Total Variation Penalty Research Articles

Convex relaxations of penalties for sparse correlated variables with bounded total variation

We study the problem of statistical estimation with a signal known to be sparse, spatially contiguous, and containing many highly correlated variables. We take inspiration from the recently introduced k-support norm, which has been successfully applied to sparse prediction problems with correlated features, but lacks any explicit structural constraints commonly found in machine learning and image processing. We address this problem by incorporating a total variation penalty in the k-support framework. We introduce the (k, s) support total variation norm as the tightest convex relaxation of the intersection of a set of sparsity and total variation constraints. We show that this norm leads to an intractable combinatorial graph optimization problem, which we prove to be NP-hard. We then introduce a tractable relaxation with approximation guarantees that scale well for grid structured graphs. We devise several first-order optimization strategies for statistical parameter estimation with the described penalty. We demonstrate the effectiveness of this penalty on classification in the low-sample regime, classification with M/EEG neuroimaging data, and image recovery with synthetic and real data background subtracted image recovery tasks. We extensively analyse the application of our penalty on the complex task of identifying predictive regions from low-sample high-dimensional fMRI brain data, we show that our method is particularly useful compared to existing methods in terms of accuracy, interpretability, and stability.

Detection of material interfaces using a regularized level set method in piezoelectric structures

An algorithm to solve the inverse problem of detecting inclusion interfaces in a piezoelectric structure is proposed. The material interfaces are implicitly represented by level sets which are identified by applying regularization using total variation penalty terms. The inverse problem is solved iteratively and the extended finite element method is used for the analysis of the structure in each iteration. The formulation is presented for three-dimensional structures and inclusions made of different materials are detected by using multiple level sets. The results obtained prove that the iterative procedure proposed can determine the location and approximate shape of material sub-domains in the presence of higher noise levels.

Iterative CBCT reconstruction using Hessian penalty

Statistical iterative reconstruction algorithms have shown potential to improve cone-beam CT (CBCT) image quality. Most iterative reconstruction algorithms utilize prior knowledge as a penalty term in the objective function. The penalty term greatly affects the performance of a reconstruction algorithm. The total variation (TV) penalty has demonstrated great ability in suppressing noise and improving image quality. However, calculated from the first-order derivatives, the TV penalty leads to the well-known staircase effect, which sometimes makes the reconstructed images oversharpen and unnatural. In this study, we proposed to use a second-order derivative penalty that involves the Frobenius norm of the Hessian matrix of an image for CBCT reconstruction. The second-order penalty retains some of the most favorable properties of the TV penalty like convexity, homogeneity, and rotation and translation invariance, and has a better ability in preserving the structures of gradual transition in the reconstructed images. An effective algorithm was developed to minimize the objective function with the majorization-minimization (MM) approach. The experiments on a digital phantom and two physical phantoms demonstrated the priority of the proposed penalty, particularly in suppressing the staircase effect of the TV penalty.

Preconditioned Douglas–Rachford Algorithms for TV- and TGV-Regularized Variational Imaging Problems

The recently introduced preconditioned Douglas---Rachford iteration (PDR) for convex---concave saddle-point problems is studied with respect to convergence rates and applied to variational imaging problems with total variation (TV) and total generalized variation (TGV) penalty. A rate of $${\mathcal {O}}(1/k)$$O(1/k) for restricted primal---dual gaps evaluated for ergodic sequences generated by the PDR iteration is established. Based on PDR, new fast iterative algorithms for TV-denoising, TV-deblurring, and TGV-denoising of second order with $$L^2$$L2 and $$L^1$$L1 discrepancy are proposed. While for denoising, symmetric (block) Red---Black Gauss---Seidel preconditioners are effective, fast Fourier transform-based preconditioners are employed for the deblurring problems. Finally, for the $$L^2$$L2-TGV-denoising problem, an effective modified primal---dual gap is developed which may serve as a stopping criterion. All algorithms are tested and compared in numerical experiments. In particular, for problems where strong convexity does not hold, it turns out that the proposed preconditioning techniques are beneficial and lead to competitive results.

