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1

Krylov solvability under perturbations of abstract inverse linear problems

Caruso Noè Angelo, Michelangeli Alessandro

Journal of Applied Analysis

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2023

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Vol. 29, nr 1

3--29

EN

When a solution to an abstract inverse linear problem on Hilbert space is approximable by finite linear combinations of vectors from the cyclic subspace associated with the datum and with the linear operator of the problem, the solution is said to be a Krylov solution. Krylov solvability of the inverse problem allows for solution approximations that, in applications, correspond to the very efficient and popular Krylov subspace methods. We study the possible behaviors of persistence, gain, or loss of Krylov solvability under suitable small perturbations of the infinite-dimensional inverse problem - the underlying motivations being the stability or instability of infinite-dimensional Krylov methods under small noise or uncertainties, as well as the possibility to decide a priori whether an infinite-dimensional inverse problem is Krylov solvable by investigating a potentially easier, perturbed problem.

2

New Binary Hausdorff Symmetry measure based seeded region growing for retinal vessel segmentation

Panda R., Puhan N. B., Panda G.

Biocybernetics and Biomedical Engineering

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2016

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Vol. 36, no. 1

119--129

EN

Automated retinal vessel segmentation plays an important role in computer-aided diagnosis of serious diseases such as glaucoma and diabetic retinopathy. This paper contributes, (1) new Binary Hausdorff Symmetry (BHS) measure based automatic seed selection, and (2) new edge distance seeded region growing (EDSRG) algorithm for retinal vessel segmentation. The proposed BHS measure directly provides a binary symmetry decision at each pixel without the computation of continuous symmetry map and image thresholding. In a multiscale mask, the BHS measure is computed using the distance sets of opposite direction angle bins with sub-pixel resolution. The computation of the BHS measure from the Hausdorff distance sets involves point set matching based geometrical interpretation of symmetry. Then, we design a new edge distance seeded region growing (EDSRG) algorithm with the acquired seeds. The performance evaluation in terms of sensitivity, specificity and accuracy is done on the publicly available DRIVE, STARE and HRF databases. The proposed method is found to achieve state-of-the-art vessel segmentation accuracy in three retinal databases; DRIVE- sensitivity (0.7337), specificity (0.9752), accuracy (0.9539); STARE-sensitivity (0.8403), specificity (0.9547), accuracy (0.9424); and HRF-sensitivity (0.8159), specificity (0.9525), accuracy (0.9420).

3

Breakthrough in Interval Data Fitting I. The Role of Hausdorff Distance

Gutowski M.W.

Prace Naukowe Politechniki Warszawskiej. Elektronika

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2009

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z. 169

63-72

EN

This is the first of two papers describing the process of fitting experimental data under interval uncertainty. Probably the most often encountered application of global optimization methods is finding the so called best fitted values of various parameters, as well as their uncertainties, based on experimental data. Here I present the methodology, designed from the very beginning as an interval-oriented tool, meant to replace to the large extent the famous Least Squares (LSQ) and other slightly less popular methods. Contrary to its classical counterparts, the presented method does not require any poorly justified prior assumptions, like smallness of experimental uncertainties or their normal (Gaussian) distribution. Using interval approach, we are able to fit rigorously and reliably not only the simple functional dependencies, with no extra effort when both variables are uncertain, but also the cases when the constitutive equation exists in implicit rather than explicit functional form. The magic word and a key to success of interval approach appears the Hausdorff distance.

4

Strong variational principles and generic well-posedness

Turinici M.

Demonstratio Mathematica

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2005

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Vol. 38, nr 4

935--941

EN

The strong version of Ekeland's variational principle [JMAA, 47(1974), 324-353] due to Georgiev [JMAA, 131(1988), 1-21] is deductible in a direct way from the standard one. In addition, a substitution between them is possible for many generic well-posed optimization problems.

5

A comparison of fundamental methods for iso surface extraction

Patera J., Skala V.

Machine Graphics and Vision

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2004

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Vol. 13, No. 4

329-343

EN

This paper compares four fundamental methods for iso surface extraction based on cell decomposition to tetrahedra. The methods are compared both on mathematically generated data sets as well as on real data sets. The comparison using mathematical data is made from different points of view such, as area approximation, or volume approximation. On the other hand, the Hausdorff distance and the root mean square are used to compare methods on real data sets. The presented comparison can be of help when deciding which of the tested methods to choose, as well as when we need to compare a newly developed method with other existing approaches.

6

3D reconstruction of parametric curves: recovering the control points

Bhuiyan M. A., Hama H.

Machine Graphics and Vision

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2004

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Vol. 13, No. 4

307-328

EN

This article introduces a new curve reconstruction method based on recovering the control points of parametric cubic curves. The method developed here has two stages: finding the 3D control points of parametric curves and reconstruction of free curves. The 3D control points of curves are computed from 2D image sequences by using projective reconstruction of the 3D control points and the bundle adjustment algorithm. The relationships among parametric curves, such as Hermite curves, Bézier curves and B-spline curves, are established so that a curve of any model can be achieved for best fitting. Some experiments are performed to show the performance and effectiveness of the algorithm. The method is based on the slope following and learning algorithm, which provides an efficient way of finding the 3D control points of any type of cubic Bézier curves. This method, which is an extension of our previous work on recovering control points of 2D Bézier curves, can automatically fit a set of data points with piecewise geometrically continuous cubic parametric curves. The experimental results demonstrate that our method is a fast and efficient way of recovering 3D control points of parametric curves, matching free curves and shape reforming.

7

Measures of noncompactness related to monotonicity

Banaś J., Olszowy L.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2001

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[Z] 41

13-23

EN

We investigate measures of noncompactness related to the monotonicity of functions. Several properties of these measures of non-compactness are derived. Particularly we give the estimates of these measures with help of the Hausdorff distance from the family of nondecreasing or nonincreasing functions. Such a result indicates some connections with approximation theory.

8

Frechet type theorem and its applications to multifunctions

Kubiś B., Kubiś W.

Bulletin of the Polish Academy of Sciences. Mathematics

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2000

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Vol. 48, no 2

165-170

EN

We state a Frechet type theorem for measurable maps with values in an almost arcwise connected metrizable space. As an application, we obtain some results on continuous approximation of measurable multifunctions.

9

Convexity induced by a set quantity

Falcon S., Sadarangani K.B.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

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2000

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[Z] 40

57-67

EN

In the geometry of Banach spaces the notion of convexity plays a very significant role and is frequently used in many branches of functional analysis. In the last few years, there have appeared some papers containing generalizations of the concept of convexity using the notion of a measure of noncompactness. The aim of this paper is to generalize the notion of convexity using the notion of a set quantity wich has been considered in. It is worthwhile mentioning that several results obtained in the geometry involving compactness conditions have counterparts in the geometry induced by a set quantity. Particularly, we introduce a modulus related to a set quantity and we obtain a generalization of a result due to Rolewicz. Moreover, we calculate this modulus for the De Blasi measures of weak noncompactness in some classics Banach spaces such as C_0, l^1, L^1 and the James space J.

10

Note on Jensen and Pexider functional equations

Smajdor W.

Demonstratio Mathematica

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1999

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Vol. 32, nr 2

363-376

EN

We determine the general solutions of the Jensen functional equation 2f (x+y):2=f(x) + f(y), x,y zawiera się M and the Pexider functional equation f(x+y)=g(x)+h(y), x,y zawiera się M , for f, g, h : M --+ S , where M is an Abelian semigroup with the division by 2 and S is an abstract convex cone satisfying the cancellation law. Some applications to set-valued versions of these equations are given.