1 Introduction

The ever-increasing demand of pixel sensors is an evident of the importance of digital imaging in various applications [20, 37]. Generally, the pixel sensor measurements follow Poisson distribution. There are various methods that are based on shrinking the pixel sensor size, increasing the image resolution at the cost of the need for denoising the Poisson-corrupted images [9, 23, 27, 28]. The presence of the noise in any imagery degrades the spatial and contrast resolution. The noise distribution can vary from Gaussian, Gamma, Poisson, or it can follow some compound distribution as well. The signal-to-noise ratio scales linearly with the Poisson intensity which is a main challenge in digital imaging [37]. Moreover, the approximation of Poisson-induced noise in images follows different models which makes image denoising algorithms to be designed for Gaussian noise. However, each resolution cell of the noise-affected image has multiple numbers of scatterers that return randomly distributed signals and cause poor visual interpretation [13]. In this paper, we have used exponential distribution [15, 19] approach to model the log-likelihood estimator [8, 22] for denoising with the advantages of approximate mean deviation, better sensitivity, better variability, and perfect reconstruction with minimized distortion index.

In all applications, the images are ultimately viewed by human beings, and therefore, every denoising algorithm must quantify visual image quality through subjective evaluation. Generally, subjective evaluation is too time-consuming and expensive process [16]. The simplest and most widely used quality metrics are peak signal-to-noise ratio (PSNR) and mean-squared error (MSE), computed by averaging the squared intensity differences of noise-induced and reference image pixels. Another important metric discussed in this paper is structural similarity (SSIM) that compares local patterns of pixel intensities that have been normalized for exposure and contrast. These parameters are mathematically convenient in the context of optimization [10, 11, 41, 42, 45]. We present experimental results from a set of three images and have compared the results among various denoising methods. Since we have considered the Poisson- and Gamma-related noise which is comparable to Bayesian-based estimation methods available in the literature [3, 14, 17, 26, 32, 38, 47, 49].

In this paper, the closed-form expressions to estimate the distortion parameter-based sensitivity and variability are derived for an uncorrupted image by minimizing the distortion. The parameters involved in the proposed method have been estimated using statistical approach with log-likelihood estimation. The uncorrupted image has been maximized for noise-free image pixels in accordance with log-likelihood estimator by employing an exponential distribution for the vectors of image with and without noise effect. This paper proposes an adaptive estimation method based on the variable exponent which gives a good estimate for Poisson- and Gamma-affected images. We have used the shape, scale, contrast, exposure, distortion, and edge strength parameters in the proposed method. The proposed method restores an image successfully by controlling the small features effectively. To assess the performance of the proposed method based denosing, we have used log-likelihood estimation. We have assessed the performance of proposed method before applying log-likelihood which is termed as proposed method before log-likelihood (PBL) and proposed method after applying log-likelihood estimation which is termed as proposed method after log-likelihood (PL).

The organisation of the paper is as follows: Sect. 2 describes the literature review. The proposed methodology is explained in Sect. 3. The performance measures are described in Sect. 4. The experimental results of the proposed method are discussed in Sect. 5. Finally, Sect. 5 concludes the paper.

2 Literature Review

In the literature, various noise reduction techniques are discussed such as resolution enhancement approach, averaging approach, and post-processing approach using filters and sliding window to estimate the statistical information of all pixels using the local mean and local variance [12, 31, 50]. Multiscale processing is the most commonly used approach in Poisson image denoising models [5]. Latest developments in this area include biased mean estimators and unbiased estimate of risk which recovers noise-free wavelet coefficients [4]. It has been shown that effective signal reconstruction can be achieved by employing shrinkage factor based on Bayesian formalism than by using the thresholding techniques [4, 36]. In [36], parameters of the mixed exponential distribution are estimated using fractional moments. We have used exponential distribution with log-likelihood approximation in the proposed method. The irregularities in Poisson data can be treated in various ways. For Poisson noise and multiplicative Gamma noise, the Bayesian maximum a posterior (MAP) likelihood estimation is presented in the literature [18, 24, 33, 35, 44, 48]. These methods follow simple model for image restoration in the presence of Poisson and multiplicative noises. But these existing methods for image restorations are designed specifically for a given type of noise, while our model can handle image restoration with mixed and unknown distribution of noises. We have compared our proposed approach with the methods presented in [24, 35]. In [24] noise of varying scales is being removed, while preserving low-contrast features in regions of low intensity which is termed as contrast-based denoising (CBD). In [35] a variational restoration model (VRM) for removing multiplicative Gamma noise is proposed using Douglas–Rachford splitting techniques. But, uniform regularization strength must be chosen to either remove high-intensity noise or to retain low-intensity features; both cannot be done using this approach. These models are not adaptive for different noise distributions, which is the advantage of our proposed method. This paper presents a novel exponential distribution using log-likelihood estimator for image denoising. A new family of exponential distribution lies between Gamma and Poisson distribution and is designed to fit the observed likelihood approximation. In our proposed method, we have used sensitivity and variation indices to offer a robust approach for exploiting the correlation and variance in any imagery for better visual quality. We have compared the test results of different image quality assessment against three sets of images with \(128\,\times \,128\) pixels and \(512\,\times \,512\) pixels.

