Dr. Stanley Chan (Purdue University) has a preprint out titled "On the Insensitivity of Bit Density to Read Noise in One-bit Quanta Image Sensors" on arXiv. This paper presents a rigorous theoretical analysis of an intuitive but curious observation that was first made in the paper by E. Fossum titled "Analog read noise and quantizer threshold estimation from Quanta Image Sensor Bit Density."
Why is the quanta image sensor bit density insensitive to read noise at high enough exposure values?
The one-bit quanta image sensor is a photon-counting device that produces binary measurements where each bit represents the presence or absence of a photon. In the presence of read noise, the sensor quantizes the analog voltage into the binary bits using a threshold value q. The average number of ones in the bitstream is known as the bit-density and is often the sufficient statistics for signal estimation. An intriguing phenomenon is observed when the quanta exposure is at unity and the threshold is q=0.5. The bit-density demonstrates a complete insensitivity as long as the read noise level does not exceeds a certain limit. In other words, the bit density stays at a constant independent of the amount of read noise. This paper provides a mathematical explanation of the phenomenon by deriving conditions under which the phenomenon happens. It was found that the insensitivity holds when some forms of the symmetry of the underlying Poisson-Gaussian distribution holds.
The paper concludes:
The insensitivity of the bit density of a 1-bit quanta image sensor is analyzed. It was found that for a quanta exposure θ = 1 and an analog voltage threshold q = 0.5, the bit density D is nearly a constant whenever the read noise satisfies the condition σ ≤ 0.4419. The proof is derived by exploiting the symmetry of the Gaussian cumulative distribution function, and the symmetry of the Poisson probability mass function at the threshold k = 0.5. An approximation scheme is introduced to provide a simplified estimate where σ ≤ 1/√2π = 0.4. In general, the analysis shows that the insensitivity of the bit density is more of a (very) special case of the 1-bit quantized Poisson-Gaussian statistics. Insensitivity can be observed when the quanta exposure θ is an integer and the threshold is q = θ−0.5. As soon as the pair (θ, q) deviates from this configuration, the insensitivity will no longer appear.
Complete article can be downloaded here: https://arxiv.org/pdf/2203.06086
An early-access version of Eric's paper is available here: https://ieeexplore.ieee.org/document/9729893
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