Modified Uniformly Redundant Array Explained

A modified uniformly redundant array (MURA) is a type of mask used in coded aperture imaging. They were first proposed by Gottesman and Fenimore in 1989.^[1]

Mathematical Construction of MURAs

MURAs can be generated in any length L that is prime and of the form

L = 4 m +1 , m = 1 , 2 , 3 ,..., the first five such values being L = 5 , 13 , 17 , 29 , 37 . The binary sequence of a linear MURA is given by A = {A _{i}} ^{L -1} ^{i =0} , where A _{i} = \begin{cases} 0 & if i = 0 , \\ 1 & if i isaquadraticresiduemodulo L, i ≠ 0 ,\\ 0 & otherwise \end{cases} These linear MURA arrays can also be arranged to form hexagonal MURA arrays. One may note that if L = 4 m + 3 and A _{0} = 1 , a uniformly redundant array(URA) is a generated.

As with any mask in coded aperture imaging, an inverse sequence must also be constructed. In the MURA case, this inverse G can be constructed easily given the original coding pattern A:

G _{i} = \begin{cases} +1 & if i = 0 , \\ +1 & if A _{i} = 1 , i ≠ 0 ,\\ -1 & if A _{i} = 0 , i ≠ 0 , \end{cases} Rectangular MURA arrays are constructed in a slightly different manner, letting A = \{A _{ij} \} ^{p -1} _{i,j =0} , whereA _{ij} = \begin{cases} 0 & if i = 0 , \\ 1 & if j = 0 , i ≠ 0 , \\ 1 & if C _{i} C _{j} = +1 , \\ 0 & otherwise, \end{cases} and C _{i} = \begin{cases} +1 & if i isaquadraticresiduemodulo p, \\ - 1 & otherwise, \end{cases} The corresponding decoding function G is constructed as follows:G _{ij} = \begin{cases} +1 & if i + j = 0 ; \\ +1 & if A _{ij} = 1 , (i + j ≠ 0 ); \\ -1 & if A _{ij} = 0 , (i + j ≠ 0 ),; \end{cases}

Notes and References

1. Fenimore. E. E.. Gottesman. Stephen R.. 1989-10-15. New family of binary arrays for coded aperture imaging. Applied Optics. EN. 28. 20. 4344–4352. 10.1364/AO.28.004344. 2155-3165.