3 Proposed Method

Statistical modeling is very important aspect in fields of scientific study and others. There are various statistical models with different type of response variables based on likelihood paradigm. Some very important generalized linearized models include Gaussian, Gamma, Binomial, Poisson. All these models belong to the exponential dispersion models (EDM) [2, 46]. We have used an intermediate model between Gaussian and Poisson distributions to evaluate first two moments, i.e., mean and variance. The variance function describes the relationship between mean and variance of response variable. The block diagram for the proposed algorithm is shown in Fig. 1. Consider random variable Y with mean \(\mu \) and variance \(\phi \mu ^{p}\) such that, \(\phi >0,p\in \left( {-\infty ,0} \right] \cup \left[ {1,\infty } \right) \) which describes \(Y=\mathrm{EDM}_p \left( {\mu ,\phi } \right) \). The mean is \(E\left( Y \right) =\mu \) and variance is written as \(\mathrm{Var}\left( Y \right) =\phi V\left( \mu \right) =\phi \mu ^{p}. \quad \mathrm{EDM}_p \left( {\mu ,\phi } \right) \) is an exponential dispersion model (EDM) of two variables \(\mu \) and \(\phi \). For \(p=0\) the EDM is Gaussian, for \(p=1,\phi =1\) the EDM is Poisson, for \(p=2,p=3\) the EDM is Gamma and inverse Gaussian and for \(1<p<2\) the EDM becomes compound Poisson distribution. The probability density function (PDF) of EDM is evaluated using numerical methods available in literature [2]. We assume that the pixel space in the image is denoted by \(\left( {i,j} \right) \), where we considered \(i,j=1,2\ldots n\) and \(n=128\) or 512. Assume that \(W_{ij} \) is a vector to be considered for all noise-free pixel values. \(N_{ij} \) is a vector that include all pixels with noise \(\left( {X_{ij} } \right) \) and without noise \(\left( {W_{ij} } \right) \) such that \(N_{ij} =W_{ij} +X_{ij}\). The noise density is represented as \(\sigma _X \) which is considered as 10, 20, 30, 40 and 50 in this paper. It is assumed that all the pixels are independent and their PDFs are Poisson-distributed. \(X_{ij}\) is independent with Poisson- and Gamma-distributed noise with mean \(\tau _{ij} .\) According to [46], \(N_{ij} \) follows an exponential compound Poisson model. The distribution of \(N_{ij} \) can also be re-parameterized in such a way that it takes the form of the exponential family such that scale parameter \(\left( p \right) \) is a function of edge strength sharpness parameter \(\left( v \right) \) as

Fig. 1
figure 1

Block diagram of the proposed algorithm

Full size image
$$\begin{aligned} p=\frac{v+2}{v+1},p\in \left( {1,2} \right) ,v>0. \end{aligned}$$
(1)

Overall mean \(\left( {\mu _{ij} } \right) \) is multiplicative in nature such that \(\mu _{ij} =\lambda _{ij} \tau _{ij}\). \(\lambda _{ij} \) is the mean of noise-free pixels. Distortion index is defined as,

$$\begin{aligned} \phi _{ij} =\frac{\lambda _{ij}^{1-p} \tau _{ij}^{2-p} }{\left( {2-p} \right) }. \end{aligned}$$
(2)

Here the probability of pixels affected by noise is zero. The probability density function [46] is written as

$$\begin{aligned} P(N_{ij}=0) = \hbox {exp} \left( -W_{ij}\lambda _{ij}\right) = \hbox {exp} \left( \frac{W_{ij}}{\phi _{ij}} \left( -k_{p}\left( \phi _{ij}\right) \right) \right) . \end{aligned}$$
(3)

For the probability that pixels affected by noise is greater than zero, the probability density function [46] is

$$\begin{aligned} f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) =C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) \exp \left( {\frac{W_{ij} }{\phi _{ij} }\left( {N_{ij} \left( \theta \right) -k_p \left( \theta \right) } \right) } \right) . \end{aligned}$$
(4)

\(C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) \) is a constant which is a function of noise-free pixels and scale parameter. It is evaluated using Gamma function [46]. Here \(\theta \) is the deviation in the mean, \(\theta =\left\{ {{\begin{array}{l} {\frac{\mu ^{1-p}}{1-p},p\ne 1} \\ {\mathrm{log} \, \mu ,p=1} \\ \end{array} }} \right. \). The exposure and contrast operator \(\left( k \right) \) in form of a cumulating generating function [2] is written as \(k=\frac{1}{2-p}\left( {\left( {1-p} \right) \theta } \right) ^{\frac{2-p}{1-p}}\). k can be represented in terms of \(\mu \) as

$$\begin{aligned} k=\left\{ {{\begin{array}{l} {\frac{\mu ^{2-p}}{2-p},\quad p\ne 2} \\ {\mathrm{log}\mu ,\quad p=2} \\ \end{array} }} \right. . \end{aligned}$$
(5)

\(C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) \) is to be evaluated numerically which is a constant which has been analyzed for two cases in this paper. It is defined with the help of gamma distribution as

$$\begin{aligned} C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) =\mathop \sum \nolimits _{i,j=1}^n \frac{1}{{i}!{j}!\Gamma \left( {vW_{ij} } \right) }\left( {\frac{W_{ij}^v \left( {\frac{W_{ij} }{\phi _{ij} }} \right) ^{v+1}}{\left( {p-1} \right) ^{v}\left( {2-p} \right) }} \right) . \end{aligned}$$
(6)

Here \(\Gamma \left( . \right) \) is Gamma function. \(N_{ij} \) has mean \(\mu _{ij} \), \(\phi _{ij} \) is the dispersion parameter. k is the exposure and contrast operator with p as scale parameter. \(p\rightarrow 1\) for over dispersed Poisson distribution and \(p\rightarrow 2\) for Gamma distribution. Our model is a bridge between Poisson and Gamma models, i.e., \(P\in \left( {1,2} \right) \). We will now calculate the constant function for these two cases.

