1 Introduction
^{†}^{†}footnotetext: Corresponding author: fuying@bit.edu.cnLight is of paramount importance to photography. Night and low light place very demanding constraints on photography due to limited photon count and inescapable noise. The natural reaction is to gather more light by, e.g., enlarging aperture setting, lengthening exposure time and opening flashlight. However, each method is a tradeoff – large aperture incurs small depth of field, and is unavailable in smartphone cameras; long exposure can induce blur due to scene variations or camera motions; flash can cause color aberrations and is useful only for nearby objects.
A practical rescue for lowlight imaging is to use burst capturing [46, 28, 42, 40]
, in which a burst of images are aligned and fused to increase the signaltonoise ratio (SNR). However, burst photography can be fragile, suffering from ghosting effect
[28, 56] when capturing dynamic scenes in the presence of vehicles, humans, etc. An emerging alternative approach is to employ a neural network to automatically learn the mapping from a lowlight noisy image to its longexposure counterpart [9]. However, such a deep learning approach generally requires a large amount of labelled training data that resembles lowlight photographs in the real world. Collecting rich highquality training samples from diverse modern camera devices is tremendously laborintensive and expensive.
In contrast, synthetic data is simple, abundant and inexpensive, but its efficacy is highly contingent upon how accurate the adopted noise formation model is. The heteroscedastic Gaussian noise model [22], instead of the commonlyused homoscedastic one, approximates well the real noise occurred in daylight or moderate lowlight settings [5, 27, 28]. However, it cannot delineate the full picture of sensor noise under severely low illuminance. An illustrative example is shown in Fig. 1, where the objectionable banding pattern artifacts, an unmodeled noise component that is exacerbated in dim environments, become clearly noticeable by human eyes.
In this paper, to avoid the effect on noise model from the image processing pipeline (ISP) [9, 5, 46] converting raw data to sRGB, we mainly focus on the noise formation model for raw images. We propose a physicsbased noise formation model for extreme lowlight raw denoising, which explicitly leverages the characteristics of CMOS photosensors to better match the physics of noise formation. As shown in Fig. 2, our proposed synthetic pipeline derives from the inherent process of electronic imaging by considering how photons go through several stages. It models sensor noise in a finegrained manner that includes many noise sources such as photon shot noise, pixel circuit noise, and quantization noise. Besides, we provide a method to calibrate the noise parameters from available digital cameras. In order to investigate the generality of our noise model, we additionally introduce an extreme lowlight denoising (ELD) dataset taken by various camera devices to evaluate our model. Extensive experiments show that the network trained only with the synthetic data from our noise model can reach the capability as if it had been trained with rich real data.
Our main contributions can be summarized as follows:

We formulate a noise model to synthesize realistic noisy images that can match the quality of real data under extreme lowlight conditions.

We present a noise parameter calibration method that can adapt our model to a given camera.

We collect a dataset with various camera devices to verify the effectiveness and generality of our model.
2 Related Work
Noise removal from a single image is an extensivelystudied yet still unresolved problem in computer vision and image processing. Single image denoising methods generally rely on the assumption that both signal and noise exhibit particular statistical regularities such that they can be separated from a single observation. Crafting an analytical regularizer associated with image priors (
e.g. smoothness, sparsity, selfsimilarity, low rank), therefore, plays a critical role in traditional design pipeline of denoising algorithms [52, 48, 18, 16, 43, 15, 6, 26]. In the modern era, most single image denoising algorithms are entirely datadriven, which consist of deep neural networks that implicitly learn the statistical regularities to infer clean images from their noisy counterparts [53, 12, 45, 61, 24, 57, 10, 27]. Although simple and powerful, these learningbased approaches are often trained on synthetic image data due to practical constraints. The most widelyused additive, white, Gaussian noise model deviates strongly from realistic evaluation scenarios, resulting in significant performance declines on photographs with real noise [49, 2].To step aside the domain gap between synthetic images and real photographs, some works have resorted to collecting paired real data not just for evaluation but for training [2, 9, 54, 8, 33]. Notwithstanding the promising results, collecting sufficient real data with groundtruth labels to prevent overfitting is exceedingly expensive and timeconsuming. Recent works exploit the use of paired (Noise2Noise [38]) or single (Noise2Void [37]) noisy images as training data instead of paired noisy and noisefree images. However, they can not substantially ease the burden of labor requirements for capturing a massive amount of realworld training data.
