Abstract. Hyperspectral images (HSIs) are often contaminated by a mixture of noise such as Gaussian noise, dead lines, stripes, and so on. In this paper, we propose a multiscale low-rank tensor regularized [Formula: see text] (MLTL2p) approach for HSI denoising and destriping, which consists of an orthogonal constrained minimization model and an iterative algorithm with convergence guarantees. The model of the proposed MLTL2p approach is built based on a new sparsity-enhanced Multiscale Low-rank Tensor regularization and a tensor [Formula: see text] norm with [Formula: see text]. The multiscale low-rank regularization for HSI denoising utilizes the global and local spectral correlation as well as the spatial nonlocal self-similarity priors of HSIs. The corresponding low-rank constraints are formulated based on independent higher-order singular value decomposition with sparsity enhancement on its core tensor to prompt more low-rankness. The tensor [Formula: see text] norm for HSI destriping is extended from the matrix [Formula: see text] norm. A proximal block coordinate descent algorithm is proposed in the MLTL2p approach to solve the resulting nonconvex nonsmooth minimization with orthogonal constraints. We show any accumulation point of the sequence generated by the proposed algorithm converges to a first-order stationary point, which is defined using three equalities of substationarity, symmetry, and feasibility for orthogonal constraints. In the numerical experiments, we compare the proposed method with state-of-the-art methods, including a deep learning based method, and test the methods on both simulated and real HSI datasets. Our proposed MLTL2p method demonstrates outperformance in terms of metrics such as mean peak signal-to-noise ratio as well as visual quality.

In the author list, "Ferry Sansoto" should be Ferry Susanto.• To reflect more accurately the contribution of the article, the title should be changed to "A permutation and equinumerosity based polynomial-time algorithm for simple undirected graph isomorphism."• In the abstract, the "Pythagorean Triples Theorem" should be removed.• In the abstract, "squared sums of elements" should be "nth power sums."• In Section 2.2, "and the sum of the individual squared elements.By checking two sums," should be ", the sum of the individual squared elements and until the sum of the nth power of the nth element in the array.By checking these sums,"• In Section 2.2, "For both vertex and edge arrays of row/column sum based on the vertex and edge adjacency matrices, if and only if one array is a permutation of another one, the corresponding two graphs are isomorphic."should be "For both the vertex and edge arrays of row/column sum based on the vertex and edge adjacency matrices, if and only if one array is a permutation of another one and the corresponding edge and vertex's adjacent relationship has been preserved, the corresponding two graphs are isomorphic."

Summary The graph isomorphism problem is to determine two finite graphs that are isomorphic which is not known with a polynomial‐time solution. This paper solves the simple undirected graph isomorphism problem with an algorithmic approach as NP=P and proposes a polynomial‐time solution to check if two simple undirected graphs are isomorphic or not. Three new representation methods of a graph as vertex/edge adjacency matrix and triple tuple are proposed. A duality of edge and vertex and a reflexivity between vertex adjacency matrix and edge adjacency matrix were first introduced to present the core idea. Beyond this, the mathematical approval is based on an equivalence between permutation and bijection. Because only addition and multiplication operations satisfy the commutative law, we propose a permutation theorem to check fast whether one of two sets of arrays is a permutation of another or not. The permutation theorem was mathematically approved by Integer Factorization Theory, Pythagorean Triples Theorem, and Fundamental Theorem of Arithmetic. For each of two n ‐ary arrays, the linear and squared sums of elements were respectively calculated to produce the results.

Euler's elastica model has been widely used in image processing. Since it is a challenging nonconvex and nonsmooth optimization model, most existing algorithms do not have convergence theory for it. In this paper, we propose a penalty relaxation algorithm with mathematical guarantee to find a stationary point of Euler's elastica model. To deal with the nonsmoothness of Euler's elastica model, we first introduce a smoothing relaxation problem, and then propose an exact penalty method to solve it. We establish the relationships between Euler's elastica model, the smoothing relaxation problem, and the penalty problem in theory regarding optimal solutions and stationary points. Moreover, we propose an efficient block coordinate descent algorithm to solve the penalty problem by taking advantage of convexity of its subproblems. We prove global convergence of the algorithm to a stationary point of the penalty problem. Finally we apply the proposed algorithm to denoise the optical coherence tomography images with real data from an optometry clinic and show the efficiency of the method for image processing using Euler's elastica model.

This paper considers the image recovery problem by taking group sparsity into account as the prior knowledge. This problem is formulated as a group sparse optimization over the intersection of a polyhedron and a possibly degenerate ellipsoid. It is a convexly constrained optimization problem with a group cardinality objective function. We use a capped folded concave function to approximate the group cardinality function and show that the solution set of the continuous approximation problem and the set of group sparse solutions are the same. Moreover, we use a penalty method to replace the constraints in the approximation problem by adding a convex nonsmooth penalty function in the objective function. We show the existence of positive penalty parameters such that the solution sets of the unconstrained penalty problem and the group sparse problem are the same. We propose a smoothing penalty algorithm and show that any accumulation point of the sequence generated by the algorithm is a directional stationary point of the continuous approximation problem. Numerical experiments for recovery of group sparse image are presented to illustrate the efficiency of the smoothing penalty algorithm with adaptive capped folded concave functions.