Case 1 When \(1<p<2\) then constant function \(C\left( \frac{W_{ij}}{\phi _{ij}};P\right) \) in Eq. (4) becomes

$$\begin{aligned} C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) =\frac{W_{ij}^{-\alpha } \left( {p-1} \right) ^{\alpha }}{\phi _{ij}^{\left( {1-\alpha } \right) } \left( {2-p} \right) ^{\alpha }i!j!\Gamma \left( {-\alpha } \right) }. \end{aligned}$$
(7a)

Here is shape parameter [2, 46] such that \(\alpha =\frac{\left( {2-p} \right) }{\left( {1-p} \right) }\). By substituting Eq. (7) in (4), we get density function as

$$\begin{aligned} f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) =\frac{W_{ij}^{-\alpha } \left( {p-1} \right) ^{\alpha }}{\phi _{ij}^{\left( {1-\alpha } \right) } \left( {2-p} \right) ^{\alpha }i!j!\Gamma \left( {-\alpha } \right) }.\exp \left( {\frac{W_{ij} }{\phi _{ij} }\left( {N_{ij} \left( \theta \right) -k_p \left( \theta \right) } \right) } \right) .\nonumber \\ \end{aligned}$$
(7b)

Case 2 When \(p>2\) then constant function \(C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) \) in Eq. (4) becomes

$$\begin{aligned} C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) =\frac{1}{\pi \left( {W_{ij} } \right) ^{\alpha }\left( {f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) } \right) }. \end{aligned}$$
(8)

After substituting probability function in terms of Gamma function and simplifying above equation, we get

$$\begin{aligned} C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) =\frac{\Gamma \left( {1+\alpha } \right) \phi ^{\left( {\alpha -1} \right) }\left( {p-1} \right) ^{\alpha }}{\left( {W_{ij} } \right) ^{\alpha }.\Gamma \left( {1+\alpha } \right) \left( {p-2} \right) ^{\alpha }y^{\alpha }}.\left( {-1} \right) ^{\alpha }\hbox {sin}\left( {-\pi \alpha } \right) \quad . \end{aligned}$$
(9a)

By substituting Eq. (9) in (4), we get density function as

$$\begin{aligned}&f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) \nonumber \\&\quad =\frac{\Gamma \left( {1+\alpha } \right) \phi ^{\left( {\alpha -1} \right) }\left( {p-1} \right) ^{\alpha }}{\left( {W_{ij} } \right) ^{\alpha }.\Gamma \left( {1+\alpha } \right) \left( {p-2} \right) ^{\alpha }y^{\alpha }}.\left( {-1} \right) ^{\alpha }\sin \left( {-\pi \alpha } \right) \nonumber \\&\qquad .\exp \left( \frac{W_{ij} }{\phi _{ij} }\left( N_{ij} \left( \theta \right) -\,k_p \left( \theta \right) \right) \right) . \end{aligned}$$
(9b)

Equations (7b) and (9b) are the equations for the probability density functions for the proposed method evaluated using exponential dispersion family of distributions. These PDFs are observed w.r.t. compound Poisson model and Gamma model to analyze sensitivity and variability using gradient approach as discussed in subsequent section.

3.1 Sensitivity and Variability Analysis

Now we estimate sensitivity and variability as a function of \(\left( {\phi ,p} \right) .\) We have used quasi-score function [21] and Pearson estimating function [1] with distortion and scale parameters. As mentioned that scale parameter provides edge strength information. Therefore, scale parameter will be useful in restoring edge information. The image sensitivity function is quasi-score function [21] of \(\left( {\phi ,p} \right) \) which is written as

$$\begin{aligned} \psi \left( {\phi ,p} \right) =\nabla \mu _{ij} C^{-1}\left( {N_{ij} -\mu _{ij} } \right) . \end{aligned}$$
(10)

The sensitivity matrix \(\left( S \right) \) of \(\psi \) is a \(n\times n\) matrix which is written as \(S=E\left( {\nabla \psi } \right) .\) The variability matrix \(\left( V \right) \) of \(\psi \) is a \(n\times n\) matrix which is written as \(V=\mathrm{Var}\left( \psi \right) \). Using Pearson function [1], we can write \(\psi \left( {\phi ,p} \right) \) as

$$\begin{aligned} \psi \left( {\phi ,p} \right) =N_{ij}^T W_{ij} -tr\left( {W_{ij} C^{-1}} \right) . \end{aligned}$$
(11)

We evaluate sensitivity and variability matrix by finding derivatives of \(\psi \) w.r.t. \(\phi \) and p. The \(\left( {n\times n} \right) \) sensitivity matrix w.r.t. \(\phi \) and p is given by

$$\begin{aligned} S\left( \phi \right) =E\left( {\frac{\delta \psi \left( {\phi ,p} \right) }{\delta \phi }} \right) , \end{aligned}$$
(12a)
$$\begin{aligned} S\left( p \right) =E\left( {\frac{\delta \psi \left( {\phi ,p} \right) }{\delta p}} \right) . \end{aligned}$$
(12b)

