Simple Harmonic Oscillator Based Reconstruction and Estimation for One-Dimensional q-Space Magnetic Resonance (1D-SHORE)

Özarslan, Evren; Koay, Cheng Guan; Basser, Peter J.

doi:10.1007/978-0-8176-8379-5_19

Cited by 59 publications

(113 citation statements)

References 34 publications

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“…The signal fitting is performed with SHORE [8], a promising signal reconstruction method suitable for q-space magnetic resonance. Within this framework the signal is represented as the linear combination of orthogonal basis functions, result of the multiplication between an exponential and an Hermite polynomial…”

Section: Signal Reconstructionmentioning

confidence: 99%

“…Our formulation of the basis functions in Eq. 5 differs from the one given in [8] with the introduction of the normalizing factor 2 u √ π, which renders the bases orthonormal. The bases are well suited for representing the signal in the complex domain: the even order basis functions are real valued and evenly symmetric whereas the odd order basis functions are imaginary and show odd symmetry, which is precisely the case of the real and the imaginary parts of the diffusion signal.…”

Section: Signal Reconstructionmentioning

confidence: 99%

“…The reconstructed signal, as the linear combination of basis functions, leads to an effective characterization of the diffusion properties and is a useful tool for measuring noise-related performances. In this respect we employed the Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) [8]. Within this framework, a Maximum Likelihood Estimator (MLE), for the reconstruction based on the Rician magnitude signal, is also provided for performance comparison.…”

Section: Introductionmentioning

confidence: 99%

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Magnitude and Complex Based Diffusion Signal Reconstruction

Pizzolato

Ghosh

Boutelier

et al. 2014

Computational Diffusion MRI

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In Diffusion Weighted Magnetic Resonance Imaging (DW-MRI) the modeling of the magnitude signal is complicated by the Rician distribution of the noise. It is well known that when dealing instead with the complex valued signal, the real and imaginary parts are affected by Gaussian distributed noise and their modeling can thus benefit from any estimation technique suitable for this noise distribution. We present a quantitative analysis of the difference between the modeling of the magnitude diffusion signal and the modeling in the complex domain. The noisy complex and magnitude diffusion signals are obtained for a physically realistic scenario in a region close to a restricting boundary. These signals are then fitted with the Simple Harmonic Oscillator based Reconstruction and Estimation (SHORE) bases and the reconstruction performances are quantitatively compared. The noisy magnitude signal is also fitted by taking into account the Rician distribution of the noise via the integration of a Maximum Likelihood Estimator (MLE) in the SHORE. We discuss the performance of the reconstructions as function of the Signal to Noise Ratio (SNR) and the sampling resolution of the diffusion signal. We show that fitting in the complex domain generally allows for quantitatively better signal reconstruction, also with a poor SNR, provided that the sampling resolution of the signal is adequate. This applies also when the reconstruction is compared to the one performed on the magnitude via the MLE. 1

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Section: Signal Reconstructionmentioning

confidence: 99%

Section: Signal Reconstructionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Magnitude and Complex Based Diffusion Signal Reconstruction

Pizzolato

Ghosh

Boutelier

et al. 2014

Computational Diffusion MRI

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“…Since Diffusion Tensor Imaging (DTI) cannot handle the complex fiber configuration, a category of reconstruction methods, named as High Angular Resolution Diffusion Imaging (HARDI), were proposed to avoid the Gaussian EAP assumption in DTI [11,7,8,1,5,10]. In HARDI, EAP and two kinds of the Orientation Distribution Functions (ODFs) defined as Φ 0 (r) = 1 Z ∞ 0 P(Rr)dR, Φ 2 (r) = ∞ 0 P(Rr)R 2 dR, are normally used to infer fiber directions, where Z in Φ 0 (r) is the normalization factor to make Φ 0 (r) as a PDF.…”

Section: P(r) Exp(−2πiqmentioning

confidence: 99%

Nonnegative Definite EAP and ODF Estimation via a Unified Multi-shell HARDI Reconstruction

