Mumford-Shah Regularization Research Articles

Mumford–Shah regularization in electrical impedance tomography with complete electrode model

In electrical impedance tomography (EIT), we aim to solve the conductivity within a target body through electrical measurements made on the surface of the target. This inverse conductivity problem is severely ill-posed, especially in real applications with only partial boundary data available. Thus regularization has to be introduced. Conventionally regularization promoting smooth features is used, however, the Mumford–Shah (M–S) regularizer familiar for image segmentation is more appropriate for targets consisting of several distinct objects or materials. It is, however, numerically challenging. We show theoretically through Γ-convergence that a modification of the Ambrosio–Tortorelli approximation of the M–S regularizer is applicable to EIT, in particular the complete electrode model of boundary measurements. With numerical and experimental studies, we confirm that this functional works in practice and produces higher quality results than typical regularizations employed in EIT when the conductivity of the target consists of distinct smoothly-varying regions.

Nonconvex and nonsmooth total variation regularization method for diffuse optical tomography based on RTE * *Submitted Funding: this work is supported by the National Natural Science Foundation of China (Grant No. 11701159).

Diffuse optical tomography with near-infrared light is a promising technique for noninvasive study of the functional characters of human tissues. Mathematically, it is a seriously ill-posed parameter identification problem. For the purpose of better providing both segmentation and piecewise constant approximation of the underlying solution, nonconvex nonsmooth total variation based regularization functional is considered in this paper. We first give a theoretical study on the well-posedness of solutions corresponding to this minimization problem in the Banach space of piecewise constant functions. Moreover, our theoretical results show that the minimizers corresponding to a sequence nonconvex nonsmooth potential functions which converge to the 0–1 functions, can be used to approximate the solution to the weak Mumford–Shah regularization. Then from the numerical side, we propose a double graduated nonconvex Gauss–Newton algorithm to solve this nonconvex nonsmooth regularization. All illustrations and numerical experiments give a flavor of the possibilities offered by the minimizers of the proposed algorithm.

FPGA Acceleration for 3-D Low-Dose Tomographic Reconstruction

X-ray computed tomography (CT) is commonly used to obtain vivo images to characterize diseases but results in radiation exposure to patients. Low-dose CT (LDCT) provides CT images of clinical quality with reduced cumulative radiation dose. Iterative image reconstruction methods with effective regularization are used for LDCT but generally require more computing resources and induce higher computational load than the conventional filtered backprojection (FBP) methods. The high computational demand of the iterative reconstruction (IR) with notably increased reconstruction time precludes its routine clinical application. In this work, we focus on the FPGA acceleration of a compute-intensive full IR (full-IR) algorithm based on the Mumford–Shah regularization. At the algorithmic level, we propose a beam-based asynchronous update algorithm to reduce the computational cost and alleviate the conflicts. At the hardware-level, we first present pipeline-friendly optimization for the original algorithm to increase the computation throughput. We then apply the LDCT-specific tiling strategy to improve the data reuse rate. The experimental results show that our implementation takes 8.5 min to reconstruct a typical physical phantom with the image quality comparable with the vendor’s result. The FPGA implementation achieves $11.6\times $ throughput against the state-of-the-art GPU version.

Image reconstruction by Mumford–Shah regularization for low-dose CT with multi-GPU acceleration

Mumford–Shah (MS) functional has emerged as a regularization technique in x-ray computed tomography (CT) recently. However, for high-resolution CT applications, the huge size of both projection data and image leads to an implementation difficulty. In this work, we propose an approach to implement and accelerate MS regularization on a multi-GPU platform to resolve the issue of data size and rich onboard memory and computing units. We have established a novel partition scheme of the 3D volume under reconstruction and corresponding multithread parallel acceleration strategy to fully utilize the computing resource of multi-GPUs. Our implementation is highly modularized and can be easily scaled with the configuration of GPUs. Experiment results with simulation data as well as real data demonstrate a superior reconstruction quality in contrast to the total variation regularization approach, especially for the ultra-low-dose case. Moreover, this is the first time that MS regularization is used for 3D reconstruction of huge images up to 30723 voxels within 12 min.

An algorithmic framework for Mumford–Shah regularization of inverse problems in imaging

The Mumford–Shah model is a very powerful variational approach for edge preserving regularization of image reconstruction processes. However, it is algorithmically challenging because one has to deal with a non-smooth and non-convex functional. In this paper, we propose a new efficient algorithmic framework for Mumford–Shah regularization of inverse problems in imaging. It is based on a splitting into specific subproblems that can be solved exactly. We derive fast solvers for the subproblems which are key for an efficient overall algorithm. Our method neither requires a priori knowledge of the gray or color levels nor of the shape of the discontinuity set. We demonstrate the wide applicability of the method for different modalities. In particular, we consider the reconstruction from Radon data, inpainting, and deconvolution. Our method can be easily adapted to many further imaging setups. The relevant condition is that the proximal mapping of the data fidelity can be evaluated a within reasonable time. In other words, it can be used whenever classical Tikhonov regularization is possible.

Regularizing properties of the Mumford–Shah functional for imaging applications

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.

Stereo Matching with Mumford-Shah Regularization and Occlusion Handling

This paper addresses the problem of correspondence establishment in binocular stereo vision. We suggest a novel spatially continuous approach for stereo matching based on the variational framework. The proposed method suggests a unique regularization term based on Mumford-Shah functional for discontinuity preserving, combined with a new energy functional for occlusion handling. The evaluation process is based on concurrent minimization of two coupled energy functionals, one for domain segmentation (occluded versus visible) and the other for disparity evaluation. In addition to a dense disparity map, our method also provides an estimation for the half-occlusion domain and a discontinuity function allocating the disparity/depth boundaries. Two new constraints are introduced improving the revealed discontinuity map. The experimental tests include a wide range of real data sets from the Middlebury stereo database. The results demonstrate the capability of our method in calculating an accurate disparity function with sharp discontinuities and occlusion map recovery. Significant improvements are shown compared to a recently published variational stereo approach. A comparison on the Middlebury stereo benchmark with subpixel accuracies shows that our method is currently among the top-ranked stereo matching algorithms.

Semi-blind image restoration via Mumford-Shah regularization

Image restoration and segmentation are both classical problems, that are known to be difficult and have attracted major research efforts. This paper shows that the two problems are tightly coupled and can be successfully solved together. Mutual support of image restoration and segmentation processes within a joint variational framework is theoretically motivated, and validated by successful experimental results. The proposed variational method integrates semi-blind image deconvolution (parametric blur-kernel), and Mumford-Shah segmentation. The functional is formulated using the T-convergence approximation and is iteratively optimized via the alternate minimization method. While the major novelty of this work is in the unified treatment of the semi-blind restoration and segmentation problems, the important special case of known blur is also considered and promising results are obtained.

PET Image Reconstruction Using Mumford-Shah Regularization Coupled with L1Data Fitting.

In positron emission tomography (PET) image reconstruction, regularization methods are usually considered to suppress noise effects in reconstructed images. In this paper, we model this reconstruction problem in a new variational framework where the Mumford-Shah (MS) regularization coupled with recently developed L1 data fidelity term is adapted. In order to simplify the numerical computation, Ambrosio and Tortorelli's Gamma-convergence approximation is also employed to substitute the irregular parts (edge set) of MS functionals with the auxiliary smooth function. In numerical studies we compare our method with the "conventional" penalized least-square (PLS) algorithm with local nonquadratic Huber penalty. Results show both the feasibility and efficiency of the proposed algorithm.