Another line of research has focused on improving the realism of synthetic training data to circumvent the difficulties in acquiring real data from cameras. By considering both photon arrival statistics (“shot” noise) and sensor readout effects (“read” noise), the works of [46, 5] employed a signaldependent heteroscedastic Gaussian model [22] to characterize the noise properties in raw sensor data. Most recently, Wang et al. [59] proposes a noise model, which considers the dynamic streak noise, color channel heterogeneous and clipping effect, to simulate the highsensitivity noise on real lowlight color images. Concurrently, a flowbased generative model, namely Noiseflow [1] is proposed to formulate the distribution of real noise using latent variables with tractable density^{1}^{1}1Note that Noiseflow requires paired real data to obtain noise data (by subtracting the ground truth images from the noisy ones) for training.. However, these approaches oversimplify the modern sensor imaging pipeline, especially the noise sources caused by camera electronics, which have been extensively studied in the electronic imaging community [36, 29, 25, 3, 17, 19, 30, 31, 4, 58, 14]. In this work, we propose a physicsbased noise formation model stemming from the essential process of electronic imaging to synthesize the noisy dataset and show that sizeable improvements of denoising performance on real data, particularly under extremely low illuminance.
3 Physicsbased Noise Formation Model
The creation of a digital sensor raw image can be generally formulated by a linear model
(1) 
where is the number of photoelectrons that is proportional to the scene irradiation, represents the overall system gain composed by analog and digital gains, and denotes the summation of all noise sources physically caused by light or camera. We focus on the single raw image denoising problem under extreme lowlight conditions. In this context, the characteristics of are formated in terms of the sensor physical process beyond the existing noise models. Deriving an optimal regularizer to tackle such noise is infeasible, as there is no analytical solver for such a noise distribution^{2}^{2}2Even if each noise component has an analytical formulation, their summation can generally be intractable.. Therefore, we rely on a learningbased neural network pipeline to implicitly learn the regularities from data. Creating training samples for this task requires careful considerations of the characteristics of raw sensor data. In the following, we first describe the detailed procedures of the physical formation of a sensor raw image as well as the noise sources introduced during the whole process. An overview of this process is shown in Fig. 2.
3.1 Sensor Raw Image Formation
Our photosensor model is primarily based upon the CMOS sensor, which is the dominating imaging sensor nowadays [50]. We consider the electronic imaging pipeline of how incident light is converted from photons to electrons, from electrons to voltage, and finally from voltage to digital numbers, to model noise.
From Photon to Electrons.
During exposure, incident lights in the form of photons hit the photosensor pixel area, which liberates photongenerated electrons (photoelectrons) proportional to the light intensity. Due to the quantum nature of light, there exists an inevitable uncertainty in the number of electrons collected. Such uncertainty imposes a Poisson distribution over this number of electrons, which follows
(2) 
where is termed as the photon shot noise and denotes the Poisson distribution. This type of noise depends on the light intensity, i.e., on the signal. Shot noise is a fundamental limitation and cannot be avoided even for a perfect sensor. There are other noise sources introduced during the photontoelectron stage, such as photo response nonuniformity and dark current noise, reported by many previous literatures [29, 25, 58, 3]. Over the last decade, technical advancements in CMOS sensor design and fabrication, e.g., onsensor dark current suppression, have led to a new generation of digital single lens reflex (DSLR) cameras with lower dark current and better photo response uniformity [23, 41]. Therefore, we assume a constant photo response and absorb the effect of dark current noise into read noise , which will be presented next.