基于非局部全变差模型和全局非零局部秩惩罚的图像去模糊

图像成像及分析模块是未来汽车应用系统中的重要组成部分，清晰的图像为后续的智能控制提供可靠保证。然而，由于成像设备自身硬件的问题，使得图像出现模糊等问题。因此，为了能够从降质图像中复原出高质量的清晰图像，并为后续的处理带来便利，本文提出一种基于非局部全变差模型和全局非零局部秩惩罚的图像去模糊方法。非局部全变差模型主要用于恢复图像中的纹理细节，而非零局部秩惩罚则主要用于约束图像的边缘，达到锐化边缘的目的。本文所提出的方法在模拟图像和真实模糊图像的去模糊上都取得了很好的效果。 The imaging and analysis module is an important part of automobile application system in the fu-ture, and clear images provide a reliable guarantee for the intelligent control system. However, due to the existing problems of imaging equipment hardware, the obtained images appear blurring. Therefore, in order to restore the clean images from the blur ones and bring convenience to the subsequent processing, this paper proposes an image deblurring method based on non-local total variation and global non-zero local rank penalty. The non-local total variation model is mainly used to restore the texture details of image, and the non-zero local rank penalty is mainly used to sharp the edge of the image. The proposed deblurring method in this paper has achieved better results on simulated images and real blurred image than other methods.

A penalized algorithm for event-specific rate models for recurrent events.

We introduce a covariate-specific total variation penalty in two semiparametric models for the rate function of recurrent event process. The two models are a stratified Cox model, introduced in Prentice and others (1981. On the regression analysis of multivariate failure time data. Biometrika 68: , 373-379.), and a stratified Aalen's additive model. We show the consistency and asymptotic normality of our penalized estimators. We demonstrate, through a simulation study, that our estimators outperform classical estimators for small-to-moderate sample sizes. Finally, an application to the bladder tumor data of Byar (1980. The veterans administration study of chemoprophylaxis for recurrent stage 1 bladder tumors: comparison of placebo, pyridoxine, and topical thiotepa. In Pavone-Macaluso, M. Smith, P. H. and F. Edsmyn (editors), Bladder Tumors and Others Topics in Urological Oncology, pp. 363-370.) is presented.

Bayesian Nonparametric Dictionary Learning for Compressed Sensing MRI

We develop a Bayesian nonparametric model for reconstructing magnetic resonance images (MRIs) from highly undersampled k -space data. We perform dictionary learning as part of the image reconstruction process. To this end, we use the beta process as a nonparametric dictionary learning prior for representing an image patch as a sparse combination of dictionary elements. The size of the dictionary and patch-specific sparsity pattern are inferred from the data, in addition to other dictionary learning variables. Dictionary learning is performed directly on the compressed image, and so is tailored to the MRI being considered. In addition, we investigate a total variation penalty term in combination with the dictionary learning model, and show how the denoising property of dictionary learning removes dependence on regularization parameters in the noisy setting. We derive a stochastic optimization algorithm based on Markov chain Monte Carlo for the Bayesian model, and use the alternating direction method of multipliers for efficiently performing total variation minimization. We present empirical results on several MRI, which show that the proposed regularization framework can improve reconstruction accuracy over other methods.

Total variation regularization in measurement and image space for PET reconstruction

The aim of this paper is to test and analyse a novel technique for image reconstruction in positron emission tomography, which is based on (total variation) regularization on both the image space and the projection space. We formulate our variational problem considering both total variation penalty terms on the image and on an idealized sinogram to be reconstructed from a given Poisson distributed noisy sinogram. We prove existence, uniqueness and stability results for the proposed model and provide some analytical insight into the structures favoured by joint regularization. For the numerical solution of the corresponding discretized problem we employ the split Bregman algorithm and extensively test the approach in comparison to standard total variation regularization on the image. The numerical results show that an additional penalty on the sinogram performs better on reconstructing images with thin structures.

