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

$A_i=\{\begin{array}{ll}0& \text{if }i=0,\\ 1& \text{if }i\text{ is a quadratic residue modulo }L,i\ne 0,\\ 0& \text{otherwise}\end{array}$

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:

$G_i=\{\begin{array}{ll}+1& \text{if }i=0,\\ +1& \text{if }{A}_i=1,i\ne 0,\\ -1& \text{if }{A}_i=0,i\ne 0,\end{array}$

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

$A_{ij}=\{\begin{array}{ll}0& \text{if }i=0,\\ 1& \text{if }j=0,i\ne 0,\\ 1& \text{if }{C}_i{C}_j=+1,\\ 0& \text{otherwise,}\end{array}$

and

$C_i=\{\begin{array}{ll}+1& \text{if }i\text{ is a quadratic residue modulo }p,\\ -1& \text{otherwise,}\end{array}$

A rectangular MURA mask of size 101

The corresponding decoding function G is constructed as follows:

$G_{ij}=\{\begin{array}{ll}+1& \text{if }i+j=0;\\ +1& \text{if }{A}_{ij}=1,(i+j\ne 0);\\ -1& \text{if }{A}_{ij}=0,(i+j\ne 0),;\end{array}$