A Simple Example of a Bayesian Uncertainty Estimate in Image Processing

Abstract:

Spatial resolution enhancement is a common goal for remote sensing 
imagery, including orthorectification. This paper discusses a 
one-dimensional example of such enhancement.

The uncertainty algorithm discretizes the original and retrieved radiances.
It uses Singular Value Decomposition (SVD) to solve the discrete linear
equations. The discretized values populate a square matrix representing
the joint probability distribution between the original radiances and the
retrieved ones. This matrix forms a finite complete probability scheme.
The algorithm first computes the marginal densities for the original radiance
bins and the retrieved ones. It then divides the joint probability 
entries by the marginal values for the retrieved values. This 
operation produces the conditional probability for the source value 
given the retrieved value. The cumulative conditional probability
then provides the basis for computing credible intervals.

A single image does not provide a large enough sample to create 
reliable estimates of credible intervals. A Monte Carlo simulation 
provides multiple similar images. These increase the populated cells 
in the joint probability matrix.

There are three interesting results from these estimates.

First, the character of the SVD spatial basis set depends on the 
overlap of the PSFs. With complete separation of the PSF kernals, 
the spatial basis acts like delta-function sampling. If 
interpolation guesses values between the samples, it may produce 
retrievals that do not follow the original spatial variation. That 
increases the width of the credible intervals. If the kernals 
overlap heavily, the SVD spatial basis vectors resemble Fourier 
series having a spatially varying frequency. The retrievals can 
exhibit Gibb's phenomenon when this happens.

Second, the prior distribution selected for the experiment is 
decidedly non-Gaussian. The marginal distribution for ``clear'' bars 
is disjoint from the distribution for ``cloudy'' ones. No original 
radiance has the mean value. The algorithm provides sensible 
credible intervals that clearly identify when the PSF's resolve the 
bars.

Third, the example retrievals illustrate Bayesian learning. New 
experimental evidence can provide the researcher a prior distribution 
based on better physics.