Orientation Distribution Function for Diffusion MRI
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1 Orientation Distribution Function for Diffusion MRI Evgeniya Balmashnova 28 October 2009
2 Diffusion Tensor Imaging
3 Diffusion MRI
4 Diffusion MRI P(r, t) = 1 (4πDt) 3/2 e 1 4t r 2 D 1 t Diffusion time D Diffusion coefficient P Probability of travel to point r in time t
5 Diffusion Tensor Imaging P(r, t) = 1 (4π D t) 3/2 e 1 4t rt D 1 r D = D xx D xy D xz D yx D yy D yz D zx D zy D zz
6 Diffusion Tensor Imaging: Application
7 Diffusion Tensor Imaging
8 Diffusion Tensor Imaging
9 MRI and Diffusion Tensor Imaging
10 MRI and Diffusion Tensor Imaging
11 MRI and Diffusion Tensor Imaging
12 MRI and Diffusion Tensor Imaging
13 MRI and Diffusion Tensor Imaging
14 High Angular Resolution Diffusion Imaging S(g) = S 0 e bd(g) (Stejskal & Tanner, 1965)
15 High Angular Resolution Diffusion Imaging S(g) = S 0 e bd(g) (Stejskal & Tanner, 1965)
16 HARDI
17 HARDI
18 HARDI
19 Diffusion Orientation Distribution Function
20 Diffusion Orientation Distribution Function
21 Diffusion Orientation Distribution Function
22 Diffusion Orientation Distribution Function
23 Alternative Decompositions Spherical harmonics N l D N (g) = c lm Y lm (g) l=0 m= l High order tensors 3 3 D N (g) =... D i 1...i N g i1... g in i 1 =1 i N =1 Hierarchial tensors N 3 3 D N (g) =... D i 1...i l g i1... g il (Florack & Balmashnova, 2008) l=0 i 1 =1 i N =1
24 Alternative Decompositions Spherical harmonics N l D N (g) = c lm Y lm (g) l=0 m= l High order tensors 3 3 D N (g) =... D i 1...i N g i1... g in i 1 =1 i N =1 Hierarchial tensors N 3 3 D N (g) =... D i 1...i l g i1... g il (Florack & Balmashnova, 2008) l=0 i 1 =1 i N =1
25 Alternative Decompositions Spherical harmonics N l D N (g) = c lm Y lm (g) l=0 m= l High order tensors 3 3 D N (g) =... D i 1...i N g i1... g in i 1 =1 i N =1 Hierarchial tensors N 3 3 D N (g) =... D i 1...i l g i1... g il (Florack & Balmashnova, 2008) l=0 i 1 =1 i N =1
26 Alternative Decompositions Spherical harmonics N l D N (g) = c lm Y lm (g) l=0 m= l High order tensors 3 3 D N (g) =... D i 1...i N g i1... g in i 1 =1 i N =1 Hierarchial tensors N 3 3 D N (g) =... D i 1...i l g i1... g il (Florack & Balmashnova, 2008) l=0 i 1 =1 i N =1
27 Spherical Harmonics Regularization Simple formula for ODF Requires bookkeeping No maxima detection High Order Tensors No straightforward regularization No straightforward ODF formulas Simple bookkeeping Maxima detection algorithms
28 Spherical Harmonics Regularization Simple formula for ODF Requires bookkeeping No maxima detection High Order Tensors No straightforward regularization No straightforward ODF formulas Simple bookkeeping Maxima detection algorithms
29 Spherical Harmonics Regularization Simple formula for ODF Requires bookkeeping No maxima detection High Order Tensors No straightforward regularization No straightforward ODF formulas Simple bookkeeping Maxima detection algorithms
30 Spherical Harmonics Regularization Simple formula for ODF Requires bookkeeping No maxima detection High Order Tensors No straightforward regularization No straightforward ODF formulas Simple bookkeeping Maxima detection algorithms
