Some Methods for Modeling Cortical Surface

1. Yuai Hua Some Methods for Modeling Cortical Surface 2010.10.29

2. What is a cortical surface like?

3. Gyral Region Gyrus Crest Line Sulcal Region Sulcal Fundi Gyral region, sulcal region, gyral crest, sulcal fundi

4. Gyral Basin Sulcal Basin Sulcal basin & Gyral basin (cross-section)

5. What will I present ? • Cortical sulcal parcellation • Cortical fundi extraction • Cortical gyral parcellation • Cortical sulcal bank segmentation • Gyral folding pattern analysis

6. Part A. Cortical sulcal parcellation sulcal region Goal: Finding sulcal basin & Key Technique: Principal direction flow field tracking method (proposed by Gang Li et al.)

7. Some Basic Concepts Principal curvatures ----- the maximum and minimum values of curvatures at a point p on a surface. Principal directions ----- the vectors along which the curvatures are principal.

8. principal directions

9. Triangulated cortical surface

10. How to find sulcal regions & sulcal basins? Stages:(1) Estimate principal curvatures and principal directions at each point;(2) Finding sulcal regions;(3) Finding sulcal basins.

11. X6 X1 X X5 X2 X4 X3 (1) Estimate principal curvatures and principal directions at each point. Step1. Calculate the normal vectors of each triangle face.

12. Step2.Calculate Weingarten Matrix in each triangle face. Weingarten Matrix • Weingarten Matrix is a symmetric matrix. • Its eigenvalues are the principal curvatures. • Its eigenvectors are the principal directions.

13. X6 X1 X X5 X2 X4 X3 Step3. Calculate Weingarten Matrix at each vertex by weighted averaging its adjacent faces. Step4.Calculate the eigenvalues and eigenvectors of each Weingarten Matrix. Those are the principal curvatures and principal directions.

14. Only the maximum principal curvatures and its corresponding principal directions are adopted.

15. Principal direction fundi Keeping in mind…… Calculating the directional derivative of maximum principal curvature along the corresponding principal direction and ensuring it decreases by choosing appropriate principal direction.

16. Thus • The principal direction points towards the sulcal fundus from the gyral crest; • The principal curvatures are large positive and negative values at gyral crown and sulcal fundi.

17. (2) Finding sulcal regions Let n = the number of the total vertices on the cortical surface

18. In order to segment cortical surface into sulcal regions and gyral regions, we should solve Problem : is unknown is a normalrandom variable Knowing: & can be estimated.

19. Hence, according to the Bayes theory and a special method (proposed by Zhang et al., 2010), we can estimateX by solving So far, a cortical surface is segmented into a series of sulcal regions and gyral regions. During this process, hidden Markov random field model, expectation maximization method and iterated algorithms are used.

20. Principal direction fundi (3) Finding sulcal basins Idea:following the maximum principal directions from a gyral crown region until to the sulcal fundus. The vertices that converge to the same fundus are grouped together, and these vertices form a sulcal basin. Thus the cortical surface are segmented into different sulcal basins.

21. Step1 Estimating flow field Problem: at a flat cortical region, the two principal curvatures might be very small, so we may not find the exactly maximum principal direction. Idea: at a sulcal fundus and gyral crown, the produced flow field should be close to the original direction field, and at flat areas, it should vary smoothly. Method: Principal direction flow field diffusion

22. Estimating flow field V(X) n: normal vector where maximum principal curvature maximum principal direction weighting parameter gradient operator Using calculus of variation to solve the equation

23. Step2Sulcal basin segmentation Method:Principal direction flow field tracking —Searching for flow trajectories Given a vertex on a flow trajectory with the principal direction the next vertex is calculated as . the one-ring adjacent vertex of

24. X6 X1 X5 =x′ X V(X) X2 X4 X3

25. X V(X) x′ V(X′) X V(X) V(X′)

26. The region at which the flow field tracking procedure stops should be a fundus. Thus every vertex flows to a fundus. Those vertices flow to the same fundus are grouped together as a sulcul basin. Cortical sulcal parcellation is over

27. Part B. A pipeline for cortical fundi extraction (Proposed by Gang Li et al)

28. Step1 Estimating curvatures and curvature derivatives Step2Detecting sulcal fundi segments The maximum principal curvature The minimum principal curvature , The principal directions. Directional derivative

