A coloring-book approach to finding coordination sequences.

An elementary method is described for finding the coordination sequences for a tiling, based on coloring the underlying graph. The first application is to the two kinds of vertices (tetravalent and trivalent) in the Cairo (or dual-32.4.3.4) tiling. The coordination sequence for a tetravalent vertex turns out, surprisingly, to be 1, 4, 8, 12, 16, …, the same as for a vertex in the familiar square (or 44) tiling. The authors thought that such a simple fact should have a simple proof, and this article is the result. The method is also used to obtain coordination sequences for the 32.4.3.4, 3.4.6.4, 4.82, 3.122 and 34.6 uniform tilings, and the snub-632 tiling. In several cases the results provide proofs for previously conjectured formulas.

Systematic errors in Hadamard transform optics.

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 eta = TWa + e, where the components of eta 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.

Masks for Hadamard transform optics, and weighing designs.

This paper gives a brief survey of the design of masks for Hadamard spectrometers and image scanners. Three different criteria are described for judging a mask, as well as techniques for choosing masks that are not too far from the optimum.

Errors in Hadamard spectroscopy or imaging caused by imperfect masks.

An analysis is given of the errors in Hadamard spectroscopy that are caused by transparent slits in the mask being systematically wider or narrower than they should be. It is shown that if the input spectrum consists of a single line, the distorted spectrum that is actually calculated consists of this line, plus four small blips. When the transparent slits are too wide, these blips are of equal height and the same sign, one pair surrounding the line, and another pair displaced a certain distance from it. When the slits are too narrow, the displaced blips have the same amplitude, but are negative. The response to an arbitrary input spectrum is then determined from this. The same method of analysis may also be used to handle other types of errors.

Doubly multiplexed dispersive spectrometers.

We analyze the performance of a dispersion instrument in which light is multiplexed both in the entrance and exit slit positions. This double multiplexing scheme allows one to recover both Fellgett's advantage and the high throughput advantage normally attributed only to interferometric spectrometers. The spectrometer's performance is evaluated for a number of binary cyclic coding schemes. Optical limitations on doubly multiplexed instruments are discussed, and we show that such spectrometers compare favorably with Michelson interferometric spectrometers. Some first results obtained with a laboratory pilot model are presented.

Codes for multiplex spectrometry.

A number of binary cyclic coding schemes for multiplex spectrometry are discussed and evaluated in terms of a linear, least mean square, unbiased estimate. The optical realization of such codes in dispersion instruments is briefly discussed. We show that there are many advantages both in the construction of the instrument and in its operation which accrue from cyclic codes.