Dynamika pohybu křivek v rovině a prostoru a její aplikace

Abstract

Cilem teto prace je hledani vhodneho matematickeho popisu krivek v pohybu a zkoumani moznych aplikaci tohoto problemu. V teoreticke casti textu, zkoumame numericke i analyticke aspekty vyvoje krivek v ramci trech ruznych pristupu. Soucasti prace je odvozeni numerickeho schematu pro popis vyvoje prostorovych krivek a take nova metoda pro popis krivek pohybujicich se na stacionarnich i promennych varietach. V prakticke casti textu jsou rozebrany ctyri ruzne aplikace metod zalozenych na teto problematice. Prvni tri aplikace, ktere spadaji do domeny pocitacoveho zpracovani obrazu, jsou obrazova segmentace, redukce sumu a pocitacove videni. Posledni zkoumanou aplikaci je matematicke modelovani migrace ricnich toku, zpusobene procesem meandrovani.

The aim of this thesis is to examine the theory of evolving curves and explore its applications. In the theoretical section of the text, both numerical and analytical aspects of the problem are investigated by three different approaches. The work contains derivation of a numerical algorithm for space curve evolution based on osculating circles and new method for describing curves in space. This method has been developed to simulate the geodesic ow on both stationary and moving surfaces. The second part of the thesis is devoted to four different applications of methods based on evolving curves. First three applications, namely image segmentation, noise reduction and shape matching, belong to the domain of image processing. In the last application, we study a mathematical model for the river channel centreline migration caused by the meandering process.