From Electrons to Voltage. After electrons are collected at each site, they are typically integrated, amplified and read out as measurable charge or voltage at the end of exposure time. Noise present during the electronstovoltage stage depends on the circuit design and processing technology used, and thus is referred to as pixel circuit noise [25]. It includes thermal noise, reset noise [36], source follower noise [39] and banding pattern noise [25]. The physical origin of these noise components can be found in the electronic imaging literatures [36, 25, 58, 39]. For instance, source follower noise is attributed to the action of traps in silicon lattice which randomly capture and emit carriers; banding pattern noise is associated with the CMOS circuit readout pattern and the amplifier.
By leveraging this knowledge, we consider the thermal noise , source follower noise and banding pattern noise in our model. The noise model of will be presented later. Here, we absorb multiple noise sources into a unified term, i.e. read noise
(3) 
Read noise
can be assumed to follow a Gaussian distribution, but the analysis of noise data (in Section
3.2) tells a longtailed nature of its shape. This can be attributed by the flicker and random telegraph signal components of source follower noise [25], or the dark spikes raised by dark current [36]. Therefore, we propose using a statistical distribution that can better characterize the longtail shape. Specifically, we model the read noise by a Tukey lambda distribution () [34], which is a distributional family that can approximate a number of common distributions (e.g., a heavytailed Cauchy distribution):
(4) 
where and indicate the shape and scale parameters respectively, while the location parameter is set to be zero given the zeromean noise assumption.
Banding pattern noise appears in images as horizontal or vertical lines. We only consider the row noise component (horizontal stripes) in our model, as the column noise component (vertical stripes) is generally negligible when measuring the noise data (Section 3.2). We simulate the row noise by sampling a value from a zeromean Gaussian distribution with a scale parameter , then adding it as an offset to the whole pixels within a single row.
From Voltage to Digital Numbers. To generate an image that can be stored in a digital storage medium, the analog voltage signal read out during last stage is quantized into discrete codes using an ADC. This process introduces quantization noise given by
(5) 
where
denotes the uniform distribution over the range
and is the quantization step.To summarize, our noise formation model consists of four major noise components:
(6) 
where , , , and denotes the overall system gain, photon shot noise, read noise, row noise and quantization noise, respectively.
3.2 Sensor Noise Evaluation
In this section, we present a noise parameter calibration method attached to our proposed noise formation model. According to Eq. (2) (4) (6), the necessary parameters to specify our noise model include overall system gain for photon shot noise ; shape and scale parameters ( and ) for read noise ; scale parameter for row noise . Given a new camera, our noise calibration method consists of two main procedures, i.e
. (1) estimating noise parameters at various ISO settings
^{3}^{3}3Noise parameters are generally stationary at a fixed ISO., and (2) modeling joint distributions of noise parameters.
Estimating noise parameters. We record two sequences of raw images to estimate and other noise parameters: flatfield frames and bias frames.
Flatfield frames are the images captured when sensor is uniformly illuminated. They can be used to derive according to the Photon Transfer method. [32]Once we have , we can firstly convert a raw digital signal into the number of photoelectrons , then impose a Poisson distribution on it, and finally revert it to – this simulates realistic photon shot noise.
Bias frames are the images captured under a lightless environment with the shortest exposure time. We took them at a dark room and the camera lens was capped on. Bias frames delineate the read noise picture independent of light, blended by the multiple noise sources aforementioned. The banding pattern noise can be tested via performing discrete Fourier transform on a bias frame. In Fig.
