Geometrical-based algorithm for variational segmentation and smoothing of vector-valued images

Abstract

An optimisation method based on a nonlinear functional is considered for segmentation and smoothing of vector-valued images. An edge-based approach is proposed to initially segment the image using geometrical properties such as metric tensor of the linearly smoothed image. The nonlinear functional is then minimised for each segmented region to yield the smoothed image. The functional is characterised with a unique solution in contrast with the Mumford-Shah functional for vector-valued images. An operator for edge detection is introduced as a result of this unique solution. This operator is analytically calculated and its detection performance and localisation are then compared with those of the DroG operator. The implementations are applied on colour images as examples of vector-valued images, and the results demonstrate robust performance in noisy environments. IntroductionVector-valued images such as colour, multi-spectral and multi-modal images can provide more valuable information than scalar images in applications ranging from satellite remote sensing to medical imaging. Vector-valued images can also be generated by extracting feature vectors from a single image as a part of image segmentation. In this paper, a variational method is considered for segmentation and smoothing of vector-valued images. Variational methods in image processing and computer vision are well established. The restoration known as 'inverse' problem was initially considered by Tikhonov and Arsenin [1] as an energy optimisation problem based on L 2 norm. The advantage of Tikhonov and Arsenin's method was that it was linear and easy to implement. Owing to its linearity, the smoothing was performed across discontinuities as well as every where else. Such a method therefore leads to blurred edges in restoration. This method was further modified by Rudin et al.[2] to introduce the notion of total variation based on L 1 norm in order to preserve edges when removing the noise. The minimisation of the functional proposed in the total variation method leads to a nonlinear differential equation whose solution was demonstrated in [2] to preserve discontinuities when it approaches to a low-pass image in regions where there are no discontinuities, hence removing the noise. This method was further generalised for vectorvalued images in [3] and applied to RGB images as an example. The generalisation of the total variation method for noise removal in textures is still a challenge. In another development, Kass et al.[4] initially introduced a contour evolution method known as the 'snake' algorithm for image segmentation based on the optimisation of a linear functional in which three terms were minimised. The first and second terms were proportional to the first and second derivatives of an affinely parametrised contour. The third term was associated with the gradient of a given input image. One of the major drawbacks of the method proposed in [4] was that its implementation was not able automatically to adjust the topology of the evolving con...