Dachun Yang

Ziwei Li, Wengu Chen, Huanmin Ge et al.

In this article, we introduce a minimization model via a non-convex transformed $\ell_p$ (TLp) penalty function with two parameters $a\in(0,\infty)$ and $p\in(0,1]$, where the case $p=1$ is known and was established by S. Zhang and J. Xin. Using the sparse convex-combination technique, we establish the exact and the stable sparse signal recovery based on the restricted isometry property (RIP). We apply a modified iteratively re-weighted least squares method and the difference of convex functions algorithm (DCA) to give the IRLSTLp algorithm for unconstrained TLp minimization and prove some related convergence results. Finally, we conduct some numerical experiments to show the robustness of the IRLSTLp and the flexibility of the TLp minimization model. The novelty of these results lies in three aspects: (i) We introduce the concept of the relaxation degree RD$_P$ of a separable penalty function $P$ to quantitatively measure how closely $P$ approaches $\ell_0$. (ii) We introduce the TLp penalty, which includes two aforementioned adjustable parameters, offering more flexibility and stronger sparsity-promotion capability of the TLp minimization model, compared with the $\ell_p$ and the TL1 minimization models. (iii) The obtained RIP upper bound for signal recovery via TLp minimization can reduce, when $p\in(0,1]$ and as $a\to \infty$, to the sharp RIP bound obtained by R. Zhang and S. Li and, especially, can recover, when $p=1$, the well-known sharp bound $δ_{2s}<\frac{\sqrt{2}}{2}$.

Zunwei Fu, Dachun Yang, Shuhui Yang

In this article we introduce the linear canonical Riesz potential (for short, LCRP) and give its symbol in terms of linear canonical transforms. Driven by image processing, we establish the convergence/divergence of these LCRPs for different kinds of functions. Concretely, for grating functions, we prove that their classical Riesz potentials diverge, whereas their LCRP converge due to the key role of chirp functions. For the characteristic function ${\mathbf 1}_P$ of a convex polygon $P$, we show that the limit of its Riesz potential at any non-boundary point $\boldsymbol{x}$ equals ${\mathbf 1}_P(\boldsymbol{x})$, but its limit at the boundaries differ from ${\mathbf 1}_P$, while it is known that, for any Schwartz function $f$, the limit of its Riesz potential at any point $\boldsymbol{x}$ always equals $f(\boldsymbol{x})$. Based on these and the inverse operator of the LCRP (namely the linear canonical Laplacian operator), we propose an asymmetric cascaded LCRP method for the multi-image encryption and create an efficient and secure cryptosystem. Systematic security evaluations, including sensitivity, statistical, noise attack, and occlusion attack analyses, demonstrate its robustness and its security. Even for a single image, the proposed method is more efficient than the known encryption approach based on the fractional Riesz potential. The novelty of these results lies in that the convergence and the divergence of LCRTs at the critical indices, respectively, for ``good" Schwartz functions and for ``bad" discrete image functions essentially affect the security of image encryption and decryption.