Non-convex Lp Research Articles

High-Fidelity compressive spectral image reconstruction through a novel Non-Convex Non-Local Low-Rank tensor approximation model

Low-quality reconstruction is one main factor that greatly limits the development of snapshot compressive imaging (SCI) systems. The common algorithms usually unfold a 3D hyperspectral image (HSI) into a vector or matrix which inevitably corrupts the intrinsic structure of the HSI and insufficiently discovers its prior information. To fully exploit the underlying structures and internal correlations of the HSIs, this paper proposes a novel non-convex model based on Non-Local low-rank Tensor Approximation with Hyper-Laplacian priori (HL), named HL-NLTA, which effectively combines local and non-local similarity information from fused spectral and spatial perspectives. Specifically, a non-local low-rank tensor model based on the minimax concave plus (MCP) penalty and log sum (LS) penalty separately is developed which can simultaneously exploit two intrinsic properties of HSI, i.e., global correlation along the spectral domain (GCAS) and non-local self-similarity along the spatial domain (NSAS). Furthermore, the Hyper-Laplacian priori regularized through a non-convex ℓp (0 < p < 1) norm can preserve well the spectral and spatial structure. An optimization algorithm based on the alternating direction multiplier method (ADMM) framework is designed and accelerated by the Generalized shrinkage/thresholding (GST) algorithm and the fast Fourier transform (FFT) to solve the new nonconvex model. Extensive simulation results on public datasets and experimental results on our real CASSI system indicate that the proposed method enables high-fidelity HSI reconstructions in terms of simultaneous spatial structure detail recovery and spectral preservation. These results verify the effectiveness of the proposed model for snapshot compressive imaging (e.g., Coded aperture snapshot spectral imaging CASSI).

A Lorentzian-[formula omitted] norm regularization based algorithm for recovering sparse signals in two types of impulsive noise

In this paper, we present a Lorentzian-ℓp(0<p<1) optimization for recovering sparse signals from measurements corrupted by two types of impulsive noise. A smoothing strategy for nonconvex ℓp norm is employed and an efficient Lorentzian based iterative algorithm is developed for solving the resulting optimization problem. We describe the detailed process of the algorithm and establish the convergence of the proposed algorithm. Numerical results show that our proposed algorithm performs effectively in recovering sparse signals from impulsive measurement noise compared with the state-of-the-art methods.

Resolution aware nonconvex quasinorm iterative digital breast tomosynthesis imaging

Digital breast tomosynthesis is a promising modality providing 3D cross sectional images of breast. However, reconstructing the image from few projections is still a challenging task. We introduce an anisotropic total variation regularized reconstruction method which exploits varying resolution along transverse and sagittal planes by adapting the weights of TV gradients. Moreover, we also investigate if non-convex ℓp quasinorm with p ∈ (0,1) can help to improve the performance of anisotropic TV in detecting micro calcifications and spheroidal masses. The contrast to noise ratio (CNR) for in-plane quality, full with at half maximum (FWHM) of artefact spread function (ASF) for resolution in the z-axis, signal difference to noise ratio (SdNR), and quality factor (QF) for a comprehensive evaluation were used in quantitative analyses and comparisons. For low contrast analysis, QF value of the proposed method for p = 0.9 is 0.18 whereas QF value of isotropic TV with ℓ1 and ℓ2 norms are 0.15 and 0.14, respectively. For high contrast features such as micro-calcifications, although the best CNR value changes with different p values, the majority of the best CNR values are calculated for values 0.8 and 0.9. Overall, the results show that the proposed method, anisotropic TV with non-convex ℓp quasinorm, outperforms isotropic TV with ℓ1 and ℓ2 norms by recovering small micro calcifications and spheroidal masses with higher CNR and increased depth resolution. The proposed method shows the best performance for p values of 0.8 ≤ p < 1.

Low-Light Enhancement Using a Plug-and-Play Retinex Model With Shrinkage Mapping for Illumination Estimation.

Low-light photography conditions degrade image quality. This study proposes a novel Retinex-based low-light enhancement method to correctly decompose an input image into reflectance and illumination. Subsequently, we can improve the viewing experience by adjusting the illumination using intensity and contrast enhancement. Because image decomposition is a highly ill-posed problem, constraints must be properly imposed on the optimization framework. To meet the criteria of ideal Retinex decomposition, we design a nonconvex Lp norm and apply shrinkage mapping to the illumination layer. In addition, edge-preserving filters are introduced using the plug-and-play technique to improve illumination. Pixel-wise weights based on variance and image gradients are adopted to suppress noise and preserve details in the reflectance layer. We choose the alternating direction method of multipliers (ADMM) to solve the problem efficiently. Experimental results on several challenging low-light datasets show that our proposed method can more effectively enhance image brightness as compared with state-of-the-art methods. In addition to subjective observations, the proposed method also achieved competitive performance in objective image quality assessments.

