All scientific instruments share a common basis when viewed as a linear system. An observing system , for example, has an input , and an output , a response (or point spread function) , and additive noise , as represented in Eq. 1:
We can characterize such a system mathematically using the definition in Eq. 1, which describes a linear instrument model in continuous space. We can define an extended instrument, as in Eq. 2, by adding another linear processing system , having a response function that produces an output given the input
The idea of an extended instrumental system such as Eq. 2 gives us a powerful way of looking at the processes that we wish to describe in this paper.
Using the extended linear system instrument model, our goal is to obtain an enhanced instrument with useful, data independent properties; may be, for example, a simple estimator of , in which case our extended instrument model can be used for resolution recovery for both spectra and images. Or the model can be used to transform from one type of point spread function to another such as a positive semi-definite estimator of in order to make possible an accurate determination of the continuum level from a sinc apodized spectrum as from an Fourier Transform Spectrometer. We will first give examples of these applications below. Then we will briefly explain our methodology in obtaining the linear filter from a novel application of a neural network, specifically a convolution network.