Speaker : Fabienne Comte (Université Paris Descartes Laboratoire MAP5)
Title : Laplace deconvolution (and its application to Dynamic Contrast Enhanced imaging)
We consider the problem of Laplace deconvolution with noisy discrete non-equally spaced observations on a finite time interval.We propose a new method for Laplace deconvolution which is based on expansions of the convolution kernel, the unknown function and the observed signal over a Laguerre functions basis (which acts as a surrogate eigenfunction basis of the Laplace convolution operator) using a regression setting. The expansion results in a small system of linear equations with the matrix of the system being triangular and Toeplitz. Because of this triangular structure, there is a common number m of terms in the function expansions to control, which is realized via a complexity penalty. The advantage of this methodology is that it leads to very fast computations, produces no boundary effects due to extension at zero and cut-off at T and provides an estimator with the risk within a logarithmic factor of m of the oracle risk. We emphasize that we consider the true observational model with possibly non-equispaced observations which are available on a finite interval of length T which appears in many different contexts, and we account for the bias associated with this model (which is not present in the case T -> infinity). The study is motivated by perfusion imaging using a short injection of contrast agent, a procedure which is applied for medical assessment of microcirculation within tissues such as cancerous tumours. The methodology is illustrated by an extensive simulation study and a real data example which confirms that the technique proposed is fast, efficient, accurate, usable from a practical point of view and very competitive.