Image Registration

1. Image Registration 박성진

2. Surface-based Registration • The 3D boundary of an anatomical object is an intuitive and easily characterized geometrical feature that can be used for registration • Surface-based methods • Determine corresponding surfaces in different images • Find the transformation that best aligns these surfaces • Point-based registration • Aligns generally small number of corresponding points • Surface-based registration • Aligns larger number of points for which correspondence is unavailable

3. Surfaces • Skin surface (air-skin interface) • Bone surface (tissue-bone interface) • Representations • Point set (collection of points on the surface) • Faceted surface, e.g., triangle set approximating surface • Implicit surface • Parametric surface, e.g., B-spline surface

4. Surface-based Registration • Disparity function • Given a set of surface points and a surface, find the rigid transformation that minimizes the mean squared distance between the points and the surface

5. Head and Hat Method

6. Iterative Closest Point Method • Initialization: • Iteratively apply the following steps, incrementing k after each loop, until convergence within a tolerance is achieved: • Compute the closest points • Compute the transformation between the initial point set and current set • Apply the transformation to produce registered points • Terminate the iterative loop when

7. Intensity-based registration • Registration based on similarity measures • Uses some measure derived from the intensity of the image directly • Assumes that there is a relationship between the image intensities of both images if the images are registered • Does not require any feature extraction, thus the registration error is not by any errors

8. Generic Intensity-based Registration Procedure Initial transformation Calculate cost function For transformation T Optimize T by maximizing cost function C Update transformation Final transformation Is new transformation an improvement?

9. Intensity-based registration • Registration-based on geometric features is independent of the modalities from which the features have been derived • Registration-based on voxel similarity measures features we must make a distinction between monomodality registration and multimodality registration

10. Monomodality image registration • Sums of Squared Differences (SSD) • Assumes an identity relationship between image intensities in both images • Optimal measure if the difference between both images is Gaussian noise • Sensitive to outliers

11. Monomodality image registration • Robust statistics can be used to reduce the influence of outliers on the registration • Sum of Absolute Differences (SAD) • Assumes an identity relationship between image intensities • Less sensitive to outliers

12. Monomodality image registration • Correlation • Assumes a linear relationship between image intensities • Sensitive to large intensity values

13. Monomodality image registration • Normalized Cross Correlation (CC) • Assumes a linear relationship between image intensities

14. Monomodality image registration • Ratio of Image Uniformity (RIU) • Normalized standard deviation

15. Registration Basis : Image Intensity • Monomodality registration • Image intensities are related by simple function • Identity : SSD, SAD • Linear : CC, RIU • Multimodality registration • Image intensities are related by some unknown function or statistical relationship • Relationship between intensities is not known a priori • Relationship between intensities can be viewed by inspecting a 2D histogram or co-occurrence matrix

16. Multimodality image registration • Partitioned image uniformity (PIU) • Used for MR-PET registration • PIU : Measure the sum of the normalized standard deviation of voxel values in image B for each intensity a in image A

17. Images as Probability Distribution • Images can be viewed as probability distributions p(a) • Marginal probability p(a) of a pixel having intensity a • Joint probability p(a,b) of a pixel having intensity a in one image and intensity b in another image • Probability distribution of an image can be estimated using • Parzen windowing • Histograms • Histograms require “binning” • Usually use 32 to 256 bins per image

18. Images as Probability Distribution

19. Intensity-based on IT • Entropy • Describes the amount of information in image A • The information content of an image is maximal if all intensities have equal probability • The information content of an image is minimal if one intensity a has a probability of one

20. Intensity-based on IT • Joint Entropy • Describes the amount of information in the combined images A and B • If A and B are totally unrelated, the joint entropy will be the sum of the entropies of A and B • If A and B are related, the joint entropy will be similar • Registration can be achieved by minimizing the joint entropy between both images

21. Intensity-based on IT • Joint Entropy is highly sensitive to the overlap of the two images • Mutual information • Describes how well one image can be explained by another images • Expressed in terms of marginal and joint probability distributions

22. Intensity-based on IT • Mutual information is still sensitive to the overlap of the two images • Normalized mutual information can be shown to be independent of the amount of overlap between images • Registration can be achieved by maximizing (normalized) Mutual Information between both images

23. Registration using Similarity Measures • Some similarity measures assume a functional relationship between intensities • Identity : SSD, SAD • Linear : CC, RIU • Nonlinear : PIU, CR • Other similarity measures only assume a statistical relationship • Joint entropy • (Normalized) Mutual Information • All similarity measures can be calculated from a 2D histogram of the images