fs(x,Xs)
Thus, we have
Key insight: Sum and product can be exchanged!
“Messages from factors to node x”
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
µfs!x(x) =X
x1
· · ·X
xM
fs(x, x1, . . . , xM )Y
m2ne(fs)\x
X
Xsm
Gm(xm, Xsm)
=X
x1
· · ·X
xM
fs(x, x1, . . . , xM )Y
m2ne(fs)\xµxm!fs(xm)
The Sum-Product Algorithm
The factors in the messages can be factorized further:
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The messages can then be computed as
“Messages from nodes to factors”
Fs(x,Xs) = fs(x, x1, . . . , xM )G1(x1, Xs1) . . . GM (xM , XsM )
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
GM (xm, Xsm) =Y
l2ne(xm)\fs
Fl(xm, Xml)
µxm!fs(xm) =Y
l2ne(xm)\fs
X
Xml
Fl(xm, Xml)
The Sum-Product Algorithm
The factors G of the neighboring nodes can again be factorized further:
!36
This results in the exact same situation as above! We can now recursively apply the derived rules:
=Y
l2ne(xm)\fs
µfl!xm(xm)
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
The Sum-Product Algorithm
!37
Summary marginalization:
1.Consider the node x as a root note 2.Initialize the recursion at the leaf nodes as:
(var) or (fac) 3.Propagate the messages from the leaves to the
root x 4.Propagate the messages back from the root to
the leaves 5.We can get the marginals at every node in the
graph by multiplying all incoming messages
µf!x(x) = 1 µx!f (x) = f(x)
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
The Max-Sum Algorithm
Sum-product is used to find the marginal distributions at every node, but: How can we find the setting of all variables that maximizes the joint probability? And what is the value of that maximal probability? Idea: use sum-product to find all marginals and then report the value for each node x that maximizes the marginal p(x) However: this does not give the overall maximum of the joint probability
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
=1
Zmaxx1
[ 1,2(x1, x2) [. . .max N�1,N (xN�1, xN )]]
The Max-Sum Algorithm
Observation: the max-operator is distributive, just like the multiplication used in sum-product:
Idea: use max instead of sum as above and exchange it with the product Chain example:
Message passing can be used as above!
!39
max(ab, ac) = amax(b, c) if a � 0
maxx
p(x) =1
Zmaxx1
. . .max[ 1,2(x1, x2) . . . N�1,N (xN�1, xN )]
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
The Max-Sum Algorithm
To find the maximum value of p(x), we start again at the leaves and propagate to the root. Two problems: • no summation, but many multiplications; this
leads to numerical instability (very small values) • when propagating back, multiple configurations
of x can maximize p(x), leading to wrong assignments of the overall maximum
Solution to the first: Transform everything into log-space and use sums
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
The Max-Sum Algorithm
Solution to the second problem: Keep track of the arg max in the forward step, i.e. store at each node which value was responsible for the maximum:
Then, in the back-tracking step we can recover the arg max by recursive substitution of:
!41
�(xn) = argmaxxn�1
[ln fn�1,n(xn�1, xn) + µxn�1!fn�1,n(xn)]
xmaxn�1 = �(xmax
n )
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Sum-Product Inference in General Graphical Models
1.Convert graph (directed or undirected) into a factor graph (there are no cycles)
2.If the goal is to marginalize at node x, then consider x as a root node
3.Initialize the recursion at the leaf nodes as: (var) or (fac)
4.Propagate messages from the leaves to x 5.Propagate messages from x to the leaves 6.Obtain marginals at every node by multiplying all incoming messages
!42
µf!x(x) = 1 µx!f (x) = f(x)
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Other Inference Algorithms• Max-Sum algorithm: used to maximize the joint
probability of all variables (no marginalization) • Junction Tree algorithm: exact inference for
general graphs (even with loops) • Loopy belief propagation: approximate
inference on general graphs (more efficient)
Special kind of undirected GM: • Conditional Random fields (e.g.: classification)
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Conditional Random Fields
• Another kind of undirected graphical model is known as Conditional Random Field (CRF). • CRFs are used for classification where labels are
represented as discrete random variables y and features as continuous random variables x • A CRF represents the conditional probability where w are parameters learned from training data.
• CRFs are discriminative and MRFs are generative
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Conditional Random Fields
Derivation of the formula for CRFs:
In the training phase, we compute parameters w that maximize the posterior:
where (x,y) is the training data and p(w) is a Gaussian prior. In the inference phase we maximize
!45
w = argmaxw
p(w | x,y) = argmaxw
p(y | x,w)p(w)
argmaxy⇤
p(y⇤ | x⇤, w)
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
CRF Training
We minimize the negative log-posterior:
Computing the likelihood is intractable, as we have to compute the partition function for each w. We can approximate the likelihood using pseudo-likelihood:
whereMarkov blanket Ci: All cliques containing yi
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Pseudo Likelihood
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Pseudo Likelihood
Pseudo-likelihood is computed only on the Markov blanket of yi and its corresp. feature nodes.
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Potential Functions
• The only requirement for the potential functions is that they are positive. We achieve that with:where f is a compatibility function that is large if the labels yC fit well to the features xC.
• This is called the log-linear model.
• The function f can be, e.g. a local classifier
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
CRF Training and Inference
Training: • Using pseudo-likelihood, training is efficient. We have
to minimize:
• This is a convex function that can be minimized using gradient descent
Inference: • Only approximatively, e.g. using loopy belief
propagation
Log-pseudo-likelihood Gaussian prior
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PD Dr. Rudolph TriebelComputer Vision Group
Machine Learning for Computer Vision
Summary• Undirected models (aka Markov random fields)
provide an intuitive representation of conditional independence
• An MRF is defined as a factorization over clique potentials and normalized globally
• Directed and undirected models have different representative power (no simple “containment”)
• Inference on undirected Markov chains is efficient using message passing
• Factor graphs are more general; exact inference can be done efficiently using sum-product
!51