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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. 1. Introduction 2. The model of image generation 3. Theoretical analysis and algorithms 4. Experimental results - PowerPoint PPT Presentation
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Statistical Region Statistical Region Merging Merging R. Nock and F. Nielsen R. Nock and F. Nielsen IEEE Transactions on pattern analysis IEEE Transactions on pattern analysis and machine intelligence and machine intelligence , Vol 26, , Vol 26, Issue 11, p.p. 1452-1458, Issue 11, p.p. 1452-1458, Nov. 2004 Nov. 2004
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Page 1: Statistical Region Merging

Statistical Region MergingStatistical Region Merging

R. Nock and F. NielsenR. Nock and F. NielsenIEEE Transactions on pattern IEEE Transactions on pattern

analysis and machine intelligenceanalysis and machine intelligence, , Vol 26, Issue 11, p.p. 1452-1458, Vol 26, Issue 11, p.p. 1452-1458,

Nov. 2004Nov. 2004

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OutlineOutline

1. Introduction1. Introduction 2. The model of image generation2. The model of image generation 3. Theoretical analysis and 3. Theoretical analysis and

algorithmsalgorithms 4. Experimental results4. Experimental results 5. Conclusion5. Conclusion

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1. Introduction1. Introduction

SegmentationSegmentation is a tantalizing and central problem is a tantalizing and central problem for image processing.for image processing.

A prominent trend in grouping focuses on A prominent trend in grouping focuses on graph graph theoremtheorem..

The authors proposed a different strategy which The authors proposed a different strategy which belongs to belongs to region growing/mergingregion growing/merging techniques. techniques. RegionsRegions are sets of pixels with are sets of pixels with homogeneous propertieshomogeneous properties

and are and are iteratively growniteratively grown by combining smaller regions. by combining smaller regions. Region growing/merging techniques usually work with a Region growing/merging techniques usually work with a

statistical test to decide the merging of regions.statistical test to decide the merging of regions. A good region merging algorithm has to find a good A good region merging algorithm has to find a good

balance between balance between preserving the perceptual unitspreserving the perceptual units and the and the risk of overmergingrisk of overmerging for the remaining for the remaining region.region.

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1. Introduction1. Introduction

A A novel model of image generationnovel model of image generation and and the segmentation approach are proposed.the segmentation approach are proposed. To To reconstruct the true regionreconstruct the true region from the from the

observed region.observed region. With high probability, it With high probability, it suffers only the suffers only the

overmergingovermerging problem in segmentation. problem in segmentation. With high probability, it has With high probability, it has small overmerging small overmerging

errorerror.. Fast and easily implementable.Fast and easily implementable. Can be used to Can be used to images with many channelsimages with many channels.. Can Can handling noise and occlusionshandling noise and occlusions..

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2 The model of image 2 The model of image generationgeneration

1. Introduction1. Introduction 2. The model of image generation2. The model of image generation

2.1 The model of image generation2.1 The model of image generation 3. Theoretical analysis and 3. Theoretical analysis and

algorithmsalgorithms 4. Experimental results4. Experimental results 5. Conclusion5. Conclusion

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2.1 The model of image 2.1 The model of image generationgeneration

The observed image, I, contains |I| pixels, The observed image, I, contains |I| pixels, each containing RGB values and each containing RGB values and belonging to the set {1,2,...,g}belonging to the set {1,2,...,g}

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2.1 The model of image 2.1 The model of image generationgeneration

The observed color channel is sampled The observed color channel is sampled from a family of Q distributions at each from a family of Q distributions at each pixel of a perfect scene, I*. (Range of the pixel of a perfect scene, I*. (Range of the Q distributions are bounded by g/Q)Q distributions are bounded by g/Q)

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2.1 The model of image 2.1 The model of image generationgenerationAn exampleAn example

Example of some true image I* Example of some true image I* (expectation) and the observed image I.(expectation) and the observed image I.

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2.1 The model of image 2.1 The model of image generationgeneration

homogeneity propertyhomogeneity property In I*, the optimal regions share a In I*, the optimal regions share a

homogeneity propertyhomogeneity property:: Inside a regionInside a region, the statistical pixels have the , the statistical pixels have the

same expectationsame expectation for every color channel. for every color channel. Different regions have different expectationsDifferent regions have different expectations

for at least one color channel.for at least one color channel. Inside a region, all distributionsInside a region, all distributions associated associated

to each pixel to each pixel can be differentcan be different, as long as , as long as the homogeneity property is satisfied.the homogeneity property is satisfied.

