Improving Entropy Registration

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}) • These probabilities make up an entropy for the image, H(N) where n is the image • 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 • H(M,N) =- ∑∑p{m, n}log(p{m,n}) • This is a volume of overlap n Є N m ЄM

5. Comparing Entropies • 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

6. Problems with the Entropy Algorithm • Noise changes probability of intensities, causing misread results • 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. • 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

8. Simple Segmentation Algorithm • The problems with entropy may be helped by segmenting the image first. • This can remove background noise by eliminating the noisy region • 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.

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

11. Simple Segmentation Algorithm • Mask examples:

12. Simple Segmentation Algorithm • 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

14. Basic Entropy Algorithm • 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

15. Basic Entropy Algorithm • Create image mask of probabilities for template and comparison images • Rotate comparison image through 360 degrees by Affine Transformation of rotation around z-axiscos Θ sin Θ 0 0- sin Θ cos Θ 0 00 0 1 00 0 0 1 • If pixel probabilities are within tolerance, add to volume • Maximum volume is maximum mutual information

16. Results • 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 • 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

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