Regularizing properties of the Mumford–Shah functional for imaging applications
Regularizing properties of the Mumford–Shah functional for imaging applications
复制标题
DOI：
10.1088/0266-5611/30/3/035007
复制
发表时间：
2014-02
期刊：
影响因子：
2.1
通讯作者：
M. Jiang;P. Maass;Thomas Page
中科院分区：
文献类型：
--
作者：
M. Jiang;P. Maass;Thomas Page
Mumford-Shah泛函最初是为了图像去噪和分割问题而引入的，并且由于它除了图像之外还提供了图像边缘的正则化而受到关注。最近，这个功能已经出现了几个成像应用，如X射线断层扫描，电阻抗断层扫描，图像去模糊和SPECT的正则化技术。在这种情况下的算子方程，有必要了解其正则化性质，并确定其适用范围。在Rondi和Santosa的方法之后，我们利用图像上的L∞约束，然而与此相反，我们不仅对图像而且对Maso等人引入的σ-收敛意义下的边缘集实现了收敛结果。分析利用了保真度项的衰减性质的假设。在上述两个条件下，我们建立了Mumford-Shah正则化对数据扰动的稳定性。此外，我们提出了一个参数选择规则，确保，重建的图像和边缘收敛到真实的图像和它的边缘的噪声水平为零。我们演示了应用Mumford-Shah正则化的一些线性和非线性成像问题，即图像去模糊，X射线层析成像和二维扩散光学层析成像。
The Mumford–Shah functional was originally introduced for image denoising and segmentation problems, and is of interest because it provides a regularization of image edges in addition to images. Recently, this functional has emerged as a regularization technique for several imaging applications, such as x-ray tomography, electric impedance tomography, image deblurring and SPECT. In this context of operator equations it is necessary to understand its regularization properties and to determine its range of applicability. Following the approach of Rondi and Santosa, we exploit an L∞-constraint on the images, however in contrast to this approach we achieve convergence results not only for the images but also for the edge sets in the sense of σ-convergence introduced by Maso et al. The analysis exploits an assumption on the decay properties of the fidelity term. Under the above two conditions, we establish the stability of the Mumford–Shah regularization for perturbations in the data. Moreover we present a parameter choice rule which ensures, that the reconstructed images and edges converge to the true image and its edges as the noise level goes to zero. We demonstrate the applications of the Mumford–Shah regularization to some linear and nonlinear imaging problems, namely image deblurring, x-ray tomography and two-dimensional diffuse optical tomography.