=Paper= {{Paper |id=Vol-3943/paper21 |storemode=property |title=Some aspects of real-time image denoising influenced by shot noise and compound Poisson noise |pdfUrl=https://ceur-ws.org/Vol-3943/paper21.pdf |volume=Vol-3943 |authors=Oleg Kobylin,Oleksandra Putiatina |dblpUrl=https://dblp.org/rec/conf/doors/KobylinP25 }} ==Some aspects of real-time image denoising influenced by shot noise and compound Poisson noise== https://ceur-ws.org/Vol-3943/paper21.pdf 
                         Oleg Kobylin et al. CEUR Workshop Proceedings                                                                                                         109–117


                         Some aspects of real-time image denoising influenced by
                         shot noise and compound Poisson noise
                         Oleg Kobylin, Oleksandra Putiatina
                         Kharkiv National University of Radioelectronics, 14 Nauki Ave., Kharkiv, 61166, Ukraine


                                      Abstract
                                      This paper introduces an alternative image model for video data that is represented as a series of images affected
                                      by shot noise. This type of noise not only disrupts the current frame in the video sequence but also goes over to
                                      subsequent frames, gradually diminishing over time until it vanishes. The shot-noise process is characterized by
                                      a sequence of jumps that decay as time passes. Under specific conditions, this process converges to a Gaussian
                                      distribution. To tackle this issue, the Kalman filter is proposed as a solution for removing noise and restoring the
                                      compromised image sequence. Numerical experiments demonstrate the effectiveness of the proposed approach
                                      in denoising videos corrupted by shot noise. The results of the proposed method were compared to the results
                                      provided by spatial Wiener filter, median filter, bilateral filter and a multilayer perceptron model. PSNR was
                                      calculated for the above methods.

                                      Keywords
                                      video, sequence of images, Gaussian process, compound Poisson noise, denoising, Kalman filter




                         1. Introduction
                         In certain applications digital devices can introduce noise to the original image or sequence of images.
                         While there are various image denoising techniques that are effective at restoring noisy images, shot
                         noise still remains mostly unexplored. Currently, no satisfactory algorithm exists that can effectively
                         denoise images sequence of images affected by shot noise.
                            We propose a novel method for restoring images affected by shot noise. Unlike noise that vanishes
                         immediately, shot noise diminishes gradually over time. This type of noise arises from defects in the
                         hardware of a device or issues within the camera sensor [1]. Shot noise, driven by a Poisson process,
                         exhibits a specific pattern of gradual decay following each occurrence.
                            When the intensity of the underlying Poisson process is high, shot noise can be effectively approxi-
                         mated by Gaussian noise. Numerous algorithms have been proposed in the literature to address various
                         types of noise. These include linear filters, such as median and mean filters, as well as non-linear filters,
                         which are discussed in sources [2, 3, 4]. Analysis and comparison of different image denoising methods
                         is discussed in [5]. Such methods can also be employed for mitigating shot noise. If the jump rate is
                         significant and the individual jumps are relatively small, the resulting shot noise increasingly resembles
                         Gaussian noise. There are several established techniques for filtering Gaussian noise, including the
                         bilinear filter, anisotropic diffusion filter, and Kernel Regression filter, as detailed in [6, 7, 8, 9].
                            The goal is to create a method for restoring images affected by shot noise by utilizing a Gaussian
                         approximation of the shot-noise process in combination with Kalman filtering.




                          doors-2025: 5th Edge Computing Workshop, April 4, 2025, Zhytomyr, Ukraine
                          " oleg.kobylin@nure.ua (O. Kobylin); oleksandra.putiatina@nure.ua (O. Putiatina)
                          ~ https://nure.ua/staff/oleg-anatoliyovich-kobilin (O. Kobylin); https://nure.ua/staff/oleksandra-ievgenivna-putjatina
                          (O. Putiatina)
                           0000-0003-0834-0475 (O. Kobylin); 0000-0003-4853-7125 (O. Putiatina)
                                     Β© 2025 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).


CEUR
                  ceur-ws.org
Workshop      ISSN 1613-0073
Proceedings

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Oleg Kobylin et al. CEUR Workshop Proceedings                                                                      109–117


2. Jump regression analysis for modeling noisy image
A standard two-dimensional grayscale image can be represented using the following jump regression
equation [10]:
                        𝐼𝑖𝑗 = 𝑓 (π‘₯𝑖 , 𝑦𝑗 ) + πœ–π‘–π‘— , 𝑖 = 1, ..., 𝑛; 𝑗 = 1, ..., π‘š.,              (1)
where

    β€’ (π‘₯𝑖 , 𝑦𝑗 ) is the (𝑖, 𝑗)π‘‘β„Ž pixel;
    β€’ 𝑓 (π‘₯𝑖 , 𝑦𝑗 ) is the true image intensity level at (π‘₯𝑖 , 𝑦𝑗 );
    β€’ πœ–π‘–π‘— is the pointwise noise;
    β€’ 𝐼𝑖𝑗 is the observed image intensity level at (π‘₯𝑖 , 𝑦𝑗 ).