Here \(\frac{\delta C}{\delta \phi }=\mathrm{diag}\left( {\mu ^{p}} \right) \) and \(\frac{\delta C}{\delta p}=\mathrm{diag}\left( {\phi \hbox {log}\left( \mu \right) \mu ^{p}} \right) \). We can show using results about characteristic function of linear and quadratic forms of non-normal variables [21] such that the entries in variability matrix is given by

$$\begin{aligned} V\left( \phi \right) =2tr\left( {N_{ij} \left( {W_{ij} C} \right) } \right) +\mathop \sum \nolimits _k k\left( {N_{ij} } \right) ^{\phi }\left( {W_{ij} } \right) ^{\phi }, \end{aligned}$$
(13a)
$$\begin{aligned} V\left( p \right) =2tr\left( {N_{ij} \left( {W_{ij} C} \right) } \right) +\mathop \sum \nolimits _k k\left( {N_{ij} } \right) ^{p}\left( {W_{ij} } \right) ^{p}. \end{aligned}$$
(13b)

To take into account the correlation matrix between vector \(\phi \) and p, we need to compute cross-sensitivity and cross-variability matrix [40]. The entries of cross-sensitivity matrix \(\left( {\mathrm{CS}} \right) \) between \(\phi \) and p are given by

$$\begin{aligned} \mathrm{CS}\left( {\phi ,p} \right) =-tr\left( {C^{-1}\frac{\delta C}{\delta \phi }.C^{-1}\frac{\delta C}{\delta p}} \right) . \end{aligned}$$
(14)

Finally, we the cross-variability matrix \(\left( {\mathrm{CV}} \right) \) between \(\phi \) and p is given by

$$\begin{aligned} \mathrm{CV}\left( {\phi ,p} \right) =2tr\left( {N_{ij} \left( {W_{ij} C} \right) } \right) +\mathop \sum \nolimits _{i,j=1}^n k\left( {N_{ij} } \right) ^{\phi }\left( {W_{ij} } \right) ^{p}. \end{aligned}$$
(15)

The sensitivity and variability analyses are an accurate way to predict the visual quality of image and image is more pleasant when cross-sensitivity (Eq. 14) and cross-variability (Eq. 15) correlation approaches to 1.0. We have also observed effect its effect on sensitivity and variability on the images under consideration. The sensitivity value is high when the noise effect is reduced and the variability value is more when noise effect is more. From Eq. (12a, 12b), it is observed that sensitivity and variability follow gradient approach w.r.t. distortion and scale parameter. Also, it is mentioned that scale parameter affects the edge strength. The values obtained for sensitivity and variability at different noise densities, i.e., \(\sigma =10,20,30,40\) and 50. The average of three images is summarized in Tables 1 and 2, respectively, for \(1<p<2\) and \(p>2\). The overall determinant of matrix \(S\left( \phi \right) \) represents sensitivity w.r.t. distortion and the overall determinant of matrix \(S\left( p \right) \) represents sensitivity w.r.t. edge strength. So, the sensitivity value should be high. The overall determinant of matrices \(V\left( \phi \right) \) represents variability w.r.t. distortion, and the overall determinant of matrix \(V\left( p \right) \) represents variability w.r.t. edge strength. So, the variability value should be low. Another important observation here is the correlation sensitivity \(\left( {\mathrm{CS}\left( {\phi ,p} \right) } \right) \) and correlation variability \(\left( {\mathrm{CV}\left( {\phi ,p} \right) } \right) \) which should be close to 1 for better visual quality.

Table 1 Calculated values for sensitivity from average of three image sets
Full size table
Table 2 Calculated values for variability from average of three image sets
Full size table

3.2 The Likelihood Function and Optimization of Distortion Parameter

The probability function \(f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) \) is an exponential probability density function. The log-likelihood function [29, 30] for an image size \(n\times n\) is given by

$$\begin{aligned} l\left( {N_{ij} ,\phi _{ij} ,p} \right) =\mathop \sum \nolimits _{i,j=1}^n \log f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) . \end{aligned}$$
(16)

We can maximize this equation w.r.t. \(\phi _{ij} \). There are various methods [1, 2, 21] for maximization process. By substituting \(f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) \) from Eq. (4), we get

$$\begin{aligned} l\left( {N_{ij} ,\phi _{ij} ,p} \right) =\mathop \sum \nolimits _{i,j=1}^n \log \left( {C\left( {\frac{W_{ij} }{\phi _{ij} },p} \right) } \right) +\frac{W_{ij} }{\phi _{ij} }\left( {N_{ij} \frac{\mu _{ij}^{1-p} }{1-p}-\frac{\mu _{ij}^{2-p} }{2-p}} \right) . \end{aligned}$$
(17)

Here the distortion index is written as

$$\begin{aligned} \phi _{ij} =\frac{-\mathop \sum \nolimits _{i,j=1}^n W_{ij} \left( {N_{ij} \frac{\mu _{ij}^{1-p} }{1-p}-\frac{\mu _{ij}^{2-p} }{2-p}} \right) }{\left( {1-v} \right) \mathop \sum \nolimits _{i,j} N_{ij} }. \end{aligned}$$
(18)

It can be estimated by maximum likelihood estimator by setting minimum and maximum values for \(\phi _{ij} \). This is achieved by substituting the derivative of Eq. (17) equal to zero. By adjusting the number of parameters, we get the distortion index as:

$$\begin{aligned} \phi _{ij} =\phi =\mathop \sum \nolimits _{i,j} \frac{2}{N-Q}\left( {N_{ij} \left( {\frac{N_{ij}^{1-p} -\mu _{ij}^{1-p} }{1-p}-\frac{N_{ij}^{2-p} -\mu _{ij}^{2-p} }{2-p}} \right) } \right) . \end{aligned}$$
(19)