31 Spherical Harmonics Regularization Simple formula for ODF Requires bookkeeping No maxima detection High Order Tensors No straightforward regularization No straightforward ODF formulas Simple bookkeeping Maxima detection algorithms
32 Spherical Harmonics Regularization Simple formula for ODF Requires bookkeeping No maxima detection High Order Tensors No straightforward regularization No straightforward ODF formulas Simple bookkeeping Maxima detection algorithms
33 1. E 0 (D 0 ) = (S(g) D 0 ) 2 dω, Ω 2. E n(d i 1...in n 1 ) = ((S(g) D i 1...i k g i1... g ik ) D i 1...in g i1... g in ) 2 dω Ω k=0 D i 1...i n g i1... g in span{y nm (g), m = n,..., n}
34 1. E 0 (D 0 ) = (S(g) D 0 ) 2 dω, Ω 2. E n(d i 1...in n 1 ) = ((S(g) D i 1...i k g i1... g ik ) D i 1...in g i1... g in ) 2 dω Ω k=0 D i 1...i n g i1... g in span{y nm (g), m = n,..., n}
35 1. E 0 (D 0 ) = (S(g) D 0 ) 2 dω, Ω 2. E n(d i 1...in n 1 ) = ((S(g) D i 1...i k g i1... g ik ) D i 1...in g i1... g in ) 2 dω Ω k=0 D i 1...i n g i1... g in span{y nm (g), m = n,..., n}
36 Regularization D N (g)= N 3 l=0 i 1 = i N =1 Di 1...i l g i1...g il D t (g)=e t g D(g)= N 3 l=0 i 1 = i N =1 Di 1...i l(t) g i1...g il D i 1...i l (t) = e tl(l+1) D i 1...i l
37 Regularization D N (g)= N 3 l=0 i 1 = i N =1 Di 1...i l g i1...g il D t (g)=e t g D(g)= N 3 l=0 i 1 = i N =1 Di 1...i l(t) g i1...g il D i 1...i l (t) = e tl(l+1) D i 1...i l
38 Regularization D N (g)= N 3 l=0 i 1 = i N =1 Di 1...i l g i1...g il D t (g)=e t g D(g)= N 3 l=0 i 1 = i N =1 Di 1...i l(t) g i1...g il D i 1...i l (t) = e tl(l+1) D i 1...i l
39 Regularization D N (g)= N 3 l=0 i 1 = i N =1 Di 1...i l g i1...g il D t (g)=e t g D(g)= N 3 l=0 i 1 = i N =1 Di 1...i l(t) g i1...g il D i 1...i l (t) = e tl(l+1) D i 1...i l
40 Regularization D N (g)= N 3 l=0 i 1 = i N =1 Di 1...i l g i1...g il D t (g)=e t g D(g)= N 3 l=0 i 1 = i N =1 Di 1...i l(t) g i1...g il D i 1...i l (t) = e tl(l+1) D i 1...i l
41 MRI and Diffusion Tensor Imaging
42 MRI and Diffusion Tensor Imaging
43 MRI and Diffusion Tensor Imaging
44 MRI and Diffusion Tensor Imaging
45 MRI and Diffusion Tensor Imaging
46 MRI and Diffusion Tensor Imaging
47 MRI and Diffusion Tensor Imaging
48 MRI and Diffusion Tensor Imaging
49 MRI and Diffusion Tensor Imaging
50 MRI and Diffusion Tensor Imaging
51 MRI and Diffusion Tensor Imaging
52 MRI and Diffusion Tensor Imaging
53 MRI and Diffusion Tensor Imaging
54 MRI and Diffusion Tensor Imaging
55 Orientation Distribution Function P(R) = S(q) exp ( 2πiq R)dq ODF(g) = 0 P(r g)dr
56 Orientation Distribution Function P(R) = S(q) exp ( 2πiq R)dq ODF(g) = 0 P(r g)dr
57 Orientation Distribution Function: Q-ball Approximation by Funk-Radon transform ODF (g) ζ[s](g) = δ(g T w)s(w)dw w =1 w =1 δ(g T w)y lm (w)dw = 2πP l (0)Y lm (g)
58 Orientation Distribution Function: Q-ball Approximation by Funk-Radon transform ODF (g) ζ[s](g) = δ(g T w)s(w)dw w =1 w =1 δ(g T w)y lm (w)dw = 2πP l (0)Y lm (g)
59 Orientation Distribution Function: Q-ball Approximation by Funk-Radon transform ODF (g) ζ[s](g) = δ(g T w)s(w)dw w =1 w =1 δ(g T w)y lm (w)dw = 2πP l (0)Y lm (g)
60 Orientation Distribution Function: Q-ball 1. Fit Nth order tensor decomposition to the signal by solving linear system of equations. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = 2π( 1)k/2 (k 1)!! k!! e k(k+1)t D i 1...i k.
