# American Institute of Mathematical Sciences

ISSN:
1930-8337

eISSN:
1930-8345

All Issues

## Inverse Problems & Imaging

December 2021 , Volume 15 , Issue 6

Special issue on modern optimization and learning-based methods in medical image computing

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2021, 15(6): Ⅰ-Ⅱ doi: 10.3934/ipi.2021070 +[Abstract](307) +[HTML](30) +[PDF](64.47KB)
Abstract:
2021, 15(6): 1287-1306 doi: 10.3934/ipi.2021019 +[Abstract](910) +[HTML](358) +[PDF](1601.74KB)
Abstract:

Limited-angle reconstruction is a very important but challenging problem in the field of computed tomography (CT) which has been extensively studied for many years. However, some difficulties still remain. Based on the theory of visible and invisible boundary developed by Quinto et.al, we propose a reconstruction model for limited-angle CT, which encodes the visible edges as priors to recover the invisible ones. The new model utilizes generalized shrinkage operators as regularizers to perform edge-preserving smoothing such that the visible edges are employed as anchors to recover piecewise-constant or piecewise-smooth reconstructions, while noises and artifacts are suppressed or removed. This work extends our previous research on limited-angle reconstruction which employs gradient \begin{document}$\ell_0$\end{document} and \begin{document}$\ell_1$\end{document} norm regularizers. The effectiveness of the proposed model and its corresponding solving algorithm shall be verified by numerical experiments with simulated data as well as real data.

2021, 15(6): 1307-1332 doi: 10.3934/ipi.2020069 +[Abstract](1072) +[HTML](512) +[PDF](7835.33KB)
Abstract:

Existing image restoration methods mostly make full use of various image prior information. However, they rarely exploit the potential of residual histograms, especially their role as ensemble regularization constraint. In this paper, we propose a residual Wasserstein regularization model (RWRM), in which a residual histogram constraint is subtly embedded into a type of variational minimization problems. Specifically, utilizing the Wasserstein distance from the optimal transport theory, this scheme is achieved by enforcing the observed image residual histogram as close as possible to the reference residual histogram. Furthermore, the RWRM unifies the residual Wasserstein regularization and image prior regularization to improve image restoration performance. The robustness of parameter selection in the RWRM makes the proposed algorithms easier to implement. Finally, extensive experiments have confirmed that our RWRM applied to Gaussian denoising and non-blind deconvolution is effective.

2021, 15(6): 1333-1346 doi: 10.3934/ipi.2020057 +[Abstract](1950) +[HTML](862) +[PDF](2745.15KB)
Abstract:

X-ray images of the lower limb bone are the most commonly used imaging modality for clinical studies, and segmentation of the femur and tibia in an X-ray image is helpful for many medical studies such as diagnosis, surgery and treatment. In this paper, we propose a new approach based on pure dilated residual U-Net for the segmentation of the femur and tibia bones. The proposed approach employs dilated convolution completely to increase the receptive field, in this way, we can make full use of the advantages of dilated convolution. We conducted experiments and evaluations on datasets provided by Tianjin hospital. Comparison with the classical U-net and FusionNet, our method has fewer parameters, higher accuracy, and converges more rapidly, which means the high performance of the proposed method.

2021, 15(6): 1347-1362 doi: 10.3934/ipi.2021017 +[Abstract](674) +[HTML](345) +[PDF](773.72KB)
Abstract:

In this paper, we propose a novel scheme for single image super resolution (SR) reconstruction. Firstly, we construct a new self-similarity framework by regarding the low resolution (LR) images as the low rank version of corresponding high resolution (HR) images. Subsequently, nuclear norm minimization (NNM) is employed to generate LR image pyramids from HR ones. The structure of our framework is beneficial to extract LR features, where we regard the quotient image, calculated between HR image and LR image at the same layer, as LR feature. This LR feature has the same dimension as LR image; however the dimension of commonly used gradient feature is 4 times than LR image. On the other hand, we employ nonlocal similar patch, within the same scale and across different scales, to generate HR and LR dictionaries. In the course of encoding, codes are calculated from both row and column of LR dictionary for each LR patch; at the same time, both low rank and sparse constraints on codes matrix give us a hand to remove coding noises. Finally, both quantitative and perceptual results demonstrate that our proposed method has a good SR performance.

