Wavelet Frames on the Sphere for Sparse Representations in High Angular Resolution Diusion Imaging. Chen Weiqiang

Transcription

1 Wavelet Frames on the Sphere for Sparse Representations in High Angular Resolution Diusion Imaging Chen Weiqiang

2 Overview 1. Introduction to High Angular Resolution Diusion Imaging (HARDI). 2. Wavelets on the Sphere for L 2 sym(s 2 ). 3. Applications of Wavelets for sparse representation of HARDI Signals. 4. Possible Future Work

3 1. Introduction to HARDI Diusion Magnetic Resonance Imaging to nd directionality of neural bers via water diusion in brain tissues.

4 1. Introduction to HARDI Diusion Magnetic Resonance Imaging to nd directionality of neural bers via water diusion in brain tissues. Directionality of neural bers useful for characterization of neuro-degenerative diseases & surgical planning.

5 1. Introduction to HARDI Diusion Magnetic Resonance Imaging to nd directionality of neural bers via water diusion in brain tissues. Directionality of neural bers useful for characterization of neuro-degenerative diseases & surgical planning. Stejskal & Tanner's model: given MR signal s, reconstruct its diusion probability density function (PDF) P via ˆ P(r) = s(q) exp(iq r) dq, r R 3. R 3

6 1. Introduction to HARDI Diusion Magnetic Resonance Imaging to nd directionality of neural bers via water diusion in brain tissues. Directionality of neural bers useful for characterization of neuro-degenerative diseases & surgical planning. Stejskal & Tanner's model: given MR signal s, reconstruct its diusion probability density function (PDF) P via ˆ P(r) = s(q) exp(iq r) dq, r R 3. R 3 For computational savings, consider High Angular Resolution Diusion Imaging (HARDI), which samples on S 2 (instead of R 3 ).

7 1. Introduction to HARDI Figure: Expected diusion PDFs in (a) isotropic, (b) single ber and (c) 2 crossing bers from a brain coronal slice.

8 1. Introduction to HARDI Compute a diusion orientation distribution function (ODF) ˆ Ψ(u) = P(ru)r 2 dr, u S 2. 0 where Ψ is a 'radial-average' of PDF P with S Ψ = 1. 2

9 1. Introduction to HARDI Compute a diusion orientation distribution function (ODF) ˆ Ψ(u) = P(ru)r 2 dr, u S 2. 0 where Ψ is a 'radial-average' of PDF P with S Ψ = 1. 2 Estimate ODF Ψ from modifying HARDI signal s s(u) = log( log s(u)), where 0 < s(u) < 1, u S 2.

10 1. Introduction to HARDI Compute a diusion orientation distribution function (ODF) ˆ Ψ(u) = P(ru)r 2 dr, u S 2. 0 where Ψ is a 'radial-average' of PDF P with S Ψ = 1. 2 Estimate ODF Ψ from modifying HARDI signal s s(u) = log( log s(u)), where 0 < s(u) < 1, u S 2. Approximate a sharper ODF via Ψ(u) 1 4π π 2 R[ b s](u), u S 2, where b is the spherical Laplacian operator, and R is the Funk Radon Transform (FRT) ˆ R[s](u) := s(q) dσ(q), u S 2, v C(u) with C(u) the great circle with normal u.

11 1. Introduction to HARDI Figure: A 3D-Diusion PDF & its corresponding ODF.

12 1. Introduction to HARDI Analytical computation of ODF Ψ 2 when modied HARDI signal ~s has representation ~s(u) R c j f j (u), u S 2, j=1 where f j are eigenfunctions of R and b, so that R[ b f j ] = λ j f j, j, giving Ψ 2 (u) 1 4π π 2 R λ j c j f j (u), u S 2. j=1

13 1. Introduction to HARDI Analytical computation of ODF Ψ 2 when modied HARDI signal ~s has representation ~s(u) R c j f j (u), u S 2, j=1 where f j are eigenfunctions of R and b, so that R[ b f j ] = λ j f j, j, giving Ψ 2 (u) 1 4π π 2 R λ j c j f j (u), u S 2. j=1 Modied spherical harmonics (modied SH) Ỹj are eigenfunctions of R and b.