N denotes total number of pixels. Q is the number of pixels used in estimating the distortion. Variance parameters have an impact on the mean parameter and vice-versa. \(p,\phi \) and \(W_{ij} \) have impact on variance of the model and less on mean. When we use the likelihood principle for estimating \(\phi \), we minimize the vector with pixels affected by noise. We will calculate the derivatives of \(W_{ij} \) w.r.t. \(\phi \). Later, we have optimized using log-likelihood criterion. By differentiating \(f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) \) w.r.t. \(\phi \), we get

$$\begin{aligned} \frac{\delta \log f\left( {\lambda _{ij} ,\tau _{ij} ,v} \right) }{\delta \phi }=\left\{ {{\begin{array}{l@{\quad }l} \frac{\mu ^{2-p}}{\phi ^{2}\left( {2-p} \right) },&{}y=0 \\ \frac{N_{ij} \mu ^{1-p}}{\phi _{ij}^2 \left( {p-1} \right) }+\frac{\mu ^{2-p}}{\phi _{ij}^2 \left( {2-p} \right) }+\frac{\delta N_{ij} }{N_{ij} \delta \phi },&{}y>0 \\ \end{array} }} \right. . \end{aligned}$$
(20)

Case 1 When \(1<p<2\) then differentiating \(W_{ij} \) w.r.t. \(\phi \), we get

$$\begin{aligned} \frac{\delta W_{ij} }{\delta \phi }=\mathop \sum \nolimits _{i,j=1}^n C\left( {\frac{W_{ij} }{\phi _{ij} };p} \right) W_{ij} , \end{aligned}$$
(21a)

Using log-likelihood estimation,

$$\begin{aligned} \frac{\delta \hbox {log}\left( {W_{ij} } \right) }{\delta \phi }\approx \phi \left( {1-\alpha } \right) -\hbox {log}\left( {2\pi } \right) -\frac{1}{2}\log \left( {-\alpha } \right) . \end{aligned}$$
(21b)

Now in order to maximize \(W_{ij} \), the derivative is set equal to zero. To do this \(W_{ij} \) is written in form of a gamma function as

$$\begin{aligned} \log \left( {W_{ij} } \right) =\phi \log \left( {\frac{N_{ij}^{-\alpha } \left( {p-1} \right) ^{\alpha }}{\phi ^{1-\alpha }\left( {2-p} \right) }} \right) -\log \Gamma \left( {1+\phi } \right) -\log \Gamma \left( {-\alpha \phi } \right) . \end{aligned}$$
(22)

Now replace the gamma function by Stirling approximation [1] and approximating \(\left( {1-\alpha \phi } \right) \) and \(\left( {-\alpha \phi } \right) \) gives

$$\begin{aligned} \log \left( {W_{ij} } \right)\approx & {} \phi \left\{ {\log \left( {\frac{N_{ij}^{-\alpha } \left( {p-1} \right) ^{\alpha }}{\phi ^{1-\alpha }\left( {2-p} \right) }} \right) +\left( {1-\alpha } \right) +\alpha \log \left( {-\alpha } \right) -\left( {1-\alpha } \right) \log \phi } \right\} \nonumber \\&-\log \left( {2\pi } \right) -\frac{1}{2}\log \left( {-\alpha } \right) -\log \phi . \end{aligned}$$
(23)

\(\alpha <0\) for \(1<p<2\), so logarithms have positive arguments. Differentiating Eq. (23) w.r.t. \(\phi \),

$$\begin{aligned} \frac{\delta \log \left( {W_{ij} } \right) }{\delta \phi }\approx \log \left( {\frac{N_{ij}^{-\alpha } \left( {p-1} \right) ^{\alpha }}{\phi ^{1-\alpha }\left( {2-p} \right) }} \right) -\frac{1}{\phi }-\log \phi +\alpha \log \left( {-\alpha \phi } \right) . \end{aligned}$$
(24)

\(\frac{\delta \log \left( {W_{ij} } \right) }{\delta \phi }=0\) at \(\phi _{\mathrm{opt}} =\frac{N_{ij}^{2-p} }{\left( {2-p} \right) \phi }.\) This approximation is good for maximum value of \(W_{ij}\) which is found as by substituting \(\phi _{\mathrm{opt}} \) in maximum likelihood equation as

$$\begin{aligned} \mathrm{max} (\log W_{ij} )=\phi _{\mathrm{opt}} \left( {\alpha -1} \right) -\log \left( {2\pi } \right) -\log \left( {\phi _{\mathrm{opt}} } \right) -\frac{1}{2}\log \left( {-\alpha } \right) . \end{aligned}$$
(25)

Case 2 When \(p>2\) then differentiating \(W_{ij} \) w.r.t. \(\phi \) we get

$$\begin{aligned} \frac{\delta W_{ij}}{\delta \phi } = \sum \nolimits _{i, j=1}^{n} C \left( \frac{W_{ij}}{\phi _{ij}};p\right) W_{ij}, \end{aligned}$$
(26a)

Approximating \(k\approx \frac{\mu ^{2-p}}{\phi \left( {p-2} \right) }\) and using log-likelihood estimation, we get

$$\begin{aligned} \frac{\delta \log \left( {W_{ij} } \right) }{\delta \phi }\approx \left( {1-\alpha } \right) k+\frac{1}{2}\log \left( \alpha \right) +\hbox {log}\left( k \right) . \end{aligned}$$
(26b)

