Hyperspherical Harmonic (HyperSPHARM) Representation

Transcription

1 Hyperspherical Harmonic (HyperSPHARM) Representation Moo K. Chung University of Wisconsin-Madison October 10, 2017, Florida State University

2 Abstracts Existing functional shape models such as the widely used spherical harmonic (SPHARM) representation assume topological invariance, so are unable to simultaneously parameterize multiple disconnected structures. In such a situation, SPHARM has to be applied separately to each individual structure. We present a novel surface parameterization technique using 4D hyperspherical harmonics (HyperSPHARM) in representing multiple disjoint objects as a single analytic form. The underlying idea behind HyperSPHARM is to project an entire collection of disconnected 3D objects onto the 4D hypersphere and simultaneously parameterize them with the 4D hyperspherical harmonics. Hence, HyperSPHARM allows for a holistic treatment of multiple disconnected structures. Although HyperSPHARM may yields similar reconstruction performance as SPHARM, HyperSPHARM can parameterize using much fewer basis functions and projection to 4D dimension obviates SPHARM s burdensome surface flattening. In addition, HyperSPHARM can handle any type of topology. The method is applied in modeling hippocampi and amygdalae of the human brain. The talk is based on paper Hosseinbor et al., 2015 Medical Image Analysis 22:89-101

3 ! Acknowledgements Pasha Hosseinbor, Nagesh Adluru, Ross Luo, Houri Voperian, Seth Pollack, Andrew Alexander, Hill Goldsmith, Richard Davidson University of Wisconsin-Madison NIH funding: EB022856, MH098098, MH061285

4 !!! Preliminary

5 Parametric shape models Fourier descriptors Spherical harmonic representation Laplace-Beltrami eigenfunction expansion

6 White matter fibers Up to half million tracts Each tract consists of about 300 control points.

7 l=0 Cosine series representation y parameterization x y x Any tract can be compactly parameterized with only 60 coefficients. basis expansion z x y z (x, y, z) = 19 β l cos(lπt)

8 Cosine series representation at various degrees

9 Tract matching Tract averaging Average of 5 tracts MATLAB: autism vs. controls wisc.edu/~chung/tracts

10 Question: Parameterize the whole white matter fibers using a single parameterization.

11 Surface parameterization 3T MRI Surface segmentation Surface flattening Spherical angle based coordinate system

12 Spherical harmonic of degree l and order m

13 Weighted-Spherical harmonics (SPHARM) Surface flattening v 1 v 2 v 3

14 SPHARM with different degrees Chung et al., 2007 IEEE Transactions on Medical Imaging 26:

15 Weighted-SPHARM heat kernel bandwidth, diffusion time Matlab: weighted-spharm/weighted-spharm.html

16 Laplace Beltrami eigenfunction expansion A C f = λf Cψ = λaψ MATLAB:

17 Laplace-Beltrami eigenfunctions on mandible

18 Heat kernel = probability distribution on manifold σ =0.2 σ = 10 K σ (p, q) = e λ jσ ψ j (p)ψ j (q) j=0

19 Heat kernel smoothing K σ X(p) = β j = j=0 e λ jσ X j ψ j (p) X(p)ψ j (p) dµ(p) X K σ X Chung, Qiu et al Medical Image Analysis 22:63-76

20 !!! Limitations

21 Existing parametric shape representations do not work for different topology Cancer growth Stroke lesions in brain Bone fusion

22 Hyoid bone fusion DS: down syndrome TD: typically developing

23 Bessel Fourier Reconstruction (BFOR)

24 2D cortical thickness Yellow: outer cortical surface Blue: inner cortical surface Chung et al NeuroImage 18:

25 Bessel Fourier reconstruction (BFOR) on cortical thickness Chung et al ISBI k=22, j=5 k=10, j=22

26 f(r, θ, ϕ) k l=0 l m= l j n=1 β lmn Z lmn (r, θ, ϕ) Z lmn (r, θ, ϕ) = S l ( λ ln r)y lm (θ, ϕ) S l (x) = π 2x J l+1/2(x)

27 Multi-shell reconstruction in diffusion weighted imaging 5 shells, 126 data points P 0 image Hosseinbor et al NeuoImage 64:

28 Hyper Spherical Harmonic (SPHARM) Representation

29 Flatland by Edwin A. Abbott, 1884 Disconnected Connected in in 3D 2D Question: Connect disconnected structures

30 Disconnected Connected in 3D 4D Question: Connect disconnected structures

31 3D stereographic projection 4D stereographic projection β (β, θ, φ) θ θ S = (S 1, S 2, S 3 )

32 4D stereographic projection

33 Hyper Spherical harmonic representation 3D coordinates S = (S 1, S 2, S 3 ) S i = N n n=0 l=0 m nl(β, θ, φ) =2 l+1/2 l m= l C i nlm Z m (β,θ,φ) nl (n + 1)Γ(n l + 1) πγ(n + l + 2) Spherical angles of a hypersphere Γ(l + 1) sin l β C l+1 m n! l (cos β) Yl (θ, φ) Gegenbauer polyonomials! 2π 0! π 0! π 0 Z m nl(ω)z m n l (Ω)sin 2 β sin θdβdθdφ = δ nn δ ll δ mm Hosseinbor et al., 2015 Medical Image Analysis 22:89-101

34 1764 parameters 140 parameters

35 Multi-shell reconstruction in diffusion weighted imaging 5 shells, 126 data points 14 parameters 30 parameters P 0 image Hosseinbor et al., 2015 Medical Image Analysis 21:15-28

36 What Next? Extremely complex multiple disconnected anatomical structures

37 Challenge: Parameterize the whole white matter fibers using HyperSPHARM.

38 Standard brain parcellation with 116 regions Precentral gyrus

39 19-layer hierarchical brain parcellation

40 Hierarchical nested connectivity

41 Extremely dense brain network nodes +0.6 billion connections HyperSPHARM Chung et al IPMI representation in R 3 R 3

42 June 22-23, 2018