Modified Uniformly Redundant Array

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.

Mathematical Construction of MURAs
MURAs can be generated in any length L that is prime and of the form
 * $$ L = 4m +1, \ \  m = 1,2,3,...,$$

the first five such values being $$L = 5,13,17,29,37$$. The binary sequence of a linear MURA is given by $$ A = {A_i}_{i=0}^{L-1}$$, where

A_i = \begin{cases} 0 & \mbox{if } i = 0, \\ 1 & \mbox{if } i \mbox{ is a quadratic residue modulo } L, i \neq 0,\\ 0  & \mbox{otherwise} \end{cases} $$ These linear MURA arrays can also be arranged to form hexagonal MURA arrays. One may note that if $$ L = 4m + 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  & \mbox{if } i = 0, \\ +1 & \mbox{if } A_i = 1, i \neq 0,\\ -1 & \mbox{if } A_i = 0, i \neq 0, \end{cases} $$ Rectangular MURA arrays are constructed in a slightly different manner, letting $$ A = \{A_{ij}\}_ {i,j =0}^{p-1}  $$, where

A_{ij} = \begin{cases} 0 & \mbox{if } i = 0, \\ 1 & \mbox{if } j = 0, i \neq 0, \\ 1 & \mbox{if } C_i C_j = +1, \\ 0 & \mbox{otherwise,} \end{cases} $$ and

C_i = \begin{cases} +1 & \mbox{if } i \mbox{ is a quadratic residue modulo }p, \\ - 1 & \mbox{otherwise,} \end{cases} $$ The corresponding decoding function G is constructed as follows:

G_{ij} = \begin{cases} +1 & \mbox{if } i + j = 0; \\ +1 & \mbox{if } A_{ij} = 1, \ (i+j \neq 0); \\ -1 & \mbox{if } A_{ij} = 0, \ (i+j \neq 0),; \end{cases} $$