61 Orientation Distribution Function: Q-ball 1. Fit Nth order tensor decomposition to the signal by solving linear system of equations. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = 2π( 1)k/2 (k 1)!! k!! e k(k+1)t D i 1...i k.
62 Orientation Distribution Function: Q-ball 1. Fit Nth order tensor decomposition to the signal by solving linear system of equations. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = 2π( 1)k/2 (k 1)!! k!! e k(k+1)t D i 1...i k.
63 Orientation Distribution Function: Q-ball 1. Fit Nth order tensor decomposition to the signal by solving linear system of equations. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = 2π( 1)k/2 (k 1)!! k!! e k(k+1)t D i 1...i k.
64 Diffusion Orientation transform (DOT) P(R) = S(q) exp ( 2πiq R)dq e 2πiq R = 4π P(R 0 g) = l l=0 m= l l l=0 m= l ( i) l j l (2πqr)Y lm (u)y lm(g) ( i) l Y lm (g) Y lm (u)i l(u)du
65 Diffusion Orientation transform (DOT) P(R) = S(q) exp ( 2πiq R)dq e 2πiq R = 4π P(R 0 g) = l l=0 m= l l l=0 m= l ( i) l j l (2πqr)Y lm (u)y lm(g) ( i) l Y lm (g) Y lm (u)i l(u)du
66 Diffusion Orientation transform (DOT) P(R) = S(q) exp ( 2πiq R)dq e 2πiq R = 4π P(R 0 g) = l l=0 m= l l l=0 m= l ( i) l j l (2πqr)Y lm (u)y lm(g) ( i) l Y lm (g) Y lm (u)i l(u)du
67 Diffusion Orientation transform (DOT)
68 Diffusion Orientation transform (DOT) Quality depends on R 0 There is no way to indicate optimal choice of R 0 Not robust to noise
69 Diffusion Orientation transform (DOT) Quality depends on R 0 There is no way to indicate optimal choice of R 0 Not robust to noise
70 Diffusion Orientation transform (DOT) Quality depends on R 0 There is no way to indicate optimal choice of R 0 Not robust to noise
71 P(R 0 g) = l,m ( i) l Y lm (g) Y lm (u)i l(u, R 0 )du I l (u, R 0 ) = 4π ODF(g) = ( i) l Y lm (g) l,m 0 J 1 (2πqR 0 )e 4π2 q 2 td(u) dq Y ( ) lm (u) I l (u, R 0 )dr 0 du 0
72 P(R 0 g) = l,m ( i) l Y lm (g) Y lm (u)i l(u, R 0 )du I l (u, R 0 ) = 4π ODF (g) = ( i) l Y lm (g) l,m 0 J 1 (2πqR 0 )e 4π2 q 2 td(u) dq Y ( ) lm (u) I l (u, R 0 )dr 0 du 0