A sparsity-enhanced periodic OGS model for weak feature extraction of rolling bearing faults

The fault symptom of rolling bearings is usually characterized by transient impulses formed at equal intervals, but the impulse signal is easily affected by noise and harmonic interferences, which increases the difficulty of extracting impulse features. In order to realize the effective extraction of weak periodic impulses under strong noise, this paper constructs a non-convex penalty function based on elastic net and Lp norm, and proposes a sparsity-enhanced periodic overlapping group shrinkage (POGS) method to detect rolling bearing faults. In the proposed sparse model, the internal function of the non-convex penalty function adopts the period-guided elastic net group sparse constraint, and the envelope autocorrelation function is used to dynamically update the period prior information to improve the extraction accuracy of highly correlated features within the group. Meanwhile, the non-convex Lp norm is introduced into the penalty function to constrain the sparsity of the overall variables, so as to guide the sparsity within and across groups (SWAG) of faults features while maintaining the weak impulse amplitudes. A comprehensive evaluation indicator is constructed as the fitness function of the moth-flame optimization (MFO) algorithm to realize automatic selection of model parameters. On the basis of the majorization-minimization (MM) algorithm and the improved soft threshold algorithm, the process of solving the objective function of the proposed model is given, and the performance of the proposed method is analyzed. The analysis results of the experimental data of rolling bearings suggest that in comparison with some existing sparse denoising methods, the proposed method exhibits better performance in the extraction of weak periodic impulses.

Sparsity-enhanced equivalent source method for acoustic source reconstruction via the Generalized Minimax-Concave penalty

The Equivalent Source Method (ESM) is a powerful technique for acoustic source reconstruction, which has been widely used in noise control and machinery fault detection. Because the inverse acoustic source reconstruction problem is typically ill-conditioned, regularization techniques are often adopted in ESM to achieve meaningful solutions. Based on the sparse distribution assumption of acoustic sources, sparse regularization can be utilized in ESM to promote the spatial resolution of reconstructed results. Existing sparse ESMs often adopt convex l1 norm or nonconvex lp norm as the regularization terms. However, l1 norm-regularized ESM suffers from an insufficient sparsity-inducing problem, which decreases the spatial resolution and reconstruction accuracy. Meanwhile, reconstructed results of lp norm-regularized ESM are often unstable due to multiple local minimas. In this paper, a sparsity-enhanced ESM is proposed, which introduces a nonconvex Generalized Minimax-Concave (GMC) penalty into ESM as the regularization term. The GMC penalty can enhance the sparsity of solutions and simultaneously maintain the convexity of the overall objective function in acoustic source reconstruction. Consequently, the GMC penalty-regularized ESM can improve the spatial resolution and reconstruction accuracy of the reconstructed results. To solve the acoustic source reconstruction problem rapidly, a computation framework based on Alternating Direction Method of Multipliers (ADMM) is derived. Simulation studies indicate that the proposed method can improve the acoustic source reconstruction accuracy in a wide frequency band, compared with existing l1 and lp norm-regularized ESMs. Two experimental cases further verify that the proposed sparsity-enhanced ESM can effectively reconstruct the acoustic sources with high spatial resolution and reconstruction accuracy at higher frequencies.

Sparsity reconstruction using nonconvex TGpV-shearlet regularization and constrained projection

In many sparsity-based image processing problems, compared with the convex ℓ1 norm approximation of the nonconvex ℓ0 quasi-norm, one can often preserve the structures better by taking full advantage of the nonconvex ℓp quasi-norm (0≤p<1). In this paper, we propose a nonconvex ℓp quasi-norm approximation in the total generalized variation (TGV)-shearlet regularization for image reconstruction. By introducing some auxiliary variables, the nonconvex nonsmooth objective function can be solved by an efficient alternating direction method of multipliers with convergence analysis. Especially, we use a generalized iterated shrinkage operator to deal with the ℓp quasi-norm subproblem, which is easy to implement. Extensive experimental results show clearly that the proposed nonconvex sparsity approximation outperforms some state-of-the-art algorithms in both the visual and quantitative measures for different sampling ratios and noise levels.