Typical bubble shape estimation in two-phase flow using inverse problem techniques

Two-phase flow models predict different arrangements of gas and liquid, which include the formation of bubbles, slugs and aerated areas. Thus, flow visualization plays a central role in development and refinement of models. Recently, high-speed cameras have enabled the recording of these flows and the challenge is often to extract relevant information from large amounts of data.This paper analyzes and proposes some schemes to estimate a typical bubble based on flow images. The key idea is to find a bubble image that represents and characterizes the flow conditions. As we show, using simple mean as estimator does not yield meaningful images and can be successfully replaced by median. Finally, observing the power spectral density of individual bubble images, we propose an inverse problem regularizer combined with total variation penalization, which significantly improves image quality and sharpens bubble boundaries.

Multimodal Image Reconstruction Using Supplementary Structural Information in Total Variation Regularization.

In this paper, we propose an iterative reconstruction algorithm which uses available information from one dataset collected using one modality to increase the resolution and signal-to-noise ratio of one collected by another modality. The method operates on the structural information only which increases its suitability across various applications. Consequently, the main aim of this method is to exploit available supplementary data within the regularization framework. The source of primary and supplementary datasets can be acquired using complementary imaging modes where different types of information are obtained (e.g. in medical imaging: anatomical and functional). It is shown by extracting structural information from the supplementary image (direction of level sets) one can enhance the resolution of the other image. Notably, the method enhances edges that are common to both images while not suppressing features that show high contrast in the primary image alone. In our iterative algorithm we use available structural information within a modified total variation penalty term. We provide numerical experiments to show the advantages and feasibility of the proposed technique in comparison to other methods.

SU‐F‐18C‐14: Hessian‐Based Norm Penalty for Weighted Least‐Square CBCT Reconstruction

Purpose:To develop a Hessian‐based norm penalty for cone‐beam CT (CBCT) reconstruction that has a similar ability in suppressing noise as the total variation (TV) penalty while avoiding the staircase effect and better preserving low‐contrast objects.Methods:We extended the TV penalty to a Hessian‐based norm penalty based on the Frobenius norm of the Hessian matrix of an image for CBCT reconstruction. The objective function was constructed using the penalized weighted least‐square (PWLS) principle. An effective algorithm was developed to minimize the objective function using a majorization‐minimization (MM) approach. We evaluated and compared the proposed penalty with the TV penalty on a CatPhan 600 phantom and an anthropomorphic head phantom, each acquired at a low‐dose protocol (10mA/10ms) and a high‐dose protocol (80mA/12ms). For both penalties, contrast‐to‐noise (CNR) in four low‐contrast regions‐of‐interest (ROIs) and the full‐width‐at‐half‐maximum (FWHM) of two point‐like objects in constructed images were calculated and compared.Results:In the experiment of CatPhan 600 phantom, the Hessian‐based norm penalty has slightly higher CNRs and approximately equivalent FWHM values compared with the TV penalty. In the experiment of the anthropomorphic head phantom at the low‐dose protocol, the TV penalty result has several artificial piece‐wise constant areas known as the staircase effect while in the Hessian‐based norm penalty the image appears smoother and more similar to that of the FDK result using the high‐dose protocol.Conclusion:The proposed Hessian‐based norm penalty has a similar performance in suppressing noise to the TV penalty, but has a potential advantage in suppressing the staircase effect and preserving low‐contrast objects.This work was supported in part by National Natural Science Foundation of China (NNSFC), under Grant Nos. 60971112 and 61375018, and Fundamental Research Funds for the Central Universities, under Grant No. 2012QN086.

