Sparse representation of 3D images for piecewise dimensionality reduction with high quality reconstruction

Abstract

Sparse representation of 3D images is considered within the context of data reduction. The goal is to produce high quality approximations of 3D images using fewer elementary components than the number of intensity points in the 3D array. This is achieved by means of a highly redundant dictionary and a dedicated pursuit strategy especially designed for low memory requirements. The benefit of the proposed framework is illustrated in the first instance by demonstrating the gain in dimensionality reduction obtained when approximating true color images as very thin 3D arrays, instead of performing an independent channel by channel approximation. The full power of the approach is further exemplified by producing high quality approximations of hyper-spectral images with a reduction of up to 371 times the number of data points in the representation.

Analysis of the self projected matching pursuit algorithm

“…Otherwise, as the number of iterations increases the accuracy of the approach would be dominated by roundoff errors. Nevertheless, a number of applications to real world signals [22][23][24][25] have already confirmed that the approach is of assistance for practical implementations of the OMP greedy strategy in situations where, due to memory requirements, direct linear algebra techniques cannot be applied.…”

“…It produces the orthogonal projection of the signal, at each iteration, by applying MP using a sub-dictionary consisting only of the already selected elements. A convenient feature of SPMP when applied in 2D (SPMP2D) [22,23] and 3D (SPMP3D) [24] is that it fully exploits the separability of dictionaries. Nevertheless, until now the method had not been analyzed.…”

Enhancing sparse representation of color images by cross channel transformation

“…, N y , k = 1, 2, 3 the processing of the 3 channels can be realised either in the pixel/intensity or in the wavelet domain. Since the representation of most images is sparser in the wavelet domain [29][30][31][32][33] we approximate in that domain and reconstruct the approximated image by the inverse wavelet transform. Thus, using a 3 × 3 matrix T, we construct the transformed arrays U and W as follows U (:, :, z) = 3 l=1 I(:, :, l)T (l, z) z = 1, 2, 3.…”

“…To this end, one possibility could be to learn the dictionary from training data [38][39][40][41][42][43][44][45]. However as demonstrated in previous works [29,32,33] a separable dictionary, which is easy to construct, is well suited for the purposes of achieving sparsity and delivers a fast implementation of the approach. Since we use that dictionary in the numerical examples, below we describe the method for constructing the atomic decomposition of the array W considering specifically a separable dictionary.…”

“…The sparsity property of some types of 3D images benefits from 3D processing [25][26][27][28]. In particular most RGB (Red-Green-Blue) images admit a sparser atomic decomposition if approximated by 3D atoms [29]. Within the 3D framework the gain in sparsity comes at expenses of computational cost though.…”