We propose a general learning based framework for solving nonsmooth and nonconvex image reconstruction problems. We model the regularization function as the composition of the $l_{2,1}$ norm and a smooth but nonconvex feature mapping parametrized as a deep convolutional neural network. We develop a descent-type algorithm to solve the nonsmooth nonconvex minimization problem by leveraging Nesterov's smoothing technique and the idea of residual learning, and learn the network parameters such that the outputs of the algorithm match the references in training data. Our method is versatile as one can employ various modern network structures into the regularization, and the resulting network inherits the convergence properties of the algorithm. We also show that the proposed network is parameter-efficient, and its performance compares favorably to the state-of-the-art methods in a variety of image reconstruction problems in practice.

We propose a novel family of connectionist models based on kernel machines and consider the problem of learning layer by layer a compositional hypothesis class (i.e., a feedforward, multilayer architecture) in a supervised setting. In terms of the models, we present a principled method to “kernelize” (partly or completely) any neural network (NN). With this method, we obtain a counterpart of any given NN that is powered by kernel machines instead of neurons. In terms of learning, when learning a feedforward deep architecture in a supervised setting, one needs to train all the components simultaneously using backpropagation (BP) since there are no explicit targets for the hidden layers (Rumelhart, Hinton, & Williams, 1986 ). We consider without loss of generality the two-layer case and present a general framework that explicitly characterizes a target for the hidden layer that is optimal for minimizing the objective function of the network. This characterization then makes possible a purely greedy training scheme that learns one layer at a time, starting from the input layer. We provide instantiations of the abstract framework under certain architectures and objective functions. Based on these instantiations, we present a layer-wise training algorithm for an [Formula: see text]-layer feedforward network for classification, where [Formula: see text] can be arbitrary. This algorithm can be given an intuitive geometric interpretation that makes the learning dynamics transparent. Empirical results are provided to complement our theory. We show that the kernelized networks, trained layer-wise, compare favorably with classical kernel machines as well as other connectionist models trained by BP. We also visualize the inner workings of the greedy kernelized models to validate our claim on the transparency of the layer-wise algorithm.

Sparse representation based classification (SRC) methods have achieved remarkable results. SRC, however, still suffer from requiring enough training samples, insufficient use of test samples, and instability of representation. In this paper, a stable inverse projection representation based classification (IPRC) is presented to tackle these problems by effectively using test samples. An IPR is first proposed and its feasibility and stability are analyzed. A classification criterion named category contribution rate is constructed to match the IPR and complete classification. Moreover, a statistical measure is introduced to quantify the stability of representation-based classification methods. Based on the IPRC technique, a robust tumor recognition framework is presented by interpreting microarray gene expression data, where a two-stage hybrid gene selection method is introduced to select informative genes. Finally, the functional analysis of candidate's pathogenicity-related genes is given. Extensive experiments on six public tumor microarray gene expression datasets demonstrate the proposed technique is competitive with state-of-the-art methods.

We propose a new two-stage joint image reconstruction method by recovering edges directly from observed data and then assembling an image using the recovered edges. More specifically, we reformulate joint image reconstruction with vectorial total-variation regularization as an $l_1$ minimization problem of the Jacobian of the underlying multimodality or multicontrast images. We provide detailed derivation of data fidelity for the Jacobian in Radon and Fourier transform domains. The new minimization problem yields an optimal convergence rate higher than that of existing primal-dual based reconstruction algorithms, and the per-iteration cost remains low by using closed-form matrix-valued shrinkages. We conducted numerical tests on a number of multicontrast CT and MR image datasets, which demonstrate that the proposed method significantly improves reconstruction efficiency and accuracy compared to the state-of-the-art joint image reconstruction methods.

Many efforts have been made for decades in order to improve the accuracy of radioactivity map in positron emission tomography (PET) images, which has important clinical implications for better diagnosis and understanding of diseases. However, there is still a challenging problem for reconstructing high resolution image with the limited acquired photon counts. In this paper, we present a nonlocal self-similar constraint for the purpose of exploiting structured sparsity within the PET reconstructed images. It is based on image patches and approached by low-rank approximation. Moreover, we adopt total variation regulation into our method to further denoise and compensate the demerits inherited in patch-based methods. These two regulation terms are firstly employed in the Poisson model, and are jointly solved in a distributed optimization framework. Experiments have presented that our proposed PNLTV method substantially outperforms existing state-of-the-art methods in PET reconstruction.