  The sequence of 2-D images can be modeled in the following way:

                                              πΌπ‘–π‘—π‘˜ = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) + πœ–π‘–π‘—π‘˜ ,                                          (2)

where

    β€’ 𝑖 = 1, ..., 𝑛; 𝑗 = 1, ..., π‘š., π‘˜ = 1, ..., 𝑙.
    β€’ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) is the (𝑖, 𝑗)π‘‘β„Ž pixel of the π‘˜ π‘‘β„Ž image;
    β€’ π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) is the true image intensity level at (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) of the π‘˜ π‘‘β„Ž image;
    β€’ πœ–π‘–π‘—π‘˜ is the pointwise noise of the π‘˜ π‘‘β„Ž image;
    β€’ πΌπ‘–π‘—π‘˜ is the observed image intensity level at (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) of the π‘˜ π‘‘β„Ž image.

  Therefore, the sequences π‘“π‘˜ (π‘₯, 𝑦), π‘˜ = 1, 2, . . . , 𝑙 are the 2-D profiles, and monitoring the image
sequence is equivalent to monitoring the 2-D profile sequence.

2.1. Shot noise process and its approximation
Shot noise is a type of random noise marked by abrupt intensity fluctuations that dissipate over time.
This phenomenon frequently appears in real-time image processing, where individual pixels may
experience sudden intensity spikes caused by defects in image recording devices. The impact of shot
noise diminishes with time, resulting in reduced interference on the same pixel in subsequent frames of
the image sequence.
  Shot noise πœ†π‘‘ is defined in the following way:
                                              𝑀𝑑
                                              βˆ‘οΈ                                                 𝑀𝑑
                                                                                                 βˆ‘οΈ
                                    βˆ’π›Ώπ‘‘                    βˆ’π›Ώ(π‘‘βˆ’π‘ π‘– )            βˆ’π›Ώπ‘‘        βˆ’π›Ώπ‘‘
                        πœ†π‘‘ = πœ†0 𝑒         +         π‘Œπ‘– 𝑒               = πœ†0 𝑒         +𝑒               π‘Œπ‘– 𝑒𝛿𝑠𝑖 ,
                                              𝑖=1                                                𝑖=1

where

    β€’ πœ†0 is the initial value of πœ†π‘‘ ;
    β€’ {π‘Œπ‘– }𝑖=1,2,... is the sequence of iid random variables with distribution function 𝐹 (𝑦) and 𝐸(π‘Œπ‘– ) =
      πœ‡1 ;
    β€’ {𝑠𝑖 }𝑖=1,2,... is the sequence representing the event times of a Poisson process 𝑀𝑑 with constant
      intensity 𝜌;
    β€’ 𝛿 is the rate of exponential decay.

  The distribution of the random variables {π‘Œπ‘– } can be arbitrary, for example normal distribution or
beta distribution [11].
  The expectation of the shot noise πœ†π‘‘ , assuming that πœ†0 is known, is as follows from [11]:
                                 πœ‡1 𝜌 (︁        πœ‡1 𝜌 )︁ βˆ’π›Ώπ‘‘    πœ‡1 𝜌
                        𝐸(πœ†π‘‘ ) =     + πœ†0 βˆ’            𝑒    β†’       π‘Žπ‘  𝑑 β†’ ∞,                     (3)
                                  𝛿              𝛿              𝛿


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and, moreover, if the initial value πœ†0 equals πœ‡1 𝜌/𝛿, then we have a stationary case and the mean value
𝐸(πœ†π‘‘ ) will be equal to πœ‡1 𝜌/𝛿 and will not depend on time 𝑑.
  The variance of the shot noise process πœ†π‘‘ is as in [11]:
                                           πœ‡2 𝜌                πœ‡2 𝜌
                             𝑉 π‘Žπ‘Ÿ(πœ†π‘‘ ) =        (1 βˆ’ π‘’βˆ’2𝛿𝑑 ) β†’      π‘Žπ‘  𝑑 β†’ ∞.                         (4)
                                           2𝛿                  2𝛿
  Consider the following linear transformation:

