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
- [math]\displaystyle{ L = 4m +1, \ \ m = 1,2,3,..., }[/math]
the first six such values being [math]\displaystyle{ L = 5,13,17,29,37 }[/math]. The binary sequence of a linear MURA is given by [math]\displaystyle{ A = {A_i}_{i=0}^{L-1} }[/math], where
- [math]\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} }[/math]
These linear MURA arrays can also be arranged to form hexagonal MURA arrays. One may note that if [math]\displaystyle{ L = 4m + 3 }[/math] and [math]\displaystyle{ A_0 = 1 }[/math], 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:
- [math]\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} }[/math]
Rectangular MURA arrays are constructed in a slightly different manner, letting [math]\displaystyle{ A = \{A_{ij}\}_ {i,j =0}^{p-1} }[/math], where
- [math]\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} }[/math]
and
- [math]\displaystyle{ C_i = \begin{cases} +1 & \mbox{if } i \mbox{ is a quadratic residue modulo }p, \\ - 1 & \mbox{otherwise,} \end{cases} }[/math]
The corresponding decoding function G is constructed as follows:
- [math]\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} }[/math]
References
- ↑ Fenimore, E. E.; Gottesman, Stephen R. (1989-10-15). "New family of binary arrays for coded aperture imaging" (in EN). Applied Optics 28 (20): 4344–4352. doi:10.1364/AO.28.004344. ISSN 2155-3165. https://www.osapublishing.org/ao/abstract.cfm?uri=ao-28-20-4344.
 |
