Multi-Tensor Fitting Guided by Orientational Distribution Function (ODF)...

Multi-Tensor Fitting Guided by OrientationalDistribution Function (ODF) Estimation

Erick Jorge Canales-Rodrıguez1,2, Lester Melie-Garcıa3, YasserIturria-Medina3, Yasser Aleman-Gomez2,4

1FIDMAG Research Foundation (Barcelona, Spain)2Centro de Investigacion Biomedica en Red de Salud Mental, CIBERSAM

(Madrid, Spain)3Cuban Neuroscience Center (Havana, Cuba)

4Hospital General Universitario Gregorio Maranon (Madrid, Spain)

May 2, 2012

E.J. Canales-Rodrıguez, ejcanalesr@gmail.com HARDI Contest, ISBI 2012

Sampling schemeDescription of the Reconstruction MethodExample

Sampling scheme

Details

N = 37 points obtained via [Appelbaum J. and Weiss Y, 1999].Method for locating N equal nonoverlapping circles on ahemisphere.

b− value = 3333 s/mm2

Two datasets: isolated voxels (IV) and structured field (SF).SNR = 5, 10, 15, 20, 25, 30, 35 and 40.(http://hardi.epfl.ch/contest.html)

References

Appelbaum J. and Weiss Y. The packing of circles on a hemisphere. Meas. Sci. Technol., 10, 10151019

(1999).

Sampling scheme

Details

References

(1999).

Sampling scheme

Details

References

(1999).

Description of the Reconstruction Method

Pipeline

1 ODF Reconstruction

2 ODF Spatial Filtering

3 ODF Deconvolution

4 ODF Maxima Extraction

5 Multi-Tensor ODF Fitting

Pipeline

3 ODF Deconvolution

Pipeline

3 ODF Deconvolution

Pipeline

3 ODF Deconvolution

Pipeline

3 ODF Deconvolution

ODF ReconstructionDescription of the Reconstruction Method

The ODF reconstruction method implemented in this work is described in[Canales-Rodrıguez E.J. et al., 2010]

and [Aganj I. et al., 2010], [Tristan-Vega A. et al., 2010].

Definition

E(q, q) = e−bD(q)

P (ρ, r) =

∫E(q, q)e−2πiρqrqq2dqdq

DOT method [Evren Ozarslan et al., 2006]

ODF (r) =

∫P (ρ, r)ρ2dρ

References

Canales-Rodrıguez E.J. et al. Diffusion orientation transform revisited.

Neuroimage, 49(2), 1326-1339 (2010).

Evren Ozarslan et al. Resolution of complex tissue microarchitecture

using the diffusion orientation transform (DOT). Neuroimage, 31(3),10861103 (2006)

Solution

ODF (r) =∞∑l=0

l∑m=−l

olmYlm(r)

olm =(−1)

πflmXl

D(q)) =

∞∑l=0

l∑m=−l

flmYlm(q)

l 0 2 4 6 8

Xl√π

840512

25201024

References

Aganj I. et al. Reconstruction of the orientation distribution function in

single- and multiple-shell q-ball imaging within constant solid angle.Magnetic Resonance in Medicine, 64(2), 554-566, (2010).

Tristan-Vega et al. A new methodology for the estimation of fiber

populations in the white matter of the brain with the Funk-Radon transform.Neuroimage, 49(2), 1301-1315 (2010)

The ODF reconstruction method implemented in this work is described in[Canales-Rodrıguez E.J. et al., 2010]

and [Aganj I. et al., 2010], [Tristan-Vega A. et al., 2010].

Definition

E(q, q) = e−bD(q)

P (ρ, r) =

ODF (r) =

∫P (ρ, r)ρ2dρ

References

Neuroimage, 49(2), 1326-1339 (2010).

Solution

ODF (r) =∞∑l=0

l∑m=−l

olmYlm(r)

olm =(−1)

πflmXl

D(q)) =

∞∑l=0

l∑m=−l

flmYlm(q)

l 0 2 4 6 8

Xl√π

840512

25201024

References

The ODF reconstruction method implemented in this work is described in[Canales-Rodrıguez E.J. et al., 2010] and [Aganj I. et al., 2010], [Tristan-Vega A. et al., 2010].

Definition

E(q, q) = e−bD(q)

P (ρ, r) =

ODF (r) =

∫P (ρ, r)ρ2dρ

References

Neuroimage, 49(2), 1326-1339 (2010).