3, the highlighted vertical pattern in the centralized Fourier spectrum reveals the existence of row noise component. To analyze the distribution of row noise, we extract the mean values of each row from raw data. These values, therefore, serve as good estimates to the underlying row noise intensities, given the zeromean nature of other noise sources. The normality of the row noise data is tested by a ShapiroWilk test [55]: the resulting value is higher than, suggesting the null hypothesis that the data are normally distributed cannot be rejected. The related scale parameter
can be easily estimated by maximizing the loglikelihood.After subtracting the estimated row noise from a bias frame, statistical models can be used to fit the empirical distribution of the residual read noise. A preliminary diagnosis (Fig. 4
Left) shows the main body of the data may follow a Gaussian distribution, but it also unveils the longtail nature of the underlying distribution. In contrast to regarding extreme values as outliers, we observe an appropriate longtail statistical distribution can characterize the noise data better.
We generate a probability plot correlation coefficient (PPCC) plot [20] to identify a statistical model from a Tukey lambda distributional family [34] that best describes the data. The Tukey lambda distribution is a family of distributions that can approximate many distributions by varying its shape parameter . It can approximate a Gaussian distribution if , or derive a heavytailed distribution if . The PPCC plot (Fig. 4 Middle) is used to find a good value of . The probability plot [60] (Fig. 4 Right) is then employed to estimate the scale parameter . The goodnessoffit can be evaluated by – the coefficient of determination w.r.t. the resulting probability plot [47]. The of the fitted Tukey Lambda distribution is much higher than the Gaussian distribution (e.g., vs. ), indicating a much better fit to the empirical data.
Although we use a unified noise model for different cameras, the noise parameters estimated from different cameras are highly diverse. Figure 4 shows the selected optimal shape parameter differs camera by camera, implying distributions with varying degree of heavy tails across cameras. The visual comparisons of real and simulated bias frames are shown in Fig. 5. It shows that our model is capable of synthesizing realistic noise across various cameras, which outperforms the Gaussian noise model both in terms of the goodnessoffit measure (i.e.,
) and the visual similarity to real noise.
Real Bias Frame  Gaussian Model  Ours  
SonyA7S2 


()  (0.961)  (0.978)  
NikonD850 

()  (0.880)  (0.972) 
Modeling joint parameter distributions. To choose noise parameters for our noise formation model, we infer the joint distributions of (, ) and (, ), from the parameter samples estimated at various ISO settings. As shown in Fig. 6, we use the linear least squares method to find the line of best fit for two sets of logscaled measurements. Our noise parameter sampling procedure is
(7)  
where denotes a uniform distribution and denotes a Gaussian distribution with mean
. and are the estimated overall system gains at the minimum and maximum ISO of a camera respectively. and indicate the fitted line’s slope and intercept respectively.is an unbiased estimator of standard deviation of the linear regression under the Gaussian error assumption. For shape parameter
, we simply sample it from the empirical distribution of the estimated parameter samples.Noisy image synthesis. To synthesize noisy images, clean images are chosen and divided by low light factors sampled uniformly from to simulate low photon count in the dark. Noise is then generated and added to the scaled clean samples, according to Eq. (6) (7). The created noisy images are finally normalized by multiplying the same low light factors to expose bright but excessively noisy contents.