Two-stage image segmentation based on nonconvex [formula omitted] approximation and thresholding

Image segmentation is of great importance in image processing. In this paper, we propose a two-stage image segmentation strategy based on the nonconvex ℓ2−ℓp approximation of the Mumford–Shah (MS) model, where we use the nonconvex ℓp (0<p<1) regularizer to approximate the Hausdorff measure and to extract more boundary information. In the first stage, we solve the nonconvex variant of the MS model efficiently via the split-Bregman algorithm. Moreover, we use a closed-form p-shrinkage operator to deal with the ℓp quasi-norm subproblem, which is easy to implement. The second stage is segmenting the u obtained in the first stage into different phases with thresholds determined by the K-means clustering method. We compare our method with several state-of-the-art methods both qualitatively and quantitatively to demonstrate the effectiveness and advantages of our strategy.

Multi-band weighted [formula omitted] norm minimization for image denoising

Low rank matrix approximation (LRMA) has a wide range of applications in computer vision and has drawn much attention in recent years. A typical nuclear norm minimization (NNM) is often used to solve a LRMA, but this is likely to overshrink the rank components due to having the same threshold. To address this problem, we propose a flexible and precise model named multi-band weighted lp norm minimization (MBWPNM). We have reformulated it into nonconvex lp norm subproblems under certain weight conditions, and we solve these subproblems via a generalized soft-thresholding algorithm. We then adopt MBWPNM for image (grayscale, color and multispectral) denoising. The proposed MBWPNM not only guarantees a more accurate approximation with a Schatten p-norm in case of a change of noise levels, but it also considers prior knowledge, for which different rank components have varying importance. Extensive experiments on additive white Gaussian noise removal and realistic noise removal demonstrate that the proposed MBWPNM achieves better performance than several state-of-the-art algorithms.

One-bit compressed sensing via ℓp(0 < p < 1)-minimization method

One-bit compressed sensing aims to recover unknown sparse signals from extremely quantized linear measurements which just capture their signs. In this paper, we propose a nonconvex ℓp(0 < p < 1) minimization model for one-bit compressed sensing problem and define the set of ℓp effectively s-sparse signals which contains genuinely s-sparse signals. Utilizing properties of covering number, we show that our method can recover the direction of ℓp effectively s-sparse signals with error order . We also employ thresholded one-bit measurements to estimate the magnitude of signals and prove that any ℓp effectively s-sparse bounded signal x can be estimated using augmented ℓp minimization model and empirical distribution function method respectively. Especially, to recover ℓp effectively s-sparse signals in practice, we introduce an adaptive binary iterative thresholding algorithm which can be utilized without knowing the sparsity of underlying signals. Numerical experiments on both synthetic and real-world data sets are conducted to demonstrate the superiority of our algorithm.

Graph-Based Non-Convex Low-Rank Regularization for Image Compression Artifact Reduction.

Block transform coded images usually suffer from annoying artifacts at low bit-rates, because of the independent quantization of DCT coefficients. Image prior models play an important role in compressed image reconstruction. Natural image patches in a small neighborhood of the high-dimensional image space usually exhibit an underlying sub-manifold structure. To model the distribution of signal, we extract sub-manifold structure as prior knowledge. We utilize graph Laplacian regularization to characterize the sub-manifold structure at patch level. And similar patches are exploited as samples to estimate distribution of a particular patch. Instead of using Euclidean distance as similarity metric, we propose to use graph-domain distance to measure the patch similarity. Then we perform low-rank regularization on the similar-patch group, and incorporate a non-convex lp penalty to surrogate matrix rank. Finally, an alternatively minimizing strategy is employed to solve the non-convex problem. Experimental results show that our proposed method is capable of achieving more accurate reconstruction than the state-of-the-art methods in both objective and perceptual qualities.

Sparse representation of classified patches for CS-MRI reconstruction

Compressed sensing MRI (CS-MRI) has demonstrated its potential in image reconstruction from undersampling k-space data to accelerate scanning time. Exact CS-MRI reconstruction is based on the image prior information of sparsity, and therefore, it relies on two important aspects of sparse domain and sparse coding. In this work, a patch based uniform model according to orthogonal dictionary learning and lp norm minimization (ODNM) is developed. First, image patches are classified into multi-classes by a use of a modified clustering method and the corresponding orthogonal dictionary is learned for each class. In addition, the non-convex lp norm regularization is employed to promote the sparsity of patch coefficients. In order to solve the proposed reconstruction model, the alternating direction method (ADM) steps are developed in which orthogonal dictionary learning, non-convex sparse coding, and image reconstruction are simultaneously conducted. In the special case of p=0.5, a fast shrinkage operator is proposed to reduce the computational complexity. Extensive experiments on real complex valued MR images under various sampling patterns demonstrate that the proposed method achieves a better performance than several state-of-the-art algorithms.

