Image Restoration Research Articles

Joint image restoration for object detection in snowy weather

AbstractAlthough existing object detectors achieve encouraging performance of object detection and localisation under real ideal conditions, the detection performance in adverse weather conditions (snowy) is very poor and not enough to cope with the detection task in adverse weather conditions. Existing methods do not deal well with the effect of snow on the identity of object features or usually ignore or even discard potential information that can help improve the detection performance. To this end, the authors propose a novel and improved end‐to‐end object detection network joint image restoration. Specifically, in order to address the problem of identity degradation of object detection due to snow, an ingenious restoration‐detection dual branch network structure combined with a Multi‐Integrated Attention module is proposed, which can well mitigate the effect of snow on the identity of object features, thus improving the detection performance of the detector. In order to make more effective use of the features that are beneficial to the detection task, a Self‐Adaptive Feature Fusion module is introduced, which can help the network better learn the potential features that are beneficial to the detection and eliminate the effect of heavy or large local snow in the object area on detection by a special feature fusion, thus improving the network's detection capability in snowy. In addition, the authors construct a large‐scale, multi‐size snowy dataset called Synthetic and Real Snowy Dataset (SRSD), and it is a good and necessary complement and improvement to the existing snowy‐related tasks. Extensive experiments on a public snowy dataset (Snowy‐weather Datasets) and SRSD indicate that our method outperforms the existing state‐of‐the‐art object detectors.

Learning from History: Task-agnostic Model Contrastive Learning for Image Restoration

Contrastive learning has emerged as a prevailing paradigm for high-level vision tasks, which, by introducing properly negative samples, has also been exploited for low-level vision tasks to achieve a compact optimization space to account for their ill-posed nature. However, existing methods rely on manually predefined and task-oriented negatives, which often exhibit pronounced task-specific biases. To address this challenge, our paper introduces an innovative method termed 'learning from history', which dynamically generates negative samples from the target model itself. Our approach, named Model Contrastive Learning for Image Restoration (MCLIR), rejuvenates latency models as negative models, making it compatible with diverse image restoration tasks. We propose the Self-Prior guided Negative loss (SPN) to enable it. This approach significantly enhances existing models when retrained with the proposed model contrastive paradigm. The results show significant improvements in image restoration across various tasks and architectures. For example, models retrained with SPN outperform the original FFANet and DehazeFormer by 3.41 and 0.57 dB on the RESIDE indoor dataset for image dehazing. Similarly, they achieve notable improvements of 0.47 dB on SPA-Data over IDT for image deraining and 0.12 dB on Manga109 for a 4x scale super-resolution over lightweight SwinIR, respectively. Code and retrained models are available at https://github.com/Aitical/MCLIR.

Omni-Kernel Network for Image Restoration

Image restoration aims to reconstruct a high-quality image from a degraded low-quality observation. Recently, Transformer models have achieved promising performance on image restoration tasks due to their powerful ability to model long-range dependencies. However, the quadratically growing complexity with respect to the input size makes them inapplicable to practical applications. In this paper, we develop an efficient convolutional network for image restoration by enhancing multi-scale representation learning. To this end, we propose an omni-kernel module that consists of three branches, i.e., global, large, and local branches, to learn global-to-local feature representations efficiently. Specifically, the global branch achieves a global perceptive field via the dual-domain channel attention and frequency-gated mechanism. Furthermore, to provide multi-grained receptive fields, the large branch is formulated via different shapes of depth-wise convolutions with unusually large kernel sizes. Moreover, we complement local information using a point-wise depth-wise convolution. Finally, the proposed network, dubbed OKNet, is established by inserting the omni-kernel module into the bottleneck position for efficiency. Extensive experiments demonstrate that our network achieves state-of-the-art performance on 11 benchmark datasets for three representative image restoration tasks, including image dehazing, image desnowing, and image defocus deblurring. The code is available at https://github.com/c-yn/OKNet.

