2.1.

FBm Model for Describing Texture Locally

To describe image texture locally, we propose to use the fBm model $B_H(t,s)$. Fractal analysis based on mathematical fBm processes has been suggested as a useful technique for characterizing real-life images, as they can describe quite complicated textures or shapes of natural scenes with a minimal parameter set.⁴¹

By definition, $B_H(t,s)$, $H\in \mathrm{[0,1]}$, is a Gaussian process [the original coordinates are at the point (0,0): $B_H(0,0)=0$] with the correlation function:⁴²

In spatial terms, the Hurst exponent $H$ describes fBm-texture roughness ($H\to 0$ for rough texture, $H\to 1$ for smooth),⁴³ and ${\sigma }_x$ describes fBm amplitude.

By $y(t,s)$, $t=1\dots N_c$, and $s=1\dots N_r$, we denote a single-component image with $N_c$ columns and $N_r$ rows (for multi-component or color images, the approach proposed here should be applied component-wise). Suppose that the image $y(t,s)$ is affected by signal-dependent noise according to the following model:

However, in the section of this paper that describes the experiment, we concentrate on a reduced-order noise model for recent CCD sensors. In this case, the noise $n[t,s,x(t,s)]$ is supposed to be the sum of two components: $n[t,s,x(t,s)]=n_{\mathrm{SI}}(t,s)+n_{\mathrm{SD}}[t,s,x(t,s)]$. The first one, $n_{\mathrm{SI}}(t,s)$, is signal-independent with variance ${\sigma }_{n.\mathrm{SI}}^2$, and the second one, $n_{\mathrm{SD}}[t,s,x(t,s)]$, is signal-dependent with variance ${\sigma }_{n.\mathrm{SD}}^2\cdot I$ (Poisson-like noise). This corresponds to a polynomial model of the first order ($n_p=1$) and, thus the coefficient vector simply reduces to $\mathbf{c}=({\sigma }_{n.\mathrm{SI}}^2,{\sigma }_{n.\mathrm{SD}}^2)$:

2.2.

Structure of $\mathrm{NI}+\mathrm{fBm}$ and $\mathrm{NI}+\mathrm{DCT}$ Signal-Independent Noise Variance Estimators

At this point, let’s briefly recall the main ideas suggested in the $\mathrm{NI}+\mathrm{fBm}$ and $\mathrm{NI}+\mathrm{DCT}$ signal-independent noise variance estimators (${\sigma }_{n.\mathrm{SD}}^2=0$) proposed by the authors in Ref. 15. Their generalized structure is recalled in Fig. 1. This structure used the fBm model and both image NI and TI maps. In the estimation process, a processed sample corresponds to a noisy textural fragment or SW of the image. The sample is described by a statistical parametrical model. The model parameters vector includes both texture parameters (fBm-model parameters) and signal-independent noise variance ${\sigma }_{n.\mathrm{SI}}^2$. After initialization (stage 1), NI/TI maps and noise standard deviation are simply assumed to be equal to initial guesses or current refined estimates in stage 2. At this stage, texture parameters are estimated for each TI SW first. Then, these estimates act as texture parameter estimates in neighboring NI SWs. This stage results in estimates of the parameter vectors for all SWs for further use in stage 3.

Fig. 1

Structure of $\mathrm{NI}+\mathrm{fBm}$ and $\mathrm{NI}+\mathrm{DCT}$ SI noise variance estimators.

In stage 3, those SWs that can be efficiently used for estimating either noise or texture parameters are identified. We propose to perform this task by considering the corresponding FI on involved parameters or, equivalently the Cramér-Rao Lower Bound (CRLB), to each SW. By setting a proper threshold on CRLB, it becomes possible to find a subset of NI/TI SWs that provide noise/texture parameters estimates with a predefined accuracy. Finally, in stage 4, the noise standard deviation is estimated for each NI SW using either a fBm-model-based ML estimator ($\mathrm{NI}+\mathrm{fBm}$) or DCT ($\mathrm{NI}+\mathrm{DCT}$). Note that by using only NI SWs ensures that all individual noise standard deviation estimates in stage 4 achieve a predefined accuracy and that there are no outliers among them. As a result, a simple nonrobust estimation procedure (for example, weighted mean) is sufficient in stage 4 to obtain the final estimate ${\widehat{\sigma }}_{n.\mathrm{SI}}^2$.