14 1. Introduction to HARDI Analytical computation of ODF Ψ 2 when modied HARDI signal ~s has representation ~s(u) R c j f j (u), u S 2, j=1 where f j are eigenfunctions of R and b, so that R[ b f j ] = λ j f j, j, giving Ψ 2 (u) 1 4π π 2 R λ j c j f j (u), u S 2. j=1 Modied spherical harmonics (modied SH) Ỹj are eigenfunctions of R and b. Wavelets are then designed from nite weighted linear combinations of modied SH Ỹ j 's.

15 1. Introduction to HARDI Denote the modied spherical harmonics (SH) index set Λ := {(m, l) Z 2N 0 : m l, l 2N 0 }, where m & l are the degree & order of modied SH Ỹl m (θ, φ), (m, l) Λ, θ [0, π] and φ [0, 2π),

16 1. Introduction to HARDI Denote the modied spherical harmonics (SH) index set Λ := {(m, l) Z 2N 0 : m l, l 2N 0 }, where m & l are the degree & order of modied SH Ỹl m (θ, φ), (m, l) Λ, θ [0, π] and φ [0, 2π), { Ỹl m Θ m l (cos θ) Φ 1,m (φ), if l m 0, (θ, φ) := Θ m l (cos θ) Φ 2,m (φ), if 0 < m l, where Θ m l are dened by the associated Legendre functions Pl m as Θ m l (cos θ) := ( 1) m (2l + 1)(l m)! Pl m (cos θ), 2(l + m)!

17 1. Introduction to HARDI Denote the modied spherical harmonics (SH) index set Λ := {(m, l) Z 2N 0 : m l, l 2N 0 }, where m & l are the degree & order of modied SH Ỹl m (θ, φ), (m, l) Λ, θ [0, π] and φ [0, 2π), { Ỹl m Θ m l (cos θ) Φ 1,m (φ), if l m 0, (θ, φ) := Θ m l (cos θ) Φ 2,m (φ), if 0 < m l, where Θ m l are dened by the associated Legendre functions Pl m as Θ m l (cos θ) := ( 1) m (2l + 1)(l m)! Pl m (cos θ), 2(l + m)! Φ 1,m (φ) := { π 1/2 cos(mφ), if l m < 0, (2π) 1/2 if m = 0, Φ 2,m (φ) := π 1/2 sin(mφ), 0 < m l.

18 1. Introduction to HARDI Figure: Diagram of modied SHs Ỹ m l, m l, l = 0, 2, 4.

19 2. Wavelets on the Sphere for L 2 sym(s 2 ) L 2 sym(s 2 ) is a Hilbert space with inner product given as L 2 sym(s 2 ) := { f L 2 (S 2 ) f ( u) = f (u), u S 2 a.e. }, ˆ f, g := f g dσ, f, g L 2 sym(s 2 ). S 2

20 2. Wavelets on the Sphere for L 2 sym(s 2 ) L 2 sym(s 2 ) is a Hilbert space with inner product given as L 2 sym(s 2 ) := { f L 2 (S 2 ) f ( u) = f (u), u S 2 a.e. }, ˆ f, g := f g dσ, f, g L 2 sym(s 2 ). S 2 Modied SH {Ỹ m l is an orthonormal basis for }(m,l) Λ L 2 sym(s 2 ) with L 2 sym(s 2 ) =H 1 H 2, H i := span {Ỹ m l : (m, l) Λ i }, Λ 1 := {(m, l) Λ : l m 0}, Λ 2 := {(m, l) Λ : 1 m l}.

21 2. Wavelet Frames on the Sphere for L 2 sym(s 2 ) A countable collection {f ν } ν I in a Hilbert space H is a frame for H if constants C & D s.t. C f 2 H ν I f, f ν H 2 D f 2 H, f H. When C = D = 1, {f ν } ν I is a tight frame for H.

22 2. Wavelet Frames on the Sphere for L 2 sym(s 2 ) A countable collection {f ν } ν I in a Hilbert space H is a frame for H if constants C & D s.t. C f 2 H ν I f, f ν H 2 D f 2 H, f H. When C = D = 1, {f ν } ν I is a tight frame for H. Frames give overcomplete, yet stable representations & superior denoising compared to bases.

