1 PoGaIN Poisson-Gaussian Image Noise Modeling from Paired Samples

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PoGaIN: Poisson-Gaussian Image Noise

Modeling from Paired Samples

Nicolas B¨

ahler∗, Majed El Helou∗,´

Etienne Objois, Kaan Okumus¸, and Sabine S¨

usstrunk, Fellow, IEEE.

Abstract—Image noise can often be accurately ﬁtted to a

Poisson-Gaussian distribution. However, estimating the distri-

bution parameters from a noisy image only is a challenging

task. Here, we study the case when paired noisy and noise-

free samples are accessible. No method is currently available

to exploit the noise-free information, which may help to achieve

more accurate estimations. To ﬁll this gap, we derive a novel,

cumulant-based, approach for Poisson-Gaussian noise modeling

from paired image samples. We show its improved performance

over different baselines, with special emphasis on MSE, effect of

outliers, image dependence, and bias. We additionally derive the

log-likelihood function for further insights and discuss real-world

applicability.

Index Terms—Image Noise, Noise Estimation, Poisson-

Gaussian Noise Modeling, Paired Samples Modeling

I. INTRODUCTION

Noise always affects image capture in any imaging pipeline.

Modeling noise distribution is thus crucial for analyzing

imaging devices, datasets [1], [2], and developing denoising

methods, especially blind ones [3]–[6]. Those approaches

include noise parameter estimation prior to the noise reduction,

and hence do not rely on the noise level being known.

Other learning-based techniques even go a step further and

are noise model-blind, meaning that no ﬁxed noise model is

imposed [7], [8]. Here, we assume a common noise model,

the Poisson-Gaussian noise model, composed of a shot and a

read noise component. The former is modeled with a Poisson

distribution, emerging from the particle nature of light whose

intensity the sensor estimates over a ﬁnite duration of time.

The latter is modeled with a Gaussian distribution, notably

for raw images that are processed by the different steps in the

image processing pipeline, which can modify the distribution.

In their seminal work, Foi et al. [9] also propose a Poisson-

Gaussian model for the noise distribution. Further, the au-

thors introduce a clever solution for ﬁtting the noise model

parameters from a noisy input image. Their algorithm begins

with local expectation and standard deviation estimates from

image parts that are assumed to depict a single underlying

intensity value. The global parametric model is then ﬁtted

through a maximum likelihood search based on the local

estimates. Multiple assumptions are made in order to reach

∗Both authors have equal contributions.

Submitted for review on October 10th, 2022, revised on November 20th,

2022, and accepted on November 26th, 2022

Work carried out in the Image and Visual Representation Lab (IVRL) at

the School of Computer and Communication Sciences, EPFL, Switzerland.

{nicolas.bahler, sabine.susstrunk}@epﬂ.ch, melhelou@ethz.ch.

Corresponding author: Nicolas B¨

ahler

All code and supplementary material at https://github.com/IVRL/PoGaIN

a ﬁnal estimate, in part due to the lack of input information

beyond the noisy image. Our premise is that when modeling

datasets or analyzing an imaging system, we may be able to

acquire paired noisy and noise-free images. We exploit this

additional information and study the problem of modeling

noise with paired samples.

We propose a novel method that estimates the parameters of

the aforementioned noise model based on noisy and noise-free

image pairs that can be used to develop new blind denoising

algorithms. The additional information of the noise-free ver-

sion of a given image enables our approach to signiﬁcantly

outperform the method introduced by Foi et al. [9]. We also

train a neural network based noise model estimator and show

that we in addition outperform this learning-based alternative.

Finally, for the sake of comparison, we introduce a variance-

based baseline method, which also takes advantage of noisy

and noise-free image pairs.

II. RELATED WORK

Denoising is one of the most fundamental tasks in image

restoration, with both theoretical impact and practical appli-

cations. Most classic denoisers, for instance PURE-LET [10],

KSVD [11], WNNM [12], BM3D [13], and EPLL [14], require

knowledge of the noise level in the input test image. Deep

learning image denoisers that have shown improved empirical

performance [15], [16] also require knowledge of noise distri-

butions, if not at test time [17], then at least for training [3],

[18]. This is due to the degradation overﬁtting of deep neural

networks [19]. Noise modeling is thus important for denoisers

at test time, but also for acquisition system analysis and

dataset modeling for training these denoisers. Past research has

focused on modeling noise from noisy images without relying

on ground truth, i.e., noise-free, information [9]. Interesting

approaches, for example Sparse Modeling [20], Dictionary

Learning [21] or non-local image denoising methods like

SAFPI [22], have been developed to push overall denoising

performance. However, none of these methods allow easy use

of noise-free data when it is available. For Poisson-Gaussian

noise modeling, for example, both FMD [1] and W2S [2] rely

on a noise modeling method that does not consider ground

truth noise-free images [9]. Hence, our approach to model

the Poisson-Gaussian Image Noise (PoGaIN) distribution ex-

ploits paired samples (noisy and noise-free images), which

signiﬁcantly improves the modeling accuracy. Our method is

based on the cumulant expansion, which is also used by other

authors to derive estimators for PoGaIN model parameters, but

for different input types, such as noisy image time series [23]

or single noisy images [24].

arXiv:2210.04866v2 [cs.CV] 19 Dec 2022

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1PoGaIN:Poisson-GaussianImageNoiseModelingfromPairedSamplesNicolasB¨ahler,MajedElHelou,´EtienneObjois,KaanOkumus¸,andSabineS¨usstrunk,Fellow,IEEE.AbstractImagenoisecanoftenbeaccuratelyttedtoaPoisson-Gaussiandistribution.However,estimatingthedistri-butionparametersfromanoisyimageonlyisachallengin...

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