Now in order to maximize \(W_{ij} \) the derivative is set equal to zero. To do this \(W_{ij} \) is written in form of a gamma function as

$$\begin{aligned} W_{ij} =\frac{\left( {\frac{\left( {p-1} \right) ^{\alpha }\phi ^{\alpha -1}}{N_{ij}^\alpha \left( {p-2} \right) }} \right) ^{k}\Gamma \left( {1+\alpha k} \right) }{\Gamma \left( {1+k} \right) }. \end{aligned}$$
(27)

By Stirling approximation [1],

$$\begin{aligned} \log \left( {W_{ij} } \right)\approx & {} k\left[ \log \left( \frac{\left( {p-1} \right) ^{\alpha }\phi ^{\alpha -1}}{N_{ij}^\alpha \left( {p-2} \right) } \right) +\left( {1-\alpha } \right) \right. \nonumber \\&\left. -\log \left( k \right) +\alpha \log \left( {\alpha k} \right) \right] +\frac{1}{2}\log \left( \alpha \right) . \end{aligned}$$
(28)

\(\frac{\delta \log \left( {W_{ij} } \right) }{\delta \phi }=0\) at \(\phi _{\mathrm{opt}} =\frac{N_{ij}^{p-2} }{\left( {p-2} \right) \phi }.\) This approximation is good for maximum value of \(W_{ij}\) which is found by substituting \(\phi _{\mathrm{opt}} \) in maximum likelihood equation as

$$\begin{aligned} \mathrm{max} (\log W_{ij} )=k\left[ {\phi _{\mathrm{opt}} \left( {1-\alpha } \right) -\log \left( k \right) -\log \left( {\phi _{\mathrm{opt}} k} \right) } \right] +\frac{1}{2}\log \left( \alpha \right) . \end{aligned}$$
(29)

Thus, the final output image \(\left( {N_{ij_{\mathrm{final}} } } \right) \) is restored from the vector of the maximized values from the noise-free image where distortion index \(\phi \) is minimized.

4 Performance Measures

The proposed method was tested on three image sets each with resolution of \(128\times 128\) pixels and \(512\times 512\) pixels. For each image, noisy observation is generated by adding the Gaussian- and Gamma-distributed noise with original image having standard deviation of 10, 20, 30, 40,  and 50 dB. All the simulations have been carried out on MATLAB R2013a on a 2.67GHz i5 processor. Mean square error (MSE), peak signal-to-noise ratio (PSNR), mean absolute error (MAE), structural similarity index metric (SSIM), image quality index (QI), and time taken during simulation are used to benchmark denoising performance.

Mean square error is computed here as parametric estimation error. Parametric error reflects the uncertainty in reliability of estimates. One good estimate is established in [25, 46] of exponential compound Poisson model as,

$$\begin{aligned} \mathrm{MSE}[N_{ij}]= & {} \sum \nolimits _{ij} \phi W_{ij} \mu _{ij}^{1} + \sum (W_{ij}\mu _{ij})^{2} \mathrm{Var} [X_{ij}\beta ]\nonumber \\&+ \sum \nolimits _{ij \in \forall (i1,j1\ne i2,j2)}(W_{i1}\mu _i1)(W_{i2}\mu _i2) \cdots (W_{ij}\mu _{ij}) \nonumber \\&\cdot \sum \nolimits _{ij\in \forall (i1,j1\ne i2,j2)} \mathrm{Cov} (X_i1\beta , X_{ij}\beta ) \mathrm{Cov} (X_1j\beta , X_{ij}\beta ). \end{aligned}$$
(30)

\(\mathrm{Cov}\left( {X_{i1} \beta ,X_{ij} \beta } \right) \) is the corresponding covariance matrix elements.

PSNR and mean absolute error (MAE) are used to measure the quality of restoration results. Mathematically PSNR is given by

$$\begin{aligned} \mathrm{PSNR}=10\mathrm{\log } \left( {\frac{255^{2}n^{2}}{\left| {N_{ij} -N_{ij_{\mathrm{final}} } } \right| ^{2}}} \right) \mathrm{dB}. \end{aligned}$$
(31)

Here, \(N_{ij} \) is the original image and \(N_{ij_{\mathrm{final}} } \) is the final restored output denoised image. Mean absolute error is written as,

$$\begin{aligned} \mathrm{MAE}=\frac{\left| {N_{ij} -N_{ij_{\mathrm{final}} } } \right| }{n^{2}}. \end{aligned}$$
(32)

PSNR can tell how well the reconstruction data match the true data and the data which is not required to restore image. PSNR can measure the intensity difference between two images. Therefore, it may not be reliably used to describe visual perception of image. So, PSNR is accurate measure for restoration of edges. We have evaluated another important visual quality assessment performance measure which is SSIM. Mathematically,

$$\begin{aligned} \mathrm{SSIM}=\frac{1}{n}\mathop \sum \nolimits _{i,j=1}^n \frac{\left( {2\mu _{N_{ij_{\mathrm{final}} } } \mu _{N_{ij} } +c_1 } \right) \left( {2\sigma _{N_{ij_{\mathrm{final}} } } +c_2 } \right) }{\left( {\mu _{N_{ij_{\mathrm{final}} } }^2 +\mu _{N_{ij} }^2 +c_1 } \right) \left( {\sigma _{N_{ij_{\mathrm{final}} } }^2 +\sigma _{N_{ij} }^2 } \right) }. \end{aligned}$$
(33)