Solution

ODF (r) =∞∑l=0

l∑m=−l

olmYlm(r)

olm =(−1)

πflmXl

D(q)) =

∞∑l=0

l∑m=−l

flmYlm(q)

l 0 2 4 6 8

Xl√π

840512

25201024

References

ODF Spatial Filtering (only in the structured-field data)Description of the Reconstruction Method

Smoothing/enhancement via Perona−Malik nonlinear diffusion

ODF (x, y, z, r) =∞∑l=0

l∑m=−l

olm(x, y, z)Ylm(r)

Secuencial smoothing [Perona and Malik](each channel/volume separately)

∂tolm = div(g(‖∇olm‖)∇olm)

olm(x, y, z, t = 0) = olm(x, y, z)

g(‖∇olm‖) = e−(‖∇olm‖

=⇒ olm

References

Perona P. and Malik J. Scale-Space and Edge Detection Using Anisotropic Diffusion. IEEE Trans. Pattern Anal. Mach. Intell.,12(7), 629-639 (1990).

ODF (x, y, z, r) =∞∑l=0

l∑m=−l

olm(x, y, z)Ylm(r)

ODF (x, y, z, r) =∞∑l=0

l∑m=−l

olm(x, y, z)Ylm(r)

olm(x, y, z, t = 0) = olm(x, y, z)

g(‖∇olm‖) = e−(‖∇olm‖

=⇒ olm

References

ODF (x, y, z, r) =∞∑l=0

l∑m=−l

olm(x, y, z)Ylm(r)

ODF (x, y, z, r) =∞∑l=0

l∑m=−l

olm(x, y, z)Ylm(r)

olm(x, y, z, t = 0) = olm(x, y, z)

g(‖∇olm‖) = e−(‖∇olm‖

=⇒ olm

References

ODF (x, y, z, r) =∞∑l=0

l∑m=−l

olm(x, y, z)Ylm(r)

ODF DeconvolutionDescription of the Reconstruction Method

Deconvolution/sharpening using the PSF of the reconstruction

The obtained ODF is a low-pass representation of the true ODF (due to the spherical harmonictruncation and the regularization).

The angular point spread function (PSF) of the reconstruction can be obtained by the

sphericalinversion

of a delta function. (see [Descoteaux M. et al., 2007]).delta PSF (Lmax = 6, λ = 0) PSF (Lmax = 6, λ = 0.1)

References

Descoteaux M. et al. Regularized, Fast and Robust Analytical Q-Ball Imaging. Magn Reson. Med., 58(3),497-510 (2007)

The true diffusion ODF can be recovered by deconvolving the obtained ODF and PSF.

The obtained ODF is a low-pass representation of the true ODF (due to the spherical harmonictruncation and the regularization).The angular point spread function (PSF) of the reconstruction can be obtained by the sphericalinversion of a delta function.

(see [Descoteaux M. et al., 2007]).

delta PSF (Lmax = 6, λ = 0) PSF (Lmax = 6, λ = 0.1)

References

The obtained ODF is a low-pass representation of the true ODF (due to the spherical harmonictruncation and the regularization).The angular point spread function (PSF) of the reconstruction can be obtained by the sphericalinversion of a delta function. (see [Descoteaux M. et al., 2007]).

References

The obtained ODF is a low-pass representation of the true ODF (due to the spherical harmonictruncation and the regularization).The angular point spread function (PSF) of the reconstruction can be obtained by the sphericalinversion of a delta function. (see [Descoteaux M. et al., 2007]).

References

ODF Maxima ExtractionDescription of the Reconstruction Method

Local fiber orientations were determined as follows:

1 All local maxima were obtained by comparing the ODF amplitudes between each point inthe grid and its nearest neighbors within and interval of 15 degrees.

2 The largest three local maxima were preserved if their amplitudes were larger than0.4×ODFmax, where ODFmax is the amplitude of the global maximum.

3 All neighbors around each maximum were used to fit an ellipsoid centered at the origin.The position of the principal direction of each ellipsoid was used to specify the local fiberorientation.

Interpolated maximum =⇒ ⇐= Discrete maximum

Multi-Tensor ODF FittingDescription of the Reconstruction Method

A non-linear fitting procedure was implemented to obtain thediffusivites corresponding to each fiber population in the voxel.(MATLAB r function: fmincon)

The algorithm computes the diffusivities that minimizes theleast-squares difference between the theoretical multi-tensor ODFand the computed ODF.

The number of fiber populations, their relative intensities andspatial orientations were assumed to be known and equal to thevalues determined in previous steps.

The reported ODF was the obtained from the multi-tensor fittedmodel.

Example2D synthetic phantom: SNR = 15, N = 37, b = 3333 s/mm2

Results: Testing SF datasetODF estimation based metrics

Multi-Tensor Fitting Guided by Orientational Distribution Function (ODF)...

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