fast and accurate hardi and its application to ... · hardi vs. dti di usion tensor imaging(dti) is...
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![Page 1: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/1.jpg)
Fast and Accurate HARDI and its Applicationto Neurological Diagnosis
Dr. Oleg Michailovich
Department of Electrical and Computer EngineeringUniversity of Waterloo
June 21, 2011
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 2: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/2.jpg)
Outline
1 Diffusion imaging and HARDI
2 Sparse approximation with spherical ridgelets
3 HARDI reconstruction as a compressed sensing problem
4 Composite compressed sensing: spatial regularization
5 Directional diffusion structure and its graph representation
6 Dimensionality reduction through isometric embedding
7 Application to first episode (FE) schizophrenia
8 Conclusions
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 3: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/3.jpg)
Diffusion MRI
The average diffusion propagator p(r) quantifies the probability ofa spin to relocate to position r + dr at experimental time τ .
p(r) =
∫R3
s(q) e−2πı (q·r)dq, r ∈ R3,q ∈ R3
where
s(q) is a normalized diffusion signal
q is the wavevector
In High Angular Resolution Diffusion Imaging, the estimation ofp(r) is superseded by estimating its radial projection known as anorientation distribution function (ODF):
Q(u) =
∫ ∞0
p(α u)α2 dα, u ∈ S2.
In this case, s(q) can be restricted to a spherical shell.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 4: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/4.jpg)
Diffusion MRI
The average diffusion propagator p(r) quantifies the probability ofa spin to relocate to position r + dr at experimental time τ .
p(r) =
∫R3
s(q) e−2πı (q·r)dq, r ∈ R3,q ∈ R3
where
s(q) is a normalized diffusion signal
q is the wavevector
In High Angular Resolution Diffusion Imaging, the estimation ofp(r) is superseded by estimating its radial projection known as anorientation distribution function (ODF):
Q(u) =
∫ ∞0
p(α u)α2 dα, u ∈ S2.
In this case, s(q) can be restricted to a spherical shell.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 5: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/5.jpg)
HARDI vs. DTI
Diffusion tensor imaging (DTI) is a well-established tool of diag-nostic dMRI.
DTI
1 Unimodal Gaussian diffusionmodel is assumed.
2 Typical number of diffusion -encoding gradients is about20 - 30.
3 Scan duration is about 5 -10 mins.
4 Incapable of discriminatingmultimodal diffusion flowswithin a voxel.
HARDI
1 No assumptions on thediffusion model are made.
2 Typical number of diffusion -encoding gradients is about80 - 100.
3 Scan duration is about 20 -30 mins.
4 Can be used to delineatemultimodal diffusion flowswithin a voxel.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 6: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/6.jpg)
HARDI vs. DTI
Diffusion tensor imaging (DTI) is a well-established tool of diag-nostic dMRI.
DTI
1 Unimodal Gaussian diffusionmodel is assumed.
2 Typical number of diffusion -encoding gradients is about20 - 30.
3 Scan duration is about 5 -10 mins.
4 Incapable of discriminatingmultimodal diffusion flowswithin a voxel.
HARDI
1 No assumptions on thediffusion model are made.
2 Typical number of diffusion -encoding gradients is about80 - 100.
3 Scan duration is about 20 -30 mins.
4 Can be used to delineatemultimodal diffusion flowswithin a voxel.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 7: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/7.jpg)
HARDI vs. DTI
Diffusion tensor imaging (DTI) is a well-established tool of diag-nostic dMRI.
DTI
1 Unimodal Gaussian diffusionmodel is assumed.
2 Typical number of diffusion -encoding gradients is about20 - 30.
3 Scan duration is about 5 -10 mins.
4 Incapable of discriminatingmultimodal diffusion flowswithin a voxel.
HARDI
1 No assumptions on thediffusion model are made.
2 Typical number of diffusion -encoding gradients is about80 - 100.
3 Scan duration is about 20 -30 mins.
4 Can be used to delineatemultimodal diffusion flowswithin a voxel.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 8: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/8.jpg)
HARDI vs. DTI: Fibre tractography
From: J. Malcolm, O. Michailovich, S. Bouix, C.-F. Westin, A. Tannenbaum, M. Shenton and Y. Rathi, “A filtered
approach to neural tractography using the Watson directional function,” Medical Image Analysis, 14(1), 2009.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 9: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/9.jpg)
HARDI vs. DTI: Fibre tractography
From: J. Malcolm, O. Michailovich, S. Bouix, C.-F. Westin, A. Tannenbaum, M. Shenton and Y. Rathi, “A filtered
approach to neural tractography using the Watson directional function,” Medical Image Analysis, 14(1), 2009.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 10: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/10.jpg)
HARDI
Thus, with r = (x , y , z) ∈ Ω ⊂ R3, HARDI measurements can bemodelled as s(u, r) : S2 × Ω→ R+.