\(\mu _{N_{ij_{\mathrm{final}} } } ,\mu _{N_{ij}}\) are the mean associated with output image and noise-induced original image, respectively. \(\sigma _{N_{ij_{\mathrm{final}} } }^2 \mathrm{and} \sigma _{N_{ij} }^2 \) are the variance associated with output image and noise-induced original image, respectively. The noise density for noisy pixels is given by \(\sigma _X \) such that

$$\begin{aligned} \sigma _X =\frac{1}{n-1}\mathop \sum \nolimits _{i,j=1}^n (N_{ij} -\mu _W )\left( {N_{ij} -\mu _{ij_{\mathrm{final}} } } \right) . \end{aligned}$$
(34)

The values of constants are chosen as \(c_1 =0.01\) and \(c_2 =0.03\) [25]. The large value of SSIM depicts better performance.

Image quality index (QI) is another parameter for visual quality assessment. Mathematically,

$$\begin{aligned} QI=\frac{4\sigma _{N_{ij} } \sigma _{N_{ij_{\mathrm{final}} } } \mu _{N_{ij} } \mu _{N_{ij_{\mathrm{final}} } } }{\left( {\sigma _{N_{ij} }^2 +\sigma _{N_{ij_{\mathrm{final}} } }^2 } \right) \left( {\mu _{N_{ij} }^2 +\mu _{N_{ij_{\mathrm{final}} } }^2 } \right) }. \end{aligned}$$
(35)

The value of image quality index should be between 0 and 1. The best value is 1. This parameter includes the effect due to loss of correlation, luminance, and contrast distortion. For accuracy assessment we have used the way to measure accuracy by using statistical technique [34]. In order to access the overall accuracy, Cohen [39] defined most widely used statistics for the estimation of the effect of change agreement called Kappa Statistic. The overall accuracy is given as \(O_{\mathrm{acc}} =\mathop \sum \limits _{i,j=1}^n \frac{N_{ij} }{T}\). Here, T is the number of pixels considered. The value of T can be considered as any value between \(1,2\ldots 512\). In our experiments the value of T is taken for 100, 300 and 500. The Kappa factor for image free pixels is \(K_W =\frac{O_{\mathrm{acc}} -p_W }{1-p_W }\) . Here, \(p_W =\frac{\mathop \sum \nolimits _{i,j=1}^n W_{ij}^2 }{n}\) is the statistic of pixels without noise. The ideal value of Kappa factor should be 1. The values for \(K_W \) are given in Table 3.

Table 3 Calculated values \(K_{W}\) w.r.t. number of pixels
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4.1 Computational Complexity

We have evaluated the computational complexity [6, 7, 43] for proposed method and compared it with the methods available in literature. The computational complexity of CBD method [24] has three parts. First part calculates pixelated Poisson noise with complexity of \(O\left( {n2^{n}} \right) \). The second part computes the prior distribution with complexity \(O\left( {2^{2n}\left( n \right) ^{O\left( n \right) }} \right) \). The third part consists of the computational load related to minimization using Euler Lagrange equation and its computational complexity is \(O\left( {2n^{2}} \right) \). Therefore, the overall complexity becomes \(O\left( {n2^{n}} \right) +O\left( {2^{2n}\left( n \right) ^{O\left( n \right) }} \right) +O\left( {2n^{2}} \right) \). The computational complexity of VRM method [35] is also divided into three parts. The first part includes calculation of discrete denoising model using gradient approach with computational complexity of \(nO\left( n \right) \). The second part calculates conjugate function using variational method with computational load of \(O\left( {n^{2}\log n} \right) \). The third part calculates the minimization function using Douglas–Rachford splitting with computational complexity of \(nO\left( {2n} \right) \). Thus, the overall complexity can be written as \(nO\left( n \right) +O\left( {n^{2}\log n} \right) +nO\left( {2n} \right) \). The proposed method without likelihood estimation has computational complexity \(O\left( {n^{2}\log n} \right) +\hbox {}nO\left( n \right) \). The first part computes PDF using EDM and the second part is for computations related to sensitivity and variability using gradient approach. The log-likelihood evaluation has additional computational complexity \(nO\left( {\log \left( {1+2n} \right) } \right) \). Therefore, proposed method with likelihood estimation has computational complexity of \(O\left( {n^{2}\log n} \right) +nO\left( n \right) +nO\left( {\log \left( {1+2n} \right) } \right) \). We have tabulated the computational time in seconds in experimental results as well. It is observed that though the computational time using proposed method is comparable to the state of art methods under comparison and the visual quality of image is better as illustrated from different performance metrics.

