An algorithm for learning sparsifying transforms of multidimensional signals

Multidimensional signals contain information of an object in more than one dimension, and usually their processing relies on complex methods in comparison with their unidimensional counterparts. In signal processing, finding a sparse representation of a signal is of great importance for compression purposes. Analytical multidimensional bases such as the Fourier, Cosine, or Wavelet Transform have been conventionally used. Recently, the use of learned dictionaries that directly adapt to the given signal are becoming popular in tasks such as image classification, image denoising, spectral unmixing, and medical image reconstruction. This paper presents an algorithm to learn transformation bases for the sparse representation of multidimensional signals. The proposed algorithm alternates between a sparse coding step solved by hard or soft thresholding strategies, and an updating dictionary step solved by a conjugate gradient method. Furthermore, the algorithm is tested using both: two-dimensional and three-dimensional patches, which are compared in terms of the sparsity performance for different types of multidimensional signals such as hyperspectral images, computerized axial tomography images and, magnetic resonance images. The attained results are compared against traditional analytical transforms and the state-of-the-art dictionary learning method: K-SVD.