Abstract. In this paper, we consider a low-rank tensor recovery problem. Based on the tensor singular value decomposition (t-SVD), we propose the ratio of the tensor nuclear norm and the tensor Frobenius norm (TNF) as a novel nonconvex surrogate of tensor’s tubal rank. The rationale of the proposed model for enforcing a low-rank structure is analyzed as its theoretical properties. Specifically, we introduce a null space property (NSP) type condition, under which a low-rank tensor is a local minimum for the proposed TNF recovery model. Numerically, we consider a low-rank tensor completion problem as a specific application of tensor recovery and employ the alternating direction method of multipliers (ADMM) to secure a model solution with guaranteed subsequential convergence under mild conditions. Extensive experiments demonstrate the superiority of our proposed model over state-of-the-art methods.

Unmixing is a crucial technique in analyzing hyperspectral imaging (HSI) data, which involves identifying the endmembers present in the data and estimating their abundance maps. Due to some practical constraints in atmospheric environment, HSI data is usually non-uniformly distributed along the spectral domain, which brings incomplete spectral information in the hyperspectral unmixing. To overcome this issue, we propose in this paper nonnegative matrix functional factorization (NMFF) which is an extension of classical nonnegative matrix factorization (NMF) for hyperspectral unmixing. In particular, we present a novel functional factorization model by incorporating the implicit neural representations (INR) to learn about endmembers. Our method effectively characterizes endmembers by learning a continuous representation through INR with positional encoding, capturing the non-uniform distribution of spectral wavelengths. This distinct approach streamlines NMFF’s iterative process for abundance extraction, bypassing the conventionally complex and cumbersome processing. When tested on various datasets, our hyperspectral unmixing approach consistently outperforms established techniques, showcasing the enhanced capabilities of our proposed model.

Integrating a low-resolution hyperspectral image and a high-resolution multispectral image is widely acknowledged as an effective approach for generating a high-resolution hyperspectral image. Recent studies have highlighted the nuclear norm as an efficient method for this problem through the utilization of low-rankness. However, the standard nuclear norm has a limitation due to treating singular values equally. To address this issue, we have incorporated the concept of the weighted nuclear norm from the image denoising problem into hyperspectral image fusion, ensuring the retention of crucial data components. Furthermore, we propose a unified framework which integrates the weighted nuclear norm, a sparse prior, and total variation regularization. This framework utilizes the ℓ1norm of coefficients to promote spatial-spectral sparsity in the fused images, while total variation is employed to preserve the spatial piecewise smooth structure. To efficiently solve the proposed model, we have designed an alternating direction method of multipliers. The experimental results show that our proposed approach surpasses the state-of-the-art methods.

Despite recent promising results on semi-supervised learning (SSL), data imbalance, particularly in the unlabeled dataset, could significantly impact the training performance of a SSL algorithm if there is a mismatch between the expected and actual class distributions. The efforts on how to construct a robust SSL framework that can effectively learn from datasets with unknown distributions remain limited. We first investigate the feasibility of adding weights to the consistency loss and then we verify the necessity of smoothed weighting schemes. Based on this study, we propose a self-adaptive algorithm, named Smoothed Adaptive Weighting (SAW). SAW is designed to enhance the robustness of SSL by estimating the learning difficulty of each class and synthesizing the weights in the consistency loss based on such estimation. We show that SAW can complement recent consistency-based SSL algorithms and improve their reliability on various datasets including three standard datasets and one gigapixel medical imaging application without making any assumptions about the distribution of the unlabeled set.

In this paper, we consider minimizing the $L_1/L_2$ term on the gradient for a limited-angle scanning problem in computed tomography (CT) reconstruction. We design a specific splitting framework for an unconstrained optimization model so that the alternating direction method of multipliers (ADMM) has guaranteed convergence under certain conditions. In addition, we incorporate a box constraint that is reasonable for imaging applications, and the convergence for the additional box constraint can also be established. Numerical results on both synthetic and experimental datasets demonstrate the effectiveness and efficiency of our proposed approach, showing significant improvements over the state-of-the-art methods in the limited-angle CT reconstruction.

We consider the high-resolution imaging problem of 3-dimensional (3D) point source image recovery from 2-dimensional data using a method based on point spread function (PSF) engineering. The method involves a new technique, recently proposed by Prasad, based on the use of a rotating PSF with a single lobe to obtain depth from defocus. The amount of rotation of the PSF encodes the depth position of the point source. Applications include high-resolution single molecule localization microscopy as well as the problem addressed in this paper on localization of space debris using a space-based telescope. The localization problem is discretized on a cubical lattice where the coordinates of nonzero entries represent the 3D locations and the values of these entries the fluxes of the point sources. Finding the locations and fluxes of the point sources is a large-scale sparse 3D inverse problem. A new non-convex regularization method with a data-fitting term based on Kullback--Leibler (KL) divergence is proposed for 3D localization for the Poisson noise model. In addition, we propose a new scheme of estimation of the source fluxes from the KL data-fitting term. Numerical experiments illustrate the efficiency and stability of the algorithms that are trained on a random subset of image data before being applied to other images. Our 3D localization algorithms can readily be applied to other kinds of depth-encoding PSFs as well.