In this paper, a new enhancing and smoothing partial differential equation coupled with time-delay regularization is presented, which is based on local geometric properties desired for image restoration. To reverse the process of image deterioration, a newly defined shock filter for edge enhancing is combined with a level set motion-based regularization equation. The balance between the backward process and the forward process is carried out by a weighting function coupled with time-delay regularization helping to identify boundary areas and homogeneous regions in a given image. The proposed model is well-posed in terms of viscosity solutions -the existence and uniqueness of periodic viscosity solution to the initial value problem of the equation is established. Numerical results for some kinds of grey-level images are demonstrated to confirm our anticipation.

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<p style='text-indent:20px;'>Abel inversion tomography plays an important role in dynamic experiments, while most known studies are started with a single Gaussian assumption. This paper proposes a mixed Poisson-Laplace-Gaussian distribution to characterize the noise in charge-coupled-device (CCD) sensed radiographic data, and develops a multi-convex optimization model to address the reconstruction problem. The proposed model is derived by incorporating varying amplitude Gaussian approximation and expectation maximization algorithm into an infimal convolution process. To solve it numerically, variable splitting and augmented Lagrangian method are integrated into a block coordinate descent framework, in which anisotropic diffusion and additive operator splitting are employed to gain edge preserving and computation efficiency. Supplementarily, a space of functions of adaptive bounded Hessian is introduced to prove the existence and uniqueness of solution to a higher-order regularized, quadratic subproblem. Moreover, a simplified algorithm with higher order regularizer is derived for Poisson noise removal. To illustrate the performance of the proposed algorithms, numerical tests on synthesized and real digital data are performed.</p>

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