À toi

Abstract

Minimal paths are built upon analogy with the theory of wave-light propagation in a medium with a refractive index, according to the principles of Pierre de Fermat. We first explain in section 1.1 the link between this minimal light paths and the refraction principle, emphasizing the interest for the use of minimal paths in image processing. Starting from the classical formulation of the active contours called snakes of Kass, Witkin, and Terzopoulos [82], we extend in section‘1.2 to the formulation of the minimal path, as presented by Cohen and Kimmel [34]. Comparing the discrete version of the minimal paths given by Dijkstra [43] with the continuous equivalent formalism of Cohen and Kimmel [34], we detail in section 1.3 several implementations of extraction techniques, and the settings of parameters involved in the model, such as the image force, named Potential. In section 1.4 we study the influence of the offset term which controls the length of the minimal path (and its curvature). 10 1 Minimal Paths in Image Processing 1.1 Minimal Paths theory 1.1.1 The minimal path in geometrical optic In order to understand the underlying law of refraction, behind the minimal path principle, let us imagine a straight seashore, separating the sea from the beach, as shown in figure 1.1. A lifeguard sitting at a point A in the beach sees a girl drowning