Estimating Norms of Commutators
We find estimates on the norm of a commutator of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular, we discuss ‖[x2, y]‖ and ‖[x1/2, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖, we can rigorously calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf(δ) for many f. In other cases, we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.
Norm Estimates for Commutators of Operators
Given two self‐adjoint Hilbert space operators A, B, and a continuous function f, we prove several inequalities of the form
An operational method for estimating signal to noise ratios from data acquired with imaging spectrometers
A method, using the concept of local means and local standard deviations of small imaging blocks and using a box counting procedure, has been developed for unsupervised estimation of the average signal to noise ratios of images in which the noise is additive. The method has been applied to simulated images with Gaussian noise, to images acquired with the Airborne Visible Infrared Imaging Spectrometer and with the Geophysical and Environmental Research Imaging Spectrometer. The method is compared with other techniques for estimating signal to noise ratios from imaging data. It is believed that the method is generally applicable to images having many small homogeneous blocks. Because the method does not take into account interband effects, it is not appropriate to use the method to estimate signal to noise ratios from images having nonnegligible interband radiometric calibration errors at spatial scales less than or equal to the size of the small imaging blocks used in the noise estimation.