Convergence analysis of active contours in image segmentation

Abstract

Active contours effectiveness in image segmentation is well known. As any adaptive system, the iterations required by the contour to delineate the target is of importance. In this process, some nodes mach their position before than others, and, due to the internal forces, the neighboring nodes evolve towards a final shape constrained to the external forces. This paper presents a signal-processing perspective of that scenario by deriving a novel frequency-based formulation. The main result of the analysis is the speed of convergence, which depends analytically on the stiffness properties, and especially on the second-order parameters and the length of the snake segments. This initial attempt to characterize the snake dynamics is supported with simulation results.

Convergence analysis of active contours

“…However, some analysis has been done about the behaviour of the active contour toward its final equilibrium situation: Giraldi and Oliveira [20] introduce a convexity analysis of the energy functional using the Hamilton's equations of the snake model and the eigenvalues of the Hessian matrix of the energy functional, which are related to the convergence rate of the snake. We presented in [21] preliminary results about the convergence when the attracted nodes of the contour are equidistant and, in [22], we proposed a procedure for speeding-up the convergence of a contour with its nodes regularly captured by external forces. This paper presents a study based on a frequency domain formulation of active contours [23] where the convergence is analyzed and splitted into their natural modes.…”

Formulación de los contornos activos en el dominio de la frecuencia y análisis de convergencia en segmentación de imagen

“…Como cualquier sistema adaptativo, las iteraciones requeridas por el contorno para delinear el objeto de interés son de gran importancia [90,249,273,274], especialmente cuando se combina con algoritmos que controlan características dependientes de la snake para prevenir un posible comportamiento inestable [266]. La velocidad de convergencia depende especialmente de los parámetros dinámicos del sistema de segundo orden y de la distancia entre los nodos atraídos [264,268]. Este Capítulo utiliza la formulación en la frecuencia para estudiar la convergencia de la snake en el ajuste iterativo de un contorno hasta que delimita el objeto de interés.…”

“…De forma general G(ω) depende de los parámetros de rigidez y de la masa total η. Esto significa que para un valor de η constante, la solución alcanzada en el estado final es la misma independientemente de la relación entre el amortiguamiento y la masa, γ. De todos modos, la velocidad de la snake hacia la situación final sí que depende de γ[268], como se verá en la siguiente sección.…”