Fast delta Hadamard transform

Abstract

In many fields (e.g., spectroscopy, imaging spectroscopy, photoacoustic imaging, coded aperture imaging) binary bit patterns known as m sequences are used to encode (by multiplexing) a series of measurements in order to obtain a larger throughput. The observed measurements must be decoded to obtain the desired spectrum (or image in the case of coded aperture imaging). Decoding in the past has used a technique called the fast Hadamard transform (FHT) whose chief advantage is that it can reduce the computational effort from N(2) multiplies to N log(2) N additions or subtractions. However, the FHT has the disadvantage that it does not readily allow one to sample more finely than the number of bits used in the m sequence. This can limit the obtainable resolution and cause confusion near the sample boundaries (phasing errors). We have developed both 1-D and 2-D methods (called fast delta Hadamard transforms, FDHT) which overcome both of the above limitations. Applications of the FDHT are discussed in the context of Hadamard spectroscopy and coded aperture imaging with uniformly redundant arrays. Special emphasis has been placed on how the FDHT can unite techniques used by both of these fields into the same mathematical basis.

Coded aperture imaging in X- and gamma-ray astronomy

“…The quality of the object reconstruction therefore depends on the choice of the aperture A(x, y) and the decoding function G (Groh etal., 1972;Brown, 1974;Cannon and Fenimore, 1980). When the modulation of the incident fluxes is obtained by means of a coded mask consisting of N elements of equal size and shape, M of which are transparent, the aperture A may be considered as a series of step functions.…”

“…In this way, the sum of the G elements being equal to zero, any d.c. background is automatically removed from the reconstructed image (Fenimore and Cannon, 1978). A different choice for G can be:…”

“…where 2 again depends on the transparency of the mask, being equal to the average height of the sidelobes in the autocorrelation function (see Section 2.3 and Fenimore, 1978). The use of this definition of G leads to the total number of counts in the image (FCFV only) being equal to the counts detected, while under Equation (2.7a) the integrated count is zero.…”

“…In the case that the object S is not point-like, but complex (several point sources, extended source) the image can be further deteriorated because the sidelobes associated with every source induce a cross-talk between different elements. This results in the superposition of a signal-dependent noise connected with the structure of the object itself, severely limiting the possibility of finding weak sources or resolving structures (Fenimore and Cannon, 1978). In Figure 2.6(a) and (b) the SPSF is shown for a 50% transparent random mask pattern for both auto-and balanced correlation as defined by Equation (2.7a).…”

“…Both the above expressions for the average noise term depend explicitly on the mask transparency. Fenimore (1978) and have studied how this parameter affects the reconstructed SNR with a view to optimizing the aperture design with respect to a wide range of observational conditions (number of sources, source intensities, background count rate). On a basis of purely statistical considerations, without taking into account any systematic effects, they have shown that in the typical conditions of high energy astronomy (few point sources, high background) the optimum transparency is very close to 50~o.…”