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--- wiki:observational_astronomy:uniformity [2023/11/09 12:04] – Roy Prouty
+++ wiki:observational_astronomy:uniformity [2023/11/09 12:55] (current) – removed Roy Prouty
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-====== Measuring Uniformity ======
-To measure the uniformity across the detector and optical system, we take a frame of a uniformly illuminated field. It is important that the field covers the entirety of the detector. Also important is that the integration time is chosen to not saturate any pixels in the frame.
-Take $C(\lambda)$ as a uniform i.e., constant value over the FoV of the detector.
-$$L_{ij}(\lambda) = \Bigl[\epsilon_{ij}(\lambda)I_{ij}(\lambda) + T_{ij}(\lambda)\Bigr]t + B_{ij}(\lambda) ~~|~~ I_{ij}(\lambda)=C(\lambda) ~ \forall ~ i,j$$
-$$L_{ij}(\lambda) = \Bigl[\epsilon_{ij}(\lambda) C(\lambda) + T_{ij}(\lambda)\Bigr]t + B_{ij}(\lambda)$$
-Solving for the homogeneity; distribute integration time
-$$L_{ij}(\lambda) = \epsilon_{ij}(\lambda) C(\lambda)t + T_{ij}(\lambda)t + B_{ij}(\lambda)$$
-Identifying the Dark Signals and subtracting from both sides
-$$L_{ij}(\lambda) - D_{ij}(\lambda) = \epsilon_{ij}(\lambda) C(\lambda)t$$
-Dividing through by the constant illumination level
-$$\frac{L_{ij}(\lambda) - D_{ij}(\lambda)}{C(\lambda)t} = \epsilon_{ij}(\lambda)$$
-Now, each pixel in $\epsilon_{ij}(\lambda)$ represents the pixel-by-pixel deviations from uniformity. Again, $\epsilon_{ij}(\lambda)$ is measured as a fraction with each pixel carrying a (to computer precision) continuous value in the range $\epsilon_{ij}(\lambda)\in \mathbb{R}_{[0,1]}$.\\ \\
-The target field can be a radiatively flat field or Lambertian surface. That is, there are no specular components to the observed radiance field. These frames are commonly dubbed **FLAT** frames. Note that the illumination constant retains a spectral dependence. This means that FLAT frames need to be taken in each filter.\\ \\
-By construction, these frames have uniform source signal. However, the time-integrated THERMAL and BIAS signals are present in these frames as well. We must therefore calibrate the FLAT frames with their own DARK frames. Again, it is ideal to collect DARK frames of the same integration time that we used to generate our FLAT frames.