Variational Regularization Technique Research Articles

Investigation of Image Enhancement Techniques for Advancing Colon Cancer Diagnosis

Colorectal cancer continues to pose a substantial worldwide health challenge, necessitating the development of advanced imaging techniques for early and accurate diagnosis. In this study, we propose a novel hybrid image enhancement approach that combines Total Variation (TV) regularization and shift-invariant filtering to improve the visibility and diagnostic quality of colon cancer images.
 The Total Variation regularization technique is employed to effectively reduce noise and enhance the edges in the input images, thereby preserving important structural details. Simultaneously, shift-invariant filtering is utilized to address spatial variations and artifacts that often arise in medical images, ensuring consistent and reliable enhancements across the entire image. Our hybrid approach synergistically integrates the strengths of both TV regularization and shift-invariant filtering, resulting in enhanced colon cancer images that offer improved contrast, reduced noise, and enhanced fine structures. This improved image quality aids medical professionals in better identifying and characterizing cancerous lesions, ultimately leading to more accurate and timely diagnoses.
 To evaluate the effectiveness of the proposed approach, we conducted extensive experimentations on a diverse dataset of colon cancer images. Quantitative and qualitative assessments demonstrate that our hybrid approach outperforms existing enhancement methods, leading to superior image quality and diagnostic accuracy. In conclusion, the hybrid image enhancement approach presented in this study offers a promising solution for enhancing colon cancer images, contributing to the early detection and effective management of this life-threatening disease. These advancements hold significant potential for improving patient outcomes and reducing the burden of colon cancer on healthcare systems worldwide.

An improved bilevel optimization approach for image super-resolution based on a fractional diffusion tensor

Variational regularization techniques are widely used to improve the quality of the super-resolved image. However, the success of these methods depends on some sensitive parameters which have to be manually tuned. Recently, an improved strategy to avoid this illness is to learn these parameters from some available data. In this work we propose an improved super-resolution method with bilevel optimization technique to learn the spatially dependent parameter λ that controls the diffusivity of the fractional tensor of the Partial Differential Equation (PDE) to remove both Gaussian and speckle noises. Also, we elaborate a mathematical well-posedness of the optimal control problem and we introduce a projected quasi-Newton algorithm to compute the HR image X and the associated parameter λ. The numerical experiments cope with our theoretical part and show a significant improvement of the restored HR image compared to some state of art super-resolution.

A Nonlinear Weighted Anisotropic Total Variation Regularization for Electrical Impedance Tomography

This paper proposes a nonlinear weighted anisotropic total variation (NWATV) regularization technique for electrical impedance tomography (EIT). The key idea is to incorporate the internal inhomogeneity information (e.g., edges of the detected objects) into the EIT reconstruction process, aiming to preserve the conductivity profiles (to be detected). We study the NWATV image reconstruction by employing a novel soft thresholding based reformulation included in the alternating direction method of multipliers (ADMM). To evaluate the proposed approach, numerical simulations and human EIT lung imaging are carried out. It is demonstrated that the properties of the internal inhomogeneity are well preserved and improved with the proposed regularization approach, in comparison to traditional total variation (TV) and recently proposed fidelity embedded regularization approaches. Owing to the simplicity of the proposed method, the computational cost is significantly decreased compared with the well established primal-dual algorithm. Precisely, with the proposed algorithm, we are able to alleviate the staircase effect arising in TV regularization and improve the reconstruction accuracy for FER does. To achieve similar accuracy as TV does, the computational times are reduced from 2.311s to 0.629s and 1.733s to 0.428s in 2D and 3D simulations, respectively, and the computational time is reduced from greater than 2s to less than 0.2s in the human experiment. Meanwhile, it was found that the proposed regularization method is quite robust to the measurement noise, which is one of the main uncertainties in EIT.

Game-Based Multitype Task Offloading Among Mobile-Edge-Computing-Enabled Base Stations

