Nikolova M, Idier J, Mohammad-Djafari A
Lab. des Signaux et Syst., CNRS, Gif-sur-Yvette, France.
IEEE Trans Image Process. 1998;7(4):571-85. doi: 10.1109/83.663502.
We propose a method for the reconstruction of signals and images observed partially through a linear operator with a large support (e.g., a Fourier transform on a sparse set). This inverse problem is ill-posed and we resolve it by incorporating the prior information that the reconstructed objects are composed of smooth regions separated by sharp transitions. This feature is modeled by a piecewise Gaussian (PG) Markov random field (MRF), known also as the weak-string in one dimension and the weak-membrane in two dimensions. The reconstruction is defined as the maximum a posteriori estimate. The prerequisite for the use of such a prior is the success of the optimization stage. The posterior energy corresponding to a PG MRF is generally multimodal and its minimization is particularly problematic. In this context, general forms of simulated annealing rapidly become intractable when the observation operator extends over a large support. In this paper, global optimization is dealt with by extending the graduated nonconvexity (GNC) algorithm to ill-posed linear inverse problems. GNC has been pioneered by Blake and Zisserman in the field of image segmentation. The resulting algorithm is mathematically suboptimal but it is seen to be very efficient in practice. We show that the original GNC does not correctly apply to ill-posed problems. Our extension is based on a proper theoretical analysis, which provides further insight into the GNC. The performance of the proposed algorithm is corroborated by a synthetic example in the area of diffraction tomography.
我们提出了一种用于重建通过具有大支撑的线性算子（例如，稀疏集上的傅里叶变换）部分观测到的信号和图像的方法。这个逆问题是不适定的，我们通过纳入重建对象由被尖锐过渡分隔的平滑区域组成的先验信息来解决它。这个特征由分段高斯（PG）马尔可夫随机场（MRF）建模，在一维中也称为弱弦，在二维中称为弱膜。重建被定义为最大后验估计。使用这种先验的前提是优化阶段的成功。对应于PG MRF的后验能量通常是多模态的，其最小化特别成问题。在这种情况下，当观测算子扩展到较大支撑时，一般形式的模拟退火很快变得难以处理。在本文中，通过将渐进非凸性（GNC）算法扩展到不适定线性逆问题来处理全局优化。GNC由Blake和Zisserman在图像分割领域率先提出。所得算法在数学上是次优的，但在实践中被证明非常有效。我们表明原始的GNC不适用于不适定问题。我们的扩展基于适当的理论分析，这为GNC提供了进一步的见解。所提出算法的性能通过衍射层析成像领域的一个合成示例得到了证实。
