NeurTV: Total Variation on the Neural Domain

Recently, we have witnessed the success of total variation (TV) for many imaging applications. However, traditional TV is defined on the original pixel domain, which limits its potential. In this work, we suggest a new TV regularization defined on the neural domain. Concretely, the discrete data is continuously and implicitly represented by a deep neural network (DNN), and we use the derivatives of DNN outputs w.r.t. input coordinates to capture local correlations of data. As compared with classical TV on the original domain, the proposed TV on the neural domain (termed NeurTV) enjoys two advantages. First, NeurTV is not limited to meshgrid but is suitable for both meshgrid and non-meshgrid data. Second, NeurTV can more exactly capture local correlations across data for any direction and any order of derivatives attributed to the implicit and continuous nature of neural domain. We theoretically reinterpret NeurTV under the variational approximation framework, which allows us to build the connection between classical TV and NeurTV and inspires us to develop variants (e.g., NeurTV with arbitrary resolution and space-variant NeurTV). Extensive numerical experiments with meshgrid data (e.g., color and hyperspectral images) and non-meshgrid data (e.g., point clouds and spatial transcriptomics) showcase the effectiveness of the proposed methods.

Revisiting Nonlocal Self-Similarity from Continuous Representation

Nonlocal self-similarity (NSS) is an important prior that has been successfully applied in multi-dimensional data processing tasks, e.g., image and video recovery. However, existing NSS-based methods are solely suitable for meshgrid data such as images and videos, but are not suitable for emerging off-meshgrid data, e.g., point cloud and climate data. In this work, we revisit the NSS from the continuous representation perspective and propose a novel Continuous Representation-based NonLocal method (termed as CRNL), which has two innovative features as compared with classical nonlocal methods. First, based on the continuous representation, our CRNL unifies the measure of self-similarity for on-meshgrid and off-meshgrid data and thus is naturally suitable for both of them. Second, the nonlocal continuous groups can be more compactly and efficiently represented by the coupled low-rank function factorization, which simultaneously exploits the similarity within each group and across different groups, while classical nonlocal methods neglect the similarity across groups. This elaborately designed coupled mechanism allows our method to enjoy favorable performance over conventional NSS methods in terms of both effectiveness and efficiency. Extensive multi-dimensional data processing experiments on-meshgrid (e.g., image inpainting and image denoising) and off-meshgrid (e.g., climate data prediction and point cloud recovery) validate the versatility, effectiveness, and efficiency of our CRNL as compared with state-of-the-art methods.

An inexact proximal majorization-minimization Algorithm for remote sensing image stripe noise removal

The stripe noise existing in remote sensing images badly degrades the visual quality and restricts the precision of data analysis. Therefore, many destriping models have been proposed in recent years. In contrast to these existing models, in this paper, we propose a nonconvex model with a DC function (i.e., the difference of convex functions) structure to remove the strip noise. To solve this model, we make use of the DC structure and apply an inexact proximal majorization-minimization algorithm with each inner subproblem solved by the alternating direction method of multipliers. It deserves mentioning that we design an implementable stopping criterion for the inner subproblem, while the convergence can still be guaranteed. Numerical experiments demonstrate the superiority of the proposed model and algorithm.

H2TF for Hyperspectral Image Denoising: Where Hierarchical Nonlinear Transform Meets Hierarchical Matrix Factorization

Recently, tensor singular value decomposition (t-SVD) has emerged as a promising tool for hyperspectral image (HSI) processing. In the t-SVD, there are two key building blocks: (i) the low-rank enhanced transform and (ii) the accompanying low-rank characterization of transformed frontal slices. Previous t-SVD methods mainly focus on the developments of (i), while neglecting the other important aspect, i.e., the exact characterization of transformed frontal slices. In this letter, we exploit the potentiality in both building blocks by leveraging the \underline{\bf H}ierarchical nonlinear transform and the \underline{\bf H}ierarchical matrix factorization to establish a new \underline{\bf T}ensor \underline{\bf F}actorization (termed as H2TF). Compared to shallow counter partners, e.g., low-rank matrix factorization or its convex surrogates, H2TF can better capture complex structures of transformed frontal slices due to its hierarchical modeling abilities. We then suggest the H2TF-based HSI denoising model and develop an alternating direction method of multipliers-based algorithm to address the resultant model. Extensive experiments validate the superiority of our method over state-of-the-art HSI denoising methods.

Low-Rank Tensor Function Representation for Multi-Dimensional Data Recovery

Since higher-order tensors are naturally suitable for representing multi-dimensional data in real-world, e.g., color images and videos, low-rank tensor representation has become one of the emerging areas in machine learning and computer vision. However, classical low-rank tensor representations can only represent data on finite meshgrid due to their intrinsical discrete nature, which hinders their potential applicability in many scenarios beyond meshgrid. To break this barrier, we propose a low-rank tensor function representation (LRTFR), which can continuously represent data beyond meshgrid with infinite resolution. Specifically, the suggested tensor function, which maps an arbitrary coordinate to the corresponding value, can continuously represent data in an infinite real space. Parallel to discrete tensors, we develop two fundamental concepts for tensor functions, i.e., the tensor function rank and low-rank tensor function factorization. We theoretically justify that both low-rank and smooth regularizations are harmoniously unified in the LRTFR, which leads to high effectiveness and efficiency for data continuous representation. Extensive multi-dimensional data recovery applications arising from image processing (image inpainting and denoising), machine learning (hyperparameter optimization), and computer graphics (point cloud upsampling) substantiate the superiority and versatility of our method as compared with state-of-the-art methods. Especially, the experiments beyond the original meshgrid resolution (hyperparameter optimization) or even beyond meshgrid (point cloud upsampling) validate the favorable performances of our method for continuous representation.

