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 $A = _^$, where
:$A_i = \begin 0 & \mbox i = 0, \\ 1 & \mbox i \mbox L, i \neq 0,\\ 0 & \mbox \end$
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'':
:$G_i = \begin +1 & \mbox i = 0, \\ +1 & \mbox A_i = 1, i \neq 0,\\ -1 & \mbox A_i = 0, i \neq 0, \end$
Rectangular MURA arrays are constructed in a slightly different manner, letting $A = \_ ^$, where
:$A_ = \begin 0 & \mbox i = 0, \\ 1 & \mbox j = 0, i \neq 0, \\ 1 & \mbox C_i C_j = +1, \\ 0 & \mbox \end$
and
:$C_i = \begin +1 & \mbox i \mboxp, \\ - 1 & \mbox \end$

The corresponding decoding function ''G'' is constructed as follows:
:$G_ = \begin +1 & \mbox i + j = 0; \\ +1 & \mbox A_ = 1, \ (i+j \neq 0); \\ -1 & \mbox A_ = 0, \ (i+j \neq 0),; \end$

References

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