Data Fidelity Term Research Articles

High-resolution lensless holographic microscopy using a physics-aware deep network.

Lensless digital inline holographic microscopy (LDIHM) is an emerging quantitative phase imaging modality that uses advanced computational methods for phase retrieval from the interference pattern. The existing end-to-end deep networks require a large training dataset with sufficient diversity to achieve high-fidelity hologram reconstruction. To mitigate this data requirement problem, physics-aware deep networks integrate the physics of holography in the loss function to reconstruct complex objects without needing prior training. However, the data fidelity term measures the data consistency with a single low-resolution hologram without any external regularization, which results in a low performance on complex biological data. We aim to mitigate the challenges with trained and physics-aware untrained deep networks separately and combine the benefits of both methods for high-resolution phase recovery from a single low-resolution hologram in LDIHM. We propose a hybrid deep framework (HDPhysNet) using a plug-and-play method that blends the benefits of trained and untrained deep models for phase recovery in LDIHM. The high-resolution phase is generated by a pre-trained high-definition generative adversarial network (HDGAN) from a single low-resolution hologram. The generated phase is then plugged into the loss function of a physics-aware untrained deep network to regulate the complex object reconstruction process. Simulation results show that the SSIM of the proposed method is increased by 0.07 over the trained and 0.04 over the untrained deep networks. The average phase-SNR is elevated by 8.2dB over trained deep models and 9.8dB over untrained deep networks on the experimental biological cells (cervical cells and red blood cells). We showed improved performance of the HDPhysNet against the unknown perturbation in the imaging parameters such as the propagation distance, the wavelength of the illuminating source, and the imaging sample compared with the trained network (HDGAN). LDIHM, combined with HDPhysNet, is a portable and technology-driven microscopy best suited for point-of-care cytology applications.

Multiobjective optimization guided by image quality index for limited-angle CT image reconstruction.

Due to the incomplete projection data collected by limited-angle computed tomography (CT), severe artifacts are present in the reconstructed image. Classical regularization methods such as total variation (TV) minimization, ℓ0 minimization, are unable to suppress artifacts at the edges perfectly. Most existing regularization methods are single-objective optimization approaches, stemming from scalarization methods for multiobjective optimization problems (MOP). To further suppress the artifacts and effectively preserve the edge structures of the reconstructed image. This study presents a multiobjective optimization model incorporates both data fidelity term and ℓ0-norm of the image gradient as objective functions. It employs an iterative approach different from traditional scalarization methods, using the maximization of structural similarity (SSIM) values to guide optimization rather than minimizing the objective function.The iterative method involves two steps, firstly, simultaneous algebraic reconstruction technique (SART) optimizes the data fidelity term using SSIM and the Simulated Annealing (SA) algorithm for guidance. The degradation solution is accepted in the form of probability, and guided image filtering (GIF) is introduced to further preserve the image edge when the degradation solution is rejected. Secondly, the result from the first step is integrated into the second objective function as a constraint, we use ℓ0 minimization to optimize ℓ0-norm of the image gradient, and the SSIM, SA algorithm and GIF are introduced to guide optimization process by improving SSIM value like the first step. With visual inspection, the peak signal-to-noise ratio (PSNR), root mean square error (RMSE), and SSIM values indicate that our approach outperforms other traditional methods. The experiments demonstrate the effectiveness of our method and its superiority over other classical methods in artifact suppression and edge detail restoration.

Pig-DTpV: A prior information guided directional TpV algorithm for orthogonal translation computed laminography

The local scanning orthogonal translation computed laminography (OTCL) has great potential for tiny fault detection of laminated structure thin-plate parts. However, it generates limited-angle and truncated projection data, which result in aliasing and truncation artifacts in the reconstructed images. The directional total variation (DTV) algorithm has been demonstrated to achieve highly accurate reconstructed images in limited-angle computed tomography (CT). However, its application in local scanning OTCL has not been explored. Based on this algorithm, we introduce the lp norm to better suppress artifacts, and prior information to further constrain the reconstructed image. Thus, we propose a prior information guided directional total p-variation (DTpV) algorithm (Pig-DTpV). The Pig-DTpV model is a constrained non-convex optimization model. The constraint term are the six DTpV terms, whereas the objective term is the data fidelity term. Then, we use the iterative reweighting strategy and the Chambolle–Pock (CP) algorithm to solve the model. The Pig-DTpV reconstruction algorithm’s performance is compared with other algorithms such as simultaneous algebraic reconstruction technique (SART), TV, reweighted anisotropic-TV (RwATV), and DTV in simulation and real data experiments. The experiment results demonstrate that the Pig-DTpV algorithm can reduce truncation and aliasing artifacts and enhance the quality of reconstructed images.