The widely used Internet of Things (IoT) mobile devices (MDs) require fast processing capability to handle a large volume of computing tasks. Mobile-edge computing (MEC) can augment the capability of IoT MDs through offloading their computing tasks to the MEC-enabled base station (MEC-BS) that covers them. Most of the existing research works only focus on the computation offloading problems for a single MEC-BS. However, the load of a MEC-BS will rise as the increase of the scale of the offloaded tasks, especially during rush hours, and further it will result in deterioration of system performance. In this article, we propose a game-based multitype task offloading scheme among MEC-BSs. The tasks offloaded from IoT MDs can be further offloaded among MEC-BSs to alleviate high-load MEC-BSs. Aiming at balancing the computing delays of the tasks on each MEC-BS, a noncooperative game is formulated to model the computation offloading for the tasks with different types, indicated by computation amount, data size, and delay tolerance. The existence and convergence of the Nash equilibrium of the game are first proved using the variational inequality and regularization techniques. Then, we design a distributed iterative algorithm to efficiently solve the game problem. Simulation results show the fast convergence of our algorithm. The reduction of total computing delay optimized by our scheme can reach 45%–50% on average in multiple scenarios, and the superiority of our scheme is also demonstrated in comparisons with reference schemes.

Total variation regularized Fréchet regression for metric-space valued data

Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparametric Fréchet regression, which refers to a regression setting where a response residing in a metric space is paired with a scalar predictor and the target is a conditional Fréchet mean. Specifically, we seek to approximate an unknown metric-space valued function by an estimator that minimizes the Fréchet version of least squares and at the same time has small total variation, appropriately defined for metric-space valued objects. We show that the resulting estimator is representable by a piece-wise constant function and establish the minimax convergence rate of the proposed estimator for metric data objects that reside in Hadamard spaces. We illustrate the numerical performance of the proposed method for both simulated and real data, including metric spaces of symmetric positive-definite matrices with the affine-invariant distance, of probability distributions on the real line with the Wasserstein distance, and of phylogenetic trees with the Billera–Holmes–Vogtmann metric.

Inexact Derivative-Free Optimization for Bilevel Learning

Variational regularization techniques are dominant in the field of mathematical imaging. A drawback of these techniques is that they are dependent on a number of parameters which have to be set by the user. A by-now common strategy to resolve this issue is to learn these parameters from data. While mathematically appealing, this strategy leads to a nested optimization problem (known as bilevel optimization) which is computationally very difficult to handle. It is common when solving the upper-level problem to assume access to exact solutions of the lower-level problem, which is practically infeasible. In this work we propose to solve these problems using inexact derivative-free optimization algorithms which never require exact lower-level problem solutions, but instead assume access to approximate solutions with controllable accuracy, which is achievable in practice. We prove global convergence and a worst-case complexity bound for our approach. We test our proposed framework on ROF denoising and learning MRI sampling patterns. Dynamically adjusting the lower-level accuracy yields learned parameters with similar reconstruction quality as high-accuracy evaluations but with dramatic reductions in computational work (up to 100 times faster in some cases).

Total variation multiscale estimators for linear inverse problems

AbstractEven though the statistical theory of linear inverse problems is a well-studied topic, certain relevant cases remain open. Among these is the estimation of functions of bounded variation ($BV$), meaning $L^1$ functions on a $d$-dimensional domain whose weak first derivatives are finite Radon measures. The estimation of $BV$ functions is relevant in many applications, since it involves minimal smoothness assumptions and gives simplified, interpretable cartoonized reconstructions. In this paper, we propose a novel technique for estimating $BV$ functions in an inverse problem setting and provide theoretical guaranties by showing that the proposed estimator is minimax optimal up to logarithms with respect to the $L^q$-risk, for any $q\in [1,\infty )$. This is to the best of our knowledge the first convergence result for $BV$ functions in inverse problems in dimension $d\geq 2$, and it extends the results of Donoho (1995, Appl. Comput. Harmon. Anal., 2, 101–126) in $d=1$. Furthermore, our analysis unravels a novel regime for large $q$ in which the minimax rate is slower than $n^{-1/(d+2\beta +2)}$, where $\beta$ is the degree of ill-posedness: our analysis shows that this slower rate arises from the low smoothness of $BV$ functions. The proposed estimator combines variational regularization techniques with the wavelet-vaguelette decomposition of operators.

A Machine Learning Approach for Removal of JPEG Compression Artifacts: A Survey

JPEG is a widely used image compression method. Though it is very efficient, it introduces certain artifacts and quantization noise. This paper is a detailed survey about various existing methods for the reduction of these artifacts. The paper explains each method and their advantages and drawbacks. Some of the methods mentioned are Weiner filtering, Image Optimization, Zero-masking, Local Edge regeneration, Multiple dictionary learning, Hybrid Filtering, Fuzzy filtering, Total Variation Regularization, Offset and Shift Technique, Post-processing et al. Also, a comparative study is made as to which method is suitable for which scenario.

