Nonnegative Tensor Patch Dictionary Approaches for Image Compression and Deblurring Applications

Sparse Nonnegative Tensor Factorization and Completion with Noisy Observations

“…where the first inequality follows from (35), the second inequality follows from |X | ijk ≤ b A ∞ , and the last inequality follows from (36). Therefore, combining (36) with (38), we get that…”

“…for a specified β ≥ 2, we construct L to be the set of all tensors A ∈ R n 1 ×r×n 3 + whose entries are discretized to one of ϑ uniformly sized bins in the range [0, 1], and D to be the set of all tensors B ∈ R r×n 2 ×n 3 + whose entries either take the value 0, or are discretized to one of ϑ uniformly sized bins in the range [0, b]. Remark 3.1 When all entries of Y are observed and Y is corrupted by additive Gaussian noise, the model ( 2) reduces to sparse NTF with tensor-tensor product, which has been applied in patch-based dictionary learning for image data [35,46]. Remark 3.2 We do not specialize the noise in model (2), and just need the joint probability density function or probability mass function of observations in (1).…”

“…Remark 4.3 From Proposition 4.1, we know that the upper bound is lower when the number of observations is larger. In particular, when we observe all entries of Y, i.e., m = n 1 n 2 n 3 , Proposition 4.1 is the upper bound of sparse NTF model with tensor-tensor product in [35,46], which has been used to construct a tensor patch dictionary prior for CT and facial images, respectively. This demonstrates that the upper bound of sparse NTF with tensor-tensor product in [35,46] is lower than that of sparse NMF in theory, where Soltani et al [46] just showed the performance of sparse NTF with tensor-tensor product is better than that of sparse NMF in experiments.…”

“…This kind of tensor-tensor product and tensor SVD has been applied in a great number of areas such as facial recognition [15], tensor completion [36,[58][59][60], and image processing [43,61]. Recently, this kind of sparse NTF models have been proposed and applied on dictionary learning problems, e.g., tomographic image reconstruction [46], image compression and image deblurring [35]. The sparse factor in this kind of NTF with tensortensor product is due to the sparse representation of patched-dictionary elements for tensor dictionary learning [46].…”

Sketch-and-project methods for tensor linear systems

“…where A ∈ R m×n×l , X ∈ R n×p×l and B ∈ R m×p×l are third-order tensors, and the operator * denotes the T-product introduced by Kilmer and Martin [1]. The problem (1.1) arises in many applications, including tensor dictionary learning [2][3][4][5][6][7], tensor neural network [8], boundary finite element method [9][10][11], etc. For T-product, it has an advantage that it can reserve the information inherent in the flattening of a tensor and, with it, many properties of numerical linear algebra can be extend to third and high order tensors [12][13][14][15][16][17][18].…”