23 2. Wavelet Frames on the Sphere for L 2 sym(s 2 ) A countable collection {f ν } ν I in a Hilbert space H is a frame for H if constants C & D s.t. C f 2 H ν I f, f ν H 2 D f 2 H, f H. When C = D = 1, {f ν } ν I is a tight frame for H. Frames give overcomplete, yet stable representations & superior denoising compared to bases. { To build } tight (wavelet) frames for L 2 sym(s 2 ), construct f 1 ν & { f 2 } ν I 1 ν forming tight frames for H 1 & H 2. ν I 2

24 2. Wavelet Frames on the Sphere for L 2 sym(s 2 ) A countable collection {f ν } ν I in a Hilbert space H is a frame for H if constants C & D s.t. C f 2 H ν I f, f ν H 2 D f 2 H, f H. When C = D = 1, {f ν } ν I is a tight frame for H. Frames give overcomplete, yet stable representations & superior denoising compared to bases. { To build } tight (wavelet) frames for L 2 sym(s 2 ), construct f 1 ν & { f 2 } ν I 1 ν forming tight frames for H 1 & H 2. ν I 2 Construct tight (wavelet) frames for H i from its orthonormal basis {Ỹ m l : (m, l) Λ i }, i = 1, 2.

25 2. Wavelets on the Sphere for L 2 sym(s 2 ) Denote Ỹ j := Ỹ m l, where j := (m, l) Λ. For i = 1, 2, k 1, wavelets in H i take the form ψk i ( ; ηi k,ν ) = µ k,ν Ỹ j (ηk,ν i )Ỹj, ν J k, j J i k where J i k Λi, η i k,ν S2, µ k,ν R\ {0},ν J k & J k is a nite set s.t. J i k J k.

26 2. Wavelets on the Sphere for L 2 sym(s 2 ) Denote Ỹ j := Ỹ m l, where j := (m, l) Λ. For i = 1, 2, k 1, wavelets in H i take the form ψk i ( ; ηi k,ν ) = µ k,ν Ỹ j (ηk,ν i )Ỹj, ν J k, j J i k where J i k Λi, η i k,ν S2, µ k,ν R\ {0},ν J k & J k is a nite set s.t. J i k J k. Regard ψ i k ( ; ηi k,ν ) as generalized translates of ψ i k = j J i k Ỹ j under T η i k,ν, i.e., ψ i k ( ; ηi k,ν ) = T η i k,ν ψi k.

27 Motivation of Generalized Translation Operator For instance, take H := L 2 [0, 2π), S := [0, 2π), Λ := Z, e j (x) := exp(ijx), x S, j Z.

28 Motivation of Generalized Translation Operator For instance, take H := L 2 [0, 2π), S := [0, 2π), Λ := Z, e j (x) := exp(ijx), x S, j Z. For ζ S, set f (x) := N 1 j=0 α je j (x), then we have T ζ f (x) = = N 1 j=0 N 1 j=0 α j e j (ζ)e j (x) α j exp(ij(x ζ)).

29 Motivation of Generalized Translation Operator For instance, take H := L 2 [0, 2π), S := [0, 2π), Λ := Z, e j (x) := exp(ijx), x S, j Z. For ζ S, set f (x) := N 1 j=0 α je j (x), then we have T ζ f (x) = = N 1 j=0 N 1 j=0 α j e j (ζ)e j (x) α j exp(ij(x ζ)). If we choose ζ := ζ l := 2πl N, l = 0,..., N 1, then, A := [e j (ζ l )] N 1 j,l=0 = [exp(i2πjl/n)]n 1 j,l=0 satises ADA = I with D = 1 N I.

30 2. Wavelets on the Sphere for L 2 sym(s 2 ) Key Construction Idea For i = 1, 2, k N 0, dene I i k := { (m, l) Λ i : 0 l L k }, Lk 2N 0.