(a) Image capture setup  (b) example images 
Model  PSNR / SSIM  PSNR / SSIM  PSNR / SSIM 

BM3D  32.92 / 0.758  29.56 / 0.686  28.88 / 0.674 
ABM3D  33.79 / 0.743  27.24 / 0.518  26.52 / 0.558 
Paired real data  38.60 / 0.912  37.08 / 0.886  36.29 / 0.874 
Noise2Noise  37.42 / 0.853  33.48 / 0.725  32.37 / 0.686 
36.10 / 0.800  31.87 / 0.640  30.99 / 0.624  
+  37.08 / 0.839  32.85 / 0.697  31.87 / 0.665 
+  38.31 / 0.884  34.39 / 0.765  33.37 / 0.730 
+  39.10 / 0.911  36.46 / 0.869  35.69 / 0.855 
++  39.23 / 0.912  36.89 / 0.877  36.01 / 0.864 
+++  39.27 / 0.914  37.13 / 0.883  36.30 / 0.872 
4 Extreme Lowlight Denoising (ELD) Dataset
To systematically study the generality of the proposed noise formation model, we collect an extreme lowlight denoising (ELD) dataset that covers 10 indoor scenes and 4 camera devices from multiple brands (SonyA7S2, NikonD850, CanonEOS70D, CanonEOS700D). We also record bias and flat field frames for each camera to calibrate our noise model. The data capture setup is shown in Fig. 7. For each scene and each camera, a reference image at the base ISO was firstly taken, followed by noisy images whose exposure time was deliberately decreased by low light factors to simulate extreme low light conditions. Another reference image then was taken akin to the first one, to ensure no accidental error (e.g. drastic illumination change or accidental camera/scene motion) occurred. We choose three ISO levels (800, 1600, 3200)^{4}^{4}4Most modern digital cameras are ISOinvariant when ISO is set higher than 3200 [13]. and two low light factors (100, 200) for noisy images to capture our dataset, resulting in 240 (32104) raw image pairs in total. The hardest example in our dataset resembles the image captured at a “pseudo” ISO up to 640000 (3200200).
5 Experiments
5.1 Experimental setting
Implementation details. A learningbased neural network pipeline is constructed to perform lowlight raw denoising. We utilize the same UNet architecture [51] as [9]. Raw Bayer images from SID Sony training dataset [9] are used to create training data. We pack the raw Bayer images into four channels (RGBG) and crop nonoverlapped regions augmented by random flipping/rotation. Our approach only use the clean raw images, as the paired noisy images are generated by the proposed noise model onthefly. Besides, we also train networks based upon other training schemes as references, including training with paired real data (short exposure and long exposure counterpart) and training with paired real noisy images (i.e., Noise2Noise [38]).
Our implementation^{5}^{5}5Code is released at https://github.com/Vandermode/NoiseModel
is based on PyTorch. We train the models with 200 epoch using
loss and Adam optimizer [35] with batch size . The learning rate is initially set to , then halved at epoch 100, and finally reduced to at epoch 180.Competing methods. To understand how accurate our proposed noise model is, we compare our method with:
(a) Noise2Noise  (b) Paired real data  (c) Ground Truth 
(d)  (e) +  (f) +++ 
Camera  Index  Nondeep  Training with real data  Training with synthetic data  

BM3D [15]  ABM3D [44]  Paired data [9]  Noise2Noise [38]  G  G+P [22]  Ours  
SonyA7S2  PSNR  
SSIM  
PSNR  
SSIM  
NikonD850  PSNR  
SSIM  
PSNR  
SSIM  
CanonEOS70D  PSNR  
SSIM  
PSNR  
SSIM  
CanonEOS700D  PSNR  
SSIM  
PSNR  
SSIM 
Input  BM3D  ABM3D  G  G+P  Noise2Noise  Paired real data  Ours 

Input  BM3D  ABM3D  G  G+P  Noise2Noise  Paired real data  Ours 

5.2 Results on SID Sony dataset
Single image raw denoising experiment is firstly conducted on images from SID Sony validation and test sets. For quantitative evaluation, we focus on indoor scenes illuminated by natural lights, to avoid flickering effect of alternating current lights [2] ^{8}^{8}8Alternating current light is not noise, but a type of illumination that breaks the irradiance constancy between short/long exposure pairs, making the quantitative evaluation inaccurate.. To account for the imprecisions of shutter speed and analog gain [2], a single scalar is calculated and multiplied into the reconstructed image to minimize the mean square error evaluated by the ground truth.
Ablation study on noise models. To verify the efficacy of the proposed noise model, we compare the performance of networks trained with different noise models developed in Section 3.1. All noise parameters are calibrated using the ELD dataset, and sampled with a process following (or similar to) Eq. (7). The results of the other methods described in Section 5.1 are also presented as references.
