Majumdar Angshul, Ward Rabab K
Department of Electrical and Computer Engineering, University of British Columbia.
Med Image Comput Comput Assist Interv. 2010;13(Pt 3):513-20. doi: 10.1007/978-3-642-15711-0_64.
This work explores the problem of solving the MR reconstruction problem when the number of K-space samples acquired in a non-Cartesian grid is considerably less than the resolution (number of pixels) of the image. Mathematically this leads to the solution of an under-determined and ill-posed inverse problem. The inverse problem can only be solved when certain additional/prior assumption is made about the solution. In this case, the prior is the sparsity of the MR image in the wavelet domain. The non-convex lp-norm () of the wavelet coefficient is a suitable metric for sparsity. Such a prior can appear in two forms--in the synthesis prior formulation, the wavelet coefficients of the image is solved for while in the analysis prior formulation the actual image is solved for. Traditionally the synthesis prior formulation is more popular. However, in this work we will show that the analysis prior formulation on redundant wavelet transform provides better MR reconstruction results compared to the synthesis prior formulation.
这项工作探讨了在非笛卡尔网格中采集的K空间样本数量远少于图像分辨率（像素数量）时解决磁共振成像（MR）重建问题。从数学角度来看，这导致了一个欠定且不适定的逆问题的求解。只有在对解做出某些额外/先验假设时，逆问题才能得到解决。在这种情况下，先验是MR图像在小波域中的稀疏性。小波系数的非凸lp范数（）是稀疏性的合适度量。这样的先验可以以两种形式出现——在合成先验公式中，求解图像的小波系数，而在分析先验公式中，求解实际图像。传统上，合成先验公式更受欢迎。然而，在这项工作中，我们将表明，与合成先验公式相比，基于冗余小波变换的分析先验公式能提供更好的MR重建结果。
