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In the present paper we consider the problem of Laplace deconvolution withnoisy discrete non-equally spaced observations on a finite time interval. Wepropose a new method for Laplace deconvolution which is based on expansions ofthe convolution kernel, the unknown function and the observed signal overLaguerre functions basis (which acts as a surrogate eigenfunction basis of theLaplace convolution operator) using regression setting. The expansion resultsin a small system of linear equations with the matrix of the system beingtriangular and Toeplitz. Due to this triangular structure, there is a commonnumber $m$ of terms in the function expansions to control, which is realizedvia complexity penalty. The advantage of this methodology is that it leads tovery fast computations, produces no boundary effects due to extension at zeroand cut-off at $T$ and provides an estimator with the risk within a logarithmicfactor of the oracle risk. We emphasize that, in the present paper, we considerthe true observational model with possibly nonequispaced observations which areavailable on a finite interval of length $T$ which appears in many differentcontexts, and account for the bias associated with this model (which is notpresent when $T\rightarrow\infty$). The study is motivated by perfusion imagingusing a short injection of contrast agent, a procedure which is applied formedical assessment of micro-circulation within tissues such as canceroustumors. Presence of a tuning parameter $a$ allows to choose the mostadvantageous time units, so that both the kernel and the unknown right handside of the equation are well represented for the deconvolution. Themethodology is illustrated by an extensive simulation study and a real dataexample which confirms that the proposed technique is fast, efficient,accurate, usable from a practical point of view and very competitive.

Laplace deconvolution on the basis of time domain data and its application to Dynamic Contrast Enhanced imaging