31 2. Wavelets on the Sphere for L 2 sym(s 2 ) Key Construction Idea For i = 1, 2, k N 0, dene I i k := { (m, l) Λ i : 0 l L k }, Lk 2N 0. Seek 1. nodes ζ i k,ν S2, ν I k with Ii k I k <, 2. weight matrices D k := diag(d k,ν ) ν I k 0, such that A i k := [Ỹj(ζ k,ν i )] j Ik i,ν I k orthogonality condition satises a discrete weighted A i k D ka i k = I. (1)

32 2. Wavelets on the Sphere for L 2 sym(s 2 ) Key Construction Idea For i = 1, 2, k N 0, dene I i k := { (m, l) Λ i : 0 l L k }, Lk 2N 0. Seek 1. nodes ζ i k,ν S2, ν I k with Ii k I k <, 2. weight matrices D k := diag(d k,ν ) ν I k 0, such that A i k := [Ỹj(ζ k,ν i )] j Ik i,ν I k orthogonality condition satises a discrete weighted A i k D ka i k = I. (1) Dene Fk i := Ai k D1/2 k, so by (1), Fk i F k i = I.

33 2. Wavelets on the Sphere for L 2 sym(s 2 ) Set [ϕ i k ( ; ζi k,ν )] ν I k := F k [Ỹ j ] j I i k, i.e., ϕ i k ( ; ζi k,ν ) = d k,ν Ỹ j (ζk,ν i ) Ỹj, ν I k. j I i k

34 2. Wavelets on the Sphere for L 2 sym(s 2 ) Set [ϕ i k ( ; ζi k,ν )] ν I k := F k [Ỹ j ] j I i k, i.e., Then, ϕ i k ( ; ζi k,ν ) = d k,ν Ỹ j (ζk,ν i ) Ỹj, ν I k. j I i k { } ϕ i k ( ; ζi k,ν ) : ν I k forms a tight frame for V i k := span {Ỹj : j I i k }, k N 0, i = 1, 2. (2)

35 2. Wavelets on the Sphere for L 2 sym(s 2 ) Set [ϕ i k ( ; ζi k,ν )] ν I k := F k [Ỹ j ] j I i k, i.e., Then, ϕ i k ( ; ζi k,ν ) = d k,ν Ỹ j (ζk,ν i ) Ỹj, ν I k. j I i k { } ϕ i k ( ; ζi k,ν ) : ν I k forms a tight frame for V i k := span {Ỹj : j I i k }, k N 0, i = 1, 2. (2) Each ϕ i k ( ; ζi k,ν ) is localized around ζi k,ν, ν I k, i.e., ϕ i k ( ; ζi k,ν ) ϕ i k (ζi k,ν ; ζi k,ν ) = min { f : f V i k, f (ζi k,ν ) = 1}.

36 2. Wavelets on the Sphere for L 2 sym(s 2 ) Lemma: { } For i = 1, 2, k N 0, dene modied Gauss-Legendre quadrature nodes ζk,ν i : ν I k ζ i k,ν := (θ k,α, φ i k,β ), ν := (α, β) I k, (3) where θ k,α := cos 1 (x k,α ), α = 0, 1,..., L k, with {x α} L k α=0 as zeros of the Legendre polynomial P Lk +1 & the zenith points are φ 1 k,β π(β + 1/2) =, φ 2 π(β + 1) k,β = L k + 1 L k + 2, β = 0, 1,..., L k.

37 2. Wavelets on the Sphere for L 2 sym(s 2 ) Lemma: { } For i = 1, 2, k N 0, dene modied Gauss-Legendre quadrature nodes ζk,ν i : ν I k ζ i k,ν := (θ k,α, φ i k,β ), ν := (α, β) I k, (3) where θ k,α := cos 1 (x k,α ), α = 0, 1,..., L k, with {x α} L k α=0 as zeros of the Legendre polynomial P Lk +1 & the zenith points are φ 1 k,β π(β + 1/2) =, φ 2 π(β + 1) k,β = L k + 1 L k + 2, β = 0, 1,..., L k. Enumerate the nodes as { } {(θ k,α, φ i k,β }(α,β) ) := (θ k,0, φ i k,0 ),..., (θ k,0, φ i k,l k ),..., (θ k,lk, φ i k,0 ),..., (θ k,l k, φ i k,l k ), & dene the diagonal weight matrix D k = 2π L k + 1 diag(w k,0,..., w k,0,..., w k,lk,..., w k,lk ), (4) where { w k,α } Lk α=0 are the Gauss-quadrature weights of P L k +1, with each unique w k,α appearing L k + 1 times in D k.