Multiple-prespecified-dictionary sparse representation for compressive sensing image reconstruction with nonconvex regularization

Multiple-prespecified-dictionary sparse representation (MSR) has shown powerful potential in compressive sensing (CS) image reconstruction, which can exploit more sparse structure and prior knowledge of images for minimization. Due to the popular L1 regularization can only achieve the suboptimal solution of L0 regularization, using the nonconvex regularization can often obtain better results in CS reconstruction. This paper proposes a nonconvex adaptive weighted Lp regularization CS framework via MSR strategy. We first proposed a nonconvex MSR based Lp regularization model, then we propose two algorithms for minimizing the resulting nonconvex Lp optimization problem. According to the fact that the sparsity levels of each regularizers are varying with these prespecified-dictionaries, an adaptive scheme is proposed to weight each regularizer for optimization by exploiting the difference of sparsity levels as prior knowledge. Simulated results show that the proposed nonconvex framework can make a significant improvement in CS reconstruction than convex L1 regularization, and the proposed MSR strategy can also outperforms the traditional nonconvex Lp regularization methodology.

A new development of non-local image denoising using fixed-point iteration for non-convex ℓp sparse optimization.

We proposed a new efficient image denoising scheme, which mainly leads to four important contributions whose approaches are different from existing ones. The first is to show the equivalence between the group-based sparse representation and the Schatten-p norm minimization problem, so that the sparsity of the coefficients for each group can be measured by estimating the underlying singular values. The second is that we construct the proximal operator for sparse optimization in ℓp space with p ∈ (0, 1] by using fixed-point iteration and obtained a new solution of Schatten-p norm minimization problem, which is more rigorous and accurate than current available results. The third is that we analyze the suitable setting of power p for each noise level σ = 20, 30, 50, 60, 75, 100, respectively. We find that the optimal value of p is inversely proportional to the noise level except for high level of noise, where the best values of p are 1 and 0.95, when the noise levels are respectively 75 and 100. Last we measure the structural similarity between two image patches and extends previous deterministic annealing-based solution to sparsity optimization problem through incorporating the idea of dictionary learning. Experimental results demonstrate that for every given noise level, the proposed Spatially Adaptive Fixed Point Iteration (SAFPI) algorithm attains the best denoising performance on the value of Peak Signal-to-Noise Ratio (PSNR) and structure similarity (SSIM), being able to retain the image structure information, which outperforms many state-of-the-art denoising methods such as Block-matching and 3D filtering (BM3D), Weighted Nuclear Norm Minimization (WNNM) and Weighted Schatten p-Norm Minimization (WSNM).

The fusion of panchromatic and multispectral remote sensing images via tensor-based sparse modeling and hyper-Laplacian prior

In this paper, we propose a tensor-based non-convex sparse modeling approach for the fusion of panchromatic and multispectral remote sensing images, and this kind of fusion is generally called pansharpening. We first upsample the low spatial-resolution multispectral image by a classical interpolation method to get an initial upsampled multispectral image. Based on the hyper-Laplacian distribution of errors between the upsampled multispectral image and the ground-truth high resolution multispectral image on gradient domain, we formulate a ℓp(0 < p < 1)-norm term to more reasonably describe the relation of these two datasets. In addition, we also model a tensor-based weighted fidelity term for the panchromatic and low resolution multispectral images, aiming to recover more spatial details. Moreover, total variation regularization is also employed to depict the sparsity of the latent high resolution multispectral image on the gradient domain. For the model solving, we design an alternating direction method of multipliers based algorithm to efficiently solve the proposed model. Furthermore, the involved non-convex ℓp subproblem is handled by an efficient generalized shrinkage/thresholding algorithm. Finally, extensive experiments on many datasets collected by different sensors demonstrate the effectiveness of our method when compared with several state-of-the-art image fusion approaches.