An Edge-Preserving Regularization Model for the Demosaicing of Noisy Color Images

This paper proposes an edge-preserving regularization technique to solve the color image demosaicing problem in the realistic case of noisy data. We enforce intra-channel local smoothness of the intensity (low-frequency components) and inter-channel local similarity of the depth of object borders and textures (high-frequency components). Discontinuities of both the low-frequency and high-frequency components are accounted for implicitly, i.e., through suitable functions of the proper derivatives. For the treatment of even the finest image details, derivatives of first, second, and third orders are considered. The solution to the demosaicing problem is defined as the minimizer of an energy function, accounting for all these constraints plus a data fidelity term. This non-convex energy is minimized via an iterative deterministic algorithm, applied to a family of approximating functions, each implicitly referring to geometrically consistent image edges. Our method is general because it does not refer to any specific color filter array. However, to allow quantitative comparisons with other published results, we tested it in the case of the Bayer CFA and on the Kodak 24-image dataset, the McMaster (IMAX) 18-image dataset, the Microsoft Demosaicing Canon 57-image dataset, and the Microsoft Demosaicing Panasonic 500-image dataset. The comparisons with some of the most recent demosaicing algorithms show the good performance of our method in both the noiseless and noisy cases.

Joint diffusion: mutual consistency-driven diffusion model for PET-MRI co-reconstruction

Objective. Positron Emission Tomography and Magnetic Resonance Imaging (PET-MRI) systems can obtain functional and anatomical scans. But PET suffers from a low signal-to-noise ratio, while MRI are time-consuming. To address time-consuming, an effective strategy involves reducing k-space data collection, albeit at the cost of lowering image quality. This study aims to leverage the inherent complementarity within PET-MRI data to enhance the image quality of PET-MRI. Approach. A novel PET-MRI joint reconstruction model, termed MC-Diffusion, is proposed in the Bayesian framework. The joint reconstruction problem is transformed into a joint regularization problem, where data fidelity terms of PET and MRI are expressed independently. The regular term, the derivative of the logarithm of the joint probability distribution of PET and MRI, employs a joint score-based diffusion model for learning. The diffusion model involves the forward diffusion process and the reverse diffusion process. The forward diffusion process adds noise to transform a complex joint data distribution into a known joint prior distribution for PET and MRI simultaneously, resembling a denoiser. The reverse diffusion process removes noise using a denoiser to revert the joint prior distribution to the original joint data distribution, effectively utilizing joint probability distribution to describe the correlations of PET and MRI for improved quality of joint reconstruction. Main results. Qualitative and quantitative improvements are observed with the MC-Diffusion model. Comparative analysis against LPLS and Joint ISAT-net on the ADNI dataset demonstrates superior performance by exploiting complementary information between PET and MRI. The MC-Diffusion model effectively enhances the quality of PET and MRI images. Significance. This study employs the MC-Diffusion model to enhance the quality of PET-MRI images by integrating the fundamental principles of PET and MRI modalities and leveraging their inherent complementarity. Furthermore, utilizing the diffusion model to learn the joint probability distribution of PET and MRI, thereby elucidating their latent correlation, facilitates a more profound comprehension of the priors obtained through deep learning, contrasting with black-box prior or artificially constructed structural similarities.

3D Poissonian image deblurring via patch-based tensor logarithmic Schatten-p minimization

In medical and biological image processing, multi-dimensional images are often corrupted by blur and Poisson noise. In this paper, we first propose a new tensor logarithmic Schatten-p (t-log-Sp ) low-rank measure and a tensor iteratively reweighted Schatten-p minimization algorithm for minimizing such measure. Furthermore, we adopt this low-rank measure to regularize the non-local tensors formed by similar 3D image patches and develop a patch-based non-local low-rank model. The data fidelity term of the model characterizes the Poisson noise distribution and blur operator. The optimization model is further solved by an alternating minimization technique combined with variable splitting. Experimental results tested on 3D fluorescence microscope images show that the proposed patch-based tensor logarithmic Schatten-p minimization method outperforms state-of-the-art methods in terms of image evaluation metrics and visual quality.