This paper considers the use of spherical designs and nonconvex minimization for recovery of sparse signals on the unit sphere ${S}^{2}$. The available information consists of low order, potentially noisy, Fourier coefficients for ${S}^{2}$. As Fourier coefficients are integrals of the product of a function and spherical harmonics, a good cubature rule is essential for the recovery. A spherical $t$-design is a set of points on ${S}^{2}$, which are nodes of an equal weight cubature rule integrating exactly all spherical polynomials of degree $\le t$. We will show that a spherical $t$-design provides a sharp error bound for the approximation signals. Moreover, the resulting coefficient matrix has orthonormal rows. In general the $\ell_1$ minimization model for recovery of sparse signals on ${S}^{2}$ using spherical harmonics has infinitely many minimizers, which means that most existing sufficient conditions for sparse recovery do not hold. To induce the sparsity, we replace the $\ell_1$-norm by the $\ell_q$-norm ($0

In this paper, we study a general optimization model, which covers a large class of existing models for many applications in imaging sciences. To solve the resulting possibly nonconvex, nonsmooth, and non-Lipschitz optimization problem, we adapt the alternating direction method of multipliers (ADMM) with a general dual step-size to solve a reformulation that contains three blocks of variables and analyze its convergence. We show that for any dual step-size less than the golden ratio, there exists a computable threshold such that if the penalty parameter is chosen above such a threshold and the sequence thus generated by our ADMM is bounded, then the cluster point of the sequence gives a stationary point of the nonconvex optimization problem. We achieve this via a potential function specifically constructed for our ADMM. Moreover, we establish the global convergence of the whole sequence if, in addition, this special potential function is a Kurdyka--Łojasiewicz function. Furthermore, we present a simple strategy for initializing the algorithm to guarantee boundedness of the sequence. Finally, we perform numerical experiments comparing our ADMM with the proximal alternating linearized minimization proposed in [Bolte, Sabach, and Teboulle, Math. Program., 146 (2014), pp. 459--494] on the background/foreground extraction problem with real data. The numerical results show that our ADMM with a nontrivial dual step-size is efficient.

Nonsmooth nonconvex optimization models have been widely used in the restoration and reconstruction of real images. In this paper, we consider a linearly constrained optimization problem with a non-Lipschitz regularization term in the objective function which includes the $l_p$ norm ($0

A neural network based on smoothing approximation is presented for a class of nonsmooth, nonconvex constrained optimization problems, where the objective function is nonsmooth and nonconvex, the equality constraint functions are linear and the inequality constraint functions are nonsmooth, convex. This approach can find a Clarke stationary point of the optimization problem by following a continuous path defined by a solution of an ordinary differential equation. The global convergence is guaranteed if either the feasible set is bounded or the objective function is level bounded. Specially, the proposed network does not require: 1) the initial point to be feasible; 2) a prior penalty parameter to be chosen exactly; 3) a differential inclusion to be solved. Numerical experiments and comparisons with some existing algorithms are presented to illustrate the theoretical results and show the efficiency of the proposed network.

Nonsmooth nonconvex regularization has remarkable advantages for the restoration of piecewise constant images. Constrained optimization can improve the image restoration using a priori information. In this paper, we study regularized nonsmooth nonconvex minimization with box constraints for image restoration. We present a computable positive constant θ for using nonconvex nonsmooth regularization, and show that the difference between each pixel and its four adjacent neighbors is either 0 or larger than θ in the recovered image. Moreover, we give an explicit form of θ for the box-constrained image restoration model with the non-Lipschitz nonconvex l(p)-norm regularization. Our theoretical results show that any local minimizer of this imaging restoration problem is composed of constant regions surrounded by closed contours and edges. Numerical examples are presented to validate the theoretical results, and show that the proposed model can recover image restoration results very well.

In this paper, a smoothing neural network (SNN) is proposed for a class of constrained non-Lipschitz optimization problems, where the objective function is the sum of a nonsmooth, nonconvex function, and a non-Lipschitz function, and the feasible set is a closed convex subset of . Using the smoothing approximate techniques, the proposed neural network is modeled by a differential equation, which can be implemented easily. Under the level bounded condition on the objective function in the feasible set, we prove the global existence and uniform boundedness of the solutions of the SNN with any initial point in the feasible set. The uniqueness of the solution of the SNN is provided under the Lipschitz property of smoothing functions. We show that any accumulation point of the solutions of the SNN is a stationary point of the optimization problem. Numerical results including image restoration, blind source separation, variable selection, and minimizing condition number are presented to illustrate the theoretical results and show the efficiency of the SNN. Comparisons with some existing algorithms show the advantages of the SNN.

Image restoration problems are often converted into large-scale, nonsmooth, and nonconvex optimization problems. Most existing minimization methods are not efficient for solving such problems. It is well known that nonlinear conjugate gradient methods are preferred to solve large-scale smooth optimization problems due to their simplicity, low storage, practical computation efficiency, and nice convergence properties. In this paper, we propose a smoothing nonlinear conjugate gradient method where an intelligent scheme is used to update the smoothing parameter at each iteration and guarantees that any accumulation point of a sequence generated by this method is a Clarke stationary point of the nonsmooth and nonconvex optimization problem. Moreover, we present a class of smoothing functions and show their approximation properties. This method is easy to implement without adding any new variables. Three image restoration problems with different pixels and different regularization terms are used in numerical tests. Experimental results and comparison with the continuation method in [M. Nikolova et al., SIAM J. Imaging Sci., 1 (2008), pp. 2–25] show the efficiency of the proposed method.