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Index
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, [email protected] HARDI Contest, ISBI 2012
Index
Index
Sampling schemeDescription of the Reconstruction MethodExample
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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).
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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).
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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).
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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)
l2
πflmXl
log(1
D(q)) =
∞∑l=0
l∑m=−l
flmYlm(q)
l 0 2 4 6 8
Xl√π
2f00
1232
6064
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)
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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)
l2
πflmXl
log(1
D(q)) =
∞∑l=0
l∑m=−l
flmYlm(q)
l 0 2 4 6 8
Xl√π
2f00
1232
6064
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)
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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)
l2
πflmXl
log(1
D(q)) =
∞∑l=0
l∑m=−l
flmYlm(q)
l 0 2 4 6 8
Xl√π
2f00
1232
6064
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)
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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‖
K)2
=⇒ 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)
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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‖
K)2
=⇒ 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)
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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‖
K)2
=⇒ 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)
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
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.
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
Example2D synthetic phantom: SNR = 15, N = 37, b = 3333 s/mm2
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Index
Example2D synthetic phantom: SNR = 15, N = 37, b = 3333 s/mm2
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Results: Testing SF datasetODF estimation based metrics
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Results: Testing SF datasetODF estimation based metrics
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012
Results: Testing SF datasetODF estimation based metrics
E.J. Canales-Rodrıguez, [email protected] HARDI Contest, ISBI 2012