Abstract-This paper shows that the solutions to various convex 1 minimization problems are unique if and only if a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as other 1 models that either minimize f (Ax − b) or impose the constraint f (Ax − b) ≤ σ, where f is a strictly convex function. For these models, this paper proves that, given a solution x * and defining I = supp(x * ) and s = sign(x * I ), x * is the unique solution if and only if AI has full column rank and there exists y such that A T I y = s and |aThis condition is previously known to be sufficient for the basis pursuit model to have a unique solution supported on I. Indeed, it is also necessary, and applies to a variety of other 1 models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically.

A great deal of interest of solving large-scale convex optimization problems has, recently, turned to gradient method and its variants. To ensure rates of linear convergence, current theory regularly assumes that the objective functions are strongly convex. This paper goes beyond the traditional wisdom by studying a strictly weaker concept than the strong convexity, named restricted strongly convexity which was recently proposed and can be satisfied by a much broader class of functions. Utilizing the restricted strong convexity, we derive rates of linear convergence for (in)exact gradient-type methods. Besides, we obtain two by-products: (1) we rederive rates of linear convergence of inexact gradient method for a class of structured smooth convex optimizations; (2) we improve the rate of linear convergence for the linearized Bregman algorithm.

In this paper, we propose a fast fixed point algorithm and apply it to total variation (TV) deblurring and segmentation. The TV-based models can be written in the form of a general minimization problem. The novel method is derived from the idea of establishing the relation between solutions of the general minimization problem and new variables, which can be obtained by a fixed point algorithm efficiently. Under gentle conditions it provides a platform to develop efficient numerical algorithms for various image processing tasks. We then specialize this fixed point methodology to the TV-based image deblurring and segmentation models, and the resulting algorithms are compared with the split Bregman method, which is a strong contender for the state-of-the-art algorithms. Numerical experiments demonstrate that the algorithm proposed here performs favorably.

The goal of image quality assessment research is to design quantitative measurements for the evaluation of image quality such that it is consistent with subjective human evaluation. Inspired by intrinsic geometric structure of nature images and characteristic of visual perception, we propose a novel geometric structural distortion model for image quality assessment in this paper, which has relatively low computational complexity and clear physical meanings. The experimental results of LIVE image database show that the proposed method is consistent with the subjective assessment of human beings and has a good performance for all distortion types.

BackgroundNephronophthisis-related ciliopathies (NPHP-RC) account for the majority of cases of monogenetically caused end-stage renal disease (ESRD) in children. Exploring the correlation between the phenotype and genotype of NPHP-RC is helpful for early diagnosis and management. We investigated the phenotype and genotype spectra of NPHP-RC in a Chinese multicentre cohort.MethodsCrosss-ectional and longitudinal data of 60 patients from 57 families with pathogenic NPHP-RC gene mutations distributed in 22 regions of China were collected into a unified, anonymous database. The mean observation time of this cohort was 3.5±3.1 years.ResultsMutations in NPHP1 and NPHP3 were the most common genetic defects. Overall, 45% of patients presented with isolated nephronophthisis (NPH), and 55% exhibited the extrarenal phenotype, which frequently involved the liver (41.7%, n=25), central nervous system (26.7%, n=16), eyes (26.7%, n=16) and skeletal system (11.7%, n=7). Accidental detection of elevated serum creatinine and non-specific symptoms caused by chronic kidney disease occurred in 65% of patients. Patients carrying NPHP1 mutations mainly presented with isolated NPH (90%, 18/20) and progressed to ESRD at a mean age of 12.9±0.5 years. The mean age of ESRD onset in the non-NPHP1 group was lower than that in the NPHP1 group (6.2±1.4 years, p<0.001), especially for patients carrying NPHP3 mutations (3.1±1.2 years), showing a heterogeneous phenotype characterised by Bardet-Biedl syndrome (12.5%, n=5), Joubert syndrome (7.5%, n=3), COACH syndrome (2.5%, n=1), Mainzer-Saldino syndrome (2.5%, n=1), short-rib thoracic dysplasia (2.5%, n=1) and unclassified symptoms (32.5%, n=13).ConclusionsThe Chinese Children Genetic Kidney Disease Database registry characterised the spectrum of the phenotype and genotype of NPHP-RC in the Chinese population. NPHP1 and NPHP3 were the most common pathogenic genes. Rapid progression to ESRD and liver involvement were noted in patients with NPHP3 mutations.