1 Improving Entropy Registration Theodor D. Richardson

2 Preliminary Results Original Rotation: 12 Entropy Result: -11 Segmented Entropy Result: -13

3 The Basic Concepts of Entropy Each pixel (or voxel) has a probability of occurrence, p{n}log(p{n}) Each pixel (or voxel) has a probability of occurrence, p{n}log(p{n}) These probabilities make up an entropy for the image, H(N) where n is the image These probabilities make up an entropy for the image, H(N) where n is the image H(N) = - ∑ p{n}log(p{n}) H(N) = - ∑ p{n}log(p{n}) n Є N

4 Comparing Entropies Two images with entropies H(M) and H(N) will have a mutual or joint entropy H(M, N) when they are overlaid Two images with entropies H(M) and H(N) will have a mutual or joint entropy H(M, N) when they are overlaid H(M,N) = - ∑∑ p{m, n}log(p{m,n}) H(M,N) = - ∑∑ p{m, n}log(p{m,n}) This is a volume of overlap This is a volume of overlap n Є Nm Є M

5 Comparing Entropies The sum of marginal entropies for this is I(M,N) = H(M) + H(N) – H(M,N) The sum of marginal entropies for this is I(M,N) = H(M) + H(N) – H(M,N) Maximizing the value of the marginal entropies is the goal of this algorithm; this means that the two images will have the most features in common Maximizing the value of the marginal entropies is the goal of this algorithm; this means that the two images will have the most features in common

6 Problems with the Entropy Algorithm Noise changes probability of intensities, causing misread results Noise changes probability of intensities, causing misread results Background of image may be a factor in alignment when it should be invariant to background Background of image may be a factor in alignment when it should be invariant to background

7 Estimating Entropies The entropy of a pixel can be estimated by the histogram intensity over the total number of pixels in the image. The entropy of a pixel can be estimated by the histogram intensity over the total number of pixels in the image. A frequently occurring pixel has less likelihood of being aligned perfectly than a rarely occurring pixel A frequently occurring pixel has less likelihood of being aligned perfectly than a rarely occurring pixel These values can be weighted by 1 – p(n) where n is the pixel intensity These values can be weighted by 1 – p(n) where n is the pixel intensity

8 Simple Segmentation Algorithm The problems with entropy may be helped by segmenting the image first. The problems with entropy may be helped by segmenting the image first. This can remove background noise by eliminating the noisy region This can remove background noise by eliminating the noisy region Watershed method was first attempted, but the gathered regions were too small Watershed method was first attempted, but the gathered regions were too small

9 Simple Segmentation Algorithm New segmentation algorithm based on region-growing from input parameters. New segmentation algorithm based on region-growing from input parameters.

10 Simple Segmentation Algorithm Find regions of image with desired intensity within tolerance bounds Find regions of image with desired intensity within tolerance bounds Create edges from connecting pixels to expand regions Create edges from connecting pixels to expand regions Select largest region Select largest region Optionally enclose region Optionally enclose region Create mask over image Create mask over image

11 Simple Segmentation Algorithm Mask examples: Mask examples:

12 Simple Segmentation Algorithm Regions outside of the mask are given a probability of 0 and are not counted in total pixels Regions outside of the mask are given a probability of 0 and are not counted in total pixels

13 Simple Segmentation Algorithm Intensity shift can adapt this segmentation method to intensity comparison alignments Intensity shift can adapt this segmentation method to intensity comparison alignments

14 Basic Entropy Algorithm The entropy (mutual information) alignment algorithm for this project makes the assumption that the image is centered already The entropy (mutual information) alignment algorithm for this project makes the assumption that the image is centered already This alignment algorithm focuses only on maximizing global mutual information This alignment algorithm focuses only on maximizing global mutual information

15 Basic Entropy Algorithm Create image mask of probabilities for template and comparison images Create image mask of probabilities for template and comparison images Rotate comparison image through 360 degrees by Affine Transformation of rotation around z-axis cos Θ sin Θ 00 - sin Θ cos Θ Rotate comparison image through 360 degrees by Affine Transformation of rotation around z-axis cos Θ sin Θ 00 - sin Θ cos Θ If pixel probabilities are within tolerance, add to volume If pixel probabilities are within tolerance, add to volume Maximum volume is maximum mutual information Maximum volume is maximum mutual information

16 Results The following is a sample of the results of the entropy algorithm with and without segmentation: The following is a sample of the results of the entropy algorithm with and without segmentation: Original Rotation: 29 Entropy Result: -25 Segmented Entropy: -28

17 Problems This entropy algorithm is not the most robust available; some use local entropy within the global information and some normalize the registration volume This entropy algorithm is not the most robust available; some use local entropy within the global information and some normalize the registration volume The assumption of a centered image is not valid for most images The assumption of a centered image is not valid for most images This entropy algorithm does not involve normalizing the joint entropy with the overall entropy This entropy algorithm does not involve normalizing the joint entropy with the overall entropy

18 Possible Future Research Expand the application of the Simple Segmentation Algorithm to other registration techniques Expand the application of the Simple Segmentation Algorithm to other registration techniques Experiment further with different mutual information algorithms and different segmentation algorithms Experiment further with different mutual information algorithms and different segmentation algorithms