2.3.

FI Distribution over Images

When solving the estimation problem for signal-independent noise, the total amount of FI on noise standard deviation contained in all noise-informative SWs determines the potential performance of an estimator (in terms of its variance).¹⁵ For signal-dependent noise, when noise standard deviation is a function of true image intensity, the distribution of FI over the image intensity range becomes of major importance. Specifically, if all noise-informative SWs have the same intensity mean $I_0$, only the value of ${\sigma }_n(I_0)$ can be estimated. This allows estimating the pure Poisson-like (${\sigma }_n^2(I)={\sigma }_{n.\mathrm{SD}}^{2}I$) or multiplicative (${\sigma }_n^2(I)={\sigma }_{\mu }^2{I}^2$) signal-dependent noise standard deviation by ${\widehat{\sigma }}_{n.\mathrm{SD}}={\widehat{\sigma }}_n(I_0)/\sqrt{I_0}$ and ${\widehat{\sigma }}_{\mu }={\widehat{\sigma }}_n(I_0)/I_0$, respectively (with accuracy that does not depend on $I_0$). If a mixture of signal-independent noise and either Poisson-like or multiplicative noise is assumed, the best accuracy can be achieved when FI is distributed equally between two intensities, $I_{\min }$ and $I_{\max }$, with $I_{\max }-I_{\min }$ being as large as possible. When no a priori information about ${\sigma }_n(I)$ is available, then FI uniformly distributed over all image intensity ranges is the best option.

Unfortunately, for a particular image, this distribution cannot be modified. What can be done, in practice, is to establish the actual distribution for a given image and to obtain the corresponding accuracy of noise parameter vector $\mathbf{c}$ estimation, taking into account available a priori information. This problem will be addressed next, and $\mathrm{NI}+\mathrm{fBm}$ and $\mathrm{NI}+\mathrm{DCT}$ estimators will be extended to estimate the ${\sigma }_n(I,\mathbf{c})$ function.

The corresponding FI about the parameter vector $\mathbf{\theta }=({\sigma }_x,H,{\sigma }_n)$ on texture parameters $({\sigma }_x,H)$ and noise standard deviation $({\sigma }_n)$ was introduced in Ref. 15 for a single SW ($N_{\mathrm{SW}}=1$) as

Assume that noise-informative SWs have been detected, and assign mean intensity ${\overline{I}}_i$ to the $i$th NI SW. For the subset $N_{\mathrm{SW}}\ge 1$ of these windows with ${\overline{I}}_i\in [I-\mathrm{\Delta }I/2,I+\mathrm{\Delta }I/2]$, we assume a signal-independent noise model with standard deviation ${\sigma }_n={\sigma }_n(I)$. Then, joint FI matrix ${\mathbf{I}}_{\mathbf{\theta }}(I,\mathrm{\Delta }I)$ on extended parameter vector $\mathbf{\theta }=({\sigma }_{x.1},H_1,{\sigma }_{x.2},H_2,\dots ,{\sigma }_{x.N_{\mathrm{SW}}},H_{N_{\mathrm{SW}}},{\sigma }_n)$ can be obtained in a similar way:

Using ${\sigma }_{{\sigma }_n}^2(I,\mathrm{\Delta }I)$, we define CRLB per unit intensity range $[I-0.5,I+0.5]$ as

To get better insight on the meaning of variables defined above, the possible shape of $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ and its relation to the image content are illustrated in Fig. 2. Figure 2(a) displays the green component of image 3 from the NED2012 color images database (Noise Estimation Database), later referred to as the test image. We created the NED2012 database especially for testing the noise parameter estimator. It is described in detail in Sec. 4.1, later in this article.

Fig. 2

(a) Green component of image #3 from the NED2012 database; (b) relative CRLB per unit intensity range, $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$, versus intensity for image #3 from the NED2012 database (test image) degraded with signal-dependent noise.

