Volume 38, Issue 5
Accurate and Efficient Image Reconstruction from Multiple Measurements of Fourier Samples

T. Scarnati & Anne Gelb

J. Comp. Math., 38 (2020), pp. 797-826.

Published online: 2020-07

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  • Abstract

Several problems in imaging acquire multiple measurement vectors (MMVs) of Fourier samples for the same underlying scene. Image recovery techniques from MMVs aim to exploit the joint sparsity across the measurements in the sparse domain. This is typically accomplished by extending the use of $\ell_1$ regularization of the sparse domain in the single measurement vector (SMV) case to using $\ell_{2,1}$ regularization so that the "jointness" can be accounted for.  Although effective, the approach is inherently coupled and therefore computationally inefficient. The method also does not consider current approaches in the SMV case that use spatially varying weighted $\ell_1$ regularization term. The recently introduced variance based joint sparsity (VBJS) recovery method uses the variance across the measurements in the sparse domain to produce a weighted MMV method that is more accurate and more efficient than the standard $\ell_{2,1}$ approach. The efficiency is due to the decoupling of the measurement vectors, with the increased accuracy resulting from the spatially varying weight. Motivated by these results, this paper introduces a new technique to even further reduce computational cost by eliminating the requirement to first approximate the underlying image in order to construct the weights. Eliminating this preprocessing step moreover reduces the amount of information lost from the data, so that our method is more accurate. Numerical examples provided in the paper verify these benefits.

  • Keywords

Multiple measurement vectors, Joint sparsity, Weighted $\ell_1$, Edge detection, Fourier data.

  • AMS Subject Headings

49N45, 68U10, 65T99

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

theresa.scarnati.1@us.af.mil (T. Scarnati)

annegelb@math.dartmouth.edu (Anne Gelb)

  • BibTex
  • RIS
  • TXT
@Article{JCM-38-797, author = {Scarnati , T. and Gelb , Anne }, title = {Accurate and Efficient Image Reconstruction from Multiple Measurements of Fourier Samples}, journal = {Journal of Computational Mathematics}, year = {2020}, volume = {38}, number = {5}, pages = {797--826}, abstract = {

Several problems in imaging acquire multiple measurement vectors (MMVs) of Fourier samples for the same underlying scene. Image recovery techniques from MMVs aim to exploit the joint sparsity across the measurements in the sparse domain. This is typically accomplished by extending the use of $\ell_1$ regularization of the sparse domain in the single measurement vector (SMV) case to using $\ell_{2,1}$ regularization so that the "jointness" can be accounted for.  Although effective, the approach is inherently coupled and therefore computationally inefficient. The method also does not consider current approaches in the SMV case that use spatially varying weighted $\ell_1$ regularization term. The recently introduced variance based joint sparsity (VBJS) recovery method uses the variance across the measurements in the sparse domain to produce a weighted MMV method that is more accurate and more efficient than the standard $\ell_{2,1}$ approach. The efficiency is due to the decoupling of the measurement vectors, with the increased accuracy resulting from the spatially varying weight. Motivated by these results, this paper introduces a new technique to even further reduce computational cost by eliminating the requirement to first approximate the underlying image in order to construct the weights. Eliminating this preprocessing step moreover reduces the amount of information lost from the data, so that our method is more accurate. Numerical examples provided in the paper verify these benefits.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2002-m2019-0192}, url = {http://global-sci.org/intro/article_detail/jcm/17284.html} }
TY - JOUR T1 - Accurate and Efficient Image Reconstruction from Multiple Measurements of Fourier Samples AU - Scarnati , T. AU - Gelb , Anne JO - Journal of Computational Mathematics VL - 5 SP - 797 EP - 826 PY - 2020 DA - 2020/07 SN - 38 DO - http://doi.org/10.4208/jcm.2002-m2019-0192 UR - https://global-sci.org/intro/article_detail/jcm/17284.html KW - Multiple measurement vectors, Joint sparsity, Weighted $\ell_1$, Edge detection, Fourier data. AB -

Several problems in imaging acquire multiple measurement vectors (MMVs) of Fourier samples for the same underlying scene. Image recovery techniques from MMVs aim to exploit the joint sparsity across the measurements in the sparse domain. This is typically accomplished by extending the use of $\ell_1$ regularization of the sparse domain in the single measurement vector (SMV) case to using $\ell_{2,1}$ regularization so that the "jointness" can be accounted for.  Although effective, the approach is inherently coupled and therefore computationally inefficient. The method also does not consider current approaches in the SMV case that use spatially varying weighted $\ell_1$ regularization term. The recently introduced variance based joint sparsity (VBJS) recovery method uses the variance across the measurements in the sparse domain to produce a weighted MMV method that is more accurate and more efficient than the standard $\ell_{2,1}$ approach. The efficiency is due to the decoupling of the measurement vectors, with the increased accuracy resulting from the spatially varying weight. Motivated by these results, this paper introduces a new technique to even further reduce computational cost by eliminating the requirement to first approximate the underlying image in order to construct the weights. Eliminating this preprocessing step moreover reduces the amount of information lost from the data, so that our method is more accurate. Numerical examples provided in the paper verify these benefits.

T. Scarnati & Anne Gelb. (2020). Accurate and Efficient Image Reconstruction from Multiple Measurements of Fourier Samples. Journal of Computational Mathematics. 38 (5). 797-826. doi:10.4208/jcm.2002-m2019-0192
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