Fundamental practical limitation
The practical value of HARDI is greatly impaired by the problem ofprohibitively long acquisition times.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 11: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/11.jpg)
HARDI
Thus, with r = (x , y , z) ∈ Ω ⊂ R3, HARDI measurements can bemodelled as s(u, r) : S2 × Ω→ R+.
Fundamental practical limitation
The practical value of HARDI is greatly impaired by the problem ofprohibitively long acquisition times.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 12: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/12.jpg)
Research questions
The main questions to be addressed are:
1 Are there tools of mathematical imaging which could be usedto speed up HARDI acquisition so as to make its requirementscomparable to those of DTI?
2 To what extent the resulting approximation errors can affectthe accuracy of subsequent diagnostic inference?
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 13: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/13.jpg)
Research questions
The main questions to be addressed are:
1 Are there tools of mathematical imaging which could be usedto speed up HARDI acquisition so as to make its requirementscomparable to those of DTI?
2 To what extent the resulting approximation errors can affectthe accuracy of subsequent diagnostic inference?
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 14: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/14.jpg)
Research questions
The main questions to be addressed are:
1 Are there tools of mathematical imaging which could be usedto speed up HARDI acquisition so as to make its requirementscomparable to those of DTI?
2 To what extent the resulting approximation errors can affectthe accuracy of subsequent diagnostic inference?
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 15: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/15.jpg)
Multifiber mixture model
At each r ∈ Ω, the HARDI signal can be modelled as
s(u, r) =M∑i=1
αi (r)
si (u,r)︷ ︸︸ ︷exp
−b (uTDi (r) u)
Observation
The energy of si (u) is supported alongside the great circles of S2.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 16: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/16.jpg)
Multifiber mixture model
At each r ∈ Ω, the HARDI signal can be modelled as
s(u, r) =M∑i=1
αi (r)
si (u,r)︷ ︸︸ ︷exp
−b (uTDi (r) u)
Observation
The energy of si (u) is supported alongside the great circles of S2.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 17: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/17.jpg)
Multifiber mixture model
At each r ∈ Ω, the HARDI signal can be modelled as
s(u, r) =M∑i=1
αi (r)
si (u,r)︷ ︸︸ ︷exp
−b (uTDi (r) u)
Observation
The energy of si (u) is supported alongside the great circles of S2.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 18: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/18.jpg)
Spherical ridgelets
Central requirement
The L2-energy of HARDI signals can be efficiently “encoded” interms of representation atoms, whose L2-energy is concentratedalongside the great circles of S2.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 19: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/19.jpg)
Spherical ridgelets
Central requirement
The L2-energy of HARDI signals can be efficiently “encoded” interms of representation atoms, whose L2-energy is concentratedalongside the great circles of S2.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 20: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/20.jpg)
Spherical ridgelets: construction
Let κ(x) = exp−ρ x (x + 1) be a Gaussian function, which we useto define:
κj(x) = κ(2−jx) = exp−ρ x
2j
( x
2j+ 1)
, j ∈ N.
The Gauss-Weierstrass scaling function χj ,v : S2 → R at resolutionj ∈ N and orientation v ∈ S2 is defined as:
χj,v(u) =∞∑n=0
2n + 1
4πκj(n) Pn(u · v), ∀u ∈ S2,
where Pn is the Legendre polynomial of order n.
Finally, the spherical ridgelets ψj ,v are obtained from χj ,v accordingto:
ψj,v =1
2πRχj+1,v − χj,v ,
where R denotes the Funk-Radon transform.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 21: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/21.jpg)
Spherical ridgelets: construction
Let κ(x) = exp−ρ x (x + 1) be a Gaussian function, which we useto define:
κj(x) = κ(2−jx) = exp−ρ x
2j
( x
2j+ 1)
, j ∈ N.
The Gauss-Weierstrass scaling function χj ,v : S2 → R at resolutionj ∈ N and orientation v ∈ S2 is defined as:
χj,v(u) =∞∑n=0
2n + 1
4πκj(n) Pn(u · v), ∀u ∈ S2,
where Pn is the Legendre polynomial of order n.
