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Abstract
As a convex surrogate of tensor multi rank, recently the tensor nuclear norm (TNN) obtains promising results in the tensor completion. However, only considering the low-tubal-rank prior is not enough for recovering the target tensor, especially when the ratio of available elements is extremely low. To address this problem, we suggest a novel low-rank tensor completion model by exploiting both low-tubal-rankness and smoothness. Especially, motivated by the capability of framelet preserving details, we characterize the spatial smoothness by framelet regularization and the smoothness of the third mode by total variation (TV) regularization. The resulting convex optimization problem is efficiently tackled by a carefully designed alternating direction method of multipliers (ADMM) algorithm. Extensive numerical results including color images, videos, and fluorescence microscope images validate the superiority of our method over the competing methods.
Highlights
Low-rank tensor completion (LRTC) is a generation of low-rank matrix completion (LRMC), which has been a hot problem of research in many fields, such as color image inpainting [1], magnetic resonance imaging (MRI) data recovery [2], video processing [3], and hyperspectral/multispectral image (HSI/MSI) processing [4]–[8]
We suggest the following LRTC model by combining framelet regularization and total variation (TV) regularization, named smooth tensor nuclear norm (TNN) (STNN) and formulated as min
THE PROPOSED MODEL AND ALGORITHM by applying TNN, framelet, and TV to the LRTC problem, we propose an LRTC model and develop an alternating direction method of multipliers (ADMM)-based algorithm to address it
Summary
Low-rank tensor completion (LRTC) is a generation of low-rank matrix completion (LRMC), which has been a hot problem of research in many fields, such as color image inpainting [1], magnetic resonance imaging (MRI) data recovery [2], video processing [3], and hyperspectral/multispectral image (HSI/MSI) processing [4]–[8]. Introducing the sum of nuclear norm (SNN) as a convex relaxation of the Tucker-rank, Liu et al [26] proposed an SNN-based LRTC model with three solving algorithms (SiLRTC, FaLR-TC, and HaLRTC). Cai et al [48] proposed a discrete wavelet frame based approach for image restoration, and they provided the piecewise linear framelets constructed by tensor product. We suggest the following LRTC model by combining framelet regularization and TV regularization, named smooth TNN (STNN) and formulated as min. The framelet regularization is used to exploit the smooth prior of the underlying tensor in the spatial domain, which can preserve the details due to the diversity of filters and the multi-level tight frame system. We suggest a novel LRTC model, which can simultaneously exploit the global low-rankness and the local smoothness of a tensor.
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