A Smoothed l0-Norm and l1-Norm Regularization Algorithm for Computed Tomography

The nonmonotone alternating direction algorithm (NADA) was recently proposed for effectively solving a class of equality-constrained nonsmooth optimization problems and applied to the total variation minimization in image reconstruction, but the reconstructed images suffer from the artifacts. Though by the l0-norm regularization the edge can be effectively retained, the problem is NP hard. The smoothed l0-norm approximates the l0-norm as a limit of smooth convex functions and provides a smooth measure of sparsity in applications. The smoothed l0-norm regularization has been an attractive research topic in sparse image and signal recovery. In this paper, we present a combined smoothed l0-norm and l1-norm regularization algorithm using the NADA for image reconstruction in computed tomography. We resolve the computation challenge resulting from the smoothed l0-norm minimization. The numerical experiments demonstrate that the proposed algorithm improves the quality of the reconstructed images with the same cost of CPU time and reduces the computation time significantly while maintaining the same image quality compared with the l1-norm regularization in absence of the smoothed l0-norm.

Regular subspaces of skew product diffusions

Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space ${E_{1}\times E_{2}}$ and expressed as$X_{t}=(X^{1}_{t},X^{2}_{A_{t}}),\quad t\geq 0,$where ${X^{i}}$ is a symmetric diffusion on ${E_{i}}$ for ${i=1,2}$, and A is a positive continuous additive functional of ${X^{1}}$. One of our main results indicates that any skew product type regular subspace of X, say$Y_{t}=(Y^{1}_{t},{Y^{2}_{\tilde{A}_{t}}}),\quad t\geq 0,$can be characterized as follows: the associated smooth measure of ${\tilde{A}}$ is equal to that of A, and ${Y^{i}}$ corresponds to a regular subspace of ${X^{i}}$ for ${i=1,2}$. Furthermore, we shall make some discussions on rotationally invariant diffusions on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$, which are special skew product diffusions on ${(0,\infty)\times S^{d-1}}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ to a new regular Dirichlet form on ${\mathbb{R}^{d}}$. More precisely, fix a regular Dirichlet form ${(\mathcal{E,F}\kern 0.569055pt)}$ on the state space ${\mathbb{R}^{d}}$. Its part Dirichlet form on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is denoted by ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. Let ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ be a regular subspace of ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. We want to find a regular subspace ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ of ${(\mathcal{E,F}\kern 0.569055pt)}$ such that the part Dirichlet form of ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is exactly ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. If ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ exists, we call it a regular extension of ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ has a unique regular extension.

Invariant measures for Anosov maps with small holes

We study Anosov diffeomorphisms on surfaces with small holes. The points that are mapped into the holes disappear and never return. In our previous paper we proved the existence of a conditionally invariant measure $\mu_+$. Here we show that the iterations of any initially smooth measure, after renormalization, converge to $\mu_+$. We construct the related invariant measure on the repeller and prove that it is ergodic and K-mixing. We prove the escape rate formula, relating the escape rate to the positive Lyapunov exponent and the entropy.

Pesin’s entropy formula for endomorphisms

In this paper we prove Pesin’s entropy formula for general C2 (or C1+α) (non-invertible) endomorphisms of a compact manifold preserving a smooth measure.