A topological asymptotic analysis for the regularized grey-level image classification problem

Abstract

Abstract. The aim of this article is to propose a new method for the grey-level image classification problem. We first present the classical variational approach without and with a regularization term in order to smooth the contours of the classified image. Then we present the general topological asymptotic analysis, and we finally introduce its application to the grey-level image classification problem.

Application of the Topological Gradient Method to Color Image Restoration

“…The model we proposed in this note is an extension to color images of the method introduced in [4] and [3]. The numerical tests show that our method can successfully remove the noise and preserve the global features of a color image.…”

“…We note here that the topological gradient process requires only 3 resolutions of a PDE (direct and adjoint with c = c 0 and then direct with c = c 1 ) for each channel. Moreover, using a discrete cosine transform for the two first resolutions (c = c 0 ) and then a preconditioned conjugate gradient for the third one (c = c 1 ), the authors already showed that the computational cost of this algorithm is in O(n. log(n)) where n is the size of the image [4,5,3]. …”

“…In [3], the authors have considered the case of the regularized classification process, and have given the expression of the topological gradient in this case. In this note, we propose another way to solve the regularized classification problem for color images.…”

“…This method is based on the topological gradient approach that has been introduced for topological optimization purpose [6,1]. The basic idea is to adapt the topological gradient approach applied to diffusive image restoration [4,5] and grey level classification problem [3]. In fact, color or multispectral images can be represented or modeled in various ways [7].…”

Image Segmentation: A Watershed Transformation Algorithm

“…By enlarging the set of admissible solutions, relaxation increases the instability of the restoration process and this could explain why the topological gradient method is so efficient. Then, using the same idea, the authors generalize in Auroux et al (2007) the topological gradient approach for classification problems and propose an extension to an unsupervised classification. A natural application of this idea is the problem of segmentation: since the identification of the main edges of the image allows us to preserve them and smooth the image outside the edges, then if the conductivity c outside edges is large enough, the regularized image is piecewise constant and provides a natural segmentation of the image.…”