As shown in Table 1, the domain gap is significant between the homoscedastic/heteroscedastic Gaussian models and the de facto noise model (characterized by the model trained with paired real data). This can be attributed to (1) the Gaussian approximation of Possion distribution is not justified under extreme low illuminance; (2) horizontal bandings are not considered in the noise model; (3) longtail nature of read noise is overlooked. By taking all these factors into account, our final model, i.e. +++
gives rise to a striking result: the result is comparable to or sometimes even better than the model trained with paired real data. Besides, training only with real lowlight noisy data is not effective enough, due to the clipping effects (that violates the zeromean noise assumption) and the large variance of corruptions (that leads to a large variance of the Noise2Noise solution)
[38]. A visual comparison of our final model and other methods is presented in Fig. 8, which shows the effectiveness of our noise formation model.Though we only quantitatively evaluate the results on indoor scenes of the SID Sony set, our method can be applied to outdoor scenes as well. The visual comparisons of both indoor and outdoor scenes from SID Sony set are presented in Fig. 9. It can be seen that the random noise can be suppressed by the model learned with heteroscedastic Gaussian noise (G+P) [22], but the resulting colors are distorted, the banding artifacts become conspicuous, and the image details are barely discernible. By contrast, our model produces visually appealing results as if it had been trained with paired real data.
5.3 Results on our ELD dataset
Method comparisons. To see whether our noise model can be applicable to other camera devices as well, we assess model performance on our ELD dataset. Table 2 and Fig. 10 summarize the results of all competing methods. It can be seen that the nondeep denoising methods, i.e. BM3D and ABM3D, fail to address the banding residuals, the color bias and the extreme values presented in the noisy input, whereas our model recovers vivid image details which can be hardly perceived on the noisy image by human observers. Moreover, our model trained with synthetic data even often outperforms the model trained with paired real data. We note the finding here conforms with the evaluation of sensor noise presented in Section 3.2, especially in Fig. 4 and 5, where we show the underlying noise distribution varies camera by camera. Consequently, training with paired real data from SID Sony camera inevitably overfits to the noise pattern merely existed on the Sony camera, leading to suboptimal results on other types of cameras. In contrast, our model relies on a very flexible noise model and a noise calibration process, making it adapts to noise characteristics of other (calibrated) camera models as well. Additional evidence can be found in Fig. 11, where we apply these two models to an image captured by a smartphone camera. Our reconstructed image is clearer and cleaner than what is restored by the model trained with paired real data.
Training with more synthesized data. A useful merit of our approach against the conventional training with paired real data, is that our model can be easily incorporated with more real clean samples to train. Fig. 11(a) shows the relative improvements of our model when training with the dataset synthesized by additional clean raw images from MIT5K dataset [7]. We find the major improvements, as shown in Fig. 13, are owing to the more accurate color and brightness restoration. By training with more raw image samples from diverse cameras, the network learns to infer picture appearances more naturally and precisely.
Sensitivity to noise calibration. Another benefit of our approach is we only need clean samples and a noise calibration process to adapt to a new camera, in contrast to capturing real noisy images accompanied with denselylabeled ground truth. Besides, the noise calibration process can be simplified once we already have a collection of parameter samples from various cameras. Fig. 11(b) shows models can reach comparable performance on target cameras without noise calibration, by simply sampling parameters from other three calibrated cameras instead.
6 Conclusion
We have presented a physicsbased noise formation model together with a noise parameter calibration method to help resolve the difficulty of extreme lowlight denoising. We revisit the electronic imaging pipeline and investigate the influential noise sources overlooked by existing noise models. This enables us to synthesize realistic noisy raw data that better match the underlying physical process of noise formation. We systematically study the efficacy of our noise formation model by introducing a new dataset that covers four representative camera devices. By training only with our synthetic data, we demonstrate a convolutional neural network can compete with or sometimes even outperform the network trained with paired real data.
Acknowledgments We thank Tianli Tao for the great help in collecting the ELD dataset. This work was partially supported by the National Natural Science Foundation of China under Grants No. 61425013 and No. 61672096.
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