Deep Linear Array Pushbroom Image Restoration: A Degradation Pipeline and Jitter-Aware Restoration Network

Linear Array Pushbroom (LAP) imaging technology is widely used in the realm of remote sensing. However, images acquired through LAP always suffer from distortion and blur because of camera jitter. Traditional methods for restoring LAP images, such as algorithms estimating the point spread function (PSF), exhibit limited performance. To tackle this issue, we propose a Jitter-Aware Restoration Network (JARNet), to remove the distortion and blur in two stages. In the first stage, we formulate an Optical Flow Correction (OFC) block to refine the optical flow of the degraded LAP images, resulting in pre-corrected images where most of the distortions are alleviated. In the second stage, for further enhancement of the pre-corrected images, we integrate two jitter-aware techniques within the Spatial and Frequency Residual (SFRes) block: 1) introducing Coordinate Attention (CoA) to the SFRes block in order to capture the jitter state in orthogonal direction; 2) manipulating image features in both spatial and frequency domains to leverage local and global priors. Additionally, we develop a data synthesis pipeline, which applies Continue Dynamic Shooting Model (CDSM) to simulate realistic degradation in LAP images. Both the proposed JARNet and LAP image synthesis pipeline establish a foundation for addressing this intricate challenge. Extensive experiments demonstrate that the proposed two-stage method outperforms state-of-the-art image restoration models. Code is available at https://github.com/JHW2000/JARNet.

Rethinking Multi-Scale Representations in Deep Deraining Transformer

Existing Transformer-based image deraining methods depend mostly on fixed single-input single-output U-Net architecture. In fact, this not only neglects the potentially explicit information from multiple image scales, but also lacks the capability of exploring the complementary implicit information across different scales. In this work, we rethink the multi-scale representations and design an effective multi-input multi-output framework that constructs intra- and inter-scale hierarchical modulation to better facilitate rain removal and help image restoration. We observe that rain levels reduce dramatically in coarser image scales, thus proposing to restore rain-free results from the coarsest scale to the finest scale in image pyramid inputs, which also alleviates the difficulty of model learning. Specifically, we integrate a sparsity-compensated Transformer block and a frequency-enhanced convolutional block into a coupled representation module, in order to jointly learn the intra-scale content-aware features. To facilitate representations learned at different scales to communicate with each other, we leverage a gated fusion module to adaptively aggregate the inter-scale spatial-aware features, which are rich in correlated information of rain appearances, leading to high-quality results. Extensive experiments demonstrate that our model achieves consistent gains on five benchmarks.

Improving Diffusion-Based Image Restoration with Error Contraction and Error Correction

Generative diffusion prior captured from the off-the-shelf denoising diffusion generative model has recently attained significant interest. However, several attempts have been made to adopt diffusion models to noisy inverse problems either fail to achieve satisfactory results or require a few thousand iterations to achieve high-quality reconstructions. In this work, we propose a diffusion-based image restoration with error contraction and error correction (DiffECC) method. Two strategies are introduced to contract the restoration error in the posterior sampling process. First, we combine existing CNN-based approaches with diffusion models to ensure data consistency from the beginning. Second, to amplify the error contraction effects of the noise, a restart sampling algorithm is designed. In the error correction strategy, the estimation-correction idea is proposed on both the data term and the prior term. Solving them iteratively within the diffusion sampling framework leads to superior image generation results. Experimental results for image restoration tasks such as super-resolution (SR), Gaussian deblurring, and motion deblurring demonstrate that our approach can reconstruct high-quality images compared with state-of-the-art sampling-based diffusion models.

Correlation Matching Transformation Transformers for UHD Image Restoration

This paper proposes UHDformer, a general Transformer for Ultra-High-Definition (UHD) image restoration. UHDformer contains two learning spaces: (a) learning in high-resolution space and (b) learning in low-resolution space. The former learns multi-level high-resolution features and fuses low-high features and reconstructs the residual images, while the latter explores more representative features learning from the high-resolution ones to facilitate better restoration. To better improve feature representation in low-resolution space, we propose to build feature transformation from the high-resolution space to the low-resolution one. To that end, we propose two new modules: Dual-path Correlation Matching Transformation module (DualCMT) and Adaptive Channel Modulator (ACM). The DualCMT selects top C/r (r is greater or equal to 1 which controls the squeezing level) correlation channels from the max-pooling/mean-pooling high-resolution features to replace low-resolution ones in Transformers, which can effectively squeeze useless content to improve the feature representation in low-resolution space to facilitate better recovery. The ACM is exploited to adaptively modulate multi-level high-resolution features, enabling to provide more useful features to low-resolution space for better learning. Experimental results show that our UHDformer reduces about ninety-seven percent model sizes compared with most state-of-the-art methods while significantly improving performance under different training sets on 3 UHD image restoration tasks, including low-light image enhancement, image dehazing, and image deblurring. The source codes will be made available at https://github.com/supersupercong/UHDformer.

