We propose a PDE-constrained optimization approach for the determination of noise distribution in total variation (TV) image denoising. An optimization problem for the determination of the weights correspondent to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.

We consider a bilevel optimisation approach for parameter learning in higher-order total variation image reconstruction models. Apart from the least squares cost functional, naturally used in bilevel learning, we propose and analyse an alternative cost based on a Huber-regularised TV seminorm. Differentiability properties of the solution operator are verified and a first-order optimality system is derived. Based on the adjoint information, a combined quasiNewton/semismooth Newton algorithm is proposed for the numerical solution of the bilevel problems. Numerical experiments are carried out to show the suitability of our approach and the improved performance of the new cost functional. Thanks to the bilevel optimisation framework, also a detailed comparison between TGV 2 and ICTV is carried out, showing the advantages and shortcomings of both regularisers, depending on the structure of the processed images and their noise level.

During the last decade, the veterinary anesthetics have gained popularity as recreational drugs. The aim of this study was to document the use of "anestecia de caballo" (xylazine) and its consequences among drug users in Puerto Rico. The study combined a cross-sectional survey with 89 drug users and two focus groups conducted in Mayagüez with frontline drug treatment providers. Drug users were recruited from communities of the San Juan metropolitan area using a variety of ethnographic and outreach strategies. A short questionnaire developed for the study collected information on sociodemographics, xylazine use, and its consequences. The two focus groups were conducted to discuss the details related to xylazine use, its consequences, and utilization awareness. The sample comprised 63 males (70.8%) and 26 females with a mean age of 37.2 years. The mean number of years of drug use was 14.3, with a mean frequency of drug use of 5.9 times daily. More than 65% reported speedball as the principal drug of use. The prevalence of xylazine use was 80.7%. More than 42% of the sample used xylazine in a mixture with speedball. The main route of administration of xylazine was injection but 14% reported the use of xylazine by inhalation. More than 35% of the sample reported skin lesions and 21.1% reported at least one overdose episode. Multiple logistic regression analysis revealed that males (OR=3.47, CI=1.10-12.00) and those who reported speedball as their main drug of use (OR=9.34, CI=2.51-34.70) were significantly more likely to be xylazine users. Focus groups revealed that drug users claimed to recognize the presence of xylaxine in a mixture of speedball based on its effects, taste, the color of the drug (dark brown), and its odor. In conclusion, the use of xylazine among drug users in Puerto Rico seems to be an emerging trend with potentially serious health consequences.

We consider the problem of image denoising in the presence of noise whose statistical properties are a combination of two different distributions. We focus on noise distributions that are frequently considered in applications, in particular mixtures of salt & pepper and Gaussian noise, and Gaussian and Poisson noise. We derive a variational image denoising model that features a total variation regularisation term and a data discrepancy that features the mixed noise as an infimal convolution of discrepancy terms of the single-noise distributions. We give a statistical derivation of this model by joint Maximum A-Posteriori (MAP) estimation, and discuss in particular its interpretation as the MAP of a so-called infinity convolution of two noise distributions. Moreover, classical singlenoise models are recovered asymptotically as the weighting parameters go to infinity. The numerical solution of the model is computed using second order Newton-type methods. Numerical results show the decomposition of the noise into its constituting components. The paper is furnished with several numerical experiments and comparisons with other existing methods dealing with the mixed noise case are shown. the Total Variation (TV) as image regulariser, which is a popular choice since the seminal works of Rudin, Osher and Fatemi [50], Chambolle and Lions [17] and Vese [58] due to its edge-preserving and arXiv:1611.00690v2 [math.OC] 20 Nov 2016Proposition 2.4. Let f ∈ L ∞ (Ω), u ∈ BV (Ω) ∩ A and λ 1 , λ 2 > 0. Let A and B be as in (2.5). Then, the minimum in the minimisation problem (2.4) is uniquely attained.Proof. We report the proof in Appendix A It is based on standard tools of calculus of variations and on the properties of Kullback-Leibler divergence recalled in Appendix B.

We review some recent learning approaches in variational imaging, based on bilevel optimisation, and emphasize the importance of their treatment in function space. The paper covers both analytical and numerical techniques. Analytically, we include results on the existence and structure of minimisers, as well as optimality conditions for their characterisation. Based on this information, Newton type methods are studied for the solution of the problems at hand, combining them with sampling techniques in case of large databases. The computational verification of the developed techniques is extensively documented, covering instances with different type of regularisers, several noise models, spatially dependent weights and large image databases.