SU‐F‐18C‐13: Low‐Dose X‐Ray CT Reconstruction Using a Hybrid First‐Order Method

Purpose:To develop a novel reconstruction method for X‐ray CT that can lead to accurate reconstruction at significantly reduced dose levels combining low X‐ray incident intensity and few views of projection data.Methods:The noise nature of the projection data at low X‐ray incident intensity was modeled and accounted by the weighted least‐squares (WLS) criterion. The total variation (TV) penalty was used to mitigate artifacts caused by few views of data. The first order primal‐dual (FOPD) algorithm was used to minimize TV in image domain, which avoided the difficulty of the non‐smooth objective function. The TV penalized WLS reconstruction was achieved by alternated FOPD TV minimization and projection onto convex sets (POCS) for data fidelity constraints. The proposed FOPD‐POCS method was evaluated using the FORBILD jaw phantom and the real cadaver head CT data.Results:The quantitative measures, root mean square error (RMSE) and contrast‐to‐noise ratio (CNR), demonstrate the superior denoising capability of WLS over LS‐based TV iterative reconstruction. The improvement of RMSE (WLS vs. LS) is 15%∼21% and that of CNR is 17%∼72% when the incident counts per ray are ranged from 1×105 to 1×103. In addition, the TV regularization can accurately reconstruct images from about 50 views of the jaw phantom. The FOPD‐POCS reconstruction reveals more structural details and suffers fewer artifacts in both the phantom and real head images. The FOPD‐POCS method also shows fast convergence at low X‐ray incident intensity.Conclusion:The new hybrid FOPD‐POCS method, based on TV penalized WLS, yields excellent image quality when the incident X‐ray intensity is low and the projection views are limited. The reconstruction is computationally efficient since the FOPD minimization of TV is applied only in the image domain. The characteristics of FOPD‐POCS can be exploited to significantly reduce radiation dose of X‐ray CT without compromising accuracy for diagnosis or treatment planning.

Sparsity-regularized image reconstruction of decomposed K-edge data in spectral CT

The development of spectral computed tomography (CT) using binned photon-counting detectors has garnered great interest in recent years and has enabled selective imaging of K-edge materials. A practical challenge in CT image reconstruction of K-edge materials is the mitigation of image artifacts that arise from reduced-view and/or noisy decomposed sinogram data. In this note, we describe and investigate sparsity-regularized penalized weighted least squares-based image reconstruction algorithms for reconstructing K-edge images from few-view decomposed K-edge sinogram data. To exploit the inherent sparseness of typical K-edge images, we investigate use of a total variation (TV) penalty and a weighted sum of a TV penalty and an ℓ1-norm with a wavelet sparsifying transform. Computer-simulation and experimental phantom studies are conducted to quantitatively demonstrate the effectiveness of the proposed reconstruction algorithms.

Generalized Higher Degree Total Variation (HDTV) Regularization

We introduce a family of novel image regularization penalties called generalized higher degree total variation (HDTV). These penalties further extend our previously introduced HDTV penalties, which generalize the popular total variation (TV) penalty to incorporate higher degree image derivatives. We show that many of the proposed second degree extensions of TV are special cases or are closely approximated by a generalized HDTV penalty. Additionally, we propose a novel fast alternating minimization algorithm for solving image recovery problems with HDTV and generalized HDTV regularization. The new algorithm enjoys a tenfold speed up compared with the iteratively reweighted majorize minimize algorithm proposed in a previous paper. Numerical experiments on 3D magnetic resonance images and 3D microscopy images show that HDTV and generalized HDTV improve the image quality significantly compared with TV.

The PyHST2 hybrid distributed code for high speed tomographic reconstruction with iterative reconstruction and a priori knowledge capabilities

We present the PyHST2 code which is in service at ESRF for phase-contrast and absorption tomography. This code has been engineered to sustain the high data flow typical of the 3rd generation synchrotron facilities (10 terabytes per experiment) by adopting a distributed and pipelined architecture. The code implements, beside a default filtered backprojection reconstruction, iterative reconstruction techniques with a priori knowledge. These latter are used to improve the reconstruction quality or in order to reduce the required data volume or the deposited dose to the sample and reach a given quality goal. The implemented a priori knowledge techniques are based on the total variation penalization and a new recently found convex functional which is based on overlapping patches. We give details of the different methods and discuss how they are implemented in the PyHST2 code, which is distributed under free license. We provide methods for estimating, in the absence of ground-truth data, the optimal parameters values for a priori techniques.

