Masking, Photobleaching, and Spreading Effects in Hadamard Transform Imaging and Spectroscopy Systems

Abstract

In analyzing the behavior of a Hadamard transform imaging spectroscopic system in an optical sectioning microscope, a previously undescribed masking effect was observed. During the process of characterizing this artifact, it was noted that while many masking errors have been reported previously in the literature, no attempt has been made to classify them or to systematically treat their effects in a variety of imaging and spectroscopy arrangements. Previous reports have documented echo artifacts in one-dimensional Hadamard mask systems based on sequences of length 2 n – 1, for which the echoes are well defined. Other valid cyclic S-sequences, such as those of prime length 4 m + 3 ≠ 2 n – 1, do not exhibit such behavior. Masking errors may be present with these sequences, but they do not appear as echoes. Recovered intensities are observed having both positive and negative magnitude distributed throughout the transform axis. These masking defects appear superficially to be “noise”, making associated errors more difficult to diagnose. Masking effects in two-dimensional systems have not been previously reported. In these, the relationship between the original image and resulting “echoes” can be quite complicated. This paper treats a variety of masking effects theoretically and presents simulations based on that treatment. Mask errors are divided into first- and second-order effects depending on whether the encoding passes through a mask once or twice. Symmetric, asymmetric, and static masking errors in one-dimensional Hadamard transform systems are treated in both first- and second-order arrangements. Where prior data exist, an attempt has been made to collect and categorize known mask-related artifacts and where appropriate provide additional documentation. Mask errors may be spatially varying or spatially invariant over the mask or within a given pixel. In systems which are spatially variant, proper sampling of the image or spectrum by the elements composing the mask is a prerequisite for successful correction of the data. Corrections applied to data from masks with spatially variant errors may cause artifacts to appear and, in some instances, complete correction may be impossible. The effects of photobleaching and mask spreading due to processes such as diffraction or aberrations in both one- and two-dimensional mask systems are investigated. Photobleaching is relatively easy to correct when an exponential decay model is applicable. In second-order systems, mask spreading gives rise to echoes or distortion even in perfectly implemented masks. Mask spreading can, in many cases, be corrected by analyzing the observed “echoes” and building a correction matrix or by using knowledge of the point, line, or other spreading function of the system. Finally, in masks of length 2 n – 1, a few simple rules greatly assist in diagnosing masking effects.

Effects of modulation defects on hadamard transform time-of-flight mass spectrometry (HT-TOFMS)

“…The modeling process involves several steps: (1) generation of an impulse vector [13] matching the peak position of TBA ϩ in the experimental spectra, (2) generation of S n , the ideal pseudorandom encoding sequence, (3) calculation of the value of the defective sequence elements at a given modulation voltage (based on the beam deflection profiles), (4) introduction of one or multiple defects in specific portions of the encoding sequence to generate S n , the defective encoding sequence, (5) encoding the impulse vector with S n , and (6) decoding the convoluted data using the inverse of S n . These operations were performed on a 700 MHz Pentium III-based PC, with 384 MB RAM.…”

“…These authors have also developed mathematical correction schemes to compensate for the effects of systematic errors. Hanley [13] recently published a review describing the source and manifestation of these errors in various optical instruments. In the same spirit, we present an analysis of the errors in HT-TOFMS arising from imperfections in the modulation scheme.…”

Highly Multiplexed Optically Sectioned Spectroscopic Imaging in a Programmable Array Microscope

“…11 In cases of spreading, the spreading function, h, of the system is applied between the application of the two masks, resulting in encoding by a matrix S s 5 S · hS that is no longer equal to S. Echoes arise when the product S 2 1 S s differs significantly from the identity matrix. Thus Eq.…”

“…Computation of the m atrix inverse (S · hS ) 2 1 used the Gauss-Jordan elimination procedure with full pivoting using published algorithm s. 34 Axial Response Simulations. Simulations of the axial response consisted of full num erical computations as de- † Here we follow the nomenclature of Hanley,11 in which h is a right circulant matrix generated from a vector consisting of an impulse and a set of coef cients, e 6 i, indexed relative to the impulse. scribed previously.…”

“…An optical arrangement in which both the illumination path and the detec- tion path are encoded, a so-called second-order arrangement, has been proposed 10 and the characteristics of such a system have been simulated. 11 Concurrent with these developm ents has been a broad effort to improve the throughput and resolution of optical sectioning m icroscopes in both single and multi-photon methods. [12][13][14][15][16][17] Am ong the m any approaches, one of the more intriguing is the use of pseudo-random modulation of the apertures.…”