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Identifier 000362967
Title Ασθενείς λύσεις για μη γραμμικές διαφορικές εξισώσεις με μερικές παραγώγους και εφαρμογές - τρισδιάστατη αναδόμηση εικόνας
Alternative Title Viscosity solutions for nonlinear partial differential equations and applications - three dimensional image reconstruction
Author Ζωχιός, Χρήστος Ευάγγελου
Thesis advisor Κοσιώρης, Γεώργιος
Abstract Viscosity solutions are a class of generalized solutions of nonlinear partial differential equations. The great value of this kind of weak solutions is the fact that very general existence, uniqueness and stability results hold for them in many problems of various fields of application. They are used in many kinds of applications, two of which are analyzed in this work. The Shallow Lake Problem, a non-standard optimal control problem derived from the combination of agricultural activities and ecological services that a shallow lake provides, is the first application studied. The optimal dynamics of the problem, necessary conditions of which are provided by the Pontryagin Maximum Principle, are studied in the beginning for a range of values of the discount factor. The number and the type of the equilibrium points are also investigated. We then prove that the value function of the Shallow Lake Problem is a viscosity solution of an Optimal Hamilton-Jacobi-Bellman (OHJB) equation, a basic for the rest of this work result. To derive this, we study the control problem on a compact control space and we prove monotonicity, semiconvexity and other, related to the subdifferential, properties for the corresponding value function, considered as a viscosity solution of a modified HJB equation. We extract many regularity results for the value (welfare) function, using this result. Furthermore, three different numerical schemes are presented, the "forward", the "backward" and the "upwind" schemes, for the approximation of the viscosity solution, based on a finite difference space discretization. Their convergence is proved using fixed point arguments. For validation of the numerical results, we compare them with the results obtained from the Simple Shooting Method, which we use as the "gold standard". The small mean relative error for all cases (different number of saddle points, different spatial step) proves the accuracy of the numerical approximations. The second application analyzed is connected with the geometric representation of the Abdominal Aortic Aneurysm (AAA), which is a localized dilatation of the aortic wall. A reliable estimate of AAA rupture risk demands accurate measurements of its geometric characteristics. So, our main objective in this part of our work is the extraction of the thrombus and outer wall boundaries from cross sections of a 3D CTA AAA image data set, using the level set framework and new geometrical methods to address the basic problem of no sufficient intensity contrast between thrombus and surrounding tissue. Tools like the inversion mapping and the convex hull of a closed curve are used to trace and reconstruct these boundaries, exploiting the presence of calcifications, which are detected by combining these tools with a thresholding technique. We also introduce three novel stopping criteria to address the leakage problem that Level Set Methods (LSM’s) present and a method for detecting the leakage regions. A Fast Marching Method (FMM) is initially used to resolve another problem of the LSM’s, namely speed, with a proper modification for images difficult to segment. In regions with few or no calcifications, an interpolation distance technique may be used to obtain the two boundaries, if required. Artificial images which simulate the real cases are then presented to test the versatility of the methods. Sensitivity to the parameter settings and reproducibility are also analyzed and segmentation time is presented. A manual segmentation, created by a medical expert, was performed in the slices of ten patient data sets (450 slices) to compare with our results. Mean distance, area overlap and relative volume error are three of the quantities used to evaluate outer wall segmentation error. For the validation of the thrombus results a method for approximating the mean wall thickness of the AAA is introduced, utilizing another tool, namely the division of a curve into sectors using its centroid. The results for the outer wall and for the mean wall thickness are comparable with previous values reported in literature, in which, however, there is no segmentation of the thrombus boundary, at least using the level set framework. These results indicate that geometrically accurate 3D reconstructions of AAA anatomy can be produced through LSM based segmentation of image data obtained from currently available imaging technology. This information is quite important for finally obtaining a reliable patient specific measure of AAA rupture risk.
Language English
Subject Abdominal Aortic Aneurysm
Calcifications
Hamilton- Jacobi-Bellman (HJB) equation
Image segmentation
Level Set Method
Optimal control
Shallow Lake Problem
Skiba point
Viscosity solution
Ανεύρυσμα Κοιλιακής Αορτής
Ασβεστώσεις
Βέλτιστος έλεγχος
Εξίσωση Hamilton-Jacobi-Bellman (HJB)
Κατάτμηση εικόνας
Λύση ιξώδους
Μέθοδος Συνόλου Στάθμης
Πρόβλημα Ρηχής Λίμνης
Σημείο Skiba
Issue date 2011-01-28
Collection   School/Department--School of Sciences and Engineering--Department of Mathematics--Doctoral theses
  Type of Work--Doctoral theses
Permanent Link https://elocus.lib.uoc.gr//dlib/c/d/8/metadata-dlib-5fd6a24bd6c67cb58b736ae5fea53d0a_1294036532.tkl Bookmark and Share
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