Finally, the spherical ridgelets ψj ,v are obtained from χj ,v accordingto:
ψj,v =1
2πRχj+1,v − χj,v ,
where R denotes the Funk-Radon transform.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 22: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/22.jpg)
Spherical ridgelets: construction
Let κ(x) = exp−ρ x (x + 1) be a Gaussian function, which we useto define:
κj(x) = κ(2−jx) = exp−ρ x
2j
( x
2j+ 1)
, j ∈ N.
The Gauss-Weierstrass scaling function χj ,v : S2 → R at resolutionj ∈ N and orientation v ∈ S2 is defined as:
χj,v(u) =∞∑n=0
2n + 1
4πκj(n) Pn(u · v), ∀u ∈ S2,
where Pn is the Legendre polynomial of order n.
Finally, the spherical ridgelets ψj ,v are obtained from χj ,v accordingto:
ψj,v =1
2πRχj+1,v − χj,v ,
where R denotes the Funk-Radon transform.Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 23: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/23.jpg)
Spherical ridgelets: construction (cont.)
Theorem
The semi-discrete set of spherical ridgelets ψj ,vj∈N,v∈S2 is a frame
for the subspace S ∈ L2(S2) of symmetric spherical functions.
In practical computations, given a set of K diffusion-encoding ori-entations ukKk=1, the ridgelet frame is discretized to result in:
s(r) = Ψc(r) + e(r), ∀r ∈ Ω,
where
s(r) = [s(u1, r), s(u2, r), . . . , s(uK , r)]T
Ψ is the ridgelet (dictionary) matrix
c(r) is the vector of ridgelet representation coefficients
e(r) is a noise term
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 24: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/24.jpg)
Spherical ridgelets: construction (cont.)
Theorem
The semi-discrete set of spherical ridgelets ψj ,vj∈N,v∈S2 is a frame
for the subspace S ∈ L2(S2) of symmetric spherical functions.
In practical computations, given a set of K diffusion-encoding ori-entations ukKk=1, the ridgelet frame is discretized to result in:
s(r) = Ψc(r) + e(r), ∀r ∈ Ω,
where
s(r) = [s(u1, r), s(u2, r), . . . , s(uK , r)]T
Ψ is the ridgelet (dictionary) matrix
c(r) is the vector of ridgelet representation coefficients
e(r) is a noise term
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 25: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/25.jpg)
Spherical ridgelets: construction (cont.)
Theorem
The semi-discrete set of spherical ridgelets ψj ,vj∈N,v∈S2 is a frame
for the subspace S ∈ L2(S2) of symmetric spherical functions.
In practical computations, given a set of K diffusion-encoding ori-entations ukKk=1, the ridgelet frame is discretized to result in:
s(r) = Ψc(r) + e(r), ∀r ∈ Ω,
where
s(r) = [s(u1, r), s(u2, r), . . . , s(uK , r)]T
Ψ is the ridgelet (dictionary) matrix
c(r) is the vector of ridgelet representation coefficients
e(r) is a noise term
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 26: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/26.jpg)
Reconstruction of HARDI signals
Bad news
Noise contamination is typically severe.
# ridgelets K
Good news
The spherical ridgelets have a low coherence w.r.t. the Diracsampling functions (µ ≈ 0.56).
Spherical ridgelets provide sparse representation of HARDIsignals (only 6÷ 8 ridgelets are needed on average).
Therefore, one can try to recover c(r) through:
minc(r)‖c(r)‖1
s.t. ‖Ψc(r)− s(r)‖2 ≤ εΨc(r) 0
which needs to be solved at each r ∈ Ω independently.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 27: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/27.jpg)
Reconstruction of HARDI signals
Bad news
Noise contamination is typically severe.
# ridgelets K
Good news
The spherical ridgelets have a low coherence w.r.t. the Diracsampling functions (µ ≈ 0.56).
Spherical ridgelets provide sparse representation of HARDIsignals (only 6÷ 8 ridgelets are needed on average).
Therefore, one can try to recover c(r) through:
minc(r)‖c(r)‖1
s.t. ‖Ψc(r)− s(r)‖2 ≤ εΨc(r) 0
which needs to be solved at each r ∈ Ω independently.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 28: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/28.jpg)
Reconstruction of HARDI signals
Bad news
Noise contamination is typically severe.
# ridgelets K
Good news
The spherical ridgelets have a low coherence w.r.t. the Diracsampling functions (µ ≈ 0.56).