Unsupervised Pan-Sharpening via Mutually Guided Detail Restoration

Pan-sharpening is a task that aims to super-resolve the low-resolution multispectral (LRMS) image with the guidance of a corresponding high-resolution panchromatic (PAN) image. The key challenge in pan-sharpening is to accurately modeling the relationship between the MS and PAN images. While supervised deep learning methods are commonly employed to address this task, the unavailability of ground-truth severely limits their effectiveness. In this paper, we propose a mutually guided detail restoration method for unsupervised pan-sharpening. Specifically, we treat pan-sharpening as a blind image deblurring task, in which the blur kernel can be estimated by a CNN. Constrained by the blur kernel, the pan-sharpened image retains spectral information consistent with the LRMS image. Once the pan-sharpened image is obtained, the PAN image is blurred using a pre-defined blur operator. The pan-sharpened image, in turn, is used to guide the detail restoration of the blurred PAN image. By leveraging the mutual guidance between MS and PAN images, the pan-sharpening network can implicitly learn the spatial relationship between the two modalities. Extensive experiments show that the proposed method significantly outperforms existing unsupervised pan-sharpening methods.

Descanning: From Scanned to the Original Images with a Color Correction Diffusion Model

A significant volume of analog information, i.e., documents and images, have been digitized in the form of scanned copies for storing, sharing, and/or analyzing in the digital world. However, the quality of such contents is severely degraded by various distortions caused by printing, storing, and scanning processes in the physical world. Although restoring high-quality content from scanned copies has become an indispensable task for many products, it has not been systematically explored, and to the best of our knowledge, no public datasets are available. In this paper, we define this problem as Descanning and introduce a new high-quality and large-scale dataset named DESCAN-18K. It contains 18K pairs of original and scanned images collected in the wild containing multiple complex degradations. In order to eliminate such complex degradations, we propose a new image restoration model called DescanDiffusion consisting of a color encoder that corrects the global color degradation and a conditional denoising diffusion probabilistic model (DDPM) that removes local degradations. To further improve the generalization ability of DescanDiffusion, we also design a synthetic data generation scheme by reproducing prominent degradations in scanned images. We demonstrate that our DescanDiffusion outperforms other baselines including commercial restoration products, objectively and subjectively, via comprehensive experiments and analyses.

Enhancing RAW-to-sRGB with Decoupled Style Structure in Fourier Domain

RAW to sRGB mapping, which aims to convert RAW images from smartphones into RGB form equivalent to that of Digital Single-Lens Reflex (DSLR) cameras, has become an important area of research. However, current methods often ignore the difference between cell phone RAW images and DSLR camera RGB images, a difference that goes beyond the color matrix and extends to spatial structure due to resolution variations. Recent methods directly rebuild color mapping and spatial structure via shared deep representation, limiting optimal performance. Inspired by Image Signal Processing (ISP) pipeline, which distinguishes image restoration and enhancement, we present a novel Neural ISP framework, named FourierISP. This approach breaks the image down into style and structure within the frequency domain, allowing for independent optimization. FourierISP is comprised of three subnetworks: Phase Enhance Subnet for structural refinement, Amplitude Refine Subnet for color learning, and Color Adaptation Subnet for blending them in a smooth manner. This approach sharpens both color and structure, and extensive evaluations across varied datasets confirm that our approach realizes state-of-the-art results. Code will be available at https://github.com/alexhe101/FourierISP.