38 2. Wavelets on the Sphere for L 2 sym(s 2 ) Then, for i = 1, 2, k N 0, A i k := [Ỹ j (ζk,ν i )] j Ik i,ν I k satises the weighted orthogonality condition (1), with diag(d k,ν ) ν I k := D k. Furthermore, for i = 1, 2, k N 0, { ϕ i k ( ; ζi k,ν ) : ν I k } yields a tight frame for V i k dened in (2), where ϕ i k ( ; ζi k,ν ) := d k,ν Ỹ j (ζk,ν i )Ỹ j, ν I k. (5) j I i k

39 2. Wavelets on the Sphere for L 2 sym(s 2 ) A sequence of nite-dimensional subspaces {V k } k 0 in H is a multiresolution analysis (MRA) of H if 1. k 0, functions ϕ k ( ; ζ k,ν ), ν I k, s.t. {ϕ k ( ; ζ k,ν ) : ν I k } spans V k. 2. k 0, V k V k k 0 V k = H. We call the functions ϕ k ( ; ζ k,ν ), ν I k, k 0, scaling functions.

40 2. Wavelets on the Sphere for L 2 sym(s 2 ) A sequence of nite-dimensional subspaces {V k } k 0 in H is a multiresolution analysis (MRA) of H if 1. k 0, functions ϕ k ( ; ζ k,ν ), ν I k, s.t. {ϕ k ( ; ζ k,ν ) : ν I k } spans V k. 2. k 0, V k V k k 0 V k = H. We call the functions ϕ k ( ; ζ k,ν ), ν I k, k 0, scaling functions. Proposition: For i = 1, 2, k N 0, let Vk i in (2) & be the subspace dened Ik i := { (m, l) Λ i } : 0 l L k, Lk 2N 0, where {L k } k N0. Then, for i = 1, 2, { Vk i } forms an k N 0 MRA for H i with scaling functions ϕ i k ( ; ζi k,ν ), ν I k, given by (5).

41 2. Wavelets on the Sphere for L 2 sym(s 2 ) Figure: Scaling functions of V 1 0,V 2 0, where V i 0 = { (m, l) Λ i : 0 l 4 }, i = 1, 2.

42 2. Wavelets on the Sphere for L 2 sym(s 2 ) Figure: Scaling functions of V 1 1, V 2 1,where V i 1 = { (m, l) Λ i : 0 l 6 }, i = 1, 2.

43 2. Wavelets on the Sphere for L 2 sym(s 2 ) For i = 1, 2, k 0, let W i k := V i k+1 \V i k & J i k := Ii k+1 \Ii k. Then where W i k := span {Ỹj : j J i k }, (6) J i k := { (m, l) Λ i : L k < l L k+1 }. (7)

44 2. Wavelets on the Sphere for L 2 sym(s 2 ) For i = 1, 2, k 0, let W i k := V i k+1 \V i k & J i k := Ii k+1 \Ii k. Then where W i k := span {Ỹj : j J i k }, (6) J i k := { (m, l) Λ i : L k < l L k+1 }. (7) Thus, we have an orthogonal decomposition H i = W 1 i W0 i W1 i..., where W 1 i := V 0 i, i = 1, 2.

45 2. Wavelets on the Sphere for L 2 sym(s 2 ) Theorem: For i = 1, 2, k 1, dene ψ i k ( ; ηi k,ν ) := ˆd k,ν Ỹl m (ηk,ν i )Ỹ l m, ηk,ν i J k (m,l) J i k where J i k is given by (7), ηi k,ν := ζi k+1,ν, ν J k := I k+1 & diag( ˆd k,ν ) := ˆD k := D k+1 given by (3) & (4).

