The Paradigm of Reconstruction as Parameter Estimation

A useful paradigm for image reconstruction is statistical parameter estimation. In its most general formulation our problem is simple. We have a vector of measured data y = ( y ₁ , , y _I) ^T. Related to the data in some way is a vector x = ( x ₁ , , x _J) ^T whose entries are parameters we wish to determine. To solve the problem, we need to describe the relationship between y and x and then use this description to solve for x. As always, the devil is in the details.

The problem as stated is so general as to include problems that lie outside our main area of interest, such as drawing inferences from census data. While we do not need to exclude such problems, to which many of the techniques discussed in this book indeed apply, we shall focus here on applications in which the relationship between data and parameters involves a physical model describing some form of remote sensing or imaging. The vector x will often represent a vectorization of a discretized two-dimensional distribution; that is, x will be a vectorized image. The data vector y in such cases may also be a vectorized image, such as a blurred version of x, or may simply be measurements, such as projections, related to x. On occasion we shall formulate our problem...