Spherical ridgelets provide sparse representation of HARDIsignals (only 6÷ 8 ridgelets are needed on average).
Therefore, one can try to recover c(r) through:
minc(r)‖c(r)‖1
s.t. ‖Ψc(r)− s(r)‖2 ≤ εΨc(r) 0
which needs to be solved at each r ∈ Ω independently.Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 29: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/29.jpg)
Composite compressed sensing
General idea
Combine the sparse constraints in the diffusion domain (u) with asmoothness constraint in the spatial domain (r).
To this end, we define:
s = [s(r1), s(r2), . . . , s(r#Ω)]T
c = [c(r1), c(r2), . . . , c(r#Ω)]T
A =
Ψ 0 . . . 00 Ψ . . . 0. . . . . . . . . . . .0 . . . 0 Ψ
and
‖Ac‖TV =K−1∑k=0
‖DkAc‖TV
where Dks[n] = s[Kn + k] is a subsampling operator.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 30: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/30.jpg)
Composite compressed sensing
General idea
Combine the sparse constraints in the diffusion domain (u) with asmoothness constraint in the spatial domain (r).
To this end, we define:
s = [s(r1), s(r2), . . . , s(r#Ω)]T
c = [c(r1), c(r2), . . . , c(r#Ω)]T
A =
Ψ 0 . . . 00 Ψ . . . 0. . . . . . . . . . . .0 . . . 0 Ψ
and
‖Ac‖TV =K−1∑k=0
‖DkAc‖TV
where Dks[n] = s[Kn + k] is a subsampling operator.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 31: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/31.jpg)
Composite compressed sensing (cont.)
The spatially constrained CS problem can be now defined as
minc
‖c‖1 + µ‖Ac‖TV
s.t. ‖Ac − s‖2 ≤ ε
Ac 0
or, equivalently, in the Lagrangian form as
minc
1
2‖Ac − s‖2
2 + λ‖c‖1 + µ‖Ac‖TV + iC (Ac).
Alternatively, one can solve
minc,u
1
2‖u − s‖2
2+λ‖c‖1 + µ‖u‖TV + iC (u)
s.t. Ac = u
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Composite compressed sensing (cont.)
The spatially constrained CS problem can be now defined as
minc
‖c‖1 + µ‖Ac‖TV
s.t. ‖Ac − s‖2 ≤ ε
Ac 0
or, equivalently, in the Lagrangian form as
minc
1
2‖Ac − s‖2
2 + λ‖c‖1 + µ‖Ac‖TV + iC (Ac).
Alternatively, one can solve
minc,u
1
2‖u − s‖2
2+λ‖c‖1 + µ‖u‖TV + iC (u)
s.t. Ac = u
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Composite compressed sensing (cont.)
The spatially constrained CS problem can be now defined as
minc
‖c‖1 + µ‖Ac‖TV
s.t. ‖Ac − s‖2 ≤ ε
Ac 0
or, equivalently, in the Lagrangian form as
minc
1
2‖Ac − s‖2
2 + λ‖c‖1 + µ‖Ac‖TV + iC (Ac).
Alternatively, one can solve
minc,u
1
2‖u − s‖2
2+λ‖c‖1 + µ‖u‖TV + iC (u)
s.t. Ac = u
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Solution by means of ADMM
The above problem can be solved iteratively as(ut+1, c t+1
)=
arg minc,u
1
2‖u − s‖2
2 + λ ‖c‖1 + µ ‖u‖TV +γ
2‖u −Ac − pt‖2
2
pt+1 = pt +
(Ac t+1 − ut+1
)
Finally, splitting the variables results in
Step 1: c t+1 = arg minc
1
2‖Ac − d t‖2
2 + α‖c‖1
Step 2: ut+1 = arg min
c
1
2‖u − d t+1‖2
2 + β‖u‖TV
Step 3: pt+1 = pt +(Ac t+1 − ut+1
)
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Solution by means of ADMM
The above problem can be solved iteratively as(ut+1, c t+1
)=
arg minc,u
1
2‖u − s‖2
2 + λ ‖c‖1 + µ ‖u‖TV +γ
2‖u −Ac − pt‖2
2
pt+1 = pt +
(Ac t+1 − ut+1
)Finally, splitting the variables results in
Step 1: c t+1 = arg minc
1
2‖Ac − d t‖2
2 + α‖c‖1
Step 2: ut+1 = arg min
c
1
2‖u − d t+1‖2
2 + β‖u‖TV
Step 3: pt+1 = pt +(Ac t+1 − ut+1
)
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Numerical aspects
Step 1 is separable in the diffusion variable, and hence it canbe executed in a “voxel-by-voxel” manner (using, e.g., FISTA)
Step 2 is separable in the spatial variable, and hence it can beexecuted in a “slice-by-slice” manner (using, e.g., Chambolle’salgorithm).