29. Criterion for fundus point : Fundus point

30. Procedure for fundi segmenting (1) Procedure for finding fundi points: in the cortical surface For each triangle ) (three vertices are

32. (2) Connect the adjacent fundi points to form fundi segments:

33. Two types of Provisional fundi segments Candidatefundi segment:if there is any candidate fundi point in it; Strict fundi segment:if there is no candidate fundi point in it;

34. (3) Linking sulcal fundi segments a) Starting from a strict fundi segment, adding the adjacent segments to it, and go on, a fundi curve will be obtained. There may be more than one fundi curve. b) Expanding again For every vertex with negative maximum curvature in a fundi curve, connect itself with its adjacent vertex which is in another fundi curve, obtain a new fundi curve.

35. c) Pruning the fundi curves less than three segments. The remain fundi curves may include some very short ones. Two types of short fundi curves: interrupted and inherent. Next, it is necessary to connect those interrupted short fundi curves to the long ones. How to tell the different kinds of the short fundi curve?

36. d) From each endpoint of the extracted fundi curve, searching the geodesic region to find whether there exists another fundi curve in the region. If any, connect the endpoints to the newly found sulcal fundi curve. e)Smoothing the extracted sulcal fundi curves. Due to the numerical error, the extracted sulcal fundi curves may exist sharp bumps, it should be smoothed. Method : minimizing the geodesic distance between the endpoints or junction points of each piece of the extracted sulcal fundi.

37. Part C.Cortical Gyral Parcellation Technique: Using probabilistic atlas and graph cuts (proposed by Gang Li et al.)

38. Characteristics of a gyrus: • Each gyral patch is a part of gyral basin • Eachgyral patchisbounded by adjacent sulcal fundi and interrupted at junctions of gyral basins • Each gyral patch belongs to only one gyral basin • Each gyral basin is composed of one or more gyral patches.

39. What we know: Every kinds of gyral patchs have been labeled by experts in the form of Probabilistic Atlas. • Probabilistic atlas is a series of maps of human brain anatomic regions. These maps were produced from a set of whole-head MRI. Each MRI was manually delineated to identify a set of 56 structures in the brain, most of which are within the cortex. • Each type of region has a label.

40. n: number of gyral patches k-th gyral structure in the Probabilitic Atlas p-th gyral patch in sulcal surface Area of k-th gyral structure in the P-Atlas Area of p-th gyral patch in sulcal surface Area of the intersection belonging to Likelihood of The label which the p-th gyral patch in sulcal surface be assigned corresponding to the gyral basin in the P-Atlas .

41. Maximum principal curvature at vertex in gyral patch . Set of all neighboring vertices between two neighboring gyral patches. represents the weight between two neighboring gyral patches.

42. . N:the set of neighboring gyral patch pairs an adjust parameter. by using graph cuts method. The cortex is segmented into different gyral basins

43. Part D.Cortical sulcal bank segmentation Sulcal bank: Each side of a sulcal basin. • A sulcal basin has two opposite sulcal banks. Goal:Segment a sulcal basin into two sulcal banks. Technique:Graph partition (proposed by Gang Li et al.)

44. Procedure Step1: Segment a cortical surface into sulcal basins. : Step2:Rough sulcal bank segmentation The triangular mesh of a sulcal basin Angular similarity between two vertices ( unit normal direction) are in the same will be large if otherwise, small. sulcal bank;

45. Distance similarity between two vertices Euclidean distance; : geodesic distance (the shortest path connecting two vertices along the triangular mesh of a sulcal basin) will be large if are in the same sulcal bank; otherwise, small.

46. Similarity between two vertices ( is a weight parameter) are in the same sulcal will be large if bank; otherwise, small.

47. Firstly, using Normal cuts method to divide the sulcal baisn into two opposing sulcal banks A and B Then set , Rough graph partition:

48. Step3: Fine sulcal bank segmentation Goal: making the boundary clearer. Technique: construct a energy function and minimize it. Cortical sulcal bank segmentation is over

49. Part E.Gyral folding pattern analysis (proposed by Kaiming Li, et al)

50. Gyral folding patterns: