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

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}}

A rectangular MURA mask of size 101

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}}

gollark: So I apparently ran out of RAM.

gollark: I'm not actually sure what I did, because it seems to be fine at placing down things in a line still but also is entirely blind to *me* doing that.

gollark: Also, it's worse.

gollark: It can. My attempt at a better one takes *seconds* per move.

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References

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.

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[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.