VQCNIR: Clearer Night Image Restoration with Vector-Quantized Codebook

Night photography often struggles with challenges like low light and blurring, stemming from dark environments and prolonged exposures. Current methods either disregard priors and directly fitting end-to-end networks, leading to inconsistent illumination, or rely on unreliable handcrafted priors to constrain the network, thereby bringing the greater error to the final result. We believe in the strength of data-driven high-quality priors and strive to offer a reliable and consistent prior, circumventing the restrictions of manual priors. In this paper, we propose Clearer Night Image Restoration with Vector-Quantized Codebook (VQCNIR) to achieve remarkable and consistent restoration outcomes on real-world and synthetic benchmarks. To ensure the faithful restoration of details and illumination, we propose the incorporation of two essential modules: the Adaptive Illumination Enhancement Module (AIEM) and the Deformable Bi-directional Cross-Attention (DBCA) module. The AIEM leverages the inter-channel correlation of features to dynamically maintain illumination consistency between degraded features and high-quality codebook features. Meanwhile, the DBCA module effectively integrates texture and structural information through bi-directional cross-attention and deformable convolution, resulting in enhanced fine-grained detail and structural fidelity across parallel decoders. Extensive experiments validate the remarkable benefits of VQCNIR in enhancing image quality under low-light conditions, showcasing its state-of-the-art performance on both synthetic and real-world datasets. The code is available at https://github.com/AlexZou14/VQCNIR.

EigenGAN: An SVD subspace-based learning for image generation using Conditional GAN

Generative adversarial networks (GANs) represent a significant advance in the field of deep learning for image generation problems. With their ability to generate highly realistic and diverse images, GANs are quickly becoming the preferred technique for a wide range of applications. However, GANs come with several limitations and challenges, chief among which are their collapse mode and instability during training. In this paper, we propose a novel generator GAN model that incorporates the singular value decomposition (SVD) process into the decoder module. The SVD integration allows the generator to recognize the underlying structures in the feature space, resulting in more robust and effective image generation. This is achieved by adjusting the singular values during the training process, effectively optimizing the SVD-based generator to produce images that match the ground truth. Another advantage of SVD integration is its ability to perform discriminative spatial decomposition that clearly reflects the differences between the generated features and the target features. By including SVD in the generator model, the resulting loss values are higher and less prone to gradient vanishing than conventional GANs, such as Pix2Pix. Our proposed SVD-based generator model can be integrated into any auto-encoder architecture, making it as a generic and versatile solution for various image-generation tasks. The effectiveness of the proposed SVD-based GAN in image generation has been validated in three challenging benchmarks for image restoration and visible-to-infrared image translation. Extensive experiments demonstrate the significant quantitative and qualitative improvements achieved by our SVD-based GAN compared to baseline GAN architectures. The overall system outperforms the current state of the art in all benchmarks tested.

Efficient Deweahter Mixture-of-Experts with Uncertainty-Aware Feature-Wise Linear Modulation

The Mixture-of-Experts (MoE) approach has demonstrated outstanding scalability in multi-task learning including low-level upstream tasks such as concurrent removal of multiple adverse weather effects. However, the conventional MoE architecture with parallel Feed Forward Network (FFN) experts leads to significant parameter and computational overheads that hinder its efficient deployment. In addition, the naive MoE linear router is suboptimal in assigning task-specific features to multiple experts which limits its further scalability. In this work, we propose an efficient MoE architecture with weight sharing across the experts. Inspired by the idea of linear feature modulation (FM), our architecture implicitly instantiates multiple experts via learnable activation modulations on a single shared expert block. The proposed Feature Modulated Expert (FME) serves as a building block for the novel Mixture-of-Feature-Modulation-Experts (MoFME) architecture, which can scale up the number of experts with low overhead. We further propose an Uncertainty-aware Router (UaR) to assign task-specific features to different FM modules with well-calibrated weights. This enables MoFME to effectively learn diverse expert functions for multiple tasks. The conducted experiments on the multi-deweather task show that our MoFME outperforms the state-of-the-art in the image restoration quality by 0.1-0.2 dB while saving more than 74% of parameters and 20% inference time over the conventional MoE counterpart. Experiments on the downstream segmentation and classification tasks further demonstrate the generalizability of MoFME to real open-world applications.

