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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
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Goal
Elucidate, non-invasively, the 3-D architectural structure of tissues,organs and organized matter: e.g., find fiber tracks
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Goal
Elucidate, non-invasively, the 3-D architectural structure of tissues,organs and organized matter: e.g., find fiber tracks
Examples
• White matter fiber tracks
• Heart muscle fiber structure
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Goal
Elucidate, non-invasively, the 3-D architectural structure of tissues,organs and organized matter: e.g., find fiber tracks
Examples
• White matter fiber tracks
• Heart muscle fiber structure
Applications
• Mapping connectivity between tissues and organs
• Monitoring structural changes in development, aging and disease
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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).
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Difficulties
• Data known only on discrete sets (sampled data)
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Difficulties
• Data known only on discrete sets (sampled data)
• Complex data (non-negative definite matrices, and vectors)
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Difficulties
• Data known only on discrete sets (sampled data)
• Complex data (non-negative definite matrices, and vectors)
• Measurements are averaged (Partial Volume Effects PVA)
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Difficulties
• Data known only on discrete sets (sampled data)
• Complex data (non-negative definite matrices, and vectors)
• Measurements are averaged (Partial Volume Effects PVA)
• Noise
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Difficulties
• Data known only on discrete sets (sampled data)
• Complex data (non-negative definite matrices, and vectors)
• Measurements are averaged (Partial Volume Effects PVA)
• Noise
• High dimensionality of data (107 values: high computational complexity)
• ...
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Tools
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Tools
• Denoising (linear, nonlinear, spatially adaptive)
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Tools
• Denoising (linear, nonlinear, spatially adaptive)
• Interpolation (spacially adaptive, non-negative definite preserving +...)
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Tools
• Denoising (linear, nonlinear, spatially adaptive)
• Interpolation (spacially adaptive, non-negative definite preserving +...)
• Numerical DE solver (fast, accurate)
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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∆.
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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∆.
• 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 ∆.
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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∆.
• 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.
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Example 1
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DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Example 2
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DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Example 3
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DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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)
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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.
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Example 5
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DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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; ....,
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
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,...
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Challenges
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Challenges
• Regularization methods in manifold of non-negative definite matrices
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Challenges
• Regularization methods in manifold of non-negative definite matrices
• Nonlinear, adaptive edge preserving filtering
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
Challenges
• Regularization methods in manifold of non-negative definite matrices
• Nonlinear, adaptive edge preserving filtering
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
• Nonlinear adaptive interpolation for edge preservation
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
• Nonlinear adaptive interpolation for edge preservation
• Methods to deal with PVA, brakes in fibers, crossing or kissing of fibers,and other singular features.
– Typeset by FoilTEX – Akram Aldroubi
DTMRI New Mathematics and Algorithms for 3-D Image Analysis
END
– Typeset by FoilTEX – Akram Aldroubi