The overall computational load scales proportionally with thenumber of processing cores.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Numerical aspects
Step 1 is separable in the diffusion variable, and hence it canbe executed in a “voxel-by-voxel” manner (using, e.g., FISTA)
Step 2 is separable in the spatial variable, and hence it can beexecuted in a “slice-by-slice” manner (using, e.g., Chambolle’salgorithm).
The overall computational load scales proportionally with thenumber of processing cores.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 38: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/38.jpg)
Numerical aspects
Step 1 is separable in the diffusion variable, and hence it canbe executed in a “voxel-by-voxel” manner (using, e.g., FISTA)
Step 2 is separable in the spatial variable, and hence it can beexecuted in a “slice-by-slice” manner (using, e.g., Chambolle’salgorithm).
The overall computational load scales proportionally with thenumber of processing cores.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 39: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/39.jpg)
In vivo experiments
O. Michailovich, Y. Rathi and S. Dolui, “Spatially regularized compressed sensing for high angular resolution diffusion
imaging,” IEEE Transactions on Medical Imaging, 30(5), 2011.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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ODF revised
• Given a recovered signal s(u, r), its corresponding ODF Q(u, r)can be computed as (Aganj et al, 2010):
Q(u, r) =1
4π+
1
16π2R∇2
b ln(− ln s(u, r))
where ∇2b stands for the Laplace-Beltrami operator.
• In practice, r belongs to a discrete set Ω:
Ω =
ri = (xi , yi , zi ) ∈ R3 | i ∈ I.
• Denoting Q(u, ri ) = Qi (u), we refer to the pair
DΩ = (rii∈I , Qi (u)i∈I)
as a directional diffusion structure (DDS).
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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ODF revised
• Given a recovered signal s(u, r), its corresponding ODF Q(u, r)can be computed as (Aganj et al, 2010):
Q(u, r) =1
4π+
1
16π2R∇2
b ln(− ln s(u, r))
where ∇2b stands for the Laplace-Beltrami operator.
• In practice, r belongs to a discrete set Ω:
Ω =
ri = (xi , yi , zi ) ∈ R3 | i ∈ I.
• Denoting Q(u, ri ) = Qi (u), we refer to the pair
DΩ = (rii∈I , Qi (u)i∈I)
as a directional diffusion structure (DDS).
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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ODF revised
• Given a recovered signal s(u, r), its corresponding ODF Q(u, r)can be computed as (Aganj et al, 2010):
Q(u, r) =1
4π+
1
16π2R∇2
b ln(− ln s(u, r))
where ∇2b stands for the Laplace-Beltrami operator.
• In practice, r belongs to a discrete set Ω:
Ω =
ri = (xi , yi , zi ) ∈ R3 | i ∈ I.
• Denoting Q(u, ri ) = Qi (u), we refer to the pair
DΩ = (rii∈I , Qi (u)i∈I)
as a directional diffusion structure (DDS).
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 43: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/43.jpg)
Graph representation of DDS
• Formally, the DDS DΩ is a subset of the probability manifoldR3 × P, where P is a set of probability densities on S2.
• DΩ is a high-dimensional space, and it is therefore imperativeto find a lower dimensional representation of DΩ in a linear metricspace.
• Such a representation should:
1 Preserve the directivity information contained in DΩ
2 Be invariant under Euclidean transformations
• To this end, we first transform DΩ into a discrete metrizable ma-nifold.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Graph representation of DDS
• Formally, the DDS DΩ is a subset of the probability manifoldR3 × P, where P is a set of probability densities on S2.
• DΩ is a high-dimensional space, and it is therefore imperativeto find a lower dimensional representation of DΩ in a linear metricspace.
• Such a representation should:
1 Preserve the directivity information contained in DΩ
2 Be invariant under Euclidean transformations
• To this end, we first transform DΩ into a discrete metrizable ma-nifold.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Graph representation of DDS
• Formally, the DDS DΩ is a subset of the probability manifoldR3 × P, where P is a set of probability densities on S2.