2021, 15(6): 1363-1379 doi: 10.3934/ipi.2020051 +[Abstract](1370) +[HTML](865) +[PDF](5013.13KB)
Abstract:

This paper proposes to learn analysis transform network for dynamic magnetic resonance imaging (LANTERN). Integrating the strength of CS-MRI and deep learning, the proposed framework is highlighted in three components: (ⅰ) The spatial and temporal domains are sparsely constrained by adaptively trained convolutional filters; (ⅱ) We introduce an end-to-end framework to learn the parameters in LANTERN to solve the difficulty of parameter selection in traditional methods; (ⅲ) Compared to existing deep learning reconstruction methods, our experimental results show that our paper has encouraging capability in exploiting the spatial and temporal redundancy of dynamic MR images. We performed quantitative and qualitative analysis of cardiac reconstructions at different acceleration factors (\begin{document}$2 \times$\end{document}-\begin{document}$11 \times$\end{document}) with different undersampling patterns. In comparison with two state-of-the-art methods, experimental results show that our method achieved encouraging performances.

2021, 15(6): 1381-1407 doi: 10.3934/ipi.2020050 +[Abstract](1671) +[HTML](765) +[PDF](11965.85KB)
Abstract:

Retinex theory is introduced to show how the human visual system perceives the color and the illumination effect such as Retinex illusions, medical image intensity inhomogeneity and color shadow effect etc.. Many researchers have studied this ill-posed problem based on the framework of the variation energy functional for decades. However, to the best of our knowledge, the existing models via the sparsity of the image based on the nonconvex \begin{document}$\ell^p$\end{document}-quasinorm were limited. To deal with this problem, this paper considers a TV\begin{document}$_p$\end{document}-HOTV\begin{document}$_q$\end{document}-based retinex model with \begin{document}$p, q\in(0, 1)$\end{document}. Specially, the TV\begin{document}$_p$\end{document} term based on the total variation(TV) regularization can describe the reflectance efficiently, which has the piecewise constant structure. The HOTV\begin{document}$_q$\end{document} term based on the high order total variation(HOTV) regularization can penalize the smooth structure called the illumination. Since the proposed model is non-convex, non-smooth and non-Lipschitz, we employ the iteratively reweighed \begin{document}$\ell_1$\end{document} (IRL1) algorithm to solve it. We also discuss some properties of our proposed model and algorithm. Experimental experiments on the simulated and real images illustrate the effectiveness and the robustness of our proposed model both visually and quantitatively by compared with some related state-of-the-art variational models.

2021, 15(6): 1409-1419 doi: 10.3934/ipi.2021067 +[Abstract](143) +[HTML](72) +[PDF](2897.95KB)
Abstract:

In recent years multi-modal data processing methods have gained considerable research interest as technological advancements in imaging, computing, and data storage have made the collection of redundant, multi-modal data more commonplace. In this work we present an image restoration method tailored for scenarios where pre-existing, high-quality images from different modalities or contrasts are available in addition to the target image. Our method is based on a novel network architecture which combines the benefits of traditional multi-scale signal representation, such as wavelets, with more recent concepts from data fusion methods. Results from numerical simulations in which T1-weighted MRI images are used to restore noisy and undersampled T2-weighted images demonstrate that the proposed network successfully utilizes information from high-quality reference images to improve the restoration quality of the target image beyond that of existing popular methods.

2021, 15(6): 1421-1450 doi: 10.3934/ipi.2020068 +[Abstract](919) +[HTML](527) +[PDF](6147.66KB)
Abstract:

For the characteristics of the degraded images with multiplicative noise, the gray level indicators for constructing adaptive total variation are proposed. Based on the new regularization term, we propose the new convex adaptive variational model. Then, considering the existence, uniqueness and comparison principle of the minimizer of the functional. The finite difference method with rescaling technique and the primal-dual method with adaptive step size are used to solve the minimization problem. The paper ends with a report on numerical tests for the denoising of images subject to multiplicative noise, the comparison with other methods is provided as well.

2021, 15(6): 1451-1469 doi: 10.3934/ipi.2021018 +[Abstract](877) +[HTML](516) +[PDF](10243.11KB)
Abstract:

In this paper, we mainly show a novel fast fractional order anisotropic diffusion algorithm for noise removal based on the recent numerical scheme called the Fast Explicit Diffusion. To balance the efficiency and accuracy of the algorithm, the truncated matrix method is used to deal with the iterative matrix in the model and its error is also estimated. In particular, we obtain the stability condition of the iteration by the spectrum analysis method. Through implementing the fast explicit format iteration algorithm with periodic change of time step size, the efficiency of the algorithm is greatly improved. At last, we show some numerical results on denoising tasks. Many experimental results confirm that the algorithm can more quickly achieve satisfactory denoising results.

2020 Impact Factor: 1.639
5 Year Impact Factor: 1.720
2020 CiteScore: 2.6