Sparse non-convex Lp regularization for cone-beam X-ray luminescence computed tomography

ABSTRACTCone-beam X-ray luminescence computed tomography (CB-XLCT) is an attractive hybrid imaging modality, and it has the potential of monitoring the metabolic processes of nanophosphors-based drugs in vivo. However, the XLCT imaging suffers from a severe ill-posed problem. In this work, a sparse nonconvex Lp (0 < p < 1) regularization was utilized for the efficient reconstruction for early detection of small tumour in CB-XLCT imaging. Specifically, we transformed the non-convex optimization problem into an iteratively reweighted scheme based on the L1 regularization. Further, an iteratively reweighted split augmented lagrangian shrinkage algorithm (IRW_SALSA-Lp) was proposed to efficiently solve the non-convex Lp (0 < p < 1) model. We studied eight different non-convex p-values (1/16, 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8) in both 3D digital mouse experiments and in vivo experiments. The results demonstrate that the proposed non-convex methods outperform L2 and L1 regularization in accurately recovering sparse targets in CB-XLCT. And among all the non-convex p-values, our Lp(1/4 < p < 1/2) methods give the best performance.

Fast multi-trace impedance inversion using anisotropic total p-variation regularization in the frequency domain

In this article, we find that the transformational matrix between the natural logarithm of the acoustic impedance and the reflection coefficient is a Toeplitz matrix. As a result, we can convert the matrix multiplication into convolution operation according to properties of the Toeplitz matrix. In this way, we calculate the convolution of matrices through a dot product operation in the frequency domain based on the convolution theorem. Thus a fast multi-trace impedance inversion method by using two dimensional fast Fourier transform in the frequency domain is proposed, which can greatly improve the inversion speed. Considering the non-convex Lp (0 ≤ p < 1) quasi-norm is more suitable for sparse optimization than the L1 norm, anisotropic total variation regularization based on the Lp quasi-norm is used to improve the inversion result. Synthetic seismic data and field data inversion results show that the proposed method improves the inversion speed more than ten times compared with the multi-trace inversion method with total variation (TV) regularization in the time domain. At the same time, the proposed method has the advantages over the inversion methods with TV regularization in the time domain of less inversion error and better depiction of thin layers.

Low Tucker rank tensor recovery via ADMM based on exact and inexact iteratively reweighted algorithms

In this paper, we establish a non-convex Lp norm relaxation model for low Tucker rank tensor recovery problem, and equivalently transform it to a non-convex minimization problem with separable structure by introducing series of auxiliary variables. In particular, we propose two alternating direction method of multipliers (ADMM) based on exact and inexact iteratively reweighted algorithms to solve the obtained non-convex relaxation problem respectively, which are proved to be convergent. We implement the proposed algorithms in numerical experiments for solving low Tucker rank tensor recovery problem on simulation data and real data, and compare them with other existing state-of-art algorithms. Numerical results show the effectiveness of the proposed algorithms for solving low rank tensor recovery problem and image recovery.

Non-Convex Compressed Sensing Using Partial Support Information

In this paper we address the recovery conditions of weighted lp minimization for signal reconstruction from compressed sensing measurements when partial support information is available. We show that weighted lp minimization with 0 < p < 1 is stable and robust under weaker suficient conditions compared to weighted lp minimization. Moreover, the suficient recovery conditions of weighted lp are weaker than those of regular lp minimization if at least 50% of the support estimate is accurate. We also review some algorithms which exist to solve the non-convex lp problem and illustrate our results with numerical experiments.

Constrained TpV Minimization for Enhanced Exploitation of Gradient Sparsity: Application to CT Image Reconstruction.

Exploiting sparsity in the image gradient magnitude has proved to be an effective means for reducing the sampling rate in the projection view angle in computed tomography (CT). Most of the image reconstruction algorithms, developed for this purpose, solve a nonsmooth convex optimization problem involving the image total variation (TV). The TV seminorm is the n}{}ell_{1} norm of the image gradient magnitude, and reducing the n}{}ell_{1} norm is known to encourage sparsity in its argument. Recently, there has been interest in employing nonconvex n}{}ell_{p} quasinorms with n}{}0<p<1 for sparsity exploiting image reconstruction, which is potentially more effective than n}{}ell_{1} because nonconvex n}{}ell_{p} is closer to n}{}ell_{0}—a direct measure of sparsity. This paper develops algorithms for constrained minimization of the total n}{}p-variation n}{}({rm T}p{rm V}), n}{}ell_{p} of the image gradient. Use of the algorithms is illustrated in the context of breast CT—an imaging modality that is still in the research phase and for which constraints on X-ray dose are extremely tight. The n}{}{rm T}p{rm V}-based image reconstruction algorithms are demonstrated on computer simulated data for exploiting gradient magnitude sparsity to reduce the projection view angle sampling. The proposed algorithms are applied to projection data from a realistic breast CT simulation, where the total X-ray dose is equivalent to two-view digital mammography. Following the simulation survey, the algorithms are then demonstrated on a clinical breast CT data set.