A subject-specific unsupervised deep learning method for quantitative susceptibility mapping using implicit neural representation

Quantitative susceptibility mapping (QSM) is an MRI-based technique that estimates the underlying tissue magnetic susceptibility based on phase signal. Deep learning (DL)-based methods have shown promise in handling the challenging ill-posed inverse problem for QSM reconstruction. However, they require extensive paired training data that are typically unavailable and suffer from generalization problems. Recent model-incorporated DL approaches also overlook the non-local effect of the tissue phase in applying the source-to-field forward model due to patch-based training constraint, resulting in a discrepancy between the prediction and measurement and subsequently suboptimal QSM reconstruction. This study proposes an unsupervised and subject-specific DL method for QSM reconstruction based on implicit neural representation (INR), referred to as INR-QSM. INR has emerged as a powerful framework for learning a high-quality continuous representation of the signal (image) by exploiting its internal information without training labels. In INR-QSM, the desired susceptibility map is represented as a continuous function of the spatial coordinates, parameterized by a fully-connected neural network. The weights are learned by minimizing a loss function that includes a data fidelity term incorporated by the physical model and regularization terms. Additionally, a novel phase compensation strategy is proposed for the first time to account for the non-local effect of tissue phase in data consistency calculation to make the physical model more accurate. Our experiments show that INR-QSM outperforms traditional established QSM reconstruction methods and the compared unsupervised DL method both qualitatively and quantitatively, and is competitive against supervised DL methods under data perturbations.

Deep-unrolling architecture for image-domain least-squares migration

Deep-image prior (DIP) is a novel approach to solving ill-posed inverse problems whose solution is parameterized with an untrained deep neural network and cascaded with the forward modeling operator. A key component to the success of such a method is represented by the choice of the network architecture, which must act as a natural prior to the inverse problem at hand and provide a strong inductive bias toward the desired solution. Inspired by the close link between neural networks and iterative algorithms in classical optimization, we apply an unrolled version of the gradient descent (GD) algorithm as our DIP network architecture, denoted as the deep-unrolling (DU) architecture. Each layer of the unrolled network is comprised of two parts: the first part corresponds to the GD step of the data-fidelity term, whereas the second part, formed by a six-layer convolutional neural network (CNN), plays the role of a regularizer in the associated objective function. The developed DU architecture is applied to the problem of image-domain least-squares migration (IDLSM) to invert migrated seismic images for their underlying reflectivity and is denoted as DU-IDLSM. As such, the DU architecture parameterizes the reflectivity, and the input of each layer of the unrolled network is the reflectivity at the previous layer. Similar to the classical DIP approach, the parameters of the DU architecture are optimized in an unsupervised fashion by minimizing the data misfit function itself. Through experiments with a part of the Sigsbee2A model and a marine field data set, we test the effectiveness of the DU-IDLSM approach and highlight two key benefits. First, the DU architecture can effectively regularize the inversion process, resulting in reflectivity estimates with fewer artifacts and higher image resolution than those produced by conventional IDLSM approaches. Second, we indicate that by including dropout layers in the CNN architecture, DU-IDLSM can produce a qualitative measure of the uncertainty associated with the least-squares migration process.

Boosting MR Image Impulse Noise Removal With Overlapping Group Sparse Fractional-Order Total Variation and Minimax Concave Penalty

The acquisition and transmission of magnetic resonance images are susceptible to noise, particularly impulse noise. Although the method based on the ℓ0-norm and overlapping group sparse total variation (ℓ0-OGSTV) is effective for impulse noise image restoration, it can only mitigate the staircase artifacts to a certain extent. To boost the impulse noise removal performance of ℓ0-OGSTV, we propose a new restoration model that consists of two terms. Specifically, in the first term, we keep using the ℓ0-norm as the data fidelity term to eliminate impulse noise. In the second term, we first introduce an overlapping group sparsity fractional-order total variation regularizer to eliminate staircase artifacts while preserving structural information. Then, we adopt the minimax-concave penalty to further accurately estimate the image edges. Finally, we employ an alternate direction method of multipliers to solve the proposed optimization model. Clinical experiments demonstrate its effectiveness in denoising medical images.