46 2. Wavelets on the Sphere for L 2 sym(s 2 ) Theorem: For i = 1, 2, k 1, dene ψ i k ( ; ηi k,ν ) := ˆd k,ν Ỹl m (ηk,ν i )Ỹ l m, ηk,ν i J k (m,l) J i k where J i k is given by (7), ηi k,ν := ζi k+1,ν, ν J k := I k+1 & diag( ˆd k,ν ) := ˆD k := D k+1 given by (3) & (4). Denote W i 1 := V i 0, J i 1 := Ii 0, ψi 1 ( ; ηi 1,ν ) := ϕi 0 ( ; ζi 0,ν ), i = 1, 2, and Γ(Ψ) := { ψ i k ( ; ηi k,ν ) : ν J k, k 1, i = 1, 2}. Then Γ(Ψ) forms a normalized tight wavelet frame for L 2 sym(s 2 ).

47 2. Wavelets on the Sphere for L 2 sym(s 2 ) Figure: Wavelet functions of W 1 0, W 2 0, where W i 0 = { (m, l) Λ i : l = 6 }, i = 1, 2.

48 3. Applications of Wavelets to HARDI Signals Key idea: ODF Ψ 2 is localized along a few radial directions, so represent ODF Ψ 2 with localized wavelets in Γ(Ψ).

49 3. Applications of Wavelets to HARDI Signals Key idea: ODF Ψ 2 is localized along a few radial directions, so represent ODF Ψ 2 with localized wavelets in Γ(Ψ). However, only HARDI signal s is known, so seek another wavelet system Γ( Ψ) to rst represent it.

50 3. Applications of Wavelets to HARDI Signals Key idea: ODF Ψ 2 is localized along a few radial directions, so represent ODF Ψ 2 with localized wavelets in Γ(Ψ). However, only HARDI signal s is known, so seek another wavelet system Γ( Ψ) to rst represent it. To reconstruct unknown ODF Ψ 2, given modied HARDI signal s(u), apply R and b as follows: Ψ 2 (u) 1 4π π 2 R[ b s](u), u S 2.

51 3. Applications of Wavelets to HARDI Signals Key idea: ODF Ψ 2 is localized along a few radial directions, so represent ODF Ψ 2 with localized wavelets in Γ(Ψ). However, only HARDI signal s is known, so seek another wavelet system Γ( Ψ) to rst represent it. To reconstruct unknown ODF Ψ 2, given modied HARDI signal s(u), apply R and b as follows: Ψ 2 (u) 1 4π π 2 R[ b s](u), u S 2. { } Thus, nd Γ( Ψ) := ψ k i ( ; ηi k,ν ) : ν J k, k 1, i = 1, 2 to represent ~s, where ψ i k ( ; ηi k,ν ) is a `pre-image' of ψi k ( ; ηi k,ν ) in Γ(Ψ) under b and R.

52 3. Applications of Wavelets to HARDI Signals To nd Γ( Ψ), recall ψk i ( ; ηi k,ν ) are nite combinations of modied SH {Ỹl m } (m,l) Λ, i.e., ψ i k ( ; ηi k,ν ) := ˆd k,ν (m,l) J i k Ỹl m (ηk,ν i )Ỹ l m, ηk,ν i J k.

53 3. Applications of Wavelets to HARDI Signals To nd Γ( Ψ), recall ψk i ( ; ηi k,ν ) are nite combinations of modied SH {Ỹl m } (m,l) Λ, i.e., ψ i k ( ; ηi k,ν ) := ˆd k,ν (m,l) J i k Ỹl m (ηk,ν i )Ỹ l m, ηk,ν i J k. {Ỹ l m } (m,l) Λ are eigenfunctions of b and R, i.e., b Ỹ m l = l(l + 1)Ỹl m, R[Ỹl m ] = 2πP l (0)Ỹ m l, (m, l) Λ.

