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Discrete Optimization in Computer Vision Nikos Komodakis Ecole des Ponts ParisTech, LIGM Traitement de l’information et vision artiﬁcielle

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Message passing algorithms for energy minimization

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Message-passing algorithms Central concept: messages These methods work by propagating messages across the MRF graph Widely used algorithms in many areas

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Message-passing algorithms But how do messages relate to optimizing the energy? Let’s look at a simple example first: we will examine the case where the MRF graph is a chain

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Message-passing on chains MRF graph

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Message-passing on chains Corresponding lattice or trellis

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Message-passing on chains Global minimum in linear time Optimization proceeds in two passes: Forward pass (dynamic programming) Backward pass

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Message-passing on chains (example on board) (algebraic derivation of messages)

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s qpr Message-passing on chains

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s qpr Forward pass (dynamic programming)

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s qpr

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s qpr

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s qpr

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s qpr

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s qpr Min-marginal for node s and label j:

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s qpr Backward pass xsxs xrxr xqxq xpxp

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Message-passing on chains How can I compute min-marginals for any node in the chain? How to compute min-marginals for all nodes efficiently? What is the running time of message-passing on chains?

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Message-passing on trees We can apply the same idea to tree- structured graphs Slight generalization from chains Resulting algorithm called: belief propagation (also called under many other names: e.g., max-product, min-sum etc.) (for chains, it is also often called the Viterbi algorithm)

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Belief propagation (BP)

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Dynamic programming: global minimum in linear time BP: Inward pass (dynamic programming) Outward pass Gives min-marginals qpr BP on a tree [Pearl’88] root leaf

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qpr Inward pass (dynamic programming)

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qpr

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qpr

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qpr

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qpr

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qpr

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qpr

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qpr Outward pass

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qpr BP on a tree: min-marginals Min-marginal for node q and label j:

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Belief propagation: message-passing on trees

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min-marginals = ???min-marginals = sum of all messages + unary potential

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What is the running time of message- passing for trees?

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Message-passing on chains Essentially, message passing on chains is dynamic programming Dynamic programming means reuse of computations

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Generalizing belief propagation Key property: min(a+b,a+c) = a+min(b,c) BP can be generalized to any operators satisfying the above property E.g., instead of (min,+), we could have: (max,*) Resulting algorithm called max-product. What does it compute? (+,*) Resulting algorithm called sum-product. What does it compute?

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Belief propagation as a distributive algorithm BP works distributively (as a result, it can be parallelized) Essentially BP is a decentralized algorithm Global results through local exchange of information Simple example to illustrate this: counting soldiers

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Counting soldiers in a line Can you think of a distributive algorithm for the commander to count its soldiers? (From David MacKay’s book “Information Theory, Inference, and Learning”)

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Counting soldiers in a line

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Counting soldiers in a tree Can we do the same for this case?

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Counting soldiers in a tree

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Counting soldiers Simple example to illustrate BP Same idea can be used in cases which are seemingly more complex: counting paths through a point in a grid probability of passing through a node in the grid In general, we have used the same idea for minimizing MRFs (a much more general problem)

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Graphs with loops How about counting these soldiers? Hmmm…overcounting?

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