# 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.[1]

## Mathematical Construction of MURAs

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

${\displaystyle L=4m+1,\ \ m=1,2,3,...,}$

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

${\displaystyle A_{i}={\begin{cases}0&{\mbox{if }}i=0,\\1&{\mbox{if }}i{\mbox{ is a quadratic residue modulo }}L,i\neq 0,\\0&{\mbox{otherwise}}\end{cases}}}$

These linear MURA arrays can also be arranged to form hexagonal MURA arrays. One may note that if ${\displaystyle L=4m+3}$ and ${\displaystyle A_{0}=1}$, 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:

${\displaystyle G_{i}={\begin{cases}+1&{\mbox{if }}i=0,\\+1&{\mbox{if }}A_{i}=1,i\neq 0,\\-1&{\mbox{if }}A_{i}=0,i\neq 0,\end{cases}}}$

Rectangular MURA arrays are constructed in a slightly different manner, letting ${\displaystyle A=\{A_{ij}\}_{i,j=0}^{p-1}}$, where

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

and

${\displaystyle C_{i}={\begin{cases}+1&{\mbox{if }}i{\mbox{ is a quadratic residue modulo }}p,\\-1&{\mbox{otherwise,}}\end{cases}}}$
A rectangular MURA mask of size 101

The corresponding decoding function G is constructed as follows:

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

## References

1. ^ Fenimore, E. E.; Gottesman, Stephen R. (1989-10-15). "New family of binary arrays for coded aperture imaging". Applied Optics. 28 (20): 4344–4352. doi:10.1364/AO.28.004344. ISSN 2155-3165.