Optimal contour approximation by deformable piecewise cubic splines and nonparametric edge location estimation.

Abstract

In this thesis, we have investigated the problem of optimal and robust estimation of edge segments and approximation of boundaries using deformable piecewise cubic splines. Our study results in an algorithm for optimal approximation of an object's boundary represented by piecewise cubic spline (PCS). The approximation uses the information generated by a nonparametric and optimal algorithm of edge detection. This thesis develops a new theory of edge detection with a new edge model and a new detection algorithm. A new technique for boundary approximation and a new deformable contour model are also proposed. In this thesis, the deformable contour is represented by a cardinal-form PCS which was originally proposed for curve interpolation but now is used for curve approximation. The approximation is optimal because the deformable contour minimizes both the internal curve constraints and external image constraints. A penalized likelihood approach is adopted to place an initial curve, which then is refined by deforming and moving the PCS. The number of knots, knot placement, and the control points of the spline model are determined automatically. The thesis analyzes effects of the internal and external constraints on the contour and proposes a solution for dynamics of the contour. In edge detection, this thesis proposes a generic edge model in a multi-dimensional image environment. This model uses a weaker condition to describe edges, in contrast with the strong conditions such as a step or other parametric functions assumed in the previous approaches. The model can be shown to be complete and unique in representing edges. A theory of nonparametric edge detection is developed and shown to have superior performance over other algorithms. This thesis also proposes a new model for estimation of contour segments. An algorithm of optimal estimation is developed using robust statistics. Our new algorithm estimates segments of edge contours from the surface of image intensity. In contrast, the previous approaches process a linking and a detection procedure separately and the linking is performed on the detected edge map. Finally, our system is designed for two dimensional images and can be extended to 3-D for surface reconstruction to determine volume boundaries.