Efficient Belief propagation for Early Vision

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Efficient Belief propagation for Early Vision

• Pedro F. Felzenszwalb University of Chicago
  Daniel P. Huttenlocher Cornell
  University
• International Journal of Computer Vision 2006.10
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perface

• Early vision problems problems based on pixels.
  Such as stereo, and image restoration
• Early vision problems can be formulated as a
  maximum a posteriori MRF (MAP-MRF) problem.
• methods to solve MAP-MRF simulated annealing ,
  which can often take an unacceptably long time
  although global optimization is achieved in theory

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outline

• Markov random field (MRF) models
• Message passing
• 3 techniques to reduce the time
• 1.Computing Messages update in linear time
• 2.BP on the bipartite graph message passing
• schedule
• 3.Multi-Grid BP

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MRF models

• be the set of pixels in an image
• be a set of labels (ex. intensities)
• the edges in the four-connected image
• grid graph
• Labeling assigns a label to
  each pixel
• energy function

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MRF models

• the cost of assigning labels
  and
• to two neighboring pixels (discontinuity
  cost) , generally based on the difference between
  labels in low-level vision problem
• so we use
• the cost of assigning label
  to pixel
• (data cost )

energy function
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Message passing

• Finding a labeling with minimum energy
  corresponds to MAP estimation problem
• max-product used to approximate the MAP
• max-product to min-sum ( with )
• less sensitive to numerical artifacts
• max-product BP works by passing messages around
  the graph defined by the
• four-connected image grid.

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Message passing

• In general  .
• Ex 1.

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Message passing

• Rules
• Or
• Ex 2. (complex)

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Message passing
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Message passing

• be the message that node p sends to
  a neighboring node q at iteration t
• are initialized to 0
• a vector
  of n_labels
• dimension
• Step 1. Compute messages

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Message passing
Step 2. Compute belief vector for each
node after T iterations
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• Step 3. minimizes individually at
  each node is selected
• time complexity
• n number of pixels
• k number of possible labels ( n_lables )
• T number of iterations

Too long time
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3 techniques

• 1. Computing a Message Update in Linear Time (
  to )
• 2. The bipartite graph message passing schedule (
  to )
• 3. Multi-Grid BP

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1.Computing Messages

• To most low-level vision application
• based on the difference
  between
• the labels and

The form of equation (3) is commonly referred to
as a min convolution
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1.Computing Messages

• Potts model

Compute fristly
time complexity turns to
go
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1.Computing Messages
(stereo and image restoration)

• truncated linear model
• First. Consider the pure linear model
• message compute

The minimization can be seen as lower envelope in
figure
go
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Ex. S1 cones root at
lower envelope
Can use two-pass algorithm to compute message
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two-pass algorithm

• initialize message with
• forward pass
• for from 1 to
• backward pass
• for from to 0

Compute messages with the linear cost
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two-pass algorithm

• Ex.
• 1.
• 2.forward pass
• for 1 to 3
• 3.backward pass
• for 2 to 0

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1.Computing Messages
second. Use Potts model to computation
pure linear model
time complexity turns to

• Truncated quadratic Model

go
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truncated linear model
Potts model
Truncated quadratic Model
Robust!
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2.BP on the Grid Graph

• BP performed more efficiently for a bipartite
  graph
• grid graph to bipartite graph every edge
  connects different nodes
• bipartite graph with nodes
• messages sent from nodes in
• depend on the messages sent from
  nodes in , vice versa

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2.BP on the Grid Graph

• Update

Without compute
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2.BP on the Grid Graph

• So we alternate update the messages from A and
  from B
• scheme when is odd , update
• when is even , update
• if is odd(even) then

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2.BP on the Grid Graph

• Moreover , store new messages in the old ones
  memory space(independent)
• The new messages are nearly the same as
• the messages in standard scheme
• when BP converges , messages converges to the
  same fixed point
• Turns to

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3.Multi-Grid BP

• BPS requires large T to produce good results
• Initialize close to a fixed value, messages will
  get convergence more rapidly (have a small fixed
  number of iterations T )
• coarse-to-fine manner
• long range interactions between pixels can be
  captured by short paths in coarse graphs

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3.Multi-Grid BP

• Details
• 0-th level the original labeling problem
• i-th level blocks of pixels
  grouped together .
• labelings where all the pixels in a block
  are assigned the same label
• level i get estimates for the messages at level
  i-1(make a good initial value)

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3.Multi-Grid BP
level 1
level 0
Illustration of two levels in the multi-grid
method
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3.Multi-Grid BP

• be the message that node p sends to the
  right at iteration t.
• , , be the messages send to
  left, up and down, respectively
• Instance

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at level i is the block containing node
at level i - 1,

level i
level i-1
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3.Multi-Grid BP

• The cost of assigning label to a block b
• (block data cost)
• Block discontinuity costs

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3.Multi-Grid BP

• Potts model the same
• truncated linear model
• Truncated quadratic Model

go
go
go
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3.Multi-Grid BP

• run BP for a small number of iterations at each
  level (between 5 and 10)
• Note that the total number of nodes in the
• hierarchy is just the number of nodes
  at the finest level

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Experiments .1

• stereo results for the Tsukuba image pair

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number of message update iterations for our
multi-grid BP method versus the standard algorithm
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running BP with all speedup techniques versus
running BP with all but one of the techniques.
min convolution method provides an important
speedup
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Experiments .2

• Restoration results with an input that has
  missing values.

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