Noise parameters estimation

December 26, 2012
The Luisier et al. mixed Poisson-Gaussian denoising method[1], discussed in a previous blog post, requires knowledge of the detector noise parameters gain, offset and noise (αδσ). Luisier estimates these parameters for an input noisy image by spatial averaging local image statistics, but when these estimates are used to denoise my subframes the results suffer from various artifacts and a loss of resolution. I have found that applying a variant of the Luisier estimation process to a set of specially prepared calibration images yields a more robust noise parameters estimation.
 
The estimation results summarized in this post are for my Quantum Scientific Imaging 683wsg, binned 2 x 2 at -20° C.
 
For each detector noise parameter, my estimation process measures a sample-statistic inside every non-overlapping N x N (typically 8 x 8) pixel block of one or more block aligned calibration images. A robust spatial average is then formed by simply computing the median of the measurements. The robustness of this process may be further increased by averaging results from multiple images or cycle-spins[2].
 
I dark subtract and flat field my light subframes to remove fixed pattern noise prior to denoising[3]. Offset (i.e. bias or pedestal) is also removed by this calibration, hence by definition the detector offset parameter δ equals 0.
 
To estimate the detector noise parameter σ, I measure block local sample-noise given by the expression
  • (0.5 σ^2{d- d2})^0.5,
where d1 and d2 are dark subframes with exposure equal to light subframe exposure and σ^2{·} is block sample-variance. The subframe difference remove fixed pattern noise and the coefficient accounts for the fact that the difference increases variance by a factor of 2. A histogram of the measurements is shown below. The median equals 14.0 DN.
 
blog 2012_12_26 noise
 
To estimate the detector gain parameter α, I measure block local sample-gain given by the expression
  • (σ^2{f1 f2} - σ^2{d1 - d2}) / (μ{f1 f2} - μ{d1 + d2}),
where f1 and f2 are flat subframes, d1 and d2 are flat-dark subframes, all with equal exposure, and σ^2{·} and μ{·} are block sample-variance and block sample-mean, respectively. This expression can be derived by solving the definitions of expectation and variance of the Luisier noise model for gain. The subframe differences employed in the sample-variance terms remove fixed pattern noise. The mean of the flat and flat-dark subframes is approximately 30,000 DN and 280 DN, respectively. The flat subframe illumination was uniform to within +/- 2.5%. A histogram of the measurements is shown below. The median equals 0.93 DN. This estimate is nearly equal to the value 0.94 DN/e- derived from a photon transfer analysis by the detector manufacturer. 
 
blog 2012_12_26 gain
 
The plot below shows the estimated photon transfer characteristics and the subframe block (sample-mean, sample-standard deviation) measurements overlaid on log-log axes. The red photon transfer curve is defined by the block sample relationship
  • σ^2{·} = α(μ{·} - δ) + σ^2,
with parameters given by the estimation. The green and brown points correspond to measurements of the calibrated dark and flat subframes used by the estimation, respectively. The blue points correspond to measurements of a set of example calibrated light subframes. Calibration consists of dark subtraction and flat fielding.
 
blog 2012_12_26

The horizontal offset between the dark and light subframe measurements is due to sky background. The light subframes have no signal-free regions. The light subframes are not photon noise limited, read noise is a non-negligible component of total noise at lower intensities.
 
The high noise tail of the dark subframe measurements is due to noise structures (e.g. hot pixels and cosmic rays) accumulated during the 2400 second exposure. The plot below shows a selection of dark subframe blocks with sample-mean equal to 50 +/- 2.5% DN. All have high sample-standard deviation due to the presence of noise structures.
 
blog 2012_12_26 dark tail
 
The high noise tail of the light subframe measurements is due to bright stars. The plot below shows a selection of light subframe blocks with sample-mean equal to 5000 +/- 2.5% DN. All have high sample-standard deviation due to the presence of bright stars or their extended image profiles.
 
blog 2012_12_26 light tail
 
The light subframe measurements with least standard deviation trend above the photon transfer curve with increasing mean. This trend is due to the presence of sky signal variations.
 
[1] Luisier et al., "Fast Interscale Wavelet Denoising of Poisson-Corrupted Images", Signal Processing, 90(2):415-427, 2010 February.
 
[2] Coifman et al., "Translation invariant de-noising", in Lecture Notes in Statistics: Wavelets and Statistics, 130:125-150, Springer Verlag, New York, 1995.
 
[3] Janesick, Photon Transfer DN → λ, SPIE, Bellingham, WA, 2007.

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