Unsupervised Deraining: Where Asymmetric Contrastive Learning Meets Self-similarity

Most of the existing learning-based deraining methods are supervisedly trained on synthetic rainy-clean pairs. The domain gap between the synthetic and real rain makes them less generalized to complex real rainy scenes. Moreover, the existing methods mainly utilize the property of the image or rain layers independently, while few of them have considered their mutually exclusive relationship. To solve above dilemma, we explore the intrinsic intra-similarity within each layer and inter-exclusiveness between two layers and propose an unsupervised non-local contrastive learning (NLCL) deraining method. The non-local self-similarity image patches as the positives are tightly pulled together, rain patches as the negatives are remarkably pushed away, and vice versa. On one hand, the intrinsic self-similarity knowledge within positive/negative samples of each layer benefits us to discover more compact representation; on the other hand, the mutually exclusive property between the two layers enriches the discriminative decomposition. Thus, the internal self-similarity within each layer (similarity) and the external exclusive relationship of the two layers (dissimilarity) serving as a generic image prior jointly facilitate us to unsupervisedly differentiate the rain from clean image. We further discover that the intrinsic dimension of the non-local image patches is generally higher than that of the rain patches. This motivates us to design an asymmetric contrastive loss to precisely model the compactness discrepancy of the two layers for better discriminative decomposition. In addition, considering that the existing real rain datasets are of low quality, either small scale or downloaded from the internet, we collect a real large-scale dataset under various rainy kinds of weather that contains high-resolution rainy images.

Uncertainty-Aware Unsupervised Image Deblurring with Deep Priors Guided by Domain Knowledge

Non-blind deblurring methods achieve decent performance under the accurate blur kernel assumption. Since the kernel error is inevitable in practice, ringing artifacts are often introduced by non-blind deblurring. Recently, semi-blind deblurring methods can handle kernel uncertainty by introducing the prior of the kernel (or induced) error. However, how to design a suitable prior for the kernel (or induced) error remains challenging. Hand-crafted prior, incorporating domain knowledge, generally performs well but may lead to poor performance when kernel (or induced) error is complex. Data-driven prior, which excessively depends on the diversity and abundance of training data, is vulnerable to out-of-distribution blurs and images. To address this challenge, we suggest a data-free deep prior for the kernel induced error (termed as residual) expressed by a customized untrained deep neural network, which allows us to flexibly adapt to different blurs and images in real scenarios. By organically integrating the respective strengths of deep priors and hand-crafted priors, we propose an unsupervised semi-blind deblurring model which recovers the latent image from the blurry image and inaccurate blur kernel. To tackle the formulated model, an efficient alternating minimization algorithm is developed. Extensive experiments demonstrate the superiority of the proposed method to both data-driven prior and hand-crafted prior based methods in terms of the image quality and the robustness to the kernel error.

Unsupervised Image Deraining: Optimization Model Driven Deep CNN

The deep convolutional neural network has achieved significant progress for single image rain streak removal. However, most of the data-driven learning methods are full-supervised or semi-supervised, unexpectedly suffering from significant performance drops when dealing with real rain. These data-driven learning methods are representative yet generalize poor for real rain. The opposite holds true for the model-driven unsupervised optimization methods. To overcome these problems, we propose a unified unsupervised learning framework which inherits the generalization and representation merits for real rain removal. Specifically, we first discover a simple yet important domain knowledge that directional rain streak is anisotropic while the natural clean image is isotropic, and formulate the structural discrepancy into the energy function of the optimization model. Consequently, we design an optimization model-driven deep CNN in which the unsupervised loss function of the optimization model is enforced on the proposed network for better generalization. In addition, the architecture of the network mimics the main role of the optimization models with better feature representation. On one hand, we take advantage of the deep network to improve the representation. On the other hand, we utilize the unsupervised loss of the optimization model for better generalization. Overall, the unsupervised learning framework achieves good generalization and representation: unsupervised training (loss) with only a few real rainy images (input) and physical meaning network (architecture). Extensive experiments on synthetic and real-world rain datasets show the superiority of the proposed method.

Non-Local Robust Quaternion Matrix Completion for Large-Scale Color Images and Videos Inpainting

The image nonlocal self-similarity (NSS) prior refers to the fact that a local patch often has many nonlocal similar patches to it across the image. In this paper we apply such NSS prior to enhance the robust quaternion matrix completion (QMC) method and significantly improve the inpainting performance. A patch group based NSS prior learning scheme is proposed to learn explicit NSS models from natural color images. The NSS-based QMC algorithm computes an optimal low-rank approximation to the high-rank color image, resulting in high PSNR and SSIM measures and particularly the better visual quality. A new joint NSS-base QMC method is also presented to solve the color video inpainting problem based quaternion tensor representation. The numerical experiments on large-scale color images and videos indicate the advantages of NSS-based QMC over the state-of-the-art methods.