Dual‐energy cone‐beam CT with a flat‐panel detector: Effect of reconstruction algorithm on material classification

Cone-beam CT (CBCT) with a flat-panel detector (FPD) is finding application in areas such as breast and musculoskeletal imaging, where dual-energy (DE) capabilities offer potential benefit. The authors investigate the accuracy of material classification in DE CBCT using filtered backprojection (FBP) and penalized likelihood (PL) reconstruction and optimize contrast-enhanced DE CBCT of the joints as a function of dose, material concentration, and detail size. Phantoms consisting of a 15 cm diameter water cylinder with solid calcium inserts (50-200 mg/ml, 3-28.4 mm diameter) and solid iodine inserts (2-10 mg/ml, 3-28.4 mm diameter), as well as a cadaveric knee with intra-articular injection of iodine were imaged on a CBCT bench with a Varian 4343 FPD. The low energy (LE) beam was 70 kVp (+0.2 mm Cu), and the high energy (HE) beam was 120 kVp (+0.2 mm Cu, +0.5 mm Ag). Total dose (LE+HE) was varied from 3.1 to 15.6 mGy with equal dose allocation. Image-based DE classification involved a nearest distance classifier in the space of LE versus HE attenuation values. Recognizing the differences in noise between LE and HE beams, the LE and HE data were differentially filtered (in FBP) or regularized (in PL). Both a quadratic (PLQ) and a total-variation penalty (PLTV) were investigated for PL. The performance of DE CBCT material discrimination was quantified in terms of voxelwise specificity, sensitivity, and accuracy. Noise in the HE image was primarily responsible for classification errors within the contrast inserts, whereas noise in the LE image mainly influenced classification in the surrounding water. For inserts of diameter 28.4 mm, DE CBCT reconstructions were optimized to maximize the total combined accuracy across the range of calcium and iodine concentrations, yielding values of ∼ 88% for FBP and PLQ, and ∼ 95% for PLTV at 3.1 mGy total dose, increasing to ∼ 95% for FBP and PLQ, and ∼ 98% for PLTV at 15.6 mGy total dose. For a fixed iodine concentration of 5 mg/ml and reconstructions maximizing overall accuracy across the range of insert diameters, the minimum diameter classified with accuracy >80% was ∼ 15 mm for FBP and PLQ and ∼ 10 mm for PLTV, improving to ∼ 7 mm for FBP and PLQ and ∼ 3 mm for PLTV at 15.6 mGy. The results indicate similar performance for FBP and PLQ and showed improved classification accuracy with edge-preserving PLTV. A slight preference for increased smoothing of the HE data was found. DE CBCT discrimination of iodine and bone in the knee was demonstrated with FBP and PLTV at 6.2 mGy total dose. For iodine concentrations >5 mg/ml and detail size ∼ 20 mm, material classification accuracy of >90% was achieved in DE CBCT with both FBP and PL at total doses <10 mGy. Optimal performance was attained by selection of reconstruction parameters based on the differences in noise between HE and LE data, typically favoring stronger smoothing of the HE data, and by using penalties matched to the imaging task (e.g., edge-preserving PLTV in areas of uniform enhancement).

Compressed-sensing-based fluorescence molecular tomographic image reconstruction with grouped sources.

