for Images A Bayesian Deformation Model

Transcription

1 Statistics in Imaging Workshop July 8, 2004 A Bayesian Deformation Model for Images Sining Chen Postdoctoral Fellow Biostatistics Division, Dept. of Oncology, School of Medicine

2 Outline 1. Introducing the problem 2. Model description 3. Application to mouse brain segmentation 4. Inter-subject registration of human brains

3 Problems to Solve Identifying abnormal tissue from a series of gated cardiac images Marking a region of interest in a large number of human brain MRIs Repeating the same operation on similar images

4 Image Registration To bring two similar images into spatial alignment, such that corresponding points of the imaged scene appear in the same position on the registered images. Also referred to as spatial normalization in certain contexts.

5 Registration Methods Curved transformations using basis functions (Ashburner and Friston 1999 Human Brain Mapping), implemented in the software package SPM. Models derived from the physics of elastic objects (Bajcsy and Kovacic 1989 Comp. Vision, Graphics & Image Processing) Spatial modeling in deformable templates...

6 Facets generalized landmarks Deformation model mechanism: place large number of facets in the volume of template image. Then locate each in the target image. template target

7 Model Heuristic A balance between preserving spatial relationships expressed in the prior matching image features expressed in the likelihood

8 Notation Template Target facet location µ x facet feature φ f The feature of a facet is a function of the image evaluated at that location: can be intensity (brightness), gradience (edgeness), laplacian (medialness), etc.

9 (a) (b) (c) (d) (e) (f) Figure 1: a) Original image; b) local rank of intensity;

10 Prior preseving spatial relationship Markov random field prior specified on a first-degree neighborhood replacements system. PSfrag replacements The spatial relationship among i and γij its neighbors i in the target image remains similar before and after deformation. µ1 µ2 x1 µ1 µ3 µ4 µ5 µ2 µ3 µ4 x1 x2 µ3 x 3 x4 γij i j x2 µ3 x 3 µ5 x4 x5 x5

11 Prior continued p(x) = 1 Z(γ) exp{ 1 2 γij((xi µi) (xj µj)) a } i,j p(xi.) exp{ 1 2 Full conditional (let a = 2): j i γij((xi µi) (xj µj)) 2 } exp{ 1 2 ( j i γij)(xi µi j i γ ij(xj µj) j i γ ij ) 2 }

12 Likelihood matching features For each facet i, feature after deformation fi should be similar to template feature φi p(xi T, φi) = 1 Ci(T, φi) exp{sim(f i, φi)} where fi = QT (xi)

13 g replacements Measure of Feature Similarity sim(fi, φi) = b(ri ρi) α cos(ui νi), α > 0 ri, ρi: quantiles of gradient magnitude, ui, νi: directions of gradients. (u, r) (ν, ρ) θ (u, r) (ν, ρ)

14 Measure of Feature Similarity continued Advantages: robust under weak assumptions on intensity ability to distinguish between meaningful and meaningless features accomodates no-match situations

15 Posterior Inference MAP (maximum a posteriori) estimate can be obtained by ICM (iterative conditional modes, Besag 1986). To obtain estimate that is closer to global maximum, we build a hierarchy of facets in scale-space.

16 Facet Tree in Scale-Space

17 Augmented Full-Conditional Spatial part only p(xi xs, s i, xp) exp{ γp((xi µi) (xp µp)) 2 + s i γis((xi µi) (xs µs)) 2 } Facet i, facet target location xi, facet template location µi, i s siblings s i, i s parent p, pre-selected parameters γp and γis

18 Automated Segmentation of Mouse Brain Images

19 Manual Segmentation of Mouse Hippocampus Goal: to investigate the relationship of Alzheimer s disease to changes in the volume of the hippocampus.

20 Automatic segmentation of another mouse MRI

21 Template Image Figure 2: Three regions of interest are marked on the template image with rectangles.

22 Figure 3: Facets before and after deformation.

23 Quantitative Assessment of Auto-segmentation PSfrag replacements mi ti ti cii log m (k) i = log ti + ɛ (k) i, ɛ N(0, 2 ) log c (k) ii = log ti + (log m (k) i log ti) + e (k) ii, e N(0, δ2 )

24 Figure 4: a) 2, b) δ 2 In our dataset, the inconsistency of auto-segmentation is approximately the same as human segmentation.

25 Inter-subject Registration of Human Brain MRIs

26 Are all brains alike?

27 Issues in evaluating inter-subject registration Problematic due to lack of information on ground truth. What was done: Qualitative assessment: visual inspection, mosaic, facet movement; Quantitative evaluation: segmentation based, cross-correlation; Done in comparison to the industrial standard SPM.

28 Mosaic before registration after facet reg. after SPM reg.

29 Facet movement Figure 5: Locating facets on subject 3

30 Human brain MRI segmentation original MRI white matter gray matter (a) (b) (b) (a) (b) (c)

31 Overlap, misclassification Figure 6: Comparison of gray matter segmentation of warped template and manually segmented target

32 Segmentation based evaluation

33 misclassify (%) facet SPM subject subject subject Table 1: Percent misclassified for white matter

34 overlap (%) facet SPM subject subject subject Table 2: Percentage overlap for white matter

35 misclassify (%) facet SPM subject subject subject Table 3: Percent misclassified for gray matter

36 overlap (%) facet SPM subject subject subject Table 4: Percentage overlap for gray matter

37 Normalized Cross-covariance Map C(f, g) = n A f ij gij A f ij A g ij n ( A f 2 + n 2 n ( 2 A f)2 )( A g2 + n 2 n ( 2 A g)2 ) (1)

38 facet SPM

39 Conclusion Used Markov random field to model prior belief of facet locations; proposed a new, robust measure of similarity; successfully performed auto-segmentation of mouse brain hippocampi given a segmented template; registered brain MRI of different individuals; quantitative evaluated performance of registration; the model performs marginally better than SPM.

40 Thank you!