• DΩ is a high-dimensional space, and it is therefore imperativeto find a lower dimensional representation of DΩ in a linear metricspace.
• Such a representation should:
1 Preserve the directivity information contained in DΩ
2 Be invariant under Euclidean transformations
• To this end, we first transform DΩ into a discrete metrizable ma-nifold.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 46: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/46.jpg)
Graph representation of DDS
• Formally, the DDS DΩ is a subset of the probability manifoldR3 × P, where P is a set of probability densities on S2.
• DΩ is a high-dimensional space, and it is therefore imperativeto find a lower dimensional representation of DΩ in a linear metricspace.
• Such a representation should:
1 Preserve the directivity information contained in DΩ
2 Be invariant under Euclidean transformations
• To this end, we first transform DΩ into a discrete metrizable ma-nifold.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 47: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/47.jpg)
Graph representation of DDS (cont.)
• DΩ can be associated with an undirected weighted graph Gω =
= (V ,E ), with no self-loops and no multiple edges.
• Each vertex vi ∈ V of Gω is related to ri ∈ Ω, while the connec-tivity on Gω is defined by means of its weights:
ωi,j =
dQ(Qi ,Qj), if ‖ri − rj‖2 = ∆√
2 dQ(Qi ,Qj), if ‖ri − rj‖2 =√
2∆
+∞, otherwise
where ∆ is a spatial resolution parameter.
• Since physiologically significant regions within the brain are to-pologically connected, Gω can be endowed with a metric:
dGω(u, v) = inf l(u, v) ,
where l(u, v) =∑n
k=1 ωk−1,k is the length of a path connecting u andv .
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Graph representation of DDS (cont.)
• DΩ can be associated with an undirected weighted graph Gω =
= (V ,E ), with no self-loops and no multiple edges.
• Each vertex vi ∈ V of Gω is related to ri ∈ Ω, while the connec-tivity on Gω is defined by means of its weights:
ωi,j =
dQ(Qi ,Qj), if ‖ri − rj‖2 = ∆√
2 dQ(Qi ,Qj), if ‖ri − rj‖2 =√
2∆
+∞, otherwise
where ∆ is a spatial resolution parameter.
• Since physiologically significant regions within the brain are to-pologically connected, Gω can be endowed with a metric:
dGω(u, v) = inf l(u, v) ,
where l(u, v) =∑n
k=1 ωk−1,k is the length of a path connecting u andv .
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Graph representation of DDS (cont.)
• DΩ can be associated with an undirected weighted graph Gω =
= (V ,E ), with no self-loops and no multiple edges.
• Each vertex vi ∈ V of Gω is related to ri ∈ Ω, while the connec-tivity on Gω is defined by means of its weights:
ωi,j =
dQ(Qi ,Qj), if ‖ri − rj‖2 = ∆√
2 dQ(Qi ,Qj), if ‖ri − rj‖2 =√
2∆
+∞, otherwise
where ∆ is a spatial resolution parameter.
• Since physiologically significant regions within the brain are to-pologically connected, Gω can be endowed with a metric:
dGω(u, v) = inf l(u, v) ,
where l(u, v) =∑n
k=1 ωk−1,k is the length of a path connecting u andv .
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 50: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/50.jpg)
Graph representation of DDS (cont.)
• To finalize the definition of Gω as a metric space, the distancedQ(Qi ,Qj) is defined to be the Jensen-Shannon divergence (akainformation radius):
dψ(Qi ,Qj) =
=
[1
2
∫S2
Qi (u) ln2 Qi (u)
Qi (u) + Qj(u)dη(u)+
+1
2
∫S2
Qj(u) ln2 Qj(u)
Qi (u) + Qj(u)dη(u)
]1/2
• Note that the Jensen-Shannon divergence defines a metric on thespace of spherical probability densities P.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Graph representation of DDS (cont.)
• To finalize the definition of Gω as a metric space, the distancedQ(Qi ,Qj) is defined to be the Jensen-Shannon divergence (akainformation radius):
dψ(Qi ,Qj) =
=
[1
2
∫S2
Qi (u) ln2 Qi (u)
Qi (u) + Qj(u)dη(u)+
+1
2
∫S2
Qj(u) ln2 Qj(u)
Qi (u) + Qj(u)dη(u)
]1/2
• Note that the Jensen-Shannon divergence defines a metric on thespace of spherical probability densities P.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Isometric embedding via MDS
• Given a DDS DΩ and its associated graph representation Gω, onecan use dGω to compute the (geodesic) distances between every pairof vertices in Gω.
