title: Modified Uniformly Redundant Array
text: # 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:
  
      $G_{ij}={\begin{cases}+1&{\mbox{if }}i+j=0;\\+1&{\mbox{if }}A_{ij}=1,\ (i+j\neq 0);\\-1&{\mbox{if }}A_{ij}=0,\ (i+j\neq 0),;\end{cases}}$
  