5 Results and Discussion

In the our experiments, we evaluated the effect of the proposed method on denoising performance. In these experiments, three images contaminated by Poisson and Gamma noises at different noise densities 10, 20, 30, 40, and 50 are used. The overall performance was quantified on a set of three images. Figures 2, 3, 4, 5, 6, and 7 show the resulting images for each denoising methods for different values of noise densities. It is observed that the visual quality of the proposed estimator is superior to other methods. Also, we have observed that proposed method without log-likelihood estimation provides performances slightly inferior in terms of SNR as compared to proposed method with log-likelihood estimation. Figures 2, 3 and 4 show the images with \(128\times 128\) pixel space and Figs. 5, 6 and 7 are for the images \(512\times 512\) pixel space. The proposed method exhibit a good performance in restoring geometrical structures of the images. We have observed that Poisson noise is minimized effectively when \(1<p<2\). For the case when \(p>2\), the distribution is more close to gamma distribution so it can be used when Gamma and inverse Gaussian noise are present in the input image. The proposed algorithm is better both visually and quantitatively as revealed by performance measures. It has been satisfied that using log-likelihood estimator the noise can be minimized using the proposed methodology. The presented approximation gives better results when the choice of the parameters is appropriate. Simulation results indicate that proposed method is able to reconstruct edges and restores the contrast very well. The noise minimized image attains the largest PSNR. The denoising performance results for MSE are (Eq. 30) given in Tables 4 and 5 for \(128\times 128\) and \(512\times 512\) pixels space, respectively. Various other performance measures are evaluated in Tables 6, 7, 8 and 9 such as PSNR (Eq. 31), MAE (Eq. 32), SSIM (Eq. 33), QI (Eq. 35) and computational time in seconds in respect of the computational complexity. The PSNR improvement brought by our approach is quite high and the visual resolution is quite remarkable. Moreover, the new algorithm with log-likelihood estimation is better than without log-likelihood estimation. It is observed that when distortion index and noise density decrease, the image quality improves. The obtained PSNR results indicate that the proposed method has better performance than others, especially at low SNRs. In our experiment, the same set of images is taken with different pixel size and noise contaminations. The obtained results for various performance measures at different noise levels are summarized in Tables 4, 5, 6, 7, 8 and 9. From the obtained results we can conclude that the proposed denoising algorithm outperforms as compared to the other methods for all noise intensity situations. When looking closer at the results, we observe that using the proposed method PSNR is improved by 19%, MAE is improved by 32% and SSIM is improved by 10% as compared to [24, 35] for different noise densities. The values of these parameters have been tabulated for different values for noise density as 10, 20, 30, 40 and 50 for both cases, i.e., \(1<p<2\) and \(p>2\). We have shown the results for small \(\left( {128\times 128} \right) \) and big \(\left( {512\times 512} \right) \) pixel space images. It is observed that even, for big size images the proposed method takes almost comparable time as taken for small size images. The proposed method outperforms other methods by resulting in more pleasant visual image. It is observed that even at high-intensity noise, all PSNR values and quality measure image quality index of our method are higher.

Fig. 2
figure 2

Qualitative results for noise density \(\sigma _{X}\) = 10 for \(128 \times 128\) pixel size. a Noisy image, b CBD [24], c VRM [35], d PBL, e PL

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Fig. 3
figure 3

Qualitative results for noise density \(\sigma _X =30\) for \(128\times 128\) pixel size. a Noisy image, b CBD [24], c VRM [35], d PBL, e PL

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Fig. 4
figure 4

Qualitative results for noise density \(\sigma _X \quad \sigma _X =50\) for \(128\times 128\) pixel size. a Noisy image, b CBD [24], c VRM [35], d PBL, e PL

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Fig. 5
figure 5

Qualitative results for noise density \(\sigma _X \quad \sigma _X =10\) for \(512\times 512\) pixel size. a Noisy image, b CBD [24], c VRM [35], d, PBL, e PL

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Fig. 6
figure 6

Qualitative results for noise density \(\sigma _X =30\) for \(512\times 512\) pixel size. a Noisy image, b CBD [24], c VRM [35], d PBL, e PL

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Fig. 7
figure 7

Qualitative results for noise density \(\sigma _X =50\) for \(512\times 512\) pixel size. a Noise image, b CBD [24], c VRM [35], d PBL, e PL

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Table 4 Calculated MSE values for \(128\,\times \,128\) pixel size using proposed method after log-likelihood estimation
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Table 5 Calculated MSE values for \(512\,\times \,512\) pixel size using proposed method after log-likelihood estimation
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Table 6 Performance measures as average of three image sets for \(1<p<2\), \(\phi = 1\) and \(k=1\) with different noise densities
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Table 7 Performance measures as average of three image sets for \(p>2\), \(\phi = 1\) and \(k=1\) with different noise densities
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Table 8 Performance measures as average of three image sets for \(1<p<2\), \(\phi =1\) and \(k=0.5\) with different noise densities
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Table 9 Performance measures as average of three image sets for \(p>2\), \(\phi =1\) and \(k=0.5\) with different noise densities
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6 Conclusion

In this paper, based on the exponential distribution function, we have proposed an adaptive image denoising model using log-likelihood approximation. In our method, we have used the mean, variance, distortion index, scale parameter, sensitivity, and variability analysis as the variable exponent to analyze and control image quality. Compared with the other methods, using the proposed method the quality of restored images is quite well. We proposed the minimization of noise using log-likelihood estimation for multiscale Poisson and Gamma induced noises in image. The denoising operation is effective to restore the image features and contrast. This paper shows that an exponential denoiser based on the log-likelihood estimation under suitable statistical conditions, is well adapted to characterize images that are effected by Poisson-distributed noise, Gaussian-distributed noise or any other compound distributed noise by varying scale parameter. In the presence of noise, the proposed multi-parameter estimator provides slightly better performance than Bayesian and variational parameter-based estimators. Experimental results on the images show the superiority of proposed denoiser compared to other denoising approaches. This suggests that the proposed method is an accurate model as it is able to restore the contrast, shape, and scale behavior of the of images; this gives the proposed estimator good denoising properties. The experimental results show that the proposed denoiser outperforms the other methods very well especially at low SNRs. The proposed algorithm may be extended to color images and video framework, which may further improve video denoising.