A Fourier-based implicit evolution scheme for active surfaces, for object segmentation in volumetric images
Active contours (AC) and active surfaces (AS) have been used extensively for segmentation and measurement in two- and three-dimensional images. The small time steps used in discretizing the evolution equation of AC/AS with the explicit scheme result in slow execution, whereas the use of the implicit evolution of AS in matrix form imposes very high memory and computational requirements. In this work we present an approach for implementing the implicit scheme for the numerical solution of the partial differential equation of the evolution of an AC/AS. The proposed approach is formulated as a deconvolution of the current contour/surface points with a one-dimensional mask that is performed using the discrete Fourier transform and it is derived using the properties of circulant matrices. The proposed scheme can handle higher accuracy numerical approximation of the discrete derivatives necessary for the method of AC/AS. It also possesses the separability property along different dimensions and it is applicable to implicit evolution of deformable surfaces, without the need to store and invert large sparse matrices. Initial results from the application of the proposed scheme to synthetic and clinical volumetric data demonstrate the correctness and applicability of the method. The computational complexity of the proposed scheme is also derived.