The variational techniques (e.g. the total variation based method) are well established and effective for image restoration, as well as many other applications, while the wavelet frame based approach is relatively new and came from a different school. This paper is designed to establish a connection between these two major approaches for image restoration. The main result of this paper shows that when spline wavelet frames of are used, a special model of a wavelet frame method, called the analysis based approach, can be viewed as a discrete approximation at a given resolution to variational methods. A convergence analysis as image resolution increases is given in terms of objective functionals and their approximate minimizers. This analysis goes beyond the establishment of the connections between these two approaches, since it leads to new understandings for both approaches. First, it provides geometric interpretations to the wavelet frame based approach as well as its solutions. On the other hand, for any given variational model, wavelet frame based approaches provide various and flexible discretizations which immediately lead to fast numerical algorithms for both wavelet frame based approaches and the corresponding variational model. Furthermore, the built-in multiresolution structure of wavelet frames can be utilized to adaptively choose proper differential operators in different regions of a given image according to the order of the singularity of the underlying solutions. This is important when multiple orders of differential operators are used in various models that generalize the total variation based method. These observations will enable us to design new methods according to the problems at hand, hence, lead to wider applications of both the variational and wavelet frame based approaches. Links of wavelet frame based approaches to some more general variational methods developed recently will also be discussed.

Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net 2.0, to discover (time-dependent) PDEs from observed dynamic data with minor prior knowledge on the underlying mechanism that drives the dynamics. The design of PDE-Net 2.0 is based on our earlier work [1] where the original version of PDE-Net was proposed. PDE-Net 2.0 is a combination of numerical approximation of differential operators by convolutions and a symbolic multi-layer neural network for model recovery. Comparing with existing approaches, PDE-Net 2.0 has the most flexibility and expressive power by learning both differential operators and the nonlinear response function of the underlying PDE model. Numerical experiments show that the PDE-Net 2.0 has the potential to uncover the hidden PDE of the observed dynamics, and predict the dynamical behavior for a relatively long time, even in a noisy environment.

15X-ray imaging dose from serial cone-beam CT (CBCT) scans raises a clinical concern in most image guided radiation therapy procedures. It is the goal of this paper to develop a fast GPU-based algorithm to reconstruct high quality CBCT images from undersampled and noisy projection data so as to lower the imaging 20 dose. For this purpose, we have developed an iterative tight frame (TF) based CBCT reconstruction algorithm. A condition that a real CBCT image has a sparse representation under a TF basis is imposed in the iteration process as regularization to the solution. To speed up the computation, a multi-grid method is employed. Our GPU implementation has achieved high computational 25 efficiency and a CBCT image of resolution can be reconstructed in ~5 min. We have tested our algorithm on a digital NCAT phantom and a physical Catphan phantom. It is found that our TF-based algorithm is able to reconstrct CBCT in the context of undersampling and low mAs levels. We have also quantitatively analyzed the reconstructed CBCT 30 image quality in terms of modulation-transfer-function and contrast-to-noise ratio under various scanning conditions. The results confirm the high CBCT image quality obtained from our TF algorithm. Moreover, our algorithm has also been validated in a real clinical context using a head-and-neck patient case. Comparisons of the developed TF algorithm and the current state-of-the-art TV 35 algorithm have also been made in various cases studied in terms of reconstructed image quality and computation efficiency.

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We propose and analyze an extremely fast, efficient and simple method for solving the problem: min{ u 1 : Au = f,u ∈ R n }. This method was first described in [1], with more details in [2] and rigorous theory given in [3] and [4]. The motivation was compressive sensing, which now has a vast and exciting history, which seems to have started with Candes, et.al. [5] and Donoho,[6]. See [2], [3] and [4] for a large set of references. Our method introduces an improvement called "kicking" of the very efficient method of [1], [2] and also applies it to the problem of denoising of undersampled signals. The use of Bregman iteration for denoising of images began in [7] and led to improved results for total variation based methods. Here we apply it to denoise signals, especially essentially sparse signals, which might even be undersampled.The Bregman distance [12], based on the convex function J, between points u and v, is defined byan element in the subgradient of J at the point v. In general D p J (u,v) = D p J (v,u) and the triangle inequality is not satisfied, so D p J (u,v) is not a distance in the usual sense. However it does measure the closeness between u and v in the sense that D p J (u,v) ≥ 0 and D p J (u,v) ≥ D p J (w,v) for all points w on the line segment connecting u and v. Moreover, if J is convex, D p J (u,v) ≥ 0, if J is strictly convex D p J (u,v) > 0 for u = v and if J is strongly convex, then there exists a constant a > 0 such that D p J (u,v) ≥ a u − v 2 .

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Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.