Topological Derivative Applied to Image Enhancement

Abstract

Medical imaging techniques like Computed Tomography (CT), Magnetic Resonance Imaging (MRI), Single Photon Emission Tomography (SPECT), Positron Emission Tomography (PET) and Ultrasound (US) have introduced a formidably powerful tool in medicine. The poor quality or high signal-to-noise ratio (SNR) of this kind of images is maior limitation for image analysis. For this reason image enhancement takes an important roll in the segmentation and analysis process. Although imaging techniques (e.g., contrast agents, biological markers) should improve the image quality this is not always enough to give a good result. Much effort has been put into the area of image enhancement. Our aim in this paper is to present a method for medical image enhancement based on the well established concept of topological sensitivity analysis and borrowing image processing techniques like anisotropic diffusion. More specifically, an appropriate functional F is associated to the image indicating the cost endowed to an specific image. Let us assume that the image being segmented is characterized by a scalar field u representing the image data. Then, the segmentation algorithm can be cast as: given the evolution equation,

$$ \frac{{\partial u}} {{\partial t}} = div\left( {k\nabla u} \right), $$

(1)

find u such that minimizes the functional F(k, u). This functional is associated to an energy norm of the image intensity u with diffusion coefficient k = k(∇u) selected using the topological derivative associated to this functional. Thus, the topological derivative, denoted as D _TF , is used as an indicator which allow us to find the best k that will be used in the evolution equation (eq. ??) ensuring that the functional decreases in every iteration. Finally, several numerical examples are presented in order to show the computational performance of this methodology.