A structured L-BFGS method and its application to inverse problems

Many inverse problems are phrased as optimization problems in which the objective function is the sum of a data-fidelity term and a regularization. Often, the Hessian of the fidelity term is computationally unavailable while the Hessian of the regularizer allows for cheap matrix-vector products. In this paper, we study an L-BFGS method that takes advantage of this structure. We show that the method converges globally without convexity assumptions and that the convergence is linear under a Kurdyka–Łojasiewicz-type inequality. In addition, we prove linear convergence to cluster points near which the objective function is strongly convex. To the best of our knowledge, this is the first time that linear convergence of an L-BFGS method is established in a non-convex setting. The convergence analysis is carried out in infinite dimensional Hilbert space, which is appropriate for inverse problems but has not been done before. Numerical results show that the new method outperforms other structured L-BFGS methods and classical L-BFGS on non-convex real-life problems from medical image registration. It also compares favorably with classical L-BFGS on ill-conditioned quadratic model problems. An implementation of the method is freely available.

Segmentation-Free Velocity Field Super-Resolution on 4D Flow MRI.

Blood flow observation is of high interest in cardiovascular disease diagnosis and assessment. For this purpose, 2D Phase-Contrast MRI is widely used in the clinical routine. 4D flow MRI sequences, which dynamically image the anatomic shape and velocity vectors within a region of interest, are promising but rarely used due to their low resolution and signal-to-noise ratio (SNR). Computational fluid dynamics (CFD) simulation is considered as a reference solution for resolution enhancement. However, its precision relies on image segmentation and a clinical expertise for the definition of the vessel borders. The main contribution of this paper is a Segmentation-Free Super-Resolution (SFSR) algorithm. Based on inverse problem methodology, SFSR relies on minimizing a compound criterion involving: a data fidelity term, a fluid mechanics term, and a spatial velocity smoothing term. The proposed algorithm is evaluated with respect to state-of-the-art solutions, in terms of quantification error and computation time, on a synthetic 3D dataset with several noise levels, resulting in a 59% RMSE improvement and factor 2 super-resolution with a noise standard deviation of 5% of the Venc. Finally, its performance is demonstrated, with a scale factor of 2 and 3, on a pulsed flow phantom dataset with more complex patterns. The application on in-vivo were achievable within the 10 min. computation time.

Low-dose CT reconstruction using dataset-free learning.

Low-Dose computer tomography (LDCT) is an ideal alternative to reduce radiation risk in clinical applications. Although supervised-deep-learning-based reconstruction methods have demonstrated superior performance compared to conventional model-driven reconstruction algorithms, they require collecting massive pairs of low-dose and norm-dose CT images for neural network training, which limits their practical application in LDCT imaging. In this paper, we propose an unsupervised and training data-free learning reconstruction method for LDCT imaging that avoids the requirement for training data. The proposed method is a post-processing technique that aims to enhance the initial low-quality reconstruction results, and it reconstructs the high-quality images by neural work training that minimizes the ℓ1-norm distance between the CT measurements and their corresponding simulated sinogram data, as well as the total variation (TV) value of the reconstructed image. Moreover, the proposed method does not require to set the weights for both the data fidelity term and the plenty term. Experimental results on the AAPM challenge data and LoDoPab-CT data demonstrate that the proposed method is able to effectively suppress the noise and preserve the tiny structures. Also, these results demonstrate the rapid convergence and low computational cost of the proposed method. The source code is available at https://github.com/linfengyu77/IRLDCT.

Physics-Informed DeepMRI: k-Space Interpolation Meets Heat Diffusion.

Recently, diffusion models have shown considerable promise for MRI reconstruction. However, extensive experimentation has revealed that these models are prone to generating artifacts due to the inherent randomness involved in generating images from pure noise. To achieve more controlled image reconstruction, we reexamine the concept of interpolatable physical priors in k-space data, focusing specifically on the interpolation of high-frequency (HF) k-space data from low-frequency (LF) k-space data. Broadly, this insight drives a shift in the generation paradigm from random noise to a more deterministic approach grounded in the existing LF k-space data. Building on this, we first establish a relationship between the interpolation of HF k-space data from LF k-space data and the reverse heat diffusion process, providing a fundamental framework for designing diffusion models that generate missing HF data. To further improve reconstruction accuracy, we integrate a traditional physics-informed k-space interpolation model into our diffusion framework as a data fidelity term. Experimental validation using publicly available datasets demonstrates that our approach significantly surpasses traditional k-space interpolation methods, deep learning-based k-space interpolation techniques, and conventional diffusion models, particularly in HF regions. Finally, we assess the generalization performance of our model across various out-of-distribution datasets. Our code are available at https://github.com/ZhuoxuCui/Heat-Diffusion.