A new iteratively total variational regularization for nonlinear inverse problems

Superior to the other well-known iterations, such as Landweber iteration, total variational regularization is a well-established method for identifying the piecewise smooth parameter and processing the image. Therefore, we propose a new iteratively total variational regularization based on total variational regularization and homotopy perturbation technique in this paper. Taking advantage of Bregman distance, we construct a nested iteration to inverse parameters. Meanwhile we prove that the error of the new method decreases monotonically, and the new iterative method with the noisy data is regular according to the discrepancy principle. In the last numerical section, compared with Landweber type total variational regularization and Runge–Kutta type regularization, numerical results of this new method indicate that this new regularization is time saving for the same accuracy, and more robust to the noisy data.

Sparse-View CT Image Recovery Using Two-Step Iterative Shrinkage-Thresholding Algorithm

We investigate an image recovery method for sparse-view computed tomography (CT) using an iterative shrinkage algorithm based on a second-order approach. The two-step iterative shrinkage-thresholding (TwIST) algorithm including a total variation regularization technique is elucidated to be more robust than other first-order methods; it enables a perfect restoration of an original image even if given only a few projection views of a parallel-beam geometry. We find that the incoherency of a projection system matrix in CT geometry sufficiently satisfies the exact reconstruction principle even when the matrix itself has a large condition number. Image reconstruction from fan-beam CT can be well carried out, but the retrieval performance is very low when compared to a parallel-beam geometry. This is considered to be due to the matrix complexity of the projection geometry. We also evaluate the image retrieval performance of the TwIST algorithm using measured projection data.

A Staggered Discontinuous Galerkin Method with Local TV Regularization for the Burgers Equation

AbstractThe staggered discontinuous Galerkin (SDG) method has been recently developed for the numerical approximation of partial differential equations. An important advantage of such methodology is that the numerical solution automatically satisfies some conservation properties which are also satisfied by the exact solution. In this paper, we will consider the numerical approximation of the inviscid Burgers equation by the SDG method. For smooth solutions, we prove that our SDG method has the properties of mass and energy conservation. It is well-known that extra care has to be taken at locations of shocks and discontinuities. In this respect, we propose a local total variation (TV) regularization technique to suppress the oscillations in the numerical solution. This TV regularization is only performed locally where oscillation is detected, and is thus very efficient. Therefore, the resulting scheme will preserve the mass and energy away from the shocks and the numerical solution is regularized locally near shocks. Detailed description of the method and numerical results are presented.

Maximum a posteriori estimates in linear inverse problems with log-concave priors are proper Bayes estimators

A frequent matter of debate in Bayesian inversion is the question of which of the two principal point-estimators, the maximum a posteriori (MAP) or the conditional mean (CM) estimate, is to be preferred. As the MAP estimate corresponds to the solution given by variational regularization techniques, this is also a constant matter of debate between the two research areas. Following a theoretical argument—the Bayes cost formalism—the CM estimate is classically preferred for being the Bayes estimator for the mean squared error cost, while the MAP estimate is classically discredited for being only asymptotically the Bayes estimator for the uniform cost function. In this article we present recent theoretical and computational observations that challenge this point of view, in particular for high-dimensional sparsity-promoting Bayesian inversion. Using Bregman distances, we present new, proper convex Bayes cost functions for which the MAP estimator is the Bayes estimator. We complement this finding with results that correct further common misconceptions about MAP estimates. In total, we aim to rehabilitate MAP estimates in linear inverse problems with log-concave priors as proper Bayes estimators.

Fast Markov chain Monte Carlo sampling for sparse Bayesian inference in high-dimensional inverse problems using L1-type priors

Sparsity has become a key concept for solving of high-dimensional inverse problems using variational regularization techniques. Recently, using similar sparsity-constraints in the Bayesian framework for inverse problems by encoding them in the prior distribution has attracted attention. Important questions about the relation between regularization theory and Bayesian inference still need to be addressed when using sparsity promoting inversion. A practical obstacle for these examinations is the lack of fast posterior sampling algorithms for sparse, high-dimensional Bayesian inversion. Accessing the full range of Bayesian inference methods requires being able to draw samples from the posterior probability distribution in a fast and efficient way. This is usually done using Markov chain Monte Carlo (MCMC) sampling algorithms. In this paper, we develop and examine a new implementation of a single component Gibbs MCMC sampler for sparse priors relying on L1-norms. We demonstrate that the efficiency of our Gibbs sampler increases when the level of sparsity or the dimension of the unknowns is increased. This property is contrary to the properties of the most commonly applied Metropolis–Hastings (MH) sampling schemes. We demonstrate that the efficiency of MH schemes for L1-type priors dramatically decreases when the level of sparsity or the dimension of the unknowns is increased. Practically, Bayesian inversion for L1-type priors using MH samplers is not feasible at all. As this is commonly believed to be an intrinsic feature of MCMC sampling, the performance of our Gibbs sampler also challenges common beliefs about the applicability of sample based Bayesian inference.