                                               (𝑝)     πœ†π‘‘ βˆ’ πœ‡1 𝜌/𝛿
                                             𝑍𝑑      = βˆšοΈ€          .                                  (5)
                                                          πœ‡2 𝜌/2𝛿
                                                        (𝑝)
   The main result of the paper [[11]] is that 𝑍𝑑 (the normalization of shot noise or linear trans-
formation) converges to some 𝑍𝑑 that is normally distributed. Assume that 𝜌 β†’ ∞ and that πœ†0 is a
random variable independent of everything else, such that (πœ†0 βˆ’ (πœ‡1 𝜌/𝛿))(πœ‡1 𝜌/2𝛿)βˆ’1/2 converges in
                           (𝑝)
distribution to 𝑍0 . Then 𝑍𝑑 converges in law to 𝑍𝑑 , where
                                                       √
                                    𝑑𝑍𝑑 = βˆ’π›Ώπ‘π‘‘ 𝑑𝑑 + 2𝛿𝑑𝐡𝑑 ,                                     (6)

where 𝐡𝑑 is standard Brownian motion.
  This implies that 𝑍𝑑 is normally distributed with mean 𝐸(𝑍𝑑 ) = 𝑍0 π‘’βˆ’π›Ώπ‘‘ β†’ 0 π‘Žπ‘  𝑑 β†’ ∞ and variance
𝑉 π‘Žπ‘Ÿ(𝑍𝑑 ) = 1 βˆ’ π‘’βˆ’2𝛿𝑑 β†’ 1 π‘Žπ‘  𝑑 β†’ ∞. If πœ†0 = πœ‡1 𝜌/𝛿, then 𝑍0 = 0 and, therefore, 𝐸(𝑍𝑑 ) = 0.
  Following the linear transformation (5) the shot-noise process πœ†π‘‘ has the following form:
                                                         βˆšοΈ‚
                                               πœ‡1 𝜌    𝜌    πœ‡2 𝜌
                                         πœ†π‘‘ =       + 𝑍𝑑         .
                                                𝛿           2𝛿

  Define πœ†Λ†π‘‘ as Gaussian approximation of πœ†π‘‘ as follows:
                                                      βˆšοΈ‚
                                      Λ†     πœ‡1 𝜌         πœ‡2 𝜌
                                      πœ†π‘‘ =       + 𝑍𝑑         .
                                              𝛿          2𝛿
  When the intensity of jumps is relatively high, shot noise can be effectively approximated as Gaussian
noise. This allows for the application of Kalman filtering techniques in real-time image restoration
processes. A series of 2-D images affected by shot noise can be represented through the following
model:
                                      πΌπ‘–π‘—π‘˜ = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) + πœ†π‘–π‘—π‘˜ ,                                 (7)
where

    β€’ 𝑖 = 1, ..., 𝑛; 𝑗 = 1, ..., π‘š., π‘˜ = 1, ..., 𝑙.;
    β€’ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) is the (𝑖, 𝑗)π‘‘β„Ž pixel of the π‘˜ π‘‘β„Ž image;
    β€’ π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) is the true image intensity level at (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) of the π‘˜ π‘‘β„Ž image;
    β€’ πœ†π‘–π‘—π‘˜ is the pointwise shot-noise of the π‘˜ π‘‘β„Ž image;
    β€’ πΌπ‘–π‘—π‘˜ is the observed image intensity level at (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) of the π‘˜ π‘‘β„Ž image.

   Applying the Gaussian approximation of the shot noise we obtain the following model of the sequence
of the 2-D images:
                                                                               βˆšοΈ‚
                                            Λ†                        πœ‡1 𝜌         πœ‡2 𝜌
                   πΌπ‘–π‘—π‘˜ = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) + πœ†π‘–π‘—π‘˜ = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) +      + 𝑍𝑑         ,            (8)
                                                                      𝛿           2𝛿
where

    β€’ 𝑍𝑑 is normally distributed with mean 0 and variance 1 as 𝑑 β†’ ∞;
    β€’ πœ‡1 and πœ‡2 are first and second initial moment of the random variable π‘Œ ;



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    β€’ 𝜌 is the intensity of the underlying Poisson process;
    β€’ 𝛿 is the rate of exponential decay.
   There occurs a correction term πœ‡π›Ώ1 𝜌 in equation (8), which is constant additive part to the image
intensity. Therefore
                                                                            βˆšοΈ‚
                                                Λ†                              πœ‡2 𝜌
                       πΌπ‘–π‘—π‘˜ = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) + πœ†π‘–π‘—π‘˜ = π‘”π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) + 𝑍𝑑         ,              (9)
                                                                               2𝛿
where
                                                                         πœ‡1 𝜌
                                   π‘”π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) +        .                        (10)
                                                                          𝛿