Efficient learning methods for large-scale optimal inversion design

In this work, we investigate various approaches that use learning from training data to solve inverse problems, following a bi-level learning approach. We consider a general framework for optimal inversion design, where training data can be used to learn optimal regularization parameters, data fidelity terms, and regularizers, thereby resulting in superior variational regularization methods. In particular, we describe methods to learn optimal $ p $ and $ q $ norms for $ {\rm L}^p-{\rm L}^q $ regularization and methods to learn optimal parameters for regularization matrices defined by covariance kernels. We exploit efficient algorithms based on Krylov projection methods for solving the regularized problems, both at training and validation stages, making these methods well-suited for large-scale problems. Our experiments show that the learned regularization methods perform well even when there is some inexactness in the forward operator, resulting in a mixture of model and measurement error.

Bi-Constraints Diffusion: A Conditional Diffusion Model with Degradation Guidance for Metal Artifact Reduction.

In recent years, score-based diffusion models have emerged as effective tools for estimating score functions from empirical data distributions, particularly in integrating implicit priors with inverse problems like CT reconstruction. However, score-based diffusion models are rarely explored in challenging tasks such as metal artifact reduction (MAR). In this paper, we introduce the BiConstraints Diffusion Model for Metal Artifact Reduction (BCDMAR), an innovative approach that enhances iterative reconstruction with a conditional diffusion model for MAR. This method employs a metal artifact degradation operator in place of the traditional metal-excluded projection operator in the data-fidelity term, thereby preserving structure details around metal regions. However, scorebased diffusion models tend to be susceptible to grayscale shifts and unreliable structures, making it challenging to reach an optimal solution. To address this, we utilize a precorrected image as a prior constraint, guiding the generation of the score-based diffusion model. By iteratively applying the score-based diffusion model and the data-fidelity step in each sampling iteration, BCDMAR effectively maintains reliable tissue representation around metal regions and produces highly consistent structures in non-metal regions. Through extensive experiments focused on metal artifact reduction tasks, BCDMAR demonstrates superior performance over other state-of-the-art unsupervised and supervised methods, both quantitatively and in terms of visual results.

Group sparse representation and saturation-value total variation based color image denoising under multiplicative noise

<abstract><p>In this article, we propose a novel group-based sparse representation (GSR) model for restoring color images in the presence of multiplicative noise. This model consists of a convex data-fidelity term, and two regularizations including GSR and saturation-value-based total variation (SVTV). The data-fidelity term is suitable for handling heavy multiplicative noise. GSR enables the retention of textures and details while sufficiently removing noise in smooth regions without producing the staircase artifacts engendered by total variation-based models. Furthermore, we introduce a multi-color channel-based GSR that involves coupling between three color channels. This avoids the generation of color artifacts caused by decoupled color channel-based methods. SVTV further improves the visual quality of restored images by diminishing certain artifacts induced by patch-based methods. To solve the proposed nonconvex model and its subproblem, we exploit the alternating direction method of multipliers, which contributes to an efficient iterative algorithm. Numerical results demonstrate the outstanding performance of the proposed model compared to other existing models regarding visual aspect and image quality evaluation values.</p></abstract>

MULTIPARENT FRACTAL IMAGE CODING-BASED METHODS FOR SALT-AND-PEPPER NOISE REMOVAL

Salt-and-pepper noise consists of outlier pixel values which significantly impair image structure and quality. Multiparent fractal image coding (MFIC) methods substantially exploit image redundancy by utilizing multiple domain blocks to approximate the range block, partially compensating for the information loss caused by noise. Motivated by this, we propose two novel image restoration methods based on MFIC to remove salt-and-pepper noise. The first method integrates Huber M-estimation into MFIC, resulting in an improved anti-salt-and-pepper noise robust fractal coding approach. The second method incorporates MFIC into a total variation (TV) regularization model, including a data fidelity term, an MFIC term and a TV regularization term. An alternative iterative method based on proximity operator is developed to effectively solve the proposed model. Experimental results demonstrate that these two proposed approaches achieve significantly enhanced performance compared to traditional fractal coding methods.

Iterative Image Reconstruction with Under-Sampled Data Assisted by a Neural Network.