BackgroundAlthough the quality of reconstructed results can be improved with the increment of the number of measurements, the scale of the matrices involved in the reconstruction of fluorescence molecular tomography (FMT) will become larger, which leads to the poor efficiency of the process of tomographic image reconstruction. In this paper, we proposed a new method for image reconstruction of FMT based on compressed sensing, in which a scheme of grouped sources is incorporated.MethodsThe forward equations are implemented using the finite element method (FEM). The reconstruction model is formulated under the framework of compressed sensing theory. The regularization term and the total variation penalty are incorporated in the objective function. During the reconstruction of FMT, the sources are divided into two groups for iteration in turn. One group of sources is employed in the first iteration of inverse problem, and the other group is employed in the next iteration.ResultsSimulation results demonstrate that the computation time and mean square error (MSE) of the reconstruction with our algorithm are less than those with the traditional method. The proposed algorithm can reconstruct the target with enhanced contrast and more accurate shape.ConclusionsThe proposed algorithm can significantly improve the speed and accuracy of the reconstruction of FMT. Furthermore, our compressed-sensing-based method can reduce the number of measurements.

Low-mAs X-ray CT image reconstruction by adaptive-weighted TV-constrained penalized re-weighted least-squares.

The negative effects of X-ray exposure, such as inducing genetic and cancerous diseases, has arisen more attentions. This paper aims to investigate a penalized re-weighted least-square (PRWLS) strategy for low-mAs X-ray computed tomography image reconstruction by incorporating an adaptive weighted total variation (AwTV) penalty term and a noise variance model of projection data. An AwTV penalty is introduced in the objective function by considering both piecewise constant property and local nearby intensity similarity of the desired image. Furthermore, the weight of data fidelity term in the objective function is determined by our recent study on modeling variance estimation of projection data in the presence of electronic background noise. The presented AwTV-PRWLS algorithm can achieve the highest full-width-at-half-maximum (FWHM) measurement, for data conditions of (1) full-view 10 mA acquisition and (2) sparse-view 80 mA acquisition. In comparison between the AwTV/TV-PRWLS strategies and the previous reported AwTV/TV-projection onto convex sets (AwTV/TV-POCS) approaches, the former can gain in terms of FWHM for data condition (1), but cannot gain for the data condition (2). In the case of full-view 10 mA projection data, the presented AwTV-PRWLS shows potential improvement. However, in the case of sparse-view 80 mA projection data, the AwTV/TV-POCS shows advantage over the PRWLS strategies.

Magnetic detection electrical impedance tomography with total variation regularization.

Magnetic detection electrical impedance tomography (MDEIT) is an imaging modality that aims to reconstruct the cross-sectional conductivity distribution of a volume from the magnetic flux density surrounding an object. The MDEIT inverse problem is inherently ill-posed, necessitating the use of regularization. The most commonly used L(2) norm regularizations generate the minimum energy solution, which blurs the sharp variations of the reconstructed image. Consequently, this paper presents the total variation (TV) regularization to preserve discontinuities and piecewise constancy of the MDEIT reconstructed image. The primal dual-interior point method (PD-IPM) is employed for minimizing the TV penalty in this paper. The proposed method is validated by MDEIT simulated data. In comparison with the performance of L(2) norm regularization, the results show that TV regularized algorithm produces sharper images and has better robustness to noise. The TV regularized algorithm preserves local smoothness and piecewise constancy, leading to improvements in the localization of the reconstructed conductivity images in MDEIT.

Solving a Linear Inverse Problem by Using an $L^{\infty}$ Fitting and a Hybrid Penalty with an Application to an Inverse Heat Conduction Problem

In this paper, we aim at solving a linear inverse problem in the case of measurement data contaminated by uniform noises. Due to ill-posedness, we use a regularization method to solve this problem. The proposed method combines an L 1 fitting with a hybrid penalty which consists of a total variation penalty and an L 2 penalty. An L 1 fitting is utilized to denoise uniform noises. We reconstruct a piecewise constant function, and hence we employ a total variation penalty to identify sharp boundaries, and use an L 2 penalty to improve the stability of the numerical algorithm. We apply the proposed method to an inverse heat conduction problem for testing the validity of the method. The numerical experiments show that the proposed method is accurate and robust.