Statistical Multiresolution Estimation for Variational Imaging: With an Application in Poisson-Biophotonics

In this paper we present a spatially-adaptive method for image reconstruction that is based on the concept of statistical multiresolution estimation as introduced in Frick et al. (Electron. J. Stat. 6:231---268, 2012). It constitutes a variational regularization technique that uses an ? ?-type distance measure as data-fidelity combined with a convex cost functional. The resulting convex optimization problem is approached by a combination of an inexact alternating direction method of multipliers and Dykstra's projection algorithm. We describe a novel method for balancing data-fit and regularity that is fully automatic and allows for a sound statistical interpretation. The performance of our estimation approach is studied for various problems in imaging. Among others, this includes deconvolution problems that arise in Poisson nanoscale fluorescence microscopy.

Image-zooming technique based on Bregmanized nonlocal total variation regularization

We study the nonlocal total variation (NL-TV) regularization technique for image zooming, which exploits the spatial interactions in images. To solve the nonlinear Euler-Lagrange equation associated with the NL-TV regularization framework, we propose a split Bregman NL-TV image-zooming method. Furthermore, we give the convergence analysis of the proposed algorithm. Experimental results illustrate the effectiveness and reliability of our method by comparing it to some previous methods.

Interpretation of the dosimetric results of three uniformity regularization methods in terms of expected treatment outcome

In IMRT treatment plan optimization there are various methods that try to regularize the variation of dose nonuniformity using purely dosimetric measures. However, although these methods can help in finding a good dose distribution, they do not provide any information regarding the expected treatment outcome. When a treatment plan optimization is performed using biological measures, the final goal should be some indication about the expected tumor control or normal tissue complications, which is the primary goal of treatment planning (the association of treatment configurations and dose prescription with the treatment outcome). In this study, this issue is analyzed distinguishing the dose-oriented treatment plan optimization from the response-oriented optimization. Three different dose distributions were obtained by using a dose-based optimization technique, an EUD-based optimization without applying any technique for regularizing the nonuniformity of the dose distribution, and an EUD-based optimization using a variational regularization technique, which controls dose nonuniformity. The clinical effectiveness of the three dose distributions was investigated by calculating the response probabilities of the tumors and organs-at-risk (OARs) involved in two head and neck and prostate cancer cases. The radiobiological models used are the linear-quadratic-Poisson and the Relative Seriality models. Furthermore, the complication-free tumor control probability and the biologically effective uniform dose (D) were used for treatment plan evaluation and comparison. The radiobiological comparison shows that the EUD-based optimization using L-curve regularization gives better results than the EUD-based optimization without regularization and dose-based optimization in both clinical cases. Concluding, it appears that the applied dose nonuniformity regularization technique is expected to improve the effectiveness of the optimized IMRT dose distributions. However, more patient cases are needed to validate the statistical significance of the results and conclusions presented in this paper.

SU-GG-T-417: Interpretation of Dosimetric Results in Terms of Expected Treatment Outcome When Optimizing Treatment Plans Using Different Methods of Regularizing Dose Inhomogeneity

Purpose: Regularization techniques for determining the optimal dose distribution have been proposed because the dose distributions produced by different IMRTtreatment planningoptimization algorithms are highly non‐uniform in the target volume. In the present work, an analysis is made about the relation of the DVH gradient and the dose to the PTV and normal tissues.Method and Materials: In this study, two head & neck and prostate cancer cases treated with IMRT were employed. Three different dose distributions were obtained by using a dose‐basedoptimization technique, an EUD‐based optimization without regularization of non‐uniformity and an EUD‐based optimization using a variational regularization technique. The clinical effectiveness of the three dose distributions was investigated by using the complication‐free tumor control probability, P + and the biologically effective uniform dose.Results: In the head & neck case, for the dose‐basedoptimization, the P + value is 32.9%, the total control probability P B is 79.6% and the total complication probability P I is 49.0%. For the EUD‐based no‐reg optimization, the P + value is 56.4%, the P B value is 71.9% and the P I value is 15.5%. For the EUD‐based reg optimization, the P + value is 67.3%, the P B value is 87.4% and the P I value is 20.1%. In the prostate case, for the dose‐basedoptimization, the P + value is 94.8%, the P B value is 97.8% and the P I value is 3.0%. For the EUD‐based no‐reg optimization, the P + value is 86.0%, the P B value is 97.3% and the P I value is 11.3%. For the EUD‐based reg optimization, the P + value is 95.3%, the P B value is 98.4% and the P I value is 3.1%. Conclusion: The radiobiological comparison shows that the EUD‐based optimization with regularization gives better results than the EUD‐based optimization without regularization and dose‐basedoptimization in both clinical cases, which indicates better clinical effectiveness.

