Local low-rank matrix recovery for hyperspectral image denoising with ℓ₀ gradient constraint

Highlights

• A novel hyperspectral image denoising method is proposed by using l0 gradient.

• A global l0 gradient constraint is utilized to recover the global smoothness of the HSI.

• An efficient algorithm is designed to solve the constrained nonconvex optimization problem.

• It is revealed that the developed algorithm is more effective than the low rank based methods in matrix recovery.

Abstract

Hyperspectral image (HSI) acquisition often suffers from the mixed noise, which greatly limits its subsequent applications. This paper proposes a novel HSI denoising method by using local low-rank matrix recovery and ℓ₀ gradient, which can simultaneously identify the low-rank structures of the clean HSI and the sparse components of the mixed noise. Specifically, the HSI is modeled locally and a scheme of rank-fixed low-rank matrix recovery is employed to separate the latent clean HSI patches from the noisy counterpart. Meanwhile, the ℓ₀ gradient constraint mechanism is utilized to characterize the piecewise smooth structure along both the spectral and spatial dimensions of the reconstructed image from the patches. Moreover, the ℓ₁ norm regularization is adopted to suppress the sparse noise, such as stripes, deadlines, impulse noise, and so on. Also, an efficient iterative schema is developed based on an augmented Lagrange algorithm. The corresponding closed-form solutions of the subproblems are derived for calculating an approximate result of the nonconvex optimization problem. Extensive experiments on both simulated and real HSIs prove that the ℓ₀ gradient constraint can preserve the edge details well and the proposed method can yield better performance than the state-of-the-art methods for HSI denoising.

Introduction

Hyperspectral images are acquired by imaging spectrometers over hundreds of bands and have rich spectral information compared with that of natural images. Unfortunately, the visual quality of HSI is inevitably damaged by different types of noise during the acquisition and transmission procedure [1], [2], which severely hinders the precision of the subsequent editing and rendering tasks, e.g., classification [3], segmentation [4], and matching [5]. Therefore, HSI denoising has become an essential preprocessing step for subsequent exploitation [6], [7], [8].

Hyperspectral images restoration is a well-studied problem [9], [10], [11], [12]. Traditional 2-D denoising methods [13] can be directly applied to restore HSI by regarding each band as one independent gray image, e.g. [14], [15]. However, the performance of the bandwise methods is limited since the strong correlation among adjacent bands are ignored. Inspired by the global spectral correlation of the clean HSI, low-rank approximation methods [16], [17], [18] have been widely adopted to HSI noise removal and achieved significant success. For example, Zhang et al. [19] employ low-rank matrix recovery (LRMR) for HSI denoising with remarkable improvement. To handle defective pixels in some channels, sparse penalty is introduced into a low-rank framework for HSI reconstruction in [20]. These methods ignore the spatial prior information of HSI. To explore the spatial structure, many traditional spatial regularizers, such as nonlocal similarity [21], total variation [22], anisotropic diffusion [23], are embedded into the low-rank based method. Motivated by the outstanding performance of the total variation (TV) regularization in image processing tasks, three-dimensional TV [24] and spatio-spectral TV (SSTV) [25] have been developed and involved in the low-rank framework for HSI mixed-noise removal, achieving state-of-the-art results.

For real-world HSIs, besides the correlations exhibited by the adjacent spectral channels, it is well known that the nearby pixels are also highly correlated. That is to say, the nonlocal similarity would effectively enhance the low-rank property of the hyperspectral imagery. Zhang et al. [19] first divide the HSI into patches and restore them sequentially. Based on the patch-wise idea, a noise-adjusted iterative mechanism is adopted in [27] to make LRMR adaptive to the different corruption levels in different bands. Lu et al. [28] introduce a novel spatial-adaptive similar pixel searching strategy to group the similar pixels. To exploit both the local similarity and the global spatial-spectral structure, He et al. [35] propose to combine the global SSTV regularization model with local patch-based rank-constrained robust PCA (RPCA) model for HSI noise removal.

Theoretically, ℓ₀ norm measures the sparsity of gradients. Furthermore, ℓ₀ gradient minimization [29], [30] can be applied to images to produce sparse solutions, e.g. as a powerful sparsity measure method. Recently, the ℓ₀ gradient minimization is used to the task of image restoration [31], [32]. Due to the discrete nature and nonconvexity of the ℓ₀ norm, minimizing ℓ₀ gradient is NP-hard and intractable. As such, a number of convex relaxation strategies have been developed [32], [33]. Inspired by the strong ability of sparsity-control of ℓ₀ gradient minimization, we combine the ℓ₀ gradient with the low-rank matrix factorization and propose a unified mixed noise removal framework named ℓ₀ gradient constrained local low-rank matrix recovery (ℓ₀-LLRMR) for HSI denoising.

We adopt the patch-based restoration schema to enhance the low-rank property and restrict the ℓ₀ gradient value of the restored image to preserve the spatial piecewise smooth information. Moreover, users can directly impose a desired sparsity of the restored image by α, i.e., the ℓ₀ gradient no more than a user-given parameter α. Here, we use the percentage of the total pixels of the observed image to determine α. In our model, the parameter α has intuitive meaning that denotes a maximum ℓ₀ gradient value of the output image. Apart from the non-convexity of the ℓ₀ gradient, the proposed model is also a constrained optimization problem. To overcome this remedy, we develop an iterative schema based on the augmented Lagrange multiplier (ALM) algorithm [34] and derive closed-form solutions of the subproblems for calculating an approximate solution of the nonconvex optimization problem. Moreover, we present the convergence tendency of the objective function.