Decoupling Degradations with Recurrent Network for Video Restoration in Under-Display Camera

Under-display camera (UDC) systems are the foundation of full-screen display devices in which the lens mounts under the display. The pixel array of light-emitting diodes used for display diffracts and attenuates incident light, causing various degradations as the light intensity changes. Unlike general video restoration which recovers video by treating different degradation factors equally, video restoration for UDC systems is more challenging that concerns removing diverse degradation over time while preserving temporal consistency. In this paper, we introduce a novel video restoration network, called D2RNet, specifically designed for UDC systems. It employs a set of Decoupling Attention Modules (DAM) that effectively separate the various video degradation factors. More specifically, a soft mask generation function is proposed to formulate each frame into flare and haze based on the diffraction arising from incident light of different intensities, followed by the proposed flare and haze removal components that leverage long- and short-term feature learning to handle the respective degradations. Such a design offers an targeted and effective solution to eliminating various types of degradation in UDC systems. We further extend our design into multi-scale to overcome the scale-changing of degradation that often occur in long-range videos. To demonstrate the superiority of D2RNet, we propose a large-scale UDC video benchmark by gathering HDR videos and generating realistically degraded videos using the point spread function measured by a commercial UDC system. Extensive quantitative and qualitative evaluations demonstrate the superiority of D2RNet compared to other state-of-the-art video restoration and UDC image restoration methods.

Synergistic Multiscale Detail Refinement via Intrinsic Supervision for Underwater Image Enhancement

Visually restoring underwater scenes primarily involves mitigating interference from underwater media. Existing methods ignore the inherent scale-related characteristics in underwater scenes. Therefore, we present the synergistic multi-scale detail refinement via intrinsic supervision (SMDR-IS) for enhancing underwater scene details, which contain multi-stages. The low-degradation stage from the original images furnishes the original stage with multi-scale details, achieved through feature propagation using the Adaptive Selective Intrinsic Supervised Feature (ASISF) module. By using intrinsic supervision, the ASISF module can precisely control and guide feature transmission across multi-degradation stages, enhancing multi-scale detail refinement and minimizing the interference from irrelevant information in the low-degradation stage. In multi-degradation encoder-decoder framework of SMDR-IS, we introduce the Bifocal Intrinsic-Context Attention Module (BICA). Based on the intrinsic supervision principles, BICA efficiently exploits multi-scale scene information in images. BICA directs higher-resolution spaces by tapping into the insights of lower-resolution ones, underscoring the pivotal role of spatial contextual relationships in underwater image restoration. Throughout training, the inclusion of a multi-degradation loss function can enhance the network, allowing it to adeptly extract information across diverse scales. When benchmarked against state-of-the-art methods, SMDR-IS consistently showcases superior performance. Our code is available at https://github.com/zhoujingchun03/SMDR-IS

SkipDiff: Adaptive Skip Diffusion Model for High-Fidelity Perceptual Image Super-resolution

It is well-known that image quality assessment usually meets with the problem of perception-distortion (p-d) tradeoff. The existing deep image super-resolution (SR) methods either focus on high fidelity with pixel-level objectives or high perception with generative models. The emergence of diffusion model paves a fresh way for image restoration, which has the potential to offer a brand-new solution for p-d trade-off. We experimentally observed that the perceptual quality and distortion change in an opposite direction with the increase of sampling steps. In light of this property, we propose an adaptive skip diffusion model (SkipDiff), which aims to achieve high-fidelity perceptual image SR with fewer sampling steps. Specifically, it decouples the sampling procedure into coarse skip approximation and fine skip refinement stages. A coarse-grained skip diffusion is first performed as a high-fidelity prior to obtaining a latent approximation of the full diffusion. Then, a fine-grained skip diffusion is followed to further refine the latent sample for promoting perception, where the fine time steps are adaptively learned by deep reinforcement learning. Meanwhile, this approach also enables faster sampling of diffusion model through skipping the intermediate denoising process to shorten the effective steps of the computation. Extensive experimental results show that our SkipDiff achieves superior perceptual quality with plausible reconstruction accuracy and a faster sampling speed.

