Susana D. Moura

Continuity envelopes of spaces of generalised smoothness, entropy and approximation numbers

We study continuity envelopes in spaces of generalised smoothness Bpq(s,Ψ) and Fpq(s,Ψ) and give some new characterisations for spaces Bpq(s,Ψ) The results are applied to obtain sharp asymptotic estimates for approximation numbers of compact embeddings of type id: Bpq(s1,Ψ) (U) → Bx-xs2 (U), Where n/p < s1-s2 < n/p+1 and U stands for the unit ball in Rn In case of entropy numbers we can prove two-sided estimates.

Complements on Growth Envelopes of Spaces with Generalized Smoothness in the Sub-Critical Case

We describe the growth envelope of Besov and Triebel-Lizorkin spaces B pq(R) and F σ pq(R) with generalized smoothness, i.e. instead of the usual scalar regularity index σ ∈ R we consider now the more general case of a sequence σ = {σj}j∈N0 . We take under consideration the range of the parameters σ, p, q which, in analogy to the classical terminology, we call sub-critical.

Limiting embeddings in smoothness Morrey spaces, continuity envelopes and applications

In this paper, the authors prove some Franke-Jawerth embedding for the Besov-type spaces B p , q s , ? ( R n ) and the Triebel-Lizorkin-type spaces F p , q s , ? ( R n ) . By using some limiting embedding properties of these spaces and the Besov-Morrey spaces N u , p , q s ( R n ) , the continuity envelopes in B p , q s , ? ( R n ) , F p , q s , ? ( R n ) and N u , p , q s ( R n ) are also worked out. As applications, the authors present some Hardy type inequalities in the scales of B p , q s , ? ( R n ) , F p , q s , ? ( R n ) and N u , p , q s ( R n ) , and also give the estimates for approximation numbers of the embeddings from B p , q s , ? ( ? ) , F p , q s , ? ( ? ) and N u , p , q s ( ? ) into C ( ? ) , where ? denotes the unit ball in R n .

Pattern Classes in Retinal Fundus Images Based on Function Norms

Retinal fundus images are widely used for screening, diagnosis and prognosis purposes in ophthalmology. Additionally these can also be used in retinal identification/recognition systems, for identification/authentication of an identity. In this paper the aim is to explain how norms in function spaces can be used to set up, automatically, classes of different retinal fundus images. These classifications rely on crucial and unique retinal features, such as the vascular network, whose location and measurement are appropriately quantified by weighted norms in function spaces. These quantifications can be understood as retinal pattern assessments and used for improving the efficiency and speed of retinal identification/recognition frameworks. The proposed methods are evaluated in a large dataset of retinal fundus images, and, besides being very fast, they achieve a reduction of the search in the dataset (for identification/recognition purposes), by 70% on average.

Automated retina identification based on multiscale elastic registration

In this work we propose a novel method for identifying individuals based on retinal fundus image matching. The method is based on the image registration of retina blood vessels, since it is known that the retina vasculature of an individual is a signature, i.e., a distinctive pattern of the individual. The proposed image registration consists of a multiscale affine registration followed by a multiscale elastic registration. The major advantage of this particular two-step image registration procedure is that it is able to account for both rigid and non-rigid deformations either inherent to the retina tissues or as a result of the imaging process itself. Afterwards a decision identification measure, relying on a suitable normalized function, is defined to decide whether or not the pair of images belongs to the same individual. The method is tested on a data set of 21721 real pairs generated from a total of 946 retinal fundus images of 339 different individuals, consisting of patients followed in the context of different retinal diseases and also healthy patients. The evaluation of its performance reveals that it achieves a very low false rejection rate (FRR) at zero FAR (the false acceptance rate), equal to 0.084, as well as a low equal error rate (EER), equal to 0.053. Moreover, the tests performed by using only the multiscale affine registration, and discarding the multiscale elastic registration, clearly show the advantage of the proposed approach. The outcome of this study also indicates that the proposed method is reliable and competitive with other existing retinal identification methods, and forecasts its future appropriateness and applicability in real-life applications.

Growth Envelopes of Anisotropic Function Spaces

The present paper is devoted to the study of growth envelopes of anisotropic function spaces. An affirmative answer is given t the question of (19, Conjecture 13), whether the growth envelopes are independent of anisotropy. As an application, related anisotropic Hardy inequalities are presented and we also discuss a connection to some anisotropic fractal sets.