Optimization of equivalent uniform dose using the L-curve criterion

Optimization of equivalent uniform dose (EUD) in inverse planning for intensity-modulated radiation therapy (IMRT) prevents variation in radiobiological effect between different radiotherapy treatment plans, which is due to variation in the pattern of dose nonuniformity. For instance, the survival fraction of clonogens would be consistent with the prescription when the optimized EUD is equal to the prescribed EUD. One of the problems in the practical implementation of this approach is that the spatial dose distribution in EUD-based inverse planning would be underdetermined because an unlimited number of nonuniform dose distributions can be computed for a prescribed value of EUD. Together with ill-posedness of the underlying integral equation, this may significantly increase the dose nonuniformity. To optimize EUD and keep dose nonuniformity within reasonable limits, we implemented into an EUD-based objective function an additional criterion which ensures the smoothness of beam intensity functions. This approach is similar to the variational regularization technique which was previously studied for the dose-based least-squares optimization. We show that the variational regularization together with the L-curve criterion for the regularization parameter can significantly reduce dose nonuniformity in EUD-based inverse planning.

SU‐FF‐T‐383: Regularization and the L‐Curve for Inverse Planning Based On Equivalent Uniform Dose

Purpose: In the dose optimization based on equivalent uniform dose (EUD), a single dose value is assigned to PTV or organ at risk. Obviously, this optimization problem is poorly defined ‐ there is an unlimited number of nonuniform dose distributions (and, correspondingly, beam intensity functions) which will satisfy the EUD‐based objective function. Together with the ill‐posedness of the underlying integral equation, this may present a challenging mathematical problem. We propose an objective function for the EUD‐based inverse planning which can solve this problem and produce stable and accurate numerical solutions. Method and Materials: We apply to EUD‐based inverse treatment planning a variational regularization technique which was previously studied for the least squares dose optimization. The regularization parameter is found using the L‐curve method which is based on minimization of the residual norm and the smoothing norm of beam intensity functions. The optimization technique is applied to the prostate cancer treatment. Results: We have compared inverse treatment planning with the standard least squares objective function and an EUD‐based objective function. The dose distributions obtained from EUD‐based optimization are extremely nonuniform with significant hot spots in the PTV. We have applied to the EUD‐based optimization the variational regularization technique with the L‐curve method for determining an optimal value of the regularization parameter. We demonstrate that the regularization method decreases dose variance and mean dose in the PTV while keeping EUD of the same value. The regularized dose distribution is stable and uniform in the PTV. Conclusion: Inclusion of a variational regularization technique into the EUD‐based dose optimization produces robust dose distributions and beam intensity functions without deterioration of optimization accuracy.

L‐curve analysis of radiotherapy optimization problems

We attempt to select an optimal value of regularization parameter in the optimization problems for intensity-modulated radiotherapy which are solved using a variational regularization technique. We apply to inverse treatment planning the L-curve method which was developed to determine the regularization parameter in the discrete ill-posed problems. The L-curve method is based on finding the regularization parameter which minimizes the residual norm which is a measure of accuracy of fit and the solution norm which is a measure of smoothness of solution. The main idea of the L-curve method is to plot the smoothing norm as a function of the residual norm for all values of the regularization parameter. This characteristic curve has an L-shaped dependence and the optimal value of regularization parameter can be found at the "corner" of the L-curve. We plot the L-curves for the optimization problems which simulate prostate radiotherapy cancer treatment with intensity-modulated beams. Different numerical methods are applied to calculate the point of maximum curvature of the L-curves which is a criterion to locate the corner. We show that the point of maximum curvature can be located in a most robust way using a formula derived from the singular value decomposition analysis.