54 3. Applications of Wavelets to HARDI Signals To nd Γ( Ψ), recall ψk i ( ; ηi k,ν ) are nite combinations of modied SH {Ỹl m } (m,l) Λ, i.e., ψ i k ( ; ηi k,ν ) := ˆd k,ν (m,l) J i k Ỹl m (ηk,ν i )Ỹ l m, ηk,ν i J k. {Ỹ l m } (m,l) Λ are eigenfunctions of b and R, i.e., b Ỹ m l = l(l + 1)Ỹl m, R[Ỹl m ] = 2πP l (0)Ỹ m l, (m, l) Λ. To obtain ψ i k ( ; ηi k,ν ) as a 'pre-image' of ψi k ( ; ηi k,ν ) under b and R, for i = 1, 2, k 1, ν J k, dene ψ i k ( ; ηi k,ν ) : = ˆd k,ν (m,l) J i k δ 1 l where δ l = l(l + 1), l 2N, and δ 0 = 1. [2πP l (0)] 1 Ỹl m (ηk,ν i )Ỹ l m,

55 3. Applications of Wavelets to HARDI Signals Then, approximate modied HARDI signal s(u) R c j ψ j (u). j=1

56 3. Applications of Wavelets to HARDI Signals Then, approximate modied HARDI signal s(u) R c j ψ j (u). j=1 ODF Ψ 2 can be reconstructed analytically: Ψ 2 (u) 1 4π π 2 = 1 4π π 2 R c j R[ b ψj ](u) j=1 R c j ψ j (u). j=1

57 3. Applications of Wavelets to HARDI Signals Then, approximate modied HARDI signal s(u) R c j ψ j (u). j=1 ODF Ψ 2 can be reconstructed analytically: Ψ 2 (u) 1 4π π 2 = 1 4π π 2 R c j R[ b ψj ](u) j=1 R c j ψ j (u). j=1 In practice, ~s is discretized with gradient directions {g k } K k=1, giving f = [~s(g k )] K k=1 & discrete wavelet matrix B where B j,k = ψ j (g k ).

58 3. Applications of Wavelets to HARDI Signals For sparse reconstruction of a single discrete ODF, 1 min c R R 2 f Bc 2 F + λ c 1, where c := [c j ] R j=1 is the coecient vector.

59 3. Applications of Wavelets to HARDI Signals For sparse reconstruction of a single discrete ODF, 1 min c R R 2 f Bc 2 F + λ c 1, where c := [c j ] R j=1 is the coecient vector. Use the fast iterative soft-thresholding algorithm (FISTA) to obtain an approximate solution c.

60 3. Applications of Wavelets to HARDI Signals For sparse reconstruction of a single discrete ODF, 1 min c R R 2 f Bc 2 F + λ c 1, where c := [c j ] R j=1 is the coecient vector. Use the fast iterative soft-thresholding algorithm (FISTA) to obtain an approximate solution c. Reconstruct a single discrete ODF Ψ 2,d by Ψ 2,d 1 4π π 2 BLPc, where L := diag[ l j (l j + 1)] R j=1, P := diag[2πp l j (0)] R j=1.

61 3. Applications of Wavelets to HARDI Signals Figure: A 2D image of HARDI ODFs.

62 3. Applications of Wavelets to HARDI Signals Neigbouring HARDI ODFs tend to be similar in diusion directionality.

63 3. Applications of Wavelets to HARDI Signals Neigbouring HARDI ODFs tend to be similar in diusion directionality. Improve HARDI reconstruction by modifying optimization models with HARDI spatial regularization.

64 3. Applications of Wavelets to HARDI Signals Neigbouring HARDI ODFs tend to be similar in diusion directionality. Improve HARDI reconstruction by modifying optimization models with HARDI spatial regularization. For each sampling gradient in {g k } K k=1, we have an N x N y Nx,Ny image of HARDI signals S(g k ):=[S(g k ) i,j ] i,j=1.

65 3. Applications of Wavelets to HARDI Signals Neigbouring HARDI ODFs tend to be similar in diusion directionality. Improve HARDI reconstruction by modifying optimization models with HARDI spatial regularization. For each sampling gradient in {g k } K k=1, we have an N x N y Nx,Ny image of HARDI signals S(g k ):=[S(g k ) i,j ] i,j=1. K Nx Ny From there, extract an R HARDI data matrix F and impose spatial regularization by adding diag(µ)wc 1 : min C 1 2 F BC 2 F + λ C 1 + diag(µ)wc 1, 4Nx Ny 1 where C : coe matrix, λ R +, µ R 0 are parameters that balance coecient sparsity & spatial regularization by the 2D-tensor Haar tight framelet lter matrix W.