The $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ function has been calculated for the test image, and it is displayed in Fig. 2(b). In this experiment, we fix $\mathrm{\Delta }I=1$ to calculate $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$. A way to calculate $\mathrm{\Delta }I$ automatically will be described in Sec. 3.1. Intensities of the test image cover range approximately from 0 to 1,200. In Fig. 2(b), four image intensity intervals with the lowest $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ can be seen, with image intensities around 10 (1), 200 (2), 400 (3), and 800 (4).

Objects in the test image with intensities falling within intervals 1–4 are marked in Fig. 2(a) with the corresponding numbers. Interval 1 corresponds to the nonintensive dark tree texture, intervals 2 and 3 relate to homogeneous house fronts, and interval 4 relate to cloudy sky. One can see that $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ indicates image areas that provide accurate noise standard deviation estimation and their localization with regard to image intensity.

3.1.

Preliminaries of the Proposed Estimators of Polynomial SD Noise Variance

While designing the signal-dependent noise parameter estimator, we assume piecewise-constant approximation to the ${\sigma }_n^2(I)$ function:

As utilizing NI SWs along with either fBm-based or DCT-based noise standard deviation estimators ($\mathrm{NI}+\mathrm{fBm}$ or $\mathrm{NI}+\mathrm{DCT}$) was proved to be efficient in Ref. 15 for signal-independent noise, we propose here to apply these two estimators to estimate ${\sigma }_{n.k}^2$ in each NI SWs subset, with mean intensity falling within a given interval $[I_{\min .k},I_{\max .k}]$.

Before using Eq. (8), the set of intervals ${\mathbf{I}}_k$ should be properly selected. This can be done by taking into account the relationship between ${\mathbf{I}}_k$ and $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$. Indeed, for noninformative image intensities with high $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$, the interval $\mathrm{\Delta }{I}_k=I_{\max .k}-I_{\min .k}$ has to be increased to provide sufficiently accurate noise standard deviation estimates. Conversely, for informative image intensities with low $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ (areas 1–4 in Fig. 2), the same accuracy can be achieved for smaller $\mathrm{\Delta }{I}_k$. It is natural to estimate noise standard deviation with a predefined accuracy ${\sigma }_{{\sigma }_n}^2(I,\mathrm{\Delta }I)={\sigma }_{n.\mathrm{rel}.\max }^2$ for each considered intensity interval ${\mathbf{I}}_k$; i.e., to require

According to Eq. (6), for an arbitrary intensity ${\overline{I}}_k=I$, equality in Eq. (9) holds for

Therefore, $\mathrm{\Delta }I(I)$ is just a scaled-down version of $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$. In the example considered in Fig. 2, we set ${\sigma }_{n.\mathrm{rel}.\max }=0.1$. For this value, $\mathrm{\Delta }I(I)$ varies from 1 to 10 in informative intensity intervals 1–4 and exceeds 200 in noninformative intervals [shown by the peaks in Fig. 2(a)]. It follows that for informative intensities, noise variance can be accurately estimated from very narrow intensity intervals, enabling fine NLF analysis.

Based on Eq. (10), the set of intervals ${\mathbf{I}}_k$ is obtained by the following procedure:

For each interval ${\mathbf{I}}_k$, ${\widehat{\sigma }}_{n.k}^2$ is estimated using either the $\mathrm{NI}+\mathrm{fBm}$ or the $\mathrm{NI}+\mathrm{DCT}$ estimator (under signal-independent hypothesis). At the same time, ${\sigma }_{{\sigma }_n.\mathrm{rel}}({\overline{I}}_k,\mathrm{\Delta }{I}_k)$ is calculated for future use. The output of the proposed signal-dependent noise estimator is the set of noise standard deviation estimates ${\widehat{\sigma }}_{n.k}^2$ obtained for each interval ${\mathbf{I}}_k$ and the corresponding set of mean intensities ${\overline{I}}_k$.

Now, the maximum likelihood estimation (MLE) of coefficients vector $\mathbf{c}$ is obtained by

3.2.

