Processing of Diﬀusion-Tensor Magnetic … of Diﬀusion-Tensor Magnetic Resonance Images ... IMA...

Processing of Diffusion-Tensor MagneticResonance Images

Akram AldroubiVanderbilt University

IMA January 2005

Collaborators:

• Peter Basser, Sinisa Pajevic,Carlo Pierpaoli (NIH)

• Adam Anderson, Zhaohua Ding, John Gore, Yonggang Lu (Vanderbilt UniversityInstitute of Imaging Science)

– Typeset by FoilTEX –

DTMRI New Mathematics and Algorithms for 3-D Image Analysis

Problem

What is DTMRI?

ρ(r|τd) =1√

‖D‖(4πτd)exp

(−rDr4τd

)(1)

– Typeset by FoilTEX – Akram Aldroubi


Goal

Elucidate, non-invasively, the 3-D architectural structure of tissues,organs and organized matter: e.g., find fiber tracks



Goal


Examples

• White matter fiber tracks

• Heart muscle fiber structure



Goal


Examples

• White matter fiber tracks

• Heart muscle fiber structure

Applications

• Mapping connectivity between tissues and organs

• Monitoring structural changes in development, aging and disease



Problem

Tracktography: Find fiber track from DT data

drds

= ε1(r), r(0) = r0

where ε1(r) is the largest eigenvector of diffusion tensor field D(r).



Difficulties

• Data known only on discrete sets (sampled data)



Difficulties


• Complex data (non-negative definite matrices, and vectors)



Difficulties



• Measurements are averaged (Partial Volume Effects PVA)



Difficulties




• Noise



Difficulties




• Noise

• High dimensionality of data (107 values: high computational complexity)

• ...



Tools



Tools

• Denoising (linear, nonlinear, spatially adaptive)



Tools


• Interpolation (spacially adaptive, non-negative definite preserving +...)



Tools


• Interpolation (spacially adaptive, non-negative definite preserving +...)

• Numerical DE solver (fast, accurate)



Shift Invariant Space Models

V∆(B) =

D(x) =∑j∈Λ

∑k∈Zn

cj(k)Bj

(x∆

− k)

: cj ∈ `2(Zn)

, (2)

where {Bj : j ∈ Λ} form a generator for the space V∆.




V∆(B) =

D(x) =∑j∈Λ

∑k∈Zn

cj(k)Bj

(x∆

− k)

: cj ∈ `2(Zn)

, (2)


• Discrete tensor data {D(k) : k ∈ Z3} can be approximated with fastfiltering algorithms by a member in D∆ ∈ V∆ defined on all R3, wherenoise is controlled by size ∆.




V∆(B) =

D(x) =∑j∈Λ

∑k∈Zn

cj(k)Bj

(x∆

− k)

: cj ∈ `2(Zn)

, (2)


• Discrete tensor data {D(k) : k ∈ Z3} can be approximated with fastfiltering algorithms by a member in D∆ ∈ V∆ defined on all R3, wherenoise is controlled by size ∆.

• Use Euler method for integration.



Example 1



Example 2



Example 3



Bayesian Approach

Euler tracking methodri+1 = ri + αεi

where α is integration step size step and εi is direction vector to beestimated. Let d = (D11D22D33D12D13D23)t and let d be the trueelement tensor vector

p(d|d) =1

(2ψ)3|Σ|1/2exp

[−1

2(d− d)tΣ−1(d− d)

]

p(d) =1

(2ψ)3|S1/2exp

[−1

2(d−m)tS−1(d−m)

]Maximize

p(d|d) =p(d|d)p(d)

p(d)



Example 4

(a,b) axial views and (c,d) sagital views. SNR 30 (a,c) and 20 (b,d).Blue= Bayesian, Red= Euler, Green=TEND.



Example 5



Other methods and remaining challenges

Methods

• Deterministic, i.e. 1-point at most 1 path between two points, e.g.Aldroubi and Basser 1998; Mori et al. 1999; Basser et al. 2000; Pouponet al. 2000; Gossl et al. 2002; Parker et al. 2002; Tench et al. 2002;Zhukov et al. 2002, Stampfli et al. 2005, Lazar et al. 2003; ....,



Other methods and remaining challenges

Methods

• Deterministic, i.e. 1-point at most 1 path between two points, e.g.Aldroubi and Basser 1998; Mori et al. 1999; Basser et al. 2000; Pouponet al. 2000; Gossl et al. 2002; Parker et al. 2002; Tench et al. 2002;Zhukov et al. 2002, Stampfli et al. 2005, Lazar et al. 2003; ....,

• Probabilistic, e.g. Hagmann et al. 2003, Koch et al. 2002, Parker et al.2003,...



Challenges



Challenges

• Regularization methods in manifold of non-negative definite matrices



Challenges

• Regularization methods in manifold of non-negative definite matrices

• Nonlinear, adaptive edge preserving filtering



• Nonlinear adaptive interpolation for edge preservation



• Nonlinear adaptive interpolation for edge preservation

• Methods to deal with PVA, brakes in fibers, crossing or kissing of fibers,and other singular features.



END


Processing of Diﬀusion-Tensor Magnetic … of Diﬀusion-Tensor Magnetic Resonance Images ... IMA...

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