I would like to bring your attention to our new paper which will appear in IEEE Trans. on Medical Imaging.
O. Lee, J.M. Kim, Y. Bresler, and J. C. Ye, "Compressive Diffuse Optical Tomography: Non-Iterative Exact Reconstruction using Joint Sparsity," IEEE Trans. Medical Imaging, Special issue on "Compressive sensing for biomedical imaging", 2011 (in press).
The novelty of this work is to demonstrate that compressed sensing is not only useful to improve resolution in diffuse optical tomography, but also helpful to overcome the nonlinearity of inverse scattering problem, which is originated from diffuse wave propagation. Accordingly, we can provide a non-iterative and exact reconstruction algorithm for diffuse optical tomography without resorting to Born-type approximation and iterative use of computationally expensive PDE solvers. The key theoretical ingredient to enable this work is a new-type of CS solver for joint sparse recovery based on recently discovered generalized MUSIC criterion. We believe that this type of approach can be extended for general inverse scattering problems such as radar, microwave imaging, optics, and etc.
I would be happy to hear any feedback from you !
Solving the nonlinear diffusion equation is a great achievement! Let me just say that I am going to read it over the week-end to digest how this implementation of MUSIC is doing the work. One tiny thing, equations (2) and (3) are approximations of the linear transport equation which itself has been looked into with some l_1 minimization approach.
Here is the paper: Compressive Diffuse Optical Tomography: Non-Iterative Exact Reconstruction using Joint Sparsity by Okkyun Lee, Jong Min Kim, , Yoram Bresler and Jong Chul Ye. The abstract reads:
Diffuse optical tomography (DOT) is a sensitive and relatively low cost imaging modality that reconstructs optical properties of a highly scattering medium. However, due to the diffusive nature of light propagation, the problem is severely illconditioned and highly nonlinear. Even though nonlinear iterative methods have been commonly used, they are computationally expensive especially for three dimensional imaging geometry. Recently, compressed sensing theory has provided a systematic understanding of high resolution reconstruction of sparse objects in many imaging problems; hence, the goal of this paper is to extend the theory to the diffuse optical tomography problem. The main contributions of this paper are to formulate the imaging problem as a joint sparse recovery problem in a compressive sensing framework and to propose a novel non-iterative and exact inversion algorithm that achieves the l0 optimality as the rank of measurement increases to the unknown sparsity level. The algorithm is based on the recently discovered generalized MUSIC criterion, which exploits the advantages of both compressive sensing and array signal processing. A theoretical criterion for optimizing the imaging geometry is provided, and simulation results conﬁrm that the new algorithm outperforms the existing algorithms and reliably reconstructs the optical inhomogeneities when we assume that the optical background is known to a reasonable accuracy
Muhammad Usman pointed out two of his recent papers: A computationally efficient OMP based compressed sensing reconstruction for dynamic MRI by Muhammad Usman, Claudia Prieto, Freddy Odille, David Atkinson, Tobias Schaeffter and Philip Batchelor. The abstract reads:
Compressed sensing (CS) methods in MRI are computationally intensive. Thus, designing novel CS algorithms that can perform faster reconstructions is crucial for everyday applications. We propose a computationally efﬁcient orthogonal matching pursuit (OMP)-based reconstruction, speciﬁcally suited to cardiac MR data. According to the energy distribution of a y–f space obtained from a sliding window reconstruction, we label the y–f space as static or dynamic. For static y–f space images, a computationally efﬁcient masked OMP reconstruction is performed, whereas for dynamic y–f space images, standard OMP reconstruction is used. The proposed method was tested on a dynamic numerical phantom and two cardiac MR datasets. Depending on the ﬁeld of view composition of the imaging data, compared to the standard OMP method, reconstruction speedup factors ranging from 1.5 to 2.5 are achieved.
K-t group sparse: a method for accelerating dynamic MRI by Muhammad Usman , Claudia Prieto, Tobias Schaeffter and Philip Batchelor. The abstract reads:
Compressed sensing (CS) is a data-reduction technique that has been applied to speed up the acquisition in MRI. However, the use of this technique in dynamic MR applications has been limited in terms of the maximum achievable reduction factor. In general, noise-like artefacts and bad temporal fidelity are visible in standard CS MRI reconstructions when high reduction factors are used. To increase the maximum achievable reduction factor, additional or prior information can be incorporated in the CS reconstruction. Here, a novel CS reconstruction method is proposed that exploits the structure within the sparse representation of a signal by enforcing the support components to be in the form of groups. These groups act like a constraint in the reconstruction. The information about the support region can be easily obtained from training data in dynamic MRI acquisitions. The proposed approach was tested in two-dimensional cardiac cine MRI with both down-sampled and undersampled data. Results show that higher acceleration factors (up to 9-fold), with improved spatial and temporal quality, can be obtained with the proposed approach in comparison to the standard CS reconstructions.