Proposed Estimators of Polynomial SD Noise Variance

Note that the signal-dependent noise variance estimate ${\sigma }_n^2(I,\widehat{\mathbf{c}})$ obtained by Eq. (11) requires availability of the values of $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ and ${\sigma }_{{\sigma }_{n.k}^2}$, that are based on true noise variance ${\sigma }_n^2(I,{\mathbf{c}}_0)$. Therefore, the signal-dependent noise variance estimation procedure should operate iteratively as described below. The structure of the proposed estimator for texture parameters and signal-dependent noise standard deviation is summarized in Fig. 3. In the following, we provide a description of this estimator and illustrate its behavior based on the test image [Fig. 2(a)] for $n_p=1$. True parameter vector ${\mathbf{c}}_0=(5.0834,0.1352)$ is obtained for this image in the experiment via a calibration procedure (see Sec. 4.1 of this article for details).

Fig. 3

Generalized scheme of proposed $\mathrm{NI}+\mathrm{fBm}$ and $\mathrm{NI}+\mathrm{DCT}$ estimators for image texture and signal-dependent noise parameters.

In the first stage, noise is assumed to be signal-independent and an initial guess ${\widehat{\sigma }}_{n.\mathrm{SI}.\mathrm{IG}}$ is calculated as specified in Ref. 15. It is equal to the minimum of sample standard deviation estimates over all image nonoverlapping SWs. For the test image, the value ${\widehat{\sigma }}_{n.\mathrm{SI}.\mathrm{IG}}^2\approx 170.60$ was obtained. The initial guess for the vector $\mathbf{c}$ assumes the initial value ${\widehat{\mathbf{c}}}_{\mathrm{IG}}=({\widehat{\sigma }}_{n.\mathrm{SI}.\mathrm{IG}}^2,0)$.

Texture and noise parameter estimates and NI and TI maps are updated in stage 2. In this stage, we fix noise parameters to be equal to either an initial guess ${\widehat{\mathbf{c}}}_{i=1}={\widehat{\mathbf{c}}}_{\mathrm{IG}}$ or the previously estimated value ${\widehat{\mathbf{c}}}_i={\widehat{\mathbf{c}}}_{i-1}$ (here, $i$ defines the iteration index). The signal-dependent noise variance for each SW (both NI and TI) is calculated according to the retained polynomial noise model (a mixture of signal-independent and Poisson-like signal-dependent noises in this case) as ${\sigma }_n^2(I,{\widehat{\mathbf{c}}}_i)$, substituting true image intensity by mean intensity over current SW. Then, fBm-model parameters for TI and NI SWs are estimated, and discrimination between texture/NI SWs can be refined as specified in Ref. 15.

Next, in stage 3, a current value of relative CRLB per unit image intensity ($\mathrm{\Delta }I=1$) $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ and the corresponding set of intervals ${\mathbf{I}}_k$ are calculated as described above. For illustration purposes, the $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ obtained in the first iteration is shown in Fig. 4(a) as a straight black line.

Fig. 4

Illustration of the signal-dependent noise estimation process by using the algorithm in Fig. 3 for the test image.

The set of intervals ${\mathbf{I}}_k$ allows the estimating of ${\sigma }_{n.k}^2$ values using either the fBm- or DCT-based estimator and refining the $\widehat{\mathbf{c}}$ estimate according to Eq. (11). The ${\widehat{\sigma }}_{n.k}^2$ estimates and ${\sigma }_n^2(I,{\widehat{\mathbf{c}}}_i)$ function are both shown in Fig. 4(a) for the first iteration of the algorithm as black dots and a solid black line, respectively.

The estimate ${\sigma }_n^2(I,{\widehat{\mathbf{c}}}_i)$ is iteratively refined by repeating stages 2–4 until convergence. The convergence of the algorithm can be observed from the comparison of Fig. 4(a) with 4(b), where the first and the final (fifth) iteration estimates of $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$, ${\widehat{\sigma }}_{n.k}^2$, and ${\sigma }_n^2(I,{\widehat{\mathbf{c}}}_i)$ for the test image are shown. Note that for each iteration, Fig. 4 shows ${\sigma }_n^2(I,{\widehat{\mathbf{c}}}_i)$ estimates at the beginning of the iteration (signal-independent noise for the first iteration), not at the end. It can be seen that ${\sigma }_{n.k}^2$ estimates are concentrated at intensities where $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ takes on minimal values. Noise variance estimate ${\sigma }_n^2(I,{\widehat{\mathbf{c}}}_i)$ for $I$ from 600 to 1200 does not change significantly with iterations. Conversely, for $I$ from 0 to 300 that corresponds to areas (1) and (2) in Fig. 2(b), the noise variance estimate decreases from about 170 [the first iteration, Fig. 4(a)] to about 5 [the last iteration, Fig. 4(b)], and the number of ${\widehat{\sigma }}_{n.k}^2$ estimates decreases significantly (a smaller number of SWs is considered NI by the algorithm).

