Medical Image Analysis Dr. Mohammad Dawood Department of Computer Science University of Münster Germany

2 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Recap

3 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Sound wavesPiezoelectric crystalsWave front formation

4 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Inverse Radon transformFiltered back projection Filtered back projection

5 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Fourier slice theoremKaczmarz Method (=ART)

6 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Image Registration

7 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration T : Transformation In this lecture Floating image: The image to be registered Target image: The stationary image

8 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Linear Transformations - Translation - Rotation - Scaling - Affine

9 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration 3D Translation

10 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration 3D Rotation

11 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration 3D Scaling

12 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Rigid registration Angles are preserved Parallel lines remain parallel

13 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Affine registration

14 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Floating Target Registration Points cloud registration

15 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration PCA = Principle Components Analysis Compute a shape-aware coordinate system for each model Origin: Centroid of all points Axes: Directions in which the model varies most or least Transform the source to align its origin/axes with the target

16 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Principal Component Analysis Set of points p 1,…,p n with centroid location c Let P be a matrix whose i-th column is vector p i – c Build the covariance matrix: c F c T

17 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Principal Component Analysis Eigenvectors of M represent principal directions of shape variation Eigenvalues indicate amount of variation along each eigenvector Eigenvector with largest (smallest) eigenvalue is the direction where the model shape varies the most (least) Eigenvector with the largest eigenvalue Eigenvector with the smallest eigenvalue

18 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Noise Axes are affected PCA result Registration Principal Component Analysis Noise Symmetries Rotation by a small anglePCA result

19 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration SVD = Single Value Decomposition Assuming that for each source point the corresponding target point is known

20 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration SVD 1. De-mean 2. Compute SVD 3. Calculate the transform

21 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Translate Rotate

22 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration SVD Stable in noise Symmetries

23 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration ICP = Iterative closest point 1. Transform the floating image by PCA-based alignment 2. For each floating point, select the closest target point 3. Align floating and target images by SVD 4. Repeat steps 2 and 3 until error below threshold.

24 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood After PCA After 10 iterAfter 1 iter

25 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood After 10 iter After 1 iter After PCA

26 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Feature Points RANSAC = Random Sample Consensus Algorithm 1.Randomly choose s samples 2.Fit a model to those samples 3.Count the number of inliers that approximately fit the model 4.Repeat N times 5.Choose the model that has the largest set of inliers

27 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Distance Measures - Sum of Squared Differences (SSD) - Root Mean Square Difference (RMSD) - Normalized Cross Correlation (NXCorr) - Mutual Information (MI)

28 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Sum of Squared Differences SSD(f,t)SSD(20f,t)

29 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Root Mean Squared Differences RMS(f,t)RMS(20f,t)

30 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Normalized Cross Correlation NXCorr(f,t)NXCorr(20f,t)

31 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Registration Mutual Information MI(f,t)MI(20f,t)

32 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Entropy for Image Registration Define a joint probability distribution: – Generate a 2-D histogram where each axis is the number of possible greyscale values in each image – each histogram cell is incremented each time a pair (I 1 (x,y), I 2 (x,y)) occurs in the pair of images If the images are perfectly aligned then the histogram is highly focused. As the images mis-align the dispersion grows recall Entropy is a measure of histogram dispersion

33 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Optical Flow

34 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Optical flow Brightness consistency constraint With Taylor expansion V : Flow(Motion)

35 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood

36 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Optical flow Lucas Kanade Algorithm: Assume locally constant flow =>

37 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Optical flow Horn Schunck Algorithm: Assume globally smooth flow

38 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Optical flow Bruhn’s Non-linear Algorithm

39 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood PET no AC – gated PET no AC – gated, MC corrected

40 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Cardiac motion Ungated data Large blur, low noise (Problem in plaque imaging) One phase Small blur, high noise

41 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood SystoleDiastole Continuity Equation:

42 Medical Image Analysis, SS-2015 Dr. Mohammad Dawood Visual result