Regularization by Denoising (RED) is a well-known method for solving image restoration problems by using learned image denoisers as priors. Since the regularization parameter in the traditional RED does not have any physical interpretation, it does not provide an approach for automatic parameter selection. This letter addresses this issue by introducing the Constrained Regularization by Denoising (CRED) method that reformulates RED as a constrained optimization problem where the regularization parameter corresponds directly to the amount of noise in the measurements. The solution to the constrained problem is solved by designing an efficient method based on alternating direction method of multipliers (ADMM). Our experiments show that CRED outperforms the competing methods in terms of stability and robustness, while also achieving competitive performances in terms of image quality.

Motivation: In fluorescence microscopy, single-molecule localization microscopy (SMLM) techniques aim at localizing with high-precision high-density fluorescent molecules by stochastically activating and imaging small subsets of blinking emitters. Super resolution plays an important role in this field since it allows to go beyond the intrinsic light diffraction limit. Results: In this work, we propose a deep learning-based algorithm for precise molecule localization of high-density frames acquired by SMLM techniques whose ℓ2-based loss function is regularized by non-negative and ℓ0-based constraints. The ℓ0 is relaxed through its continuous exact ℓ0 (CEL0) counterpart. The arising approach, named DeepCEL0, is parameter-free, more flexible, faster and provides more precise molecule localization maps if compared to the other state-of-the-art methods. We validate our approach on both simulated and real fluorescence microscopy data. Availability and implementation: DeepCEL0 code is freely accessible at https://github.com/sedaboni/DeepCEL0.

We consider a variational model for single-image super-resolution based on the assumption that the gradient of the target image is sparse. We enforce this assumption by considering both an isotropic and an anisotropic $\ell^0$ regularisation on the image gradient combined with a quadratic data fidelity, similarly as studied in \cite{Storath14} for signal recovery problems. For the numerical realisation of the model, we propose a novel efficient ADMM splitting algorithm whose substeps solutions are computed efficiently by means of hard-thresholding and standard conjugate-gradient solvers. We test our model on highly-degraded synthetic and real-world data and quantitatively compare our results with several sparsity-promoting variational approaches as well as with state-of-the-art deep-learning techniques. Our experiments show that thanks to the $\ell^0$ smoothing on the gradient, the super-resolved images can be used to improve the accuracy of standard segmentation algorithms for applications like QR codes and cell detection and land-cover classification problems.

The relatively poor spatial resolution of thermal images is a limitation for many thermal remote sensing applications. A possible solution to mitigate this problem is super-resolution, which should preserve the radiometric content of the original data and should be applied to both the cases where a single image or multiple images of the target surface are available. In this perspective, we propose a new super-resolution algorithm, which can handle either single or multiple images. It is based on a total variation regularization approach and implements a fully automated choice of all the parameters, without any training dataset nor a priori information. Through simulations, the accuracy of the generated super-resolution images was assessed, in terms of both global statistical indicators and analysis of temperature errors at hot and cold spots. The algorithm was tested and applied to aerial and terrestrial thermal images. Results and comparisons with state-of-the-art methods confirmed an excellent compromise between the quality of the high-resolution images obtained and the required computational time.

Deep learning methods have state-of-the-art performances in many image restoration tasks. Their effectiveness is mostly related to the size of the dataset used for the training. Deep image prior (DIP) is an energy-function framework which eliminates the dependency on the training set, by considering the structure of a neural network as an handcrafted prior offering high impedance to noise and low impedance to signal. In this paper, we analyze and compare the use of different optimization schemes inside the DIP framework for the denoising task.