Now let us comment on the algorithm behavior for the textured area of the test image with an intensity close to 60 (dark trees in the lower-right part of the test image). With iterations, noise variance estimates are decreasing for this area. Consequently, the signal-to-noise ratio (measured locally) is constantly increasing. As a result, this area progressively becomes less and less noise informative [this is reflected by the increase of $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ function clearly seen in Fig. 4]. Finally, in iteration 5 [Fig. 4(b)], this area is excluded from the noise variance estimation process (it becomes related to the TI map).

The results obtained at final iteration confirm that, given a degraded image, $\mathrm{\Delta }{\sigma }_{{\sigma }_n.\mathrm{rel}}^2(I)$ indicates that image areas that can provide accurate noise variance estimation and their localization with regard to image intensity. It can be clearly seen that the estimated curve ${\sigma }_n^2(I,\widehat{\mathbf{c}})$ is very close to noise variance ${\sigma }_n^2(I,{\mathbf{c}}_0)$ obtained by the calibration procedure. For the considered test image, there are enough NI intensities to provide a high estimation accuracy of signal-dependent noise variance. For images with not enough NI (homogeneous) areas for estimating noise parameters, ${\sigma }_n^2(I,\widehat{\mathbf{c}})$ may become inaccurate. Based on our experiments, we have found that this extreme case can always be detected by checking the accuracy of $\widehat{\mathbf{c}}$ at the current iteration or the entries of the ${\mathbf{\sigma }}_{\mathbf{c}}^2$ matrix [see Eq. (12) for details].

4.1.

Image Database for Testing Signal-Dependent Noise Estimation Algorithms: NED2012

A key item in testing noise variance estimators is the availability of images with known noise parameters. Artificial noise-free images with synthetic noise raise questions about the applicability of the obtained results to the real images corrupted by real noise. Another possibility lies in using real-life images with a low level of noise, with synthetic noise added for test purposes. In this area, the TID2008⁴⁴ database has been extensively used.⁴⁵ However, we have identified four drawbacks of TID2008 that prevent us from exploiting it in our study:

To overcome these drawbacks, we have decided to base our study on 12-bit raw images from the Nikon D80 DSLR camera with a 10.2-Mpx CCD sensor. No extra noise was generated; we only dealt with the parameter estimation for noise that was originally present in D80 images. Our main assumption is that the noise parameters remain the same with time and camera operational conditions. Absence of noise spatial correlation is also assumed. Our experiments have shown no violation of these assumptions at the attained level of accuracy.

To accurately estimate true noise parameters of Nikon D80 sensor for ISO100, we have used the following semi-automatic calibration procedure:

Visual analysis of noise variance dependence on image intensity shows notable deviation from the theoretic linear shape (first-order polynomial), especially for the blue component (Fig. 5). To take this into account, the order of the approximation polynomial was set to $n_p=2$. The obtained estimates are given in Table 1 (we call them calibration lines). The noise parameters obtained via the calibration procedure will be marked with index “0” below.

Fig. 5

Nikon D80 noise variance dependence on image intensity for red, green, and blue channels.

Table 1

Sensor noise parameters for the Nikon D80 digital SLR camera (statistical error at one sigma level is specified)

It can be seen that the green channel is the least noisy one, followed by the red and blue channels. Quadratic terms for all channels are nonzero. These values are statistically significant (more than $0.704·{10}^{-5}/0.04924·{10}^{-5}$ or 14.3 sigma) and cannot be neglected. They probably appear due to the internal regulations of the camera.