• These distances can be arranged into an N × N matrix
δ = δi,j = dGω(vi , vj)
with N = #Ω.
• A lower dimensional representation of DΩ can be found by meansof multidimensional scaling (MDS) which finds a configuration of Npoints XN = tkNk=1 in, e.g., Rd that minimizes the stress
E(XN) =∑i<j
(‖ti − tj‖2 − δi,j)2
• In this work, we use d = 3.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Isometric embedding via MDS
• Given a DDS DΩ and its associated graph representation Gω, onecan use dGω to compute the (geodesic) distances between every pairof vertices in Gω.
• These distances can be arranged into an N × N matrix
δ = δi,j = dGω(vi , vj)
with N = #Ω.
• A lower dimensional representation of DΩ can be found by meansof multidimensional scaling (MDS) which finds a configuration of Npoints XN = tkNk=1 in, e.g., Rd that minimizes the stress
E(XN) =∑i<j
(‖ti − tj‖2 − δi,j)2
• In this work, we use d = 3.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 54: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/54.jpg)
Isometric embedding via MDS
• Given a DDS DΩ and its associated graph representation Gω, onecan use dGω to compute the (geodesic) distances between every pairof vertices in Gω.
• These distances can be arranged into an N × N matrix
δ = δi,j = dGω(vi , vj)
with N = #Ω.
• A lower dimensional representation of DΩ can be found by meansof multidimensional scaling (MDS) which finds a configuration of Npoints XN = tkNk=1 in, e.g., Rd that minimizes the stress
E(XN) =∑i<j
(‖ti − tj‖2 − δi,j)2
• In this work, we use d = 3.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 55: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/55.jpg)
Isometric embedding via MDS
• Given a DDS DΩ and its associated graph representation Gω, onecan use dGω to compute the (geodesic) distances between every pairof vertices in Gω.
• These distances can be arranged into an N × N matrix
δ = δi,j = dGω(vi , vj)
with N = #Ω.
• A lower dimensional representation of DΩ can be found by meansof multidimensional scaling (MDS) which finds a configuration of Npoints XN = tkNk=1 in, e.g., Rd that minimizes the stress
E(XN) =∑i<j
(‖ti − tj‖2 − δi,j)2
• In this work, we use d = 3.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 56: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/56.jpg)
Euclidean invariant biomarkers
• The low-dimensional representation XN of a DDS can be cha-racterized by its moments
ξp,q,r =1
N
N∑k=1
(t1k)p(t2
k)q(t3k)r ,
with p +q +r ≤ P and the first-order moments ξ0,0,1, ξ0,1,0 and ξ1,1,0
set to zero.
• Given two DDSs D1Ω and D2
Ω and their corresponding momentsξ1
p,q,r and ξ2p,q,r, the distance between these structures can be
defined to be:
dD(D1Ω,D
2Ω) =
√ ∑p+q+r≤P
∣∣ξ1p,q,r − ξ2
p,q,r
∣∣2• In this work, P is set to be equal to 3, which results in a total of16 moments of the 2nd and 3rd orders.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Euclidean invariant biomarkers
• The low-dimensional representation XN of a DDS can be cha-racterized by its moments
ξp,q,r =1
N
N∑k=1
(t1k)p(t2
k)q(t3k)r ,
with p +q +r ≤ P and the first-order moments ξ0,0,1, ξ0,1,0 and ξ1,1,0
set to zero.
• Given two DDSs D1Ω and D2
Ω and their corresponding momentsξ1
p,q,r and ξ2p,q,r, the distance between these structures can be
defined to be:
dD(D1Ω,D
2Ω) =
√ ∑p+q+r≤P
∣∣ξ1p,q,r − ξ2
p,q,r
∣∣2
• In this work, P is set to be equal to 3, which results in a total of16 moments of the 2nd and 3rd orders.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Euclidean invariant biomarkers
• The low-dimensional representation XN of a DDS can be cha-racterized by its moments
ξp,q,r =1
N
N∑k=1
(t1k)p(t2
k)q(t3k)r ,
with p +q +r ≤ P and the first-order moments ξ0,0,1, ξ0,1,0 and ξ1,1,0
set to zero.
