A Review on Deep Learning in Medical Image Reconstruction

Medical imaging is crucial in modern clinics to guide the diagnosis and treatment of diseases. Medical image reconstruction is one of the most fundamental and important components of medical imaging, whose major objective is to acquire high-quality medical images for clinical usage at the minimal cost and risk to the patients. Mathematical models in medical image reconstruction or, more generally, image restoration in computer vision, have been playing a prominent role. Earlier mathematical models are mostly designed by human knowledge or hypothesis on the image to be reconstructed, and we shall call these models handcrafted models. Later, handcrafted plus data-driven modeling started to emerge which still mostly relies on human designs, while part of the model is learned from the observed data. More recently, as more data and computation resources are made available, deep learning based models (or deep models) pushed the data-driven modeling to the extreme where the models are mostly based on learning with minimal human designs. Both handcrafted and data-driven modeling have their own advantages and disadvantages. One of the major research trends in medical imaging is to combine handcrafted modeling with deep modeling so that we can enjoy benefits from both approaches. The major part of this article is to provide a conceptual review of some recent works on deep modeling from the unrolling dynamics viewpoint. This viewpoint stimulates new designs of neural network architectures with inspirations from optimization algorithms and numerical differential equations. Given the popularity of deep modeling, there are still vast remaining challenges in the field, as well as opportunities which we shall discuss at the end of this article.Comment: 31 pages, 6 figures. Survey pape

Correspondence between Multiwavelet Shrinkage/Multiple Wavelet Frame Shrinkage and Nonlinear Diffusion

There are numerous methodologies for signal and image denoising. Wavelet, wavelet frame shrinkage, and nonlinear diffusion are effective ways for signal and image denoising. Also, multiwavelet transforms and multiple wavelet frame transforms have been used for signal and image denoising. Multiwavelets have important property that they can possess the orthogonality, short support, good performance at the boundaries, and symmetry simultaneously. The advantage of multiwavelet transform for signal and image denoising was illustrated by Bui et al. in 1998. They showed that the evaluation of thresholding on a multiwavelet basis has produced good results. Further, Strela et al. have showed that the decimated multiwavelet denoising provides superior results than decimated conventional (scalar) wavelet denoising. Mrazek, Weickert, and Steidl in 2003 examined the association between one-dimensional nonlinear diffusion and undecimated Haar wavelet shrinkage. They proved that nonlinear diffusion could be presented by using wavelet shrinkage. High-order nonlinear diffusion in terms of one-dimensional frame shrinkage and two-dimensional frame shrinkage were presented in 2012 by Jiang, and in 2013 by Dong, Jiang, and Shen, respectively. They obtained that the correspondence between both approaches leads to a different form of diffusion equation that mixes benefits from both approaches. The objective of this dissertation is to study the correspondence between one-dimensional multiwavelet shrinkage and high-order nonlinear diffusion, and to study high-order nonlinear diffusion in terms of one-dimensional multiple frame shrinkage also well. Further, this dissertation formulates nonlinear diffusion in terms of 2D multiwavelet shrinkage and 2D multiple wavelet frame shrinkage. From the experiment results, it can be inferred that nonlinear diffusion in terms of multiwavelet shrinkage/multiple frame shrinkage gives better results than a scalar case. On the whole, this dissertation expands nonlinear diffusion in terms of wavelet shrinkage and nonlinear diffusion in terms of frame shrinkage from the scalar wavelets and frames to the multiwavelets and multiple frames

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

Image restoration: Wavelet frame shrinkage, nonlinear evolution PDEs, and beyond

In the past few decades, mathematics based approaches have been widely adopted in various image restoration problems; the partial differential equation (PDE) based approach (e.g., the total variation model [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259-268] and its generalizations, nonlinear diffusions [P. Perona and J. Malik, IEEE Trans. Pattern Anal. Mach. Intel., 12 (1990), pp. 629-639; F. Catte et al., SIAM J. Numer. Anal., 29 (1992), pp. 182-193], etc.) and wavelet frame based approach are some successful examples. These approaches were developed through different paths and generally provided understanding from different angles of the same problem. As shown in numerical simulations, implementations of the wavelet frame based approach and the PDE based approach quite often end up solving a similar numerical problem with similar numerical behaviors, even though different approaches have advantages in different applications. Since wavelet frame based and PDE based approaches have all been modeling the same types of problems with success, it is natural to ask whether the wavelet frame based approach is fundamentally connected with the PDE based approach when we trace them all the way back to their roots. A fundamental connection of a wavelet frame based approach with a total variation model and its generalizations was established in [J. Cai, B. Dong, S. Osher, and Z. Shen, J. Amer. Math. Soc., 25 (2012), pp. 1033-1089]. This connection gives the wavelet frame based approach a geometric explanation and, at the same time, it equips a PDE based approach with a time frequency analysis. Cai et al. showed that a special type of wavelet frame model using generic wavelet frame systems can be regarded as an approximation of a generic variational model (with the total variation model as a special case) in the discrete setting. A systematic convergence analysis, as the resolution of the image goes to infinity, which is the key step in linking the two approaches, is also given in Cai et al. Motivated by Cai et al. and [Q. Jiang, Appl. Numer. Math., 62 (2012), pp. 51-66], this paper establishes a fundamental connection between the wavelet frame based approach and nonlinear evolution PDEs, provides interpretations and analytical studies of such connections, and proposes new algorithms for image restoration based on the new understandings. Together with the results in [J. Cai et al., J. Amer. Math. Soc., 25 (2012), pp. 1033-1089], we now have a better picture of how the wavelet frame based approach can be used to interpret the general PDE based approach (e.g., the variational models or nonlinear evolution PDEs) and can be used as a new and useful tool in numerical analysis to discretize and solve various variational and PDE models. To be more precise, we shall establish the following: (1) The connections between wavelet frame shrinkage and nonlinear evolution PDEs provide new and inspiring interpretations of both approaches that enable us to derive new PDE models and (better) wavelet frame shrinkage algorithms for image restoration. (2) A generic nonlinear evolution PDE (of parabolic or hyperbolic type) can be approximated by wavelet frame shrinkage with properly chosen wavelet frame systems and carefully designed shrinkage functions. (3) The main idea of this work is beyond the scope of image restoration. Our analysis and discussions indicate that wavelet frame shrinkage is a new way of solving PDEs in general, which will provide a new insight that will enrich the existing theory and applications of numerical PDEs, as well as those of wavelet frames