Transcript Document

Statistical Region Merging
R. Nock and F. Nielsen
IEEE Transactions on pattern analysis
and machine intelligence, Vol 26, Issue
11, p.p. 1452-1458, Nov. 2004
Outline
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1.
2.
3.
4.
5.
Introduction
The model of image generation
Theoretical analysis and algorithms
Experimental results
Conclusion
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1. Introduction
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Segmentation is a tantalizing and central problem for
image processing.
A prominent trend in grouping focuses on graph theorem.
The authors proposed a different strategy which belongs
to region growing/merging techniques.
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Regions are sets of pixels with homogeneous properties and are
iteratively grown by combining smaller regions.
Region growing/merging techniques usually work with a
statistical test to decide the merging of regions.
A good region merging algorithm has to find a good
balance between preserving the perceptual units and the
risk of overmerging for the remaining region.
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1. Introduction
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A novel model of image generation and the
segmentation approach are proposed.
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To reconstruct the true region from the observed
region.
With high probability, it suffers only the overmerging
problem in segmentation.
With high probability, it has small overmerging error.
Fast and easily implementable.
Can be used to images with many channels.
Can handling noise and occlusions.
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2 The model of image generation
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1. Introduction
2. The model of image generation
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2.1 The model of image generation
3. Theoretical analysis and algorithms
4. Experimental results
5. Conclusion
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2.1 The model of image generation
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The observed image, I, contains |I| pixels, each
containing RGB values and belonging to the set
{1,2,...,g}
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2.1 The model of image generation
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The observed color channel is sampled from a
family of Q distributions at each pixel of a
perfect scene, I*. (Range of the Q distributions
are bounded by g/Q)
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2.1 The model of image generation
An example
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Example of some true image I* (expectation)
and the observed image I.
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2.1 The model of image generation
homogeneity property
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In I*, the optimal regions share a
homogeneity property:
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Inside a region, the statistical pixels have the
same expectation for every color channel.
Different regions have different expectations
for at least one color channel.
Inside a region, all distributions associated
to each pixel can be different, as long as
the homogeneity property is satisfied.
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3. Theoretical analysis and
algorithms
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1. Introduction
2. The model of image generation
3. Theoretical analysis and algorithms
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3.1 Theoretical analysis
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Merging predicate
Order in merging
3.2 Other properties of the proposed approach
3.3 Proposed algorithm: SRM
4. Experimental results
5. Conclusion
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3.1 Theoretical analysis and
algorithms
Theoretical analysis
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Two essential components in defining a
region merging algorithm:
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Merging predicate: define how to merge to
undetermined region.
Order in merging: define an order to be
followed to check the merging predicate.
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3.1 Theoretical analysis and
algorithms
Merging predicate
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Theorem 1 (The independent bounded
difference inequality). Let X   X1 , X 2 ,..., X n 
be a vector of n R.V.s. Suppose the real-valued
function f satisfies f ( x)  f ( x' )  ck whenever
vectors x and x’ differ only in kth coordinate.
Then, for any   0 ,


  2 2
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Pr f ( X )       exp
2
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
 (c )
k
k
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where  is the expected value of the R.V. f(X)
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3.1 Theoretical analysis and
algorithms
Merging predicate
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From thm 1, we obtain the result on the
deviation of observed differences between
regions of I.
Corollary 1. Consider a fixed couple (R,R’) of
regions of I.0    1, the probability is no more
than  that
1  1
1  2
(R  R ' )  E(R  R ' )  g
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ln  b( R, R' )
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2Q  R R'  
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3.1 Theoretical analysis and
algorithms
Merging predicate
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In the same statistical region, E( R  R ' )  0 and
with a high probability  1  ( N ) that R  R '
does not exceed b( R, R' ) .
Merging predicate : merge R and R’ iff
R  R '  b( R, R ' )
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3.1 Theoretical analysis and
algorithms
Order in merging
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Ideally, the order to test the merging of
regions is:
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when any test between two true regions
occurs, that means that all tests inside each
of the two true regions have previously
occurred.
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3.2 Theoretical analysis and
algorithms
Other properties of the proposed
approach
The proposed approach is proved that only
overmerging occurs, with high probability.
The proposed approach has been shown to have
an upperbound on the error incurred w.r.t. the
optimal sementation, with high probability.
The proposed approach is easily extended to
numerical channels, such as RGB.
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3.3 Theoretical analysis and
algorithms
Proposed algorithm: SRM
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To choose a merging predicate and order in
merging to approximate the ideal segmentation
method.
Merging predicate: merge R and R’ iff
R  R '  b( R, R ' )
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Order in merging: choose a real-valued
function f and radix sort f(.,.) to approximate
the order in merging. ( O(|I|*log(g)) )
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4. Experimental results
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1.
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3.
4.
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Introduction
The model of image generation
Theoretical analysis and algorithms
Experimental results
4.1
4.2
4.3
4.4
4.5
Choice of f
Noise handling
Enhance the noise handling ability
Handling occlusions
Controlling the scale of the segmentation
5. Conclusion
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4.1 Experimental results
Choice of f
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Choose f a ( p, p' )  pa  pa ' , where pa and pa ' are
the pixel channel values.
The preordering can manage dramatic
improvements over conventional scanning.
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4.1 Experimental results
Choice of f
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A second choice of f is to use  x and  y in
Sobel filters, where smoothing filter is performed
by [1 2 1] and derivative filter is [-1 0 0 1].
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4.1 Experimental results
Choice of f
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Comparison of the
two choices of f
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4.2 Experimental results
Noise handling
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Two noise types to be
handled:
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Transmission noise t(q):
chosen uniformly in {1,2,...,g}
Salt and pepper noise s(q):
chosen uniformly in {1,g}
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4.3 Experimental results
Enhance the noise handling ability
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By integrating the moving average operators,
the first kind of f: f a ( p, p' )  pa  pa ' is replaced
by f a ( p, p' )  N p ( p) a  N p ( p' ) a
For the second kind of f, the smoothing filter is
extended to be [1 2 ... △+1 △ ... 1], and the
derivative filter is extended to be [-△ -△+1 ...
△].
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4.3 Experimental results
Enhance the noise handling ability
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Noise handling ability
of the extended SRM
methods.
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4.4 Experimental results
Handling occlusions
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First run SRM as already presented.
In a second stage, run SRM again with the
modification of f a (.,.)  pa  pa ' to f a ( R, R' )  R a  R'a ,
and 4-connexity to clique connexity.
Radix sorting with f has an overall time
complexity O( (|I|+k2)logg) ).
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4.4 Experimental results
Handling occlusions
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SRM with occlusion
handling.
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4.5 Experimental results
Controlling the scale of the
segmentation
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The objective of multiscale segmentation
is to get a hierarchy of segmentations at
different scales.
In SRM, scale is controlled by tuning of
parameter Q: as Q increases, the regions
found are getting smaller.
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5. Conclusion
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1.
2.
3.
4.
5.
Introduction
The model of image generation
Theoretical analysis and algorithms
Experimental results
Conclusion
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5. Conclusion
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A novel model of image generation is proposed, which
captures the idea that grouping is an inference problem.
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A simple merging predicate and ordering in merging are
provided.
SRM suffers only overmerging problems and achieves
low error in segmentation, both with high probability.
SRM is very fast (segments a 512x512 image is in about
one second on an Intel Pentium 4 2.4G processor)
SRM is able to cope with significant noise corruption,
handling occlusions, and perform scale-sensitive
segmentations.
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