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.
TWO-DIMENSIONAL METRIC AND TETRAD GRAVITIES AS CONSTRAINED SECOND-ORDER SYSTEMS