Hyperspectral images are often contaminated with noise which degrades the quality of data. Recently, tensor robust principal component analysis (TRPCA) has been utilized to remove noise from hyperspectral images, improving classification accuracy. However, the high dimensionality and size of hyperspectral data present  computational challenges both in terms of storage and processing power, especially in the case of TRPCA. The situation is exacerbated when the data is too large to fit in available memory. In this paper, we propose a tensor-robust CUR (TRCUR) algorithm for hyperspectral data compression and denoising. We heavily downsample the input hyperspectral image to form small subtensors; and perform TRPCA on the small subtensors. The desired hyperspectral image is recovered by combining the low-rank solution of the subtensors using tensor CUR reconstruction. We provide a theoretical guarantee to show that the desired low-rank tensor can be exactly recovered using our proposed TRCUR method. Numerical experiments indicate that our method is up to 23 times faster than performing TRPCA on the original input data while maintaining the classification accuracy.

Removing noise from hyperspectral images can be very beneficial for improving classification accuracy. Recently, tensor robust principal component analysis (TRPCA) has been successfully employed to reduce noise in hyperspectral images. In TRPCA, a minimization involving a tensor nuclear norm and a ℓ1-norm is employed to separate the low-rank hyperspectral image from the sparse noise. Tensor nuclear norm minimization is solved by iteratively performing tensor singular value thresholding (T-SVT). However, TRPCA possesses high computational complexity primarily due to the implementation of the T-SVT operator. The conventional approach for T-SVT is first to perform full tensor singular value decomposition (T-SVD), and then to shrink the singular values of the frontal slices in the frequency domain. However, when the solution of tensor nuclear norm minimization is a low-rank tensor, a good strategy is to incrementally find the singular values until they fall below the threshold. In this letter, we propose a randomized blocked algorithm for computing tensor singular value thresholding, and we leverage the compression attained by the fast Fourier transform (FFT) to accelerate TRPCA. Numerical experiments indicate that our method is significantly faster performing than TRPCA via the full T-SVD while maintaining classification accuracy.