For testing the two proposed estimators, we selected 25 D80 images taken from the same camera during a two-year interval and organized them into the NED2012 database. The database includes images with different content. Some of them have large homogeneous areas (e.g., sky), while others are quite textural, and defocused areas are present (Fig. 6). All images are presented as a CFA array of $2611\times 3900\text{pixels}$ in size. Red, green, and blue channel data were extracted from the CFA array by subsampling. Images have different ISO values (from 100 to 320), different shutter speeds (from $1/1250\mathrm{s}$ to $1/30\mathrm{s}$), different apertures (from f/4 to f/14) and different focal lengths (from 18 mm to 135 mm). From this set of parameters, it is the ISO parameter that directly affects noise parameters for both signal-independent and -dependent components. In order to compensate for this influence and convert all images to the reference ISO100, we have simply normalized each image by a factor of 100/ISO before processing.

Fig. 6

Images of the NED2012 database.

We will first prove that estimation results obtained by the $\mathrm{NI}+\mathrm{DCT}$ method agree with the calibration data shown above. For this goal, we applied the $\mathrm{NI}+\mathrm{DCT}$ method to all images from the NED2012 database simultaneously by processing 1,500 SWs of $9\times 9\text{pixels}$ randomly selected from each NED2012 image (a total of $1,500\times 25=37,500$ windows). In this manner, very high estimation accuracy of the $c_0$, $c_1$, and $c_2$ coefficients can be reached. Estimation results for all three channels are shown in Table 1 (the $\mathrm{NI}+\mathrm{DCT}$ line). Figure 7 illustrates these results for the blue channel.

Fig. 7

Comparison of $\mathrm{NI}+\mathrm{DCT}$ and calibration noise variance estimates for the blue channel of the NED2012 database.

One can clearly see that there is no statistically significant difference between estimates of noise parameters obtained for two different datasets (the NED2012 database and the set of 17 calibration images) by the calibration procedure and the $\mathrm{NI}+\mathrm{DCT}$ method (they differ by less than $4\sigma$). It is worth noting that the accuracy provided by the $\mathrm{NI}+\mathrm{DCT}$ method is only slightly worse than the one obtained by the calibration procedure. In this test, the $\mathrm{NI}+\mathrm{DCT}$ method shows its potential ability to deal with signal-dependent noise that is more complex than mixtures of additive and Poisson/multiplicative noises.

In the next experiment, we restricted ourselves to the case of a mixture of signal-independent and Poisson-like signal-dependent noises. The overall noise variance is thus signal-dependent: ${\sigma }_n^2(I)={\sigma }_{n.\mathrm{SI}}^2+I\cdot {\sigma }_{n.\mathrm{SD}}^2$. As was shown above, the noise in the Nikon D80 images does not strictly follow this linear model. Therefore, in order to nullify quadratic term $c_2$ before estimation, we normalized the image intensity in each SW ${\mathbf{I}}_{\mathrm{SW}}$ by

4.2.

Accuracy Analysis of Signal-Dependent Noise Parameter Estimation

The considered set of four estimators, $\mathrm{NI}+\mathrm{fBm}$, $\mathrm{NI}+\mathrm{DCT}$, ASRPFR, and ClipPoisGaus_stdEst2D, were applied to each of the 25 images of the NED2012 database in a component-wise manner (red, green, and blue components were processed independently). A SW of $9\times 9\text{pixels}$ in size was selected for both $\mathrm{NI}+\mathrm{DCT}$ and $\mathrm{NI}+\mathrm{fBm}$. The choice of this particular SW size is justified in this subsection. For each component, the two noise variance components ${\widehat{\sigma }}_{n.\mathrm{SI}}^2$ and ${\widehat{\sigma }}_{n.\mathrm{SD}}^2$ were estimated. Overall, 25 estimates were obtained for each channel. Their empirical probability density functions (pdfs), pdf (${\widehat{\sigma }}_{n.\mathrm{SI}}^2$) and pdf (${\widehat{\sigma }}_{n.\mathrm{SD}}^2$), are shown in Figs. 8 and 9, respectively. The main statistical characteristics of these estimates [mean $M(\cdot )$, bias and standard deviation $\mathrm{STD}(\cdot )$] are given in Table 2. The bias measure is calculated as $\text{bias}=100\%[M(\widehat{a})-a_0]/a_0$, where $a_0$ is the true value of $a$.

Fig. 8

Experimental pdfs of signal-independent noise component variance ${\widehat{\sigma }}_{n.\mathrm{SI}}^2$ estimates by (a) $\mathrm{NI}+\mathrm{fBm}$, (b) $\mathrm{NI}+\mathrm{DCT}$, (c) ASRPFR, and (d) ClipPoisGaus_stdEst2D. Calibration variances ${\sigma }_{n.\mathrm{SI}.0}^2$ are marked by vertical dot-dashed lines.