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3. Theoretical analysis and 3. Theoretical analysis and algorithmsalgorithms

1. Introduction1. Introduction 2. The model of image generation2. The model of image generation 3. Theoretical analysis and algorithms3. Theoretical analysis and algorithms

3.1 Theoretical analysis 3.1 Theoretical analysis Merging predicateMerging predicate Order in mergingOrder in merging

3.2 Other properties of the proposed approach3.2 Other properties of the proposed approach 3.3 Proposed algorithm: SRM3.3 Proposed algorithm: SRM

4. Experimental results4. Experimental results 5. Conclusion5. Conclusion

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3.1 Theoretical analysis and 3.1 Theoretical analysis and algorithmsalgorithms

Theoretical analysisTheoretical analysis Two essential components in defining Two essential components in defining

a region merging algorithm:a region merging algorithm: Merging predicate: define how to merge Merging predicate: define how to merge

to undetermined region.to undetermined region. Order in merging: define an order to be Order in merging: define an order to be

followed to check the merging followed to check the merging predicate.predicate.

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3.1 Theoretical analysis and 3.1 Theoretical analysis and algorithmsalgorithms

Merging predicateMerging predicate Theorem 1 (The independent bounded Theorem 1 (The independent bounded

difference inequality).difference inequality). Let Let be a vector of n R.V.s. Suppose the real- be a vector of n R.V.s. Suppose the real-valued function f satisfies valued function f satisfies whenever vectors x and x’ differ only in kth whenever vectors x and x’ differ only in kth coordinate. Then, for any ,coordinate. Then, for any ,

where is the expected value of the R.V. where is the expected value of the R.V. f(X)f(X)

nXXXX ,...,, 21

kkc

Xf 2

2

)(2exp)(Pr

kcxfxf )'()(

0

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3.1 Theoretical analysis and 3.1 Theoretical analysis and algorithmsalgorithms

Merging predicateMerging predicate From thm 1, we obtain the result on the From thm 1, we obtain the result on the

deviation of observed differences between deviation of observed differences between regions of I.regions of I.

Corollary 1.Corollary 1. Consider a fixed couple Consider a fixed couple (R,R’) of regions of I. , the (R,R’) of regions of I. , the probability is no more than that probability is no more than that

10

)',(2

ln'

11

2

1)'()'( RRb

RRQgRRERR

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3.1 Theoretical analysis and 3.1 Theoretical analysis and algorithmsalgorithms

Merging predicateMerging predicate In the same statistical region, In the same statistical region,

and with a high probability that and with a high probability that does not exceed . does not exceed .

Merging predicate :Merging predicate : merge R and R’ iff merge R and R’ iff

0)'( RRE)(1 N 'RR

)',( RRb

)',(' RRbRR

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3.1 Theoretical analysis and 3.1 Theoretical analysis and algorithmsalgorithms

Order in mergingOrder in merging Ideally, the order to test the merging Ideally, the order to test the merging

of regions is:of regions is: when any test between two true regions when any test between two true regions

occurs, that means that all tests inside occurs, that means that all tests inside each of the two true regions have each of the two true regions have previously occurred.previously occurred.

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3.2 Theoretical analysis and 3.2 Theoretical analysis and algorithmsalgorithms

Other properties of the Other properties of the proposed approachproposed approach The proposed approach is proved that The proposed approach is proved that only only

overmerging occursovermerging occurs, with high probability., with high probability. The proposed approach has been shown to The proposed approach has been shown to

have an upperbound on the errorhave an upperbound on the error incurred incurred w.r.t. the optimal sementation, with high w.r.t. the optimal sementation, with high probability.probability.

The proposed approach is The proposed approach is easily extended easily extended to numerical channelsto numerical channels, such as RGB., such as RGB.

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3.3 Theoretical analysis and 3.3 Theoretical analysis and algorithmsalgorithms

Proposed algorithm: SRMProposed algorithm: SRM To choose a merging predicate and order To choose a merging predicate and order

in merging to approximate the ideal in merging to approximate the ideal segmentation method.segmentation method.

Merging predicate:Merging predicate: merge R and R’ iff merge R and R’ iff

Order in merging:Order in merging: choose a real-valued choose a real-valued function f and radix sort f(.,.) to function f and radix sort f(.,.) to approximate the order in merging. ( O(|I|approximate the order in merging. ( O(|I|*log(g)) )*log(g)) )

)',(' RRbRR

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4. Experimental results4. Experimental results

1. Introduction1. Introduction 2. The model of image generation2. The model of image generation 3. Theoretical analysis and algorithms3. Theoretical analysis and algorithms 4. Experimental results4. Experimental results

4.1 Choice of f4.1 Choice of f 4.2 Noise handling4.2 Noise handling 4.3 Enhance the noise handling ability4.3 Enhance the noise handling ability 4.4 Handling occlusions4.4 Handling occlusions 4.5 Controlling the scale of the segmentation4.5 Controlling the scale of the segmentation

5. Conclusion5. Conclusion

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4.1 Experimental results4.1 Experimental resultsChoice of fChoice of f

Choose , where and Choose , where and are the pixel channel values.are the pixel channel values.