Cheng

Jiang

Deriche

2012

Medical Image Computing and Computer-Assisted Intervention – MICCAI 2012

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Abstract. In High Angular Resolution Diffusion Imaging (HARDI), Orientation Distribution Function (ODF) and Ensemble Average Propagator (EAP) are two important Probability Density Functions (PDFs) which reflect the water diffusion and fiber orientations. Spherical Polar Fourier Imaging (SPFI) is a recent modelfree multi-shell HARDI method which estimates both EAP and ODF from the diffusion signals with multiple b values. As physical PDFs, ODFs and EAPs are nonnegative definite respectively in their domains S 2 and R 3 . However, existing ODF/EAP estimation methods like SPFI seldom consider this natural constraint. Although some works considered the nonnegative constraint on the given discrete samples of ODF/EAP, the estimated ODF/EAP is not guaranteed to be nonnegative definite in the whole continuous domain. The Riemannian framework for ODFs and EAPs has been proposed via the square root parameterization based on pre-estimated ODFs and EAPs by other methods like SPFI. However, there is no work on how to estimate the square root of ODF/EAP called as the wavefuntion directly from diffusion signals. In this paper, based on the Riemannian framework for ODFs/EAPs and Spherical Polar Fourier (SPF) basis representation, we propose a unified model-free multi-shell HARDI method, named as Square Root Parameterized Estimation (SRPE), to simultaneously estimate both the wavefunction of EAPs and the nonnegative definite ODFs and EAPs from diffusion signals. The experiments on synthetic data and real data showed SRPE is more robust to noise and has better EAP reconstruction than SPFI, especially for EAP profiles at large radius.

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“…Spherical Polar Fourier Imaging (SPFI) [1,3] models the signal in terms of an orthonormal basis comprising spherical harmonics (SH) and Gaussian-Laguerre polynomials, and was proposed to sparsely represent E(q). The authors in [10] expanded the diffusion signal in terms of another orthonormal basis that appears in the 3D quantum mechanical harmonic oscillator problem. Their basis is closely related to the SPFI basis.…”

Section: Introductionmentioning

confidence: 99%

Bessel Fourier Orientation Reconstruction: An Analytical EAP Reconstruction Using Multiple Shell Acquisitions in Diffusion MRI

Hosseinbor

Chung

et al. 2011

Lecture Notes in Computer Science

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Abstract. The estimation of the ensemble average propagator (EAP) directly from q-space DWI signals is an open problem in diffusion MRI. Diffusion spectrum imaging (DSI) is one common technique to compute the EAP directly from the diffusion signal, but it is burdened by the large sampling required. Recently, several analytical EAP reconstruction schemes for multiple q-shell acquisitions have been proposed. One, in particular, is Diffusion Propagator Imaging (DPI) which is based on the Laplace's equation estimation of diffusion signal for each shell acquisition. Viewed intuitively in terms of the heat equation, the DPI solution is obtained when the heat distribution between temperatuere measurements at each shell is at steady state.We propose a generalized extension of DPI, Bessel Fourier Orientation Reconstruction (BFOR), whose solution is based on heat equation estimation of the diffusion signal for each shell acquisition. That is, the heat distribution between shell measurements is no longer at steady state. In addition to being analytical, the BFOR solution also includes an intrinsic exponential smootheing term. We illustrate the effectiveness of the proposed method by showing results on both synthetic and real MR datasets.

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Simple Harmonic Oscillator Based Reconstruction and Estimation for One-Dimensional q-Space Magnetic Resonance (1D-SHORE)

Cited by 59 publications

References 34 publications

Magnitude and Complex Based Diffusion Signal Reconstruction

Magnitude and Complex Based Diffusion Signal Reconstruction

Nonnegative Definite EAP and ODF Estimation via a Unified Multi-shell HARDI Reconstruction

Bessel Fourier Orientation Reconstruction: An Analytical EAP Reconstruction Using Multiple Shell Acquisitions in Diffusion MRI

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