• Given two DDSs D1Ω and D2
Ω and their corresponding momentsξ1
p,q,r and ξ2p,q,r, the distance between these structures can be
defined to be:
dD(D1Ω,D
2Ω) =
√ ∑p+q+r≤P
∣∣ξ1p,q,r − ξ2
p,q,r
∣∣2• In this work, P is set to be equal to 3, which results in a total of16 moments of the 2nd and 3rd orders.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 59: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/59.jpg)
Diagnosis of FES
1 The subject pool consisted of 20 FE patients (16 males, 4females, average age: 21.21 ± 4.56 years) and 20 normalcontrols (15 males, 5 females, average age: 22.47 ± 3.48years) with the p-value for age being 0.34.
2 HARDI data was acquired using a 3T MRI scanner with spa-tial resolution of 1.667× 1.667× 1.7 mm3 and a b-value of900 s/mm2.
3 For each subject, expert segmentation was carried out toidentify the brain regions corresponding to the left (Ωi
L) andright (Ωi
R) centrum semiovale (with 1 ≤ i ≤ 20).
4 For each subject in the FE and normal control groups, thepairwise distances between X L
Niand XR
Niwere computed and
used as a diagnostic biomarker.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 60: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/60.jpg)
Diagnosis of FES
1 The subject pool consisted of 20 FE patients (16 males, 4females, average age: 21.21 ± 4.56 years) and 20 normalcontrols (15 males, 5 females, average age: 22.47 ± 3.48years) with the p-value for age being 0.34.
2 HARDI data was acquired using a 3T MRI scanner with spa-tial resolution of 1.667× 1.667× 1.7 mm3 and a b-value of900 s/mm2.
3 For each subject, expert segmentation was carried out toidentify the brain regions corresponding to the left (Ωi
L) andright (Ωi
R) centrum semiovale (with 1 ≤ i ≤ 20).
4 For each subject in the FE and normal control groups, thepairwise distances between X L
Niand XR
Niwere computed and
used as a diagnostic biomarker.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 61: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/61.jpg)
Diagnosis of FES
1 The subject pool consisted of 20 FE patients (16 males, 4females, average age: 21.21 ± 4.56 years) and 20 normalcontrols (15 males, 5 females, average age: 22.47 ± 3.48years) with the p-value for age being 0.34.
2 HARDI data was acquired using a 3T MRI scanner with spa-tial resolution of 1.667× 1.667× 1.7 mm3 and a b-value of900 s/mm2.
3 For each subject, expert segmentation was carried out toidentify the brain regions corresponding to the left (Ωi
L) andright (Ωi
R) centrum semiovale (with 1 ≤ i ≤ 20).
4 For each subject in the FE and normal control groups, thepairwise distances between X L
Niand XR
Niwere computed and
used as a diagnostic biomarker.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 62: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/62.jpg)
Diagnosis of FES
1 The subject pool consisted of 20 FE patients (16 males, 4females, average age: 21.21 ± 4.56 years) and 20 normalcontrols (15 males, 5 females, average age: 22.47 ± 3.48years) with the p-value for age being 0.34.
2 HARDI data was acquired using a 3T MRI scanner with spa-tial resolution of 1.667× 1.667× 1.7 mm3 and a b-value of900 s/mm2.
3 For each subject, expert segmentation was carried out toidentify the brain regions corresponding to the left (Ωi
L) andright (Ωi
R) centrum semiovale (with 1 ≤ i ≤ 20).
4 For each subject in the FE and normal control groups, thepairwise distances between X L
Niand XR
Niwere computed and
used as a diagnostic biomarker.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
![Page 63: Fast and Accurate HARDI and its Application to ... · HARDI vs. DTI Di usion tensor imaging(DTI) is a well-established tool of diag-nostic dMRI. DTI 1 Unimodal Gaussian di usion model](https://reader033.vdocuments.us/reader033/viewer/2022052815/60a3fd3de397c221d30a31a3/html5/thumbnails/63.jpg)
Diagnosis of FES (cont.)
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis
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Conclusions
The sparsifying properties of spherical ridgelets are crucial forthe CS-based reconstruction of HARDI signals.
Adding the spatial regularization and positivity constraints cansubstantially improve the reconstruction accuracy.
HARDI can be performed using as few diffusion gradients as itis required by a standard DTI (i.e. K = 20÷ 30).
A new method for low-dimensional representation of HARDIsignals based on isometric embedding was formulated.
It was demonstrated that the performance of the method re-mains reliable, when the sampling density is reduced by a fac-tor of 4 with respect to its conventional value.
Oleg Michailovich: [email protected] Fast and accurate HARDI & neurological diagnosis