The preordering can manage dramatic The preordering can manage dramatic improvements over conventional scanning.improvements over conventional scanning.

')',( aaa ppppf ap 'ap

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4.1 Experimental results4.1 Experimental resultsChoice of fChoice of f

A second choice of f is to use and in A second choice of f is to use and in Sobel filters, where smoothing filter is Sobel filters, where smoothing filter is performed by [1 2 1] and derivative filter performed by [1 2 1] and derivative filter is [-1 0 0 1].is [-1 0 0 1].

x y

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4.1 Experimental results4.1 Experimental resultsChoice of fChoice of f

Comparison of Comparison of the two choices the two choices of fof f

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4.2 Experimental results4.2 Experimental resultsNoise handlingNoise handling

Two noise types to be Two noise types to be handled:handled: Transmission noise t(q): Transmission noise t(q):

chosen uniformly in chosen uniformly in {1,2,...,g}{1,2,...,g}

Salt and pepper noise Salt and pepper noise s(q): chosen uniformly in s(q): chosen uniformly in {1,g}{1,g}

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4.3 Experimental results4.3 Experimental results Enhance the noise handling Enhance the noise handling

abilityability By integrating the moving average By integrating the moving average

operators, the first kind of f: operators, the first kind of f: is replaced by is replaced by

For the second kind of f, the smoothing For the second kind of f, the smoothing filter is extended to be [1 2 ... △+1 △ ... 1], filter is extended to be [1 2 ... △+1 △ ... 1], and the derivative filter is extended to be and the derivative filter is extended to be [-△ -△+1 ... △]. [-△ -△+1 ... △].

')',( aaa ppppf

apapa pNpNppf )'()()',(

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4.3 Experimental results4.3 Experimental results Enhance the noise handling Enhance the noise handling

abilityability Noise handling Noise handling

ability of the ability of the extended SRM extended SRM methods.methods.

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4.4 Experimental results4.4 Experimental results Handling occlusionsHandling occlusions

First run SRM as already presented.First run SRM as already presented. In a second stage, run SRM again with the In a second stage, run SRM again with the

modification of to modification of to , and 4-connexity to clique connexity. , and 4-connexity to clique connexity.

Radix sorting with f has an overall time Radix sorting with f has an overall time complexity O( (|I|+kcomplexity O( (|I|+k22)log)loggg) ).) ).

'(.,.) aaa ppf aaa RRRRf ')',(

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4.4 Experimental results4.4 Experimental results Handling occlusionsHandling occlusions

SRM with occlusion SRM with occlusion handling.handling.

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4.5 Experimental results4.5 Experimental results Controlling the scale of the Controlling the scale of the

segmentationsegmentation The objective of The objective of multiscale multiscale

segmentationsegmentation is to get a is to get a hierarchy of hierarchy of segmentationssegmentations at different scales. at different scales.

In SRM, scale is controlled by tuning In SRM, scale is controlled by tuning of parameter Q: as of parameter Q: as Q increasesQ increases, the , the regionsregions found are getting found are getting smallersmaller..

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5. Conclusion5. Conclusion

1. Introduction1. Introduction 2. The model of image generation2. The model of image generation 3. Theoretical analysis and 3. Theoretical analysis and

algorithmsalgorithms 4. Experimental results4. Experimental results 5. Conclusion5. Conclusion

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5. Conclusion5. Conclusion

A novel model of image generationA novel model of image generation is proposed, which is proposed, which captures the idea that grouping is an inference captures the idea that grouping is an inference problem.problem. A simple merging predicate and ordering in merging are A simple merging predicate and ordering in merging are

provided.provided. SRM suffers only SRM suffers only overmergingovermerging problems and achieves problems and achieves

low errorlow error in segmentation, both in segmentation, both with high probabilitywith high probability. . SRM is very fastSRM is very fast ( (segments a 512x512 image is in segments a 512x512 image is in

about one second on an Intel Pentium 4 2.4G about one second on an Intel Pentium 4 2.4G processorprocessor))

SRM is able to SRM is able to cope with significant noise corruptioncope with significant noise corruption, , handling occlusionshandling occlusions, and perform , and perform scale-sensitive scale-sensitive segmentationssegmentations..


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