The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

Hendrickson, Aaron

doi:10.4310/19-sii564

Cited by 3 publications

(9 citation statements)

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“…Indeed, Section 3.5.1 we will take this claim a step further by using the theoretical foundation laid in the previous chapter to show that this dominance can be achieved even at illumination levels near zero if we adopt a notation of optimal sample sizes. This observation opens the door to not only justifying the estimation of g and its confidence intervals under low illumination but also provides a pathway for design and control of experiment of g-estimation at low illumination; thus, addressing the low-illumination problem of conversion gain estimation introduced in [12]. Finally, this chapter will end by demonstrating the effectiveness of the theoretical results derived herein with an experiment measuring g under low illumination conditions.…”

Section: Photon Transfer Conversion Gain Estimationmentioning

confidence: 72%

“…The distribution of this estimator was derived in the form of the centralized inverse-Fano distribution and it was noted that this distribution had no finite moments due the density of P being nonzero at the origin. In an effort to gain some insight about this estimator, further investigations were carried out in [12] to derive the first moment in the sense of the Cauchy principal value…”

Section: Previous Workmentioning

confidence: 99%

“…In this scenario, one must capture increasingly large samples to force P(| P | < ) ≈ 0 and produce a well-behaved estimate of P −1 . This tension between the need to measure g at low-illumination and the large sample sizes it entails ultimately led to the low-illumination problem of conversion gain measurement [12].…”

Section: Introductionmentioning

confidence: 99%

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A novel approach to photon transfer conversion gain estimation

Hendrickson

2021

Preprint

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Nonuniformities in the imaging characteristics of modern image sensors are a primary factor in the push to develop a pixel-level generalization of the photon transfer characterization method. In this paper, we seek to develop a body of theoretical results leading toward a comprehensive approach for tackling the biggest obstacle in the way of this goal: a means of pixel-level conversion gain estimation. This is accomplished by developing an estimator for the reciprocal-difference of normal variances and then using this to construct a novel estimator of the conversion gain. The first two moments of this estimator are derived and used to construct exact and approximate confidence intervals for its absolute relative bias and absolute coefficient of variation, respectively. A means of approximating and computing optimal sample sizes are also discussed and used to demonstrate the process of pixel-level conversion gain estimation for a real image sensor.

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Section: Photon Transfer Conversion Gain Estimationmentioning

confidence: 72%

Section: Previous Workmentioning

confidence: 99%

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A novel approach to photon transfer conversion gain estimation

Hendrickson

2021

Preprint

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“…This result can also be obtained as a consequence of (A6) and (A7). Mathematically equivalent result expressed via the beta function can be found as Lemma 2.1 in [17].…”

Section: Definition 21 (Gamma Difference Distribution)mentioning

confidence: 99%

“…Another reason consists in the fact that the special functions can have different computer implementations with respect to speed or reliability of computations. Simultaneously we introduce a more compact and readable notation compared to Klar [3], inspired by Hendrickson [4,17].…”

Section: Definition 21 (Gamma Difference Distribution)mentioning

confidence: 99%

A practical, effective calculation of gamma difference distributions with open data science tools

Hančová,

Gajdoš,

Hanč

2021

Preprint

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At present, there is still no officially accepted and extensively verified implementation of computing the gamma difference distribution allowing unequal shape parameters. We explore four computational ways of the gamma difference distribution with the different shape parameters resulting from time series kriging, a forecasting approach based on the best linear unbiased prediction, and linear mixed models. The results of our numerical study, with emphasis on using open data science tools, demonstrate that our open tool implemented in high-performance Python(with Numba) is exponentially fast, highly accurate, and very reliable. It combines numerical inversion of the characteristic function and the trapezoidal rule with the double exponential oscillatory transformation (DE quadrature). At the double 53-bit precision, our tool outperformed the speed of the analytical computation based on Tricomi's U (a, b, z) function in CAS software (commercial Mathematica, open SageMath) by 1.5-2 orders. At the default precision of scientific numerical computational tools, it exceeded open SciPy, NumPy, and commercial MATLAB 5-10 times. The potential future application of our tool for a mixture of characteristic functions could open new possibilities for fast data analysis based on exact probability distributions in areas like multidimensional statistics, measurement uncertainty analysis in metrology as well as in financial mathematics and risk analysis. KEYWORDS numerical inversion of the characteristic function; double exponential quadrature; computational tools; high-performance Python; econometrics; time series; kriging CONTACT Martina Hančová.

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On the optimal measurement of conversion gain in the presence of dark noise

Hendrickson¹,

Haefner

Preece

2022

J. Opt. Soc. Am. A

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Working from a model of Gaussian pixel noise, we present and unify over 25 years of developments in the statistical analysis of the photon transfer conversion gain measurement. We then study a two-sample estimator of the conversion gain that accounts for the general case of non-negligible dark noise. The moments of this estimator are ill-defined (their integral representations diverge), and so we propose a method for assigning pseudomoments, which are shown to agree with actual sample moments under mild conditions. A definition of optimal sample size pairs for this two-sample estimator is proposed and used to find approximate optimal sample size pairs that allow experimenters to achieve a predetermined measurement uncertainty with as little data as possible. The conditions under which these approximations hold are also discussed. Design and control of experiment procedures are developed and used to optimally estimate a per-pixel conversion gain map of a real image sensor. Experimental results show excellent agreement with theoretical predictions and are backed up with Monte Carlo simulation. The per-pixel conversion gain estimates are then applied in a demonstration of per-pixel read noise estimation of the same image sensor. The results of this work open the door to a comprehensive pixel-level adaptation of the photon transfer method.

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The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

Cited by 3 publications

References 0 publications

A novel approach to photon transfer conversion gain estimation

A novel approach to photon transfer conversion gain estimation

A practical, effective calculation of gamma difference distributions with open data science tools

On the optimal measurement of conversion gain in the presence of dark noise

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