Improving Entropy Registration

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Improving Entropy Improving Entropy RegistrationRegistration

Theodor D. RichardsonTheodor D. Richardson

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Preliminary ResultsPreliminary ResultsOriginal Rotation: 12Entropy Result: -11Segmented Entropy Result: -13

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The Basic Concepts of The Basic Concepts of EntropyEntropy

• Each pixel (or voxel) has a Each pixel (or voxel) has a probability of occurrence, probability of occurrence,

p{n}log(p{n})p{n}log(p{n})• These probabilities make up an These probabilities make up an

entropy for the image, H(N) where n entropy for the image, H(N) where n is the imageis the image

• H(N) = -H(N) = -∑∑p{n}log(p{n})p{n}log(p{n})n Є N

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Comparing EntropiesComparing Entropies

• Two images with entropies H(M) and Two images with entropies H(M) and H(N) will have a mutual or joint entropy H(N) will have a mutual or joint entropy H(M, N) when they are overlaidH(M, N) when they are overlaid

•H(M,N) =H(M,N) = - ∑∑- ∑∑p{m, n}log(p{m,n})p{m, n}log(p{m,n})

•This is a volume of overlapThis is a volume of overlap

n Є N m Є M

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Comparing EntropiesComparing Entropies

• The sum of marginal entropies for The sum of marginal entropies for this is this is I(M,N) = H(M) + H(N) – H(M,N)I(M,N) = H(M) + H(N) – H(M,N)

• Maximizing the value of the marginal Maximizing the value of the marginal entropies is the goal of this entropies is the goal of this algorithm; this means that the two algorithm; this means that the two images will have the most features in images will have the most features in commoncommon

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Problems with the Entropy Problems with the Entropy AlgorithmAlgorithm

• Noise changes probability of Noise changes probability of intensities, causing misread resultsintensities, causing misread results

• Background of image may be a Background of image may be a factor in alignment when it should be factor in alignment when it should be invariant to backgroundinvariant to background

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Estimating EntropiesEstimating Entropies

• The entropy of a pixel can be The entropy of a pixel can be estimated by the histogram intensity estimated by the histogram intensity over the total number of pixels in the over the total number of pixels in the image.image.

• A frequently occurring pixel has less A frequently occurring pixel has less likelihood of being aligned perfectly likelihood of being aligned perfectly than a rarely occurring pixelthan a rarely occurring pixel

• These values can be weighted by 1 – These values can be weighted by 1 – p(n) where n is the pixel intensityp(n) where n is the pixel intensity

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Simple Segmentation Simple Segmentation AlgorithmAlgorithm•The problems with entropy may be The problems with entropy may be

helped by segmenting the image helped by segmenting the image first.first.

•This can remove background noise This can remove background noise by eliminating the noisy regionby eliminating the noisy region

•Watershed method was first Watershed method was first attempted, but the gathered attempted, but the gathered regions were too smallregions were too small

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Simple Segmentation Simple Segmentation AlgorithmAlgorithm

•New segmentation algorithm New segmentation algorithm based on region-growing from based on region-growing from input parameters.input parameters.

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Simple Segmentation Simple Segmentation AlgorithmAlgorithm•Find regions of image with desired Find regions of image with desired

intensity within tolerance boundsintensity within tolerance bounds•Create edges from connecting Create edges from connecting

pixels to expand regionspixels to expand regions•Select largest regionSelect largest region•Optionally enclose regionOptionally enclose region•Create mask over imageCreate mask over image

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•Mask examples:Mask examples:

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•Regions outside of the mask are Regions outside of the mask are given a probability of 0 and are given a probability of 0 and are not counted in total pixelsnot counted in total pixels

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Simple Segmentation Simple Segmentation AlgorithmAlgorithm• Intensity shift can adapt this segmentation Intensity shift can adapt this segmentation

method to intensity comparison method to intensity comparison alignmentsalignments

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Basic Entropy AlgorithmBasic Entropy Algorithm

•The entropy (mutual information) The entropy (mutual information) alignment algorithm for this project alignment algorithm for this project makes the assumption that the makes the assumption that the image is centered alreadyimage is centered already

•This alignment algorithm focuses This alignment algorithm focuses only on maximizing global mutual only on maximizing global mutual information information

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Basic Entropy AlgorithmBasic Entropy Algorithm

• Create image mask of probabilities for template Create image mask of probabilities for template and comparison imagesand comparison images

• Rotate comparison image through 360 degrees by Rotate comparison image through 360 degrees by Affine Transformation of rotation around z-axisAffine Transformation of rotation around z-axis

cos cos ΘΘ sin sin ΘΘ 00 00- sin - sin ΘΘ cos cos ΘΘ 00 0000 00 11 0000 00 00 11

• If pixel probabilities are within tolerance, add to If pixel probabilities are within tolerance, add to volumevolume

• Maximum volume is maximum mutual informationMaximum volume is maximum mutual information

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ResultsResults• The following is a sample of the results of the entropy The following is a sample of the results of the entropy

algorithm with and without segmentation:algorithm with and without segmentation:

Original Rotation: 29Entropy Result: -25Segmented Entropy: -28

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ProblemsProblems

• This entropy algorithm is not the This entropy algorithm is not the most robust available; some use most robust available; some use local entropy within the global local entropy within the global information and some normalize the information and some normalize the registration volumeregistration volume

• The assumption of a centered image The assumption of a centered image is not valid for most imagesis not valid for most images

• This entropy algorithm does not This entropy algorithm does not involve normalizing the joint entropy involve normalizing the joint entropy with the overall entropywith the overall entropy

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Possible Future ResearchPossible Future Research

•Expand the application of the Expand the application of the Simple Segmentation Algorithm Simple Segmentation Algorithm to other registration techniquesto other registration techniques

•Experiment further with different Experiment further with different mutual information algorithms mutual information algorithms and different segmentation and different segmentation algorithmsalgorithms

Improving Entropy Registration

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