2.2. Compound Poisson process and its approximation
Compound Poisson noise process is defined in the following way:
                                                         𝑀𝑑
                                                         βˆ‘οΈ
                                                  𝐽𝑑 =         π‘Œπ‘– .
                                                         𝑖=1

This definition is a special case of the shot-noise process with 𝛿 = 0.
We recall the result from previous subsection:

                                    (𝜌)       𝐽𝑑 βˆ’ πœ‡1 𝜌
                                  𝑉𝑑      =    √        β†’ 𝐡𝑑 , π‘Žπ‘  𝜌 β†’ ∞,
                                                 πœ‡2 𝜌

where 𝐽𝑑 = 𝑀 𝑖=1 π‘Œπ‘– , 𝑀𝑑 is Poisson process with intensity 𝜌 and 𝐡𝑑 is Brownian motion.
          βˆ‘οΈ€ 𝑑
  This means that compound Poisson process

                                       𝐽𝑑 = πœ‡2 𝜌/π‘‰π‘‘πœŒ + πœ‡1 𝜌
                                            βˆšοΈ€


can be approximated by
                                                       √
                                              𝐽ˆ 𝑑 = 𝐡𝑑 πœ‡2 𝜌 + πœ‡1 𝜌
.
  Applying the Gaussian approximation of the compound Poisson noise process we obtain the following
approximate model of the sequence of the 2-D images. Therefore
                                                                             √
                       πΌπ‘–π‘—π‘˜ = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) + 𝐽ˆ π‘–π‘—π‘˜ = π‘”π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) + 𝐡𝑑 πœ‡2 𝜌,             (11)

where
                                                                   πœ‡1 𝜌
                                   π‘”π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) = π‘“π‘˜ (π‘₯π‘–π‘˜ , π‘¦π‘—π‘˜ ) +  .                             (12)
                                                                    𝛿
   It can be noticed that both shot-noise and compound Poisson noise process both have Gaussian
approximations, but these noise processes have different impact on the images. Shot noise decays and
disappears as time passes. But the Compound Poisson noise process does not disappear, its effect stays
at that part of the image where it occurred.
   Both types of noise can be filtered out using Kalman filter if the intensity of the underlying Poisson
process is high and thus the Gaussian approximation holds.

2.3. Kalman filtering for image restoration
Kalman filtering is used to estimate the variables of the control system subject to stochastic disturbances
caused by noisy measurements of the input variables [12, 13, 14, 15]. There are two kinds of equations
in Kalman filter: time update equations and measurement update equations. The time update equations
obtain a priory estimates of the state and covariance matrix. The measurement update equations
improve the estimate of the state by using new measurement.



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  We assume that the image sequence is captured by several cameras and the noise interferes during
the image acquisition process. Then the image sequence is processed for further analysis. The aim is to
construct the filter that would efficiently denoise the image sequence and would produce acceptable
results with such kind of noise. The representation of the image for the Kalman filter is pixelwise.
  The noise free pixel value is π‘“π‘˜ , which is assumed to be the first order autoregression model. The
process looks as follows:
                                            π‘“π‘˜+1 = π›Όπ‘“π‘˜ + π‘£π‘˜ ,                                       (13)
where

    β€’ 𝛼 is a constant and depends on signal parameters;
    β€’ π‘£π‘˜ is Gaussian noise with zero mean and 𝜎 2 variance.

  Such a model is often used to represent pixel values in video signals.
  The measured signal is given by the following equation:

                                                πΌπ‘˜ = π‘“π‘˜ + π‘€π‘˜ ,                                         (14)

where π‘€π‘˜ the independent of π‘£π‘˜ additive zero mean Gaussian white noise with variance 𝜎 2 .
We need to construct a linear unbiased estimate 𝑓ˆ π‘˜ of π‘“π‘˜ having the observations 𝐼1 , 𝐼2 , ..., πΌπ‘˜ . The
estimation error is denoted by π‘“Λœ π‘˜ = π‘“π‘˜ βˆ’ 𝑓ˆ π‘˜ . The variance of the error is denoted by 𝑝
                                                                                          Λœπ‘˜ and defined as
follows:
                                              Λœπ‘˜ = 𝐸[(π‘“Λœ π‘˜ )2 ].
                                              𝑝
  The discrete Kalman filter is assessed through the iterative application of the following equations:

    β€’ filter equations;
                                                   π‘“π‘˜* = π‘Ž2 𝑓ˆ π‘˜βˆ’1 ,                                   (15)
                                           𝑓ˆ π‘˜ = π‘“π‘˜* + πΎπ‘˜ {πΌπ‘˜ βˆ’ π‘“π‘˜* };                                (16)
    β€’ variance of the estimation error and the coefficient 𝐾;

                                                𝑝*π‘˜ = π‘Ž2 𝑝
                                                         Λœπ‘˜βˆ’1 + πœŽπ‘£2 ,                                  (17)

                                            πΎπ‘˜ = 𝑝*π‘˜ {𝑝*π‘˜ + πœŽπ‘€
                                                             2 βˆ’1
                                                               } ,                                     (18)
                                                 Λœπ‘˜ = 𝑝*π‘˜ βˆ’ πΎπ‘˜ 𝑝*π‘˜ .
                                                 𝑝                                                     (19)

  Note that 𝑝*π‘˜+1 = 𝐸[(π‘Ž2 π‘“Λœ π‘˜ + πœŽπ‘£ )2 ]. This algorithm is applied to each pixel and at each time instant.
  The estimated pixel value 𝑓ˆ π‘˜ the output for each iteration π‘˜. It provides the filtered image for the
further analysis.
  We consider the following parameters for the shot-noise process:

    β€’ intensity of the Poisson process equals 1.04,
    β€’ jump size y is normally distributed N(0, 0.007).

   The shot-noise and compound Poisson process look as shown in figures 1 and 2.
   The example consists of a sequence of 2D RGB images, where certain pixels in the video are affected
by a shot noise process, leading to corruption as noted in [16]. A Kalman filter is then employed to
process the sequence of images corrupted by shot noise, effectively restoring them. Figures 3 and 4
illustrate the example.
   Figure 3 shows one image taken from the original video sequence of images. The original image
sequence is not corrupted by any noise. Figure 4(a) shows the same image as in figure 3 but corrupted
by the additive shot-noise. One can see that not the whole image is corrupted but only a part of it (the
upper part). This means that not every pixel is noisy but only the pixels on the upper side. In real
life this may occur due to the problems with the camera sensor. Next, the shot-noise corrupted image



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Figure 1: Compound Poisson process.




Figure 2: Shot noise process.


sequence is denoised using the Kalman filtering techniques. In figure 4(b) the corresponding denoised
image is shown. By comparing figures 3 and 4(b) (calculating the MSE), one can evaluate the quality of
restoration. Although, the noise is not removed completely, the quality of the image becomes much
better. The similar results concerning compound Poisson noise are show in figures 4(c) and 4(d).
    The results of the performance of other denoising techniques, such as spatial Wiener filter, median
filter, bilateral filter, multilayer perceptron model, comparing to the proposed approach are provided in
table 1.
    The higher PSNR value value means the better image filtration quality, therefore, the use of proposed
filtering approach (Kalman filtering) gave better results amoung other filters.



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Figure 3: Original image.




            (a)                       (b)                      (c)                      (d)
Figure 4: (a) shot noise corrupted image, (b) restored image without shot noise, compound Poisson noise
corrupted image, (d) restored image without compound Poisson noise removed.




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Oleg Kobylin et al. CEUR Workshop Proceedings                                                    109–117


Table 1
Comparison of results in PSNR(dB).
                                          Filters                PSNR(dB) value
                             The proposed filtering approach           39.36
                                  Spatial Wiener filter                35.71
                                     Median filter                     37.93
                                     Bilateral filter                  36.55
                              Multilayer perceptron model              38.97


3. Conclusion
In this paper we have solved the problem of filtering the pixels of the video corrupted by shot-noise
and compound Poisson noise. Shot noise brings the effect of instant jump of the pixel value that
slowly decays as time passes. Compound Poisson noise gives the effect of the instant jump of the pixel
intensity that does not decay as time passes. Therefore it is very important to filter this kind of noise.
Standard filtering techniques would not work well for such type of noise. Therefore, we considered
approximating shot-noise and compound Poisson noise with Gaussian process and applying Kalman
filtering to remove shot noise from the sequence of images. Kalman filtering uses the filtered image
from the previous step to denoise (filter) the image on the current step. The graphical illustration was
shown on several figures. In average, the shot noise corrupted image sequence is 90% restored using
Kalman filter comparing to the original image sequence. For compound Poisson process the filtering
quality is 80% on average.
   The central result of this paper allows to model the effects of the random shot-noise and compound
Poisson jumps that corrupts the video data, to approximate the both kinds of noise by Gaussian noise
and to apply the Kalman filtering techniques for the video restoration.
Declaration on Generative AI: The authors have not employed any generative AI tools.


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