We present a novel framework, namely, accelerated alternating direction method of multipliers (AADMM), for acceleration of linearized ADMM. The basic idea of AADMM is to incorporate a multistep acceleration scheme into linearized ADMM. We demonstrate that for solving a class of convex composite optimization with linear constraints, the rate of convergence of AADMM is better than that of linearized ADMM, in terms of their dependence on the Lipschitz constant of the smooth component. Moreover, AADMM is capable of dealing with the situation when the feasible region is unbounded, as long as the corresponding saddle point problem has a solution. A backtracking algorithm is also proposed for practical performance.

We develop a variational model and a faster and robust numerical algorithm for simultaneous sensitivity map estimation and image reconstruction in partially parallel MR imaging with significantly under-sampled data. The proposed model uses a maximum likelihood approach to minimizing the residue of data fitting in the presence of independent Gaussian noise. The usage of maximum likelihood estimation dramatically reduces the sensitivity to the selection of model parameter, and increases the accuracy and robustness of the algorithm. Moreover, variable splitting based on the specific structure of the objective function, and alternating direction method of multipliers (ADMM) are used to accelerate the computation. The preliminary results indicate that the proposed method resulted in fast and robust reconstruction.

This paper presents a variational model for simultaneous multiphase segmentation and intensity bias estimation for images corrupted by strong noise and intensity inhomogeneity. Since the pixel intensities are not reliable samples for region statistics due to the presence of noise and intensity bias, we use local information based on the joint density within image patches to perform image partition. Hence, the pixel intensity has a multiplicative distribution structure. Then, the maximum-a-posteriori (MAP) principle with those pixel density functions generates the model. To tackle the computational problem of the resultant nonsmooth nonconvex minimization, we relax the constraint on the characteristic functions of partition regions, and apply primal-dual alternating gradient projections to construct a very efficient numerical algorithm. We show that all the variables have closed-form solutions in each iteration, and the computation complexity is very low. In particular, the algorithm involves only regular convolutions and pointwise projections onto the unit ball and canonical simplex. Numerical tests on a variety of images demonstrate that the proposed algorithm is robust, stable, and attains significant improvements on accuracy and efficiency over the state-of-the-arts.

A new algorithm is presented for efficiently solving image reconstruction problems that arise in partially parallel magnetic resonance imaging. This algorithm minimizes an objective function of the form φ(Bu) + 1/2||FpSu - f||2, where φ is the regularization term which may be nonsmooth. In image reconstruction, the φ term corresponds to total variation smoothing and/or L1 regularization term. The least square term 1/2||FpSu - f||2is the fidelity term. In our application, f represents undersampled data from a partially parallel imaging (PPI) system. The proposed algorithm is a generalization of the Bregman operator splitting algorithm with variable stepsize (BOSVS) in which the previous Barzilai-Borwein (BB) step is replaced by a cyclic BB (CBB) step, and an L1 term Ψ is added to the energy function. Experimental results on clinical partially parallel imaging data are given.

Joint image deblurring and denoising has long been an interesting problem. Traditional deconvolution methods (like the ROF model) only work for Gaussian noise. Median-based approaches are generally concerned with the removal of impulse noise, which are more likely to hamper the deblurring process. In this paper, we propose a spareland model for deblurring images corrupted by impulse noise. The key point is to approximate the probability density function by two different randomly mixed Gaussian distributions. Experimental results are provided at the end of this paper to demonstrate the effectiveness of the proposed method.

In this paper, we present a multiphase segmentation model for MR images in the presence of strong intensity inhomogeneity. The problem is formalized as a constraint min-max optimization problem that consists both primal and dual variables. We use the primal dual hybrid gradient (PDHG) algorithm to alternately solve for the optimal solutions. The proposed algorithm is quite efficient in that all the subproblems have closed form solutions. Moreover, the computational complexity is shown to be linear with respect to the size of the image. Numerical experiments on various images demonstrated that our algorithm outperforms recently developed methods in terms of efficiency and accuracy.

This paper presents two fast algorithms for total variation–based image reconstruction in a magnetic resonance imaging technique known as partially parallel imaging (PPI), where the inversion matrix is large and ill-conditioned. These algorithms utilize variable splitting techniques to decouple the original problem into more easily solved subproblems. The first method reduces the image reconstruction problem to an unconstrained minimization problem, which is solved by an alternating proximal minimization algorithm. One phase of the algorithm solves a total variation (TV) denoising problem, and the second phase solves an ill-conditioned linear system. Linear and sublinear convergence results are given, and an implementation based on a primal-dual hybrid gradient (PDHG) scheme for the TV problem and on a Barzilai–Borwein scheme for the linear inversion is proposed. The second algorithm exploits the special structure of the PPI reconstruction problem by decomposing it into one subproblem involving Fourier transforms and another subproblem that can be treated by the PDHG scheme. Numerical results and comparisons with recently developed methods indicate the efficiency of the proposed algorithms.