Image reconstruction with under-sampled data is usually achieved by an iterative algorithm, which minimizes an objective function. The objective function commonly contains a data fidelity term and one or more Bayesian terms. A popular Bayesian term is the total variation (TV) norm of the image. This paper suggests an addition Bayesian term that is generated by a neural network. This neural network is essentially a classifier. This classifier can recognize the artifacts caused by the incomplete data. This classifier is trained by patient images reconstructed by complete and incomplete data sets. This newly introduced Bayesian term is referred to as the CNN score, which is a real number in (-∞, ∞). Patient studies show the good correlation between the CNN score and the severeness of the artifacts due to the incomplete measurements. A neural network can extract features from images that are suffering from incomplete measurements and convert the features into a CNN score. An iterative image reconstruction algorithm can be developed to minimize this CNN score to suppress the artifacts in the reconstructed image.

Alternating direction method of multipliers for displacement estimation in ultrasound strain elastography.

Ultrasound strain imaging, which delineates mechanical properties to detect tissue abnormalities, involves estimating the time delay between two radio-frequency (RF) frames collected before and after tissue deformation. The existing regularized optimization-based time-delay estimation (TDE) techniques suffer from at least one of the following drawbacks: (1) The regularizer is not aligned with the tissue deformation physics due to taking only the first-order displacement derivative into account; (2) The -norm of the displacement derivatives, which oversmooths the estimated time-delay, is utilized as the regularizer; (3) The modulus function defined mathematically should be approximated by a smooth function to facilitate the optimization of -norm. Our purpose is to develop a novel TDE technique that resolves the aforementioned shortcomings of the existing algorithms. Herein, we propose employing the alternating direction method of multipliers (ADMM) for optimizing a novel cost function consisting of -norm data fidelity term and -norm first- and second-order spatial continuity terms. ADMM empowers the proposed algorithm to use different techniques for optimizing different parts of the cost function and obtain high-contrast strain images with smooth backgrounds and sharp boundaries. We name our technique ADMM for totaL variaTion RegUlarIzation in ultrasound STrain imaging (ALTRUIST). ALTRUIST's efficacy is quantified using absolute error (AE), Structural SIMilarity (SSIM), signal-to-noise ratio (SNR), contrast-to-noise ratio (CNR), and strain ratio (SR) with respect to GLUE, OVERWIND, and -SOUL, three recently published energy-based techniques, and UMEN-Net, a state-of-the-art deep learning-based algorithm. Analysis of variance (ANOVA)-led multiple comparison tests and paired -tests at overall significance level were conducted to assess the statistical significance of our findings. The Bonferroni correction was taken into account in all statistical tests. Two simulated layer phantoms, three simulated resolution phantoms, one hard-inclusion simulated phantom, one multi-inclusion simulated phantom, one experimental breast phantom, and three in vivo liver cancer datasets have been used for validation experiments. We have published the ALTRUIST code at http://code.sonography.ai. ALTRUIST substantially outperforms the four state-of-the-art benchmarks in all validation experiments, both qualitatively and quantitatively. ALTRUIST yields up to , , and SNR improvements and , , and CNR improvements over -SOUL, its closest competitor, for simulated, phantom, and in vivo liver cancer datasets, respectively, where the asterisk (*) indicates statistical significance. In addition, ANOVA-led multiple comparison tests and paired -tests indicate that ALTRUIST generally achieves statistically significant improvements over GLUE, UMEN-Net, OVERWIND, and -SOUL in terms of AE, SSIM map, SNR, andCNR. A novel ultrasonic displacement tracking algorithm named ALTRUIST has been developed. The principal novelty of ALTRUIST is incorporating ADMM for optimizing an -norm regularization-based cost function. ALTRUIST exhibits promising performance in simulation, phantom, and in vivoexperiments.

A duality approach to regularized learning problems in Banach spaces

Regularized learning problems in Banach spaces, which often minimize the sum of a data fidelity term in one Banach norm and a regularization term in another Banach norm, is challenging to solve. We construct a direct sum space based on the Banach spaces for the fidelity term and the regularization term, and recast the objective function as the norm of a quotient space of the direct sum space. We then express the original regularized problem as an optimization problem in the dual space of the direct sum space. It is to find the maximum of a linear function on a convex polytope, which may be solved by linear programming. A solution of the original problem is then obtained by using related extremal properties of norming functionals from a solution of the dual problem. Numerical experiments demonstrate that the proposed duality approach is effective for solving the regularization learning problems.