Segmentation, Classification and Denoising of a Time Series Field by a Variational Method

Abstract

Many real life problems can be represented by an ordered sequence of digital images. At a given pixel a specific time course is observed which is morphologically related to the time courses at neighbor pixels. Useful information can be usually extracted from a set of such observations if we are able to classify pixels in groups, according to some features of interest for the final user. Moreover parameters with a physical meaning can be extracted from the time courses. In a continuous setting we can formalize the problem by assuming to observe a noisy version of a positive real function defined on a bounded set ${\mathcal{T}}\times\Omega\subset\mathbb{R}\times\mathbb{R}^{2}$ , parameterized by a vector of unknown functions defined on ℝ² with discontinuities along regular curves in Ω which separate regions with different features. Suitable regularity conditions on the parameters are also assumed in order to take into account the physical constraints. The problem consists in estimating the parameter functions, segmenting Ω in subsets with regular boundaries and assigning to each subset a label according to the values that the parameters assume on the subset. A global model is proposed which allows to address all of the above subproblems in the same framework. A variational approach is then adopted to compute the solution and an algorithm has been developed. Some numerical results obtained by using the proposed method to solve a dynamic Magnetic Resonance imaging problem are reported.