73 P(R 0 g) = l,m ( i) l Y lm (g) Y lm (u)i l(u, R 0 )du I l (u, R 0 ) = 4π ODF (g) = ( i) l Y lm (g) l,m 0 J 1 (2πqR 0 )e 4π2 q 2 td(u) dq Y ( ) lm (u) I l (u, R 0 )dr 0 du 0
74 R l l+3 0Γ( 2 I l (u, R 0 ) = ) 2 l+3 π 3/2 (D(u)t) (l+3)/2 Γ(l + 3/2) 1 F 1 ( l ; l ; R2 0 4D(u)t ) 1 1F 1 (a, b, z) = t a 1 (1 t) b a 1 e zt dt 0 where 0 I l (u, R 0 )dr 0 = 1 Ĩ l D(u)t Ĩ l = Γ( l+1 2 ) 4lπ 3/2 Γ(l/2)
75 R l l+3 0Γ( 2 I l (u, R 0 ) = ) 2 l+3 π 3/2 (D(u)t) (l+3)/2 Γ(l + 3/2) 1 F 1 ( l ; l ; R2 0 4D(u)t ) 1 1F 1 (a, b, z) = t a 1 (1 t) b a 1 e zt dt 0 where 0 I l (u, R 0 )dr 0 = 1 Ĩ l D(u)t Ĩ l = Γ( l+1 2 ) 4lπ 3/2 Γ(l/2)
76 R l l+3 0Γ( 2 I l (u, R 0 ) = ) 2 l+3 π 3/2 (D(u)t) (l+3)/2 Γ(l + 3/2) 1 F 1 ( l ; l ; R2 0 4D(u)t ) 1 1F 1 (a, b, z) = t a 1 (1 t) b a 1 e zt dt 0 where 0 I l (u, R 0 )dr 0 = 1 Ĩ l D(u)t Ĩ l = Γ( l+1 2 ) 4lπ 3/2 Γ(l/2)
77 R l l+3 0Γ( 2 I l (u, R 0 ) = ) 2 l+3 π 3/2 (D(u)t) (l+3)/2 Γ(l + 3/2) 1 F 1 ( l ; l ; R2 0 4D(u)t ) 1 1F 1 (a, b, z) = t a 1 (1 t) b a 1 e zt dt 0 where 0 I l (u, R 0 )dr 0 = 1 Ĩ l D(u)t Ĩ l = Γ( l+1 2 ) 4lπ 3/2 Γ(l/2)
78 1 D(u)t = l,m α lm Y lm (u) ODF(g) = l,m ( 1) l/2 Ĩ l α lm Y lm (g)
79 1 D(u)t = l,m α lm Y lm (u) ODF(g) = l,m ( 1) l/2 Ĩ l α lm Y lm (g)
80 1 1. Fit Nth order tensor decomposition to D(u)t. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = ( 1)k/2 Γ( k+1 2 ) 4kπ 3/2 Γ(k/2) e k(k+1)t D i 1...i k.
81 1 1. Fit Nth order tensor decomposition to D(u)t. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = ( 1)k/2 Γ( k+1 2 ) 4kπ 3/2 Γ(k/2) e k(k+1)t D i 1...i k.
82 1 1. Fit Nth order tensor decomposition to D(u)t. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = ( 1)k/2 Γ( k+1 2 ) 4kπ 3/2 Γ(k/2) e k(k+1)t D i 1...i k.
83 1 1. Fit Nth order tensor decomposition to D(u)t. This yields D i 1...i k for k = 0,... N. 2. Compute the ODF coefficients D i 1...i k ODF (t) = ( 1)k/2 Γ( k+1 2 ) 4kπ 3/2 Γ(k/2) e k(k+1)t D i 1...i k.
84 Hardware phantom
85 Synthetic data Multi-tensor model n S(g) = p k e bgt D k g k=1
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88 Finsler Metric Riemannian metric Finsler metric.
89 Open Questions Voxel classification Scales selection Reality check
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