Alexandru Ioan Mitrea

Iterative Methods for Obtaining Energy-Minimizing Parametric Snakes with Applications to Medical Imaging

After a brief survey on the parametric deformable models, we develop an iterative method based on the finite difference schemes in order to obtain energy-minimizing snakes. We estimate the approximation error, the residue, and the truncature error related to the corresponding algorithm, then we discuss its convergence, consistency, and stability. Some aspects regarding the prosthetic sugical methods that implement the above numerical methods are also pointed out.

On the unbounded divergence in the best approximation on equidistant nodes

Abstract In this work, we point out a superdense (meaning residual, dense and uncountable) set X 0 in the Banach space of all functions f : [ − 1 , 1 ] → R possessing r th continuous derivatives ( r ∈ N ) such that for each function in X 0 the discrete best approximation polynomials associated with the equidistant nodes in [ − 1 , 1 ] unboundedly diverge on a superdense set in [ − 1 , 1 ] of full measure.

Multi-resolution analysis, numerical methods and reverse dynamics, integrated in software instruments for performance improving in the satellite remote sensing

In this paper we present the general conception and the mathematical background of a software environment aimed to provide enhanced remotely-sensed images processing capabilities. Integral and discrete transforms, multi-resolution analysis, reverse dynamics, symbolic computation principles and tools are involved. As mathematical background, this paper presents some details concerning discrete transforms, multi-resolution analysis, and reverse dynamics

Diseased tissue area detection and delimitation, by fusion between finite difference methods and textural analysis

The basic goals of this paper target on the development and implementation of algorithms which provide the energy-minimizing snakes in parametric form, then in applying them to textural analysis based medical diagnosis. In order to derive these algorithms, we focus on Finite Differences Methods (Explicit and Crank-Nicolson Finite Difference Schemes), widely used in medical image processing applications. We examine the consistency, stability, and convergence rate, proving their increased quality able to provide maximum accuracy when determining the diseased anatomic tissue delimitation in the context of medical images.

Mathematical Methods for Assessing the Prognostic of Fixed Partial Dentures Resulting from Evaluating a Group of Dental Patients in Romania

Based on some mathematical and statistical approaches, our study leads to some conclusions concerning the procedures related to the orodental prosthetics. Occlusal equilibration in orodental prosthetics is a major issue because besides motivating patients for a regular daily oral hygiene, it could significantly increase the longevity of FPR. More dental hygiene information should be given after prosthetic treatment and patients should be motivated to attend recalls on a regular basis for professional teeth-cleaning. Interdental cleaning aids should be explained and the patients have to be motivated to use them at least once a day and the using technique should be individualized. Regarding the application of the deformable models theory, implemented in the context of an expert type software environment, it is known that the fact that modelling by advanced methods and techniques based on the deformable surfaces theory increases the efficiency of the dentofacial prosthetics procedures is a domain of great interest in the actual medical research.

Finite difference based algorithms for optimal snakes used in medical diagnosis

In this paper, we describe briefly the 2D parametric deformable models, then we infer an algorithm, via Crank-Nicolson finite difference scheme for finding the so-called energy-minimizing (i.e. optimal) snake. The residue, the truncature - error and the order of accuracy related to this algorithm are estimated, in order to establish its consistency. We prove that the algorithm leads to better results regarding its convergence than the algorithm developed by the explicit finite difference scheme. Some aspects concerning the computerized assistance, that implement this algorithm in both prosthetic surgical methods and medical treatment monitoring in Metabolic Syndrome, are also pointed out.

On the Approximation-Error of Some Numerical Methods for Obtaining the Optimal Deformable Model

This paper deals with two approximation methods for obtaining the optimal deformable model: a discretization scheme by finite differences that generates an algorithm providing the approximating solution for energy-minimizing surfaces and a reconstruction method based on the Chebyshev discrete best approximation which approximates a plane deformable model represented by a finite set of points. Estimates for the approximation-error of these methods and results concerning their convergence or the topological structure of the set of unbounded divergence are presented, too.

On the Stability and Convergence Rate of Some Discretized Schemes for Parametric Deformable Models Used in Medical Image Analysis

In order to find the energy-minimizing surface and to reduce the computational requirements, we consider an associated simplified model, [1], and we derive an algorithm for solving numerically the corresponding Euler-Gauss-Ostrogradsky equation of Calculus of Variations. The stability and the convergence of the algorithm are discussed, together with some aspects regarding the statistical modeling, applied in medical imaging.

Computer assisted prosthetic surgery, based on deformable surfaces and statistical modeling

This paper is devoted to investigate numerical optimization algorithms of the theory of deformable surfaces and to emphasize some of their computerized applications in medical image analysis and prosthetic surgery. After defining the 3D variational deformable model, both in static and dynamic form, an algorithm for finding the optimal deformable surface is presented, together with estimations for its approximation error and a condition for its convergence. Finally, a statistical modeling and a corresponding algorithm applied in surgery are presented.

Deformable Variational Models Used in Medical Imaging

This paper is devoted to point out some mathematical methods and techniques in the theory of deformable models, with applications in medical image analysis. After presenting the definition of 3D-deformable model, some reconstruction methods based on projection operators and discretization methods related to the corresponding evolution equation, are pointed out. Finally, some applications in echographic images are given.