Asymptotic pseudounitary stacking operators

THEORETICAL DEFINITION OF A STACKING OPERATOR

In practice, integration of discrete data is performed by stacking. In theory, it is convenient to represent a stacking operator in the form of a continuous integral:

$$S(t,y)= {\bf A}\left[M(z,x)\right]= \int\limits_{\Omega} w(x;t,y) M(\theta(x;t,y),x) dx\;.$$ (1)

Function $M(z,x)$ is the input of the operator, $S(t,y)$ is the output, $\Omega$ is the summation aperture, $\theta$ represents the summation path, and $w$ stands for the weighting function. The range of integration (the operator aperture) may also depend on $t$ and $y$ . Allowing $x$ to be a two-dimensional variable, we can use definition (1) to represent an operator applied to three-dimensional data. Throughout this paper, I assume that $t$ and $z$ belong to a one-dimensional space, and that $x$ and $y$ have the same number of dimensions.

The goal of inversion is to reconstruct some function $\widehat{M}(z,x)$ for a given $S(t,y)$ , so that $\widehat{M}$ is in some sense close to $M$ in equation (1).

Asymptotic pseudounitary stacking operators