Fast decoding algorithms for coded aperture systems

Abstract

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. a b s t r a c t Fast decoding algorithms are described for a number of established coded aperture systems. The fast decoding algorithms for all these systems offer significant reductions in the number of calculations required when reconstructing images formed by a coded aperture system and hence require less computation time to produce the images. The algorithms may therefore be of use in applications that require fast image reconstruction, such as near real-time nuclear medicine and location of hazardous radioactive spillage. Experimental tests confirm the efficacy of the fast decoding techniques.

Application of fast cross‐correlation algorithms

“…The prescription for the construction of A * and G * is given in [3]. If we have n primitive systems, each of order p i where i = 1, 2, …, n then the cross‐correlation of A * with G * using the fast algorithm of [10] means that the required number of multiplications N is given by where ( − 1/ p i ) is the Legendre symbol of − 1 with respect to p i [10, equation (18)]. The quantity N is less than v 2 , and in most cases significantly less, giving a reduction in the number of required multiplications, and hence a faster cross‐correlation when using product sequences over that of the standard method.…”

“…This Letter outlines a simple strategy that enables applications with parameters similar to those of the difference sets to benefit from fast cross‐correlation by means of the methods given in [10].…”

“…Although such systems find many applications, performing the crosscorrelation of (1), hereafter referred to as the standard cross-correlation method, can be very time consuming, particularly for large values of v. The standard method of (1) requires v 2 multiplications. To this end, methods for speeding up the process have been demonstrated for the URAs by Roques [9] and for the product systems by Byard [10]. In both of these cases the special ways in which the sequences are constructed enables a rearrangement of the summation of (1) in such a way as to allow fewer multiplications to be required and hence a speed up of the whole process.…”

“…The prescription for the construction of A* and G* is given in [3]. If we have n primitive systems, each of order p i where i = 1, 2, …, n then the cross-correlation of A* with G* using the fast algorithm of [10] means that the required number of multiplications N is given by…”

“…Conclusion: Many applications require the cross-correlation of binary sequences. For those sequences based on the product systems, a fast algorithm has been demonstrated that speeds up the process while preserving the linear cross-correlation approach [10]. However, for sequences derived from difference sets, no such general fast algorithm is known.…”