Fig. 9

Experimental pdfs of signal-dependent noise component variance ${\widehat{\sigma }}_{n.\mathrm{SD}}^2$ estimates by (a) $\mathrm{NI}+\mathrm{fBm}$, (b) $\mathrm{NI}+\mathrm{DCT}$, (c) ASRPFR, and (d) ClipPoisGaus_stdEst2D. Calibration variances ${\sigma }_{n.\mathrm{SD}.0}^2$ are marked by vertical dot-dashed lines.

Table 2

Mean value M(·), bias and standard deviation STD(·) of noise variance estimates on the NED2012 database.

Preliminary conclusions can be drawn from these results. Among these four estimators, ASRPFR and ClipPoisGaus_stdEst2D provide worse performance than the $\mathrm{NI}+\mathrm{fBm}$ and $\mathrm{NI}+\mathrm{DCT}$ with regard to both signal-independent and -dependent components. The main factor that degraded the performance of the ASRPFR and ClipPoisGaus_stdEst2D estimators is the significant number of outliers. However, they form a pronounced mode in the vicinity of the true value of signal-independent and -dependent noise component variances anyway. Therefore, we have decided to characterize the performance of these estimators by calling on extra robust measures. Specifically, median is used instead of mean value, and median absolute deviation (MAD) instead of standard deviation.

Estimates of signal-independent noise component variance are biased for all four estimators. The absolute value of the bias is the smallest for the $\mathrm{NI}+\mathrm{DCT}$ and $\mathrm{NI}+\mathrm{fBm}$ methods (i.e., less than 6%). It increases to more than 30% for the ASRPFR method and to more than 12% for the ClipPoisGaus_stdEst2D. The estimates of the signal-dependent noise component variance are practically unbiased for the $\mathrm{NI}+\mathrm{DCT}$ method, negatively biased by about 6% for the NI+fBm method, and exhibit a positive bias less than $25\%$ for the ASRPFR and ClipPoisGaus_stdEst2D methods.

It is worth highlighting that the $\mathrm{NI}+\mathrm{DCT}$ method provides the best estimation accuracy on both signal-independent and -dependent components. More precisely, with regard to the standard deviation of the noise variance estimates (specified in Table 2), it outperforms the $\mathrm{NI}+\mathrm{fBm}$ method by approximately 1.25 to 2.6 times, the ASRPFR method by 3.6 to 10 times, and by an even greater degree for the ClipPoisGaus_stdEst2D. We explain the reduced performance of the $\mathrm{NI}+\mathrm{fBm}$ estimator with regard to the $\mathrm{NI}+\mathrm{DCT}$ one by the sensitivity of the former to errors on Hurst exponent estimation. A second possible reason could be deviations of real image texture from the assumed fBm-model.

It is important to mention here that both ASRPFR and ClipPoisGaus_stdEst2D have been applied to NED2012 images without normalization [Eq. (13)] for quadratic term $c_2$ compensation. The reason for this is that such normalization operates at the SW level and depends on image partitioning during processing. We had no opportunity to modify the original implementation of ASRPFR and ClipPoisGaus_stdEst2D to take this into account. Therefore, in an additional experiment, we quantified the noncompensated $c_2$ term influence on $\mathrm{NI}+\mathrm{DCT}$ (Table 2). Overall, it led to negative bias of signal-independent component variance and positive bias of signal-dependent component variance. For red and green channels, this additional bias was negligible and does not exceed 5% in magnitude. For the blue channel, this bias was more significant, with a magnitude of about 15%. We thus believe this is evidence that the noncompensated $c_2$ term cannot explain the decreased performance of ASRPFR and ClipPoisGaus_stdEst2D.

