In image processing, classical methods minimize a suitable functional that balances between computational feasibility (convexity of the functional is ideal) and suitable penalties reflecting the desired image decomposition. The fact that algorithms derived from such minimization problems can be used to construct (deep) learning architectures has spurred the development of algorithms that can be trained for a specifically desired image decomposition, e.g., into cartoon and texture. While many such methods are very successful, theoretical guarantees are only scarcely available. To this end, in this contribution, we formalize a general class of intersection point problems encompassing a wide range of (learned) image decomposition models, and we give an existence result for a large subclass of such problems, i.e., giving the existence of a fixed point of the corresponding algorithm. This class generalizes classical model-based variational problems, such as the TV-$\ell^2$-model or the more general TV-Hilbert model. To illustrate the potential for learned algorithms, novel (nonlearned) choices within our class show comparable results in denoising and texture removal.

Automated biometric identification systems are inherently challenged to optimize false (non-)match rates. This can be addressed either by directly improving comparison subsystems, or indirectly by allowing only “good quality” biometric queries to be compared. We are interested in the latter, where the challenge lies in relating the “good quality” of a query to its utility with respect to a comparison subsystem. First, we propose a new general robust biometric quality validation scheme (RBQ VS) that, mimicking the use-case, robustly quantifies comparison improvement obtained by employing a specific quality estimator. For this purpose, we robustify an existing validation scheme by repeated random subsampling cross-validation. Second, specifically for the task of fingerprint comparison, we propose a novel biometric feature for quality estimation. Since comparison subsystems based on fingerprint minutiae, which are ridge endings and bifurcations, appear to miss minutiae or detect spurious minutiae, especially in the presence of smudge noise, we propose an algorithm aiming at measuring corruption by smudge. To this end, we employ a recently developed three parts image-decomposition and link our new smudge noise quality estimator (SNoQE) to the structure of the texture part found. At last, using the FVC databases and an NIST database, we compare the SNoQE with the popular NFIQ 2.0 estimator, and its predecessor. Experimental results show that the single-feature SNoQE can compete with the multi-feature NFIQ 2.0 and, in fact, adds new information not sufficiently reproduced by the NFIQ 2.0. Indeed, a simple combination of SNoQE and NFIQ 2.0 tends to outperform on all databases included in the comparison study. An implementation of the RBQ VS and the SNoQE can be found online.