In this paper, we present a fast numerical algorithm for solving total variation and l(1) (TVL1) based image reconstruction with application in partially parallel magnetic resonance imaging. Our algorithm uses variable splitting method to reduce computational cost. Moreover, the Barzilai-Borwein step size selection method is adopted in our algorithm for much faster convergence. Experimental results on clinical partially parallel imaging data demonstrate that the proposed algorithm requires much fewer iterations and/or less computational cost than recently developed operator splitting and Bregman operator splitting methods, which can deal with a general sensing matrix in reconstruction framework, to get similar or even better quality of reconstructed images.

Both acquisition and reconstruction speed are crucial for magnetic resonance (MR) imaging in clinical applications. In this paper, we present a fast reconstruction algorithm for SENSE in partially parallel MR imaging with arbitrary k-space trajectories. The proposed method is a combination of variable splitting, the classical penalty technique and the optimal gradient method. Variable splitting and the penalty technique reformulate the SENSE model with sparsity regularization as an unconstrained minimization problem, which can be solved by alternating two simple minimizations: One is the total variation and wavelet based denoising that can be quickly solved by several recent numerical methods, whereas the other one involves a linear inversion which is solved by the optimal first order gradient method in our algorithm to significantly improve the performance. Comparisons with several recent parallel imaging algorithms indicate that the proposed method significantly improves the computation efficiency and achieves state-of-the-art reconstruction quality.

This paper developed a new soft multiphase segmentation model. Different from most maximum-likelihood based and Bayesian-estimation based methods, the proposed model introduced a geometrical constraint- ¿the length term¿ into the model which makes the model more rigorous in analysis while still flexible in implementation. Moreover, the model used mixed Gaussian with different parameters for different patterns. As a result, it is more robust to noise. The experiments demonstrated its high efficiency.

In this paper, we propose a segmentation assisted registration model. It partitions the domain of images into several regions such that the residue image in each region is identically distributed with zero mean and variance to be optimized. In this model, we minimize an energy that combines negative log-likelihood of the residue in each region, smoothness of the deformation field and length of the partition curve. It can be viewed as a generalization of the sum of squared difference model and global Gaussian model where the variance is a constant in the entire domain. By taking different variances in different regions, the registration becomes more efficient and accurate, which are demonstrated by the experiments on synthetic and clinical data.

We have developed a geometric deformable model that employs neighborhood influence to achieve robust segmentation for noisy and broken edges. The fundamental power of this strategy rests with the explicitly combination of regional inter-point constraints, image forces, and a priori boundary information for each geometric contour point within its adaptively determined local influence domain. This formulation thus naturally unifies the essences of the geometric and parametric snakes through automatic local scale selection, and exhibits their respective fundamental strengths of allowing stable boundary detection when the edge information is weak and possibly discontinuous, while maintaining the abilities to handle topological changes during front evolution. In particular, this paper presents an implementation of the method through local integration of the level set function and the image/prior-driven evolution forces, where the resulting partial differential equation is solved numerically using standard finite difference method. Experimental results on synthetic and real images demonstrate its superior performance.

Piecewise constant and piecewise smooth Mumford-Shah (MS) models have been widely studied and used for image segmentation. More complicated than piecewise constant MS, global Gaussian intensity distribution within each partitioned region has also been studied. However, all these frameworks are limited in power and robustness in finding objects whose interiors have high noise level and/or complex multi-modal intensity distribution. To overcome these drawbacks,we propose a non-parametric kernel based model which is able to simultaneously segment and smooth images without adding extra smoothing terms. At each point within each partitioned smooth region, a Gaussian kernel with mean the intensity of the given to-be-segmented image at this point and a small local variance depending on the location is applied to create a nonparametric intensity statistics approximation. To save computation, a quadratic kernel involving simple calculation could replace the Gaussian kernel that involves expensive exponential calculation. We demonstrate the superiority of proposed model over other models by showing segmentation results from various images with different levels and types of noise.

We propose a novel variational approach to automatically soft-segment medical images into a fixed number of classes. Our method combines fuzzy classification and active contours in a single variational framework. This approach allows the use of tools from both de-formable geometry and clustering in a well-defined setting and provides a useful, unsupervised segmentation technique. The model was tested on synthetic and MRI brain data, with promising results.

