Systematic errors in Hadamard transform optics

Abstract

This paper analyzes the systematic errors in Hadamard transform optical instruments caused by moving masks, incorrect mask alignment, faulty mask fabrication, missing data, diffraction, etc. and describes techniques for reducing or eliminating these errors. In a great many cases the behavior of the instrument can be characterized by a single matrix equation of the form 71 = TWa + e, where the components of 71 are the measurements, T is a matrix characterizing the instrument, W specifies the mask configurations, a is a vector containing the unknown spectral intensities, and the components of e are small random errors.

What are the advantages of ghost imaging? Multiplexing for x-ray and electron imaging

“…The reconstruction procedure is exactly the same as for the multiplex scan. ment [13,14]. Our presentation generalizes these examples and many others into a single framework.…”

“…This is well known in the field of spectroscopy, where the n −1/2 reduction of the noise as a function of multiple channels is known as the Felgett advantage, one motivation behind Fourier transform and Hadamard spectroscopies. See [14] for a comprehensive study of noise in multiplex spectroscopy.…”

“…For our discussion, the unframiliar reader need only know that S n is a deterministic matrix where approximately half of the elements are 1, the rest are 0, and the inverse of S n is given by 2(n + 1) −1 · (2S T n − J n ) = S T . By J n we mean an n × n matrix with all elements 1 [13,14].…”

“…2. Hadamard sampling can be considered optimal in terms of maximizing Felgett's advantage [14]. It is interesting, however, that the performance of a random Bernoulli sampling is nearly identical to the carefully designed Hadamard one.…”

“…The factor 1.66 in the half Gaussian distribution approximates (1 − 2/π) −1/2 . Note that all these values assume A T A is not singular, which can only occur when m ≥ n. ‡ Hadamard error estimate derived in [14].…”