# Sensitivity and the Single Source

For a coded aperture camera illuminated by a single source and background, the expected number of counts in each detector element is given by: where:

\$I\$ is the source intensity.
\$r\$ is the rato of pinhole area to detector area.
\$N\$ is the number of elements in the position senstive detector.
\$A\$ is the detector area.
\$T\$ is the integration time.
\$B sub d\$ is the diffuse background rate.
\$B sub p\$ is the intrinsic detector background rate.
\$OMEGA\$ is the solid angle of the detector, as seen from the aperture.
The vector \$bold s\$ could be called the "shadow" vector: in the ideal case \$s sub i\$ is \$1\$ where the source illuminates the detector through the aperture, and \$0\$ elsewhere.

In the case of a dominant, known background, the matched filter method can be used to estimate \$I\$ from the data. The total background per detector element is: To implement the matched filter, the background must be subtracted from \$bold h\$, yielding \$bold h prime\$: The matched filter vector is then given by: and the estimated intensity is: where: \$S\$ is the "power" in the shadow pattern (in a mathematical rather than a physical sense).

Assuming that the background dominates the noise, the signal to noise ratio may be approximated by: If the projected aperture lies entirely within the boundary of the detector and the observed shadow pattern is perfectly sharp, \$S ~=~ r\$. If in addition the particle background is a negligible contributer to the noise, \$r\$ drops out of the signal to noise equation: \$PSI sub 0\$ represents an upper limit on the sensitivity of the coded aperture system.

If the background is uncertain, the expected detector response may be written as: where: I'll call the first column of the detector response matrix the "background vector", or \$bold b\$; \$b sub i = 1\$ for all \$i\$. The least squares estimates of \$I\$ and \$beta\$ may be obtained by solving the equation: At this point, it is convenient to introduce two more definitions:  \$R\$ is the average value in the shadow pattern \$bold s\$; thus \$R sup 2\$ is the portion of the power in the shadow pattern due to its nonzero average (the "DC" power), and \$S tilde\$ is the rest of the power in the shadow pattern (the "AC" power). Rewriting the least squares equation in terms of \$R\$ and \$S\$ (also noting that \$bold b cdot bold b = N\$): The solution is: The estimated source intensity is: where: [Contents] [Previous]
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