We present novel reconstruction and stability analysis methodologies for two-dimensional, multi-coil MRI, based on analytic continuation ideas. We show that the 2-D, limited-data MRI inverse problem, whereby the missing parts of $\textbf{k}$-space (Fourier space) are lines parallel to either $k_1$ or $k_2$ (i.e., the $\textbf{k}$-space axis), can be reduced to a set of 1-D Fredholm type inverse problems. The Fredholm equations are then solved to recover the 2-D image on 1-D line profiles (``slice-by-slice" imaging). The technique is tested on a range of medical in vivo images (e.g., brain, spine, cardiac), and phantom data. Our method is shown to offer optimal performance, in terms of structural similarity, when compared against similar methods from the literature, and when the $\textbf{k}$-space data is sub-sampled at random so as to simulate motion corruption. In addition, we present a Singular Value Decomposition (SVD) and stability analysis of the Fredholm operators, and compare the stability properties of different $\textbf{k}$-space sub-sampling schemes (e.g., random vs uniform accelerated sampling).
翻译:我们根据分析性的连续性想法为二维、多焦的MRI提供了新的重建和稳定分析方法。 我们显示, 2D、 有限数据MRI 反向问题, 即$\ textbf{k} $- space (四空间) 的缺失部分与 $k_ 1美元或 $k_ 2美元( 即$\ textbf{k} 美元- 空间轴) 的线平行线平行( 美元, 即 $\ textbf{k} 美元- 空间轴) 可以降低为一套 1D 加速型的反向问题。 然后, Fredholm 方程式将解答, 以恢复 1D 线剖面上的 2D 图像( 切除/ 切除/ 切除” 成像) 。 该技术将测试在维沃图像( 如脑、 脊柱、 心) 或 phantom 数据中的一系列医疗图象( 如 ) 。 我们的方法显示, 在结构相似性方面提供最佳性表现, 最佳性表现, 如果与文献中的类似方法比较, 。 $\ textf{k}, 当我们的数据数据被随机地被复制,, 和 以随机地标为VD 变换成为 变换成为 。