An efficient multi‐scale transformer for satellite image dehazing

AbstractGiven the impressive achievement of convolutional neural networks (CNNs) in grasping image priors from extensive datasets, they have been widely utilized for tasks related to image restoration. Recently, there is been significant progress in another category of neural architectures—Transformers. These models have demonstrated remarkable performance in natural language tasks and higher‐level vision applications. Despite their ability to address some of CNNs limitations, such as restricted receptive fields and adaptability issues, Transformer models often face difficulties when processing images with a high level of detail. This is because the complexity of the computations required increases significantly with the image's spatial resolution. As a result, their application to most high‐resolution image restoration tasks becomes impractical. In our research, we introduce a novel Transformer model, named DehFormer, by implementing specific design modifications in its fundamental components, for example, the multi‐head attention and feed‐forward network. Specifically, the proposed architecture consists of the three modules, that is, (a) multi‐scale feature aggregation network (MSFAN), (b) the gated‐Dconv feed‐forward network (GFFN), (c) and the multi‐Dconv head transposed attention (MDHTA). For the MDHTA module, our objective is to scrutinize the mechanics of scaled dot‐product attention through the utilization of per‐element product operations, thereby bypassing the need for matrix multiplications and operating directly in the frequency domain for enhanced efficiency. For the GFFN module, which enables only the relevant and valuable information to advance through the network hierarchy, thereby enhancing the efficiency of information flow within the model. Extensive experiments are conducted on the SateHazelk, RS‐Haze, and RSID datasets, resulting in performance that significantly exceeds that of existing methods.

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.

Focus-aware and deep restoration network with transformer for multi-focus image fusion

Multi-focus image fusion (MFIF) aims to break the limitations of physical devices and integrate different focused regions from multiple source images into a single fully focused image. However, existing fusion methods suffer from two issues when designing the fusion framework: over-reliance on post-processing operations or underestimation of their impact, leading to unsatisfactory visual results. To this end, we propose a fusion model that combines focus-aware and deep restoration. Specifically, we first employ a designed focus-aware module to integrate spatial frequency information from various levels of depth features in the source images. This integration produces an initial decision map that reflects the focus information. Then, we treat the optimization of the decision map as an image restoration task, rather than relying on commonly used post-processing operations in the MFIF field. In the restoration task, the initial decision map is passed as input along with the gradient information of the source images to the deep restoration network combined with the transformer for obtaining the final decision map. To facilitate the training of the reconstruction network, we also propose a large dataset with labels and a new set of loss functions. Extensive experimental results conclusively demonstrate that the proposed method produces fusion results of superior visual quality. This is substantiated by an assessment across eight objective metrics, which encompass normalized mutual information (1.193), nonlinear correlation information entropy (0.846), gradient-based similarity measurement (0.727), phase congruency-based measurements (0.844), structural similarity-based assessments (0.902), human perception-inspired evaluations (0.808), peak signal-to-noise ratio (74.55), and correlation coefficients (0.971). These metrics collectively affirm the outstanding fusion performance of our method. The code will be able to be found at https://github.com/govenda/FARfusion.

Forward-backward-half forward splitting algorithm with deviations

ABSTRACT In this paper, we present a forward-backward-half forward splitting algorithm with deviations for solving the structured monotone inclusion problem composed of a maximally monotone operator, a maximally monotone and Lipschitz continuous operator and a cocoercive operator. The weak and linear convergence of the proposed algorithms is established. The aim of introducing deviations is to improve the performance of the forward-backward-half forward splitting algorithm through suitable choices of the deviations. A numerical example is given to show that the two-step inertial variant of the forward-backward-half forward splitting algorithms with deviations outperforms the existing methods in Briceño-Arias and Davis [Forward-backward-half forward algorithm for solving monotone inclusions, SIAM J Optimiz, 2017;28(4):2839–2871. doi: 10.1137/17M1120099], Zong et al. [An accelerated forward-backward-half forward splitting algorithm for monotone inclusion with applications to image restoration, Optimization, 2022;73(2):401–428. doi: 10.1080/02331934.2022.2107926] and Fan et al. [Convergence of an inertial shadow Douglas–Rachford splitting algorithm for monotone inclusions, Numer Func Anal Optim, 2021;42(14):1627–1644. doi: 10.1080/01630563.2021.2001749].