We present a novel level set representation and front propagation scheme for active contours where the analysis/evolution domain is sampled by unstructured point cloud. These sampling points are adaptively distributed according to both local data and level set geometry, hence allow extremely convenient enhancement/reduction of local front precision by simply putting more/fewer points on the computation domain without grid refinement (as the cases in finite difference schemes) or remeshing (typical infinite element methods). The front evolution process is then conducted on the point-sampled domain, without the use of computational grid or mesh, through the precise but relatively expensive moving least squares (MLS) approximation of the continuous domain, or the faster yet coarser generalized finite difference (GFD) representation and calculations. Because of the adaptive nature of the sampling point density, our strategy performs fast marching and level set local refinement concurrently. We have evaluated the performance of the method in image segmentation and shape recovery applications using real and synthetic data.

In this paper, we use the cumulative distribution of a random variable to define its information content and thereby develop an alternative measure of uncertainty that extends Shannon entropy to random variables with continuous distributions. We call this measure cumulative residual entropy (CRE). The salient features of CRE are as follows: 1) it is more general than the Shannon entropy in that its definition is valid in the continuous and discrete domains, 2) it possesses more general mathematical properties than the Shannon entropy, and 3) it can be easily computed from sample data and these computations asymptotically converge to the true values. The properties of CRE and a precise formula relating CRE and Shannon entropy are given in the paper. Finally, we present some applications of CRE to reliability engineering and computer vision.

In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion-weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of L(P) norms, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The complex valued nonlinear form leads to a more accurate (when compared to the linearized version) estimate of the tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of Cholesky factors and estimated. The constrained variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Experiments with complex-valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm.

Diffusion tensor magnetic resonance imaging (DT-MRI) is a relatively new imaging modality in the field of medical imaging. This modality of imaging allows one to capture the structural connectivity if any between functionally meaningful regions for example, in the brain. The data however can be noisy and requires restoration. In this paper, we present a unified model for simultaneous smoothing and estimation of diffusion tensor field from DT-MRI. The diffusion tensor field is estimated directly from the raw data with L/sup P/ smoothness and positive definiteness constraints. The data term we employ is from the original Stejskal-Tanner equation instead of the linearized version as usually done in literature. In addition, we use Cholesky decomposition to ensure positive definiteness of the diffusion tensor. The unified model is discretized and solved numerically using limited memory quasi-Newton method. Both synthetic and real data experiments are shown to depict the algorithm performance.

In this note we present a coupled optimization model for boundary determination. One part of the model incorporates a prior shape into a geometric active contour model with a fixed parameter. The second part determines the 'best' parameter used in the first part by maximizing the mutual information of the image geometry between the prior and an aligned novel image over all the alignments that are the solutions of the first part corresponding to different parameters. We also present an alternative method, which generates an intensity model formed as the average of a set of aligned training images. Experimental results on cardiac ultrasound images are presented. These results indicate that the proposed model provides close agreement with expert traced borders, and the parameter determined in this model for one image can be used for images with similar properties. The existence of a solution to the proposed minimization problem is also discussed.

We use the cumulative distribution of a random variable to define the information content in it and use it to develop a novel measure of information that parallels Shannon entropy, which we dub cumulative residual entropy (CRE). The key features of CRE may be summarized as, (1) its definition is valid in both the continuous and discrete domains, (2) it is mathematically more general than the Shannon entropy and (3) its computation from sample data is easy and these computations converge asymptotically to the true values. We define the cross-CRE (CCRE) between two random variables and apply it to solve the uni- and multimodal image alignment problem for parameterized (rigid, affine and projective) transformations. The key strengths of the CCRE over using the now popular mutual information method (based on Shannon's entropy) are that the former has significantly larger noise immunity and a much larger convergence range over the field of parameterized transformations. These strengths of CCRE are demonstrated via experiments on synthesized and real image data.

Image denoising and segmentation are fundamental problems in the field of image processing and computer vision with numerous applications. We propose a partial differential equation (PDE) based smoothing and segmentation framework wherein the image data are smoothed via an evolution equation that is controlled by a vector field describing a viscous fluid flow. Image segmentation in this framework is defined by locations in the image where the fluid velocity is a local maximum. The nonlinear image smoothing is selectively achieved to preserve edges in the image. The novelty of this approach lies in the fact that the selective term is derived from a nonlinearly regularized image gradient field unlike most earlier techniques which either used a constant (with respect to time) selective term or a time varying nonlinearly smoothed scalar valued term. Implementation results on synthetic and real images are presented to depict the performance of the technique in comparison to methods recently reported in literature.