66 3. Applications of Wavelets to HARDI Signals Solved with split Bregman method by setting Q = WC and P = C, giving 1 min F C,P,Q 2 BC 2 F +λ P 1 + diag(µ)q 1 s.t. Q = WC & P = C.

67 3. Applications of Wavelets to HARDI Signals Solved with split Bregman method by setting Q = WC and P = C, giving 1 min F C,P,Q 2 BC 2 F +λ P 1 + diag(µ)q 1 s.t. Q = WC & P = C. Denoting Lagrange multipliers as Λ 1, Λ 2 & ρ 1, ρ 2 as positive parameters, dene an augmented Lagrangian L ρ1,ρ 2 (C, Q, P; Λ 1, Λ 2 ) := 1 2 F BC 2 F + λ P 1 + µ diag(µ)q 1 ρ 1 Λ 1, Q WC + ρ 1 2 Q WC 2 F ρ 2 Λ 2, P C + ρ 2 2 P C 2 F.

68 3. Applications of Wavelets to HARDI Signals Then, for a xed (C k, Q k, P k, Λ k 1, Λk 2 ), we update (C k+1, Q k+1, P k+1, Λ k+1 1, Λ2 k+1 ) by a Gauss-Seidel scheme on the above augmented Lagrangian as follows: 1. C k+1 = arg min C L ρ1,ρ 2 (C, Q k, P k ; Λ k 1, Λk 2 ); 2. Q k+1 = arg min Q L ρ1,ρ 2 (C k+1, Q, P k ; Λ k 1, Λk 2 ); 3. P k+1 = arg min P L ρ1,ρ 2 (C k+1, Q k+1, P; Λ k 1, Λk 2 ); 4. Λ k+1 1 = Λ k 1 (Qk+1 W (C k+1 ) ); 5. Λ k+1 2 = Λ k 2 (Pk+1 C k+1 ).

69 3. Applications of Wavelets to HARDI Signals Spatially regularized wavelet frame method for CSA QBI Input: HARDI data matrix F and tolerance level ɛ. Initialization: set C 0 = [B B + λi ] 1 B F, Q 0 = W (C 0 ), P 0 = C 0, Λ 1, Λ 2 = 0. For k = 0, 1,..., while C k+1 C k > ɛ, perform: 1. C k+1 = [B B +(ρ 1 +ρ 2 )I ] 1 [B F +ρ 1 (Q k Λ k 1 ) W +ρ 2 (P k Λ 2 )]; 2. Q k+1 = T 1 µ/ρ 1 [W (C k+1 ) + Λ k 1 ]; 3. P k+1 = T 1 λ/ρ 2 [C k+1 + Λ k 2 ]; 4. Λ k+1 1 = Λ k 1 (Qk+1 W (C k+1 ) ); 5. Λ k+1 2 = Λ k 2 (Pk+1 C k+1 ). Output: Coecient matrix C := C K to reconstruct multi-voxel version of discrete ODF Ψ 2,d.

70 3. Applications of Wavelets to HARDI Signals Figure: Structured Field Testing Dataset: voxel set Ω d : Data set simulates a realistic 3D conguration of tracts: it comprises 5 dierent ber bundles (Sub-plots A-E). In each voxel, the directions are color-coded based on their orientation (x-axis, y -axis, z-axis). Subplot F shows a representative slice with the ODF orientations.

71 3. Applications of Wavelets to HARDI Signals Method / SNR (db) Spherical Harmonics (SH) Spherical Ridgelets (SR) Wavelet Frames (WF) Table: NMSE values of SH, SR, WF based methods (with spatial regularization) for HARDI testing data for SNR levels 5, 10, 20 db. Normalized mean square error (NMSE): 1 2 Ψ est(r) Ψ true(r) 2 NMSE :=. N xn y N z Ψ r Ω true(r) 2 2 d

72 Possible Future Work Extend the above wavelet frame representation to handle HARDI deconvolution problems. Design other wavelet frame systems for other cases of Hilbert spaces for practical applications. Consider integration of above techniques for feature extraction with deep learning.