# 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 six 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:

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