Laplace Deconvolution on the Basis of Time Domain Data and its Application to Dynamic Contrast-Enhanced Imaging
提出一种基于拉盖尔函数展开的拉普拉斯反卷积新方法，处理有限时间区间上非等距噪声观测数据，通过复杂度惩罚控制展开项数，实现快速无边界效应的估计，适用于灌注成像等医学应用。
Summary 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 → ∞). 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 presence of a tuning parameter a allows the choice of the most advantageous time units, so that both the kernel and the unknown right-hand side of the equation are well represented for the deconvolution. 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.