In Fig. 10, we detail the signal-independent and -dependent noise variance estimates obtained with the best $\mathrm{NI}+\mathrm{DCT}$ estimator on each component (red, green, and blue) of all images from the NED2012 database. For most of the images, very accurate estimates were obtained. For these images, the potential estimation accuracy on the signal-independent noise component variance is about 0.2–1, and it is about $0.8-2.2·{10}^{-3}$ on the signal-independent noise component variance. But for some of the images, namely 13, 15, 16, 22, and 23, an increased estimation error was observed. This is reflected by a corresponding increase in the potential estimation accuracy for signal-independent and -dependent noise components to about 2 and $8\times {10}^{-3}$, respectively.

Fig. 10

(a) Signal-independent and (b) signal-dependent noise component variance according to image index $i_{\mathrm{NED}}$ estimated by the $\mathrm{NI}+\mathrm{DCT}$ on red, green, and blue components of each image from the NED2012 database. The true noise parameters are shown as dashed horizontal lines.

Let us now assess the efficiency of the analyzed estimators with regard to the diagonal terms $({\sigma }_{{\sigma }_{n.\mathrm{SI}}}^2,{\sigma }_{{\sigma }_{n.\mathrm{SD}}}^2)$ of the correlation matrix ${\mathbf{\sigma }}_{\mathbf{c}}^2$ of coefficient vector $\mathbf{c}$ defined above by Eq. (12). For this purpose, two normalized errors, one for ${\widehat{\sigma }}_{n.\mathrm{SI}}^2$ and another one for ${\widehat{\sigma }}_{n.\mathrm{SD}}^2$, respectively, are to be considered:

Note that for an efficient estimator, both ${\widehat{\sigma }}_{n.\mathrm{SI}.\text{norm}}^2$ and ${\widehat{\sigma }}_{n.\mathrm{SD.}\text{norm}}^2$ should be distributed close to the normal distribution $N(0,1)$. The statistical efficiency of the analyzed estimators with regard to these bounds can be estimated as

Figure 11 displays ${\widehat{\sigma }}_{n.\mathrm{SI}.\text{norm}}^2$ and ${\widehat{\sigma }}_{n.\mathrm{SD.}\text{norm}}^2$ pdfs, respectively, for the best $\mathrm{NI}+\mathrm{DCT}$ method (with $N=9$). The theoretical pdf $N(0,1)$ is added for comparison purposes. Efficiencies $\widehat{e}$ for both noise components are shown in Table 3 for four different window sizes: $N=7$, 9, 11, and 13.

Fig. 11

Empirical pdfs of the normalized estimates of signal-independent and signal-dependent noise components variances by the $\mathrm{NI}+\mathrm{DCT}$ estimator $N=9$. The theoretical pdf $N(0,1)$ is shown as a dashed black curve.

Table 3

Statistical characteristics of normalized estimates for the NED2012 database (tree channels).

As is shown, the $\mathrm{NI}+\mathrm{DCT}$ estimator exhibits rather high efficiency for different SW sizes, with a value of about 10 for both noise components. Note that for the $\mathrm{NI}+\mathrm{fBm}$ estimator, the corresponding efficiency (not included in Table 3) is about 2%, and it is less than 0.1% for both the ASRPFR and ClipPoisGaus_stdEst2D estimators. More detailed analysis shows that the performance of the $\mathrm{NI}+\mathrm{DCT}$ estimator is highest for $N=9$ and 11. It is slightly degrading for $N=13$ and the degradation is much more significant for $N=7$. This can be explained by the influence of two factors compensating each other. On one side, the accuracy of texture parameter estimation reduces for smaller sized SWs. This factor is responsible for the decline in performance for $N=7$. On the other hand, the image texture becomes more heterogeneous for larger windows, making the fBm model less adequate. This factor is responsible for the slight decrease in performance for $N=13$. Taking into account that the processing time is increasing fast with the SW size, we suggest $N=9$ as the best setting.

It is important to note that even for the $\mathrm{NI}+\mathrm{DCT}$ estimator, there exists an essential gap between the current level of accuracy and potential accuracy ${\mathbf{\sigma }}_{\mathbf{c}}^2$. This gap can be attributed equally to inherent shortcomings of the $\mathrm{NI}+\mathrm{DCT}$ estimator and to the bound ${\mathbf{\sigma }}_{\mathbf{c}}^2$ itself. For example, the latter does not properly take into account errors associated with Hurst exponent interpolation from TI map to NI map. Further research needs to be undertaken to reduce this gap and to better assess the potential of blind noise parameter estimation.