Current Issue

Images are contaminated mostly during acquisition from the image sensor. The raw-data acquired by sensors undergoes many image processing stages. Estimating the characteristics of noise is vital for improving the image quality in subsequent image processing stages. Most of the denoising methods assume additive white Gaussian noise(AWGN), which can successfully model thermal and electrical noise. The variance of noise under AWGN is fixed and signal-independent. However, due to the significant size reduction of pixel units, modern high-resolution imaging sensors are more sensitive to the photon noise[1].

Therefore, images acquired with modern imaging sensors are better modeled as signal-dependent, where the variance of noise is a function of signal intensity. The objective of the signal-dependent noise estimation methods is to predict the dependency of the noise variance on signal intensity, which is formally known as the noise level function.

Generally, the existing signal-dependent noise estimation methods consist of four steps: 1) image segmentation, 2) smooth region detection, 3) noise variance estimation, and 4) fitting obtained data samples into a global parametric model. To estimate the noise variance – intensity pairs, most common practice is to cluster noisy image into several segments with nearly constant intensities[1][2].

The main assumption is that on the homogeneous regions of each segment the intensity variations are mainly caused by noise. To detect such image regions some methods utilize high pass filters[3][4], edge and texture extractors or local statistics[5][6].

In this manner, for each intensity segment a set of homogeneous patches are obtained by means of hard segmentation. To estimate unbiased noise variance, an input image is transformed into other domains, such as discrete cosine transform(DCT)[7][8], and wavelet transform[1], where noise can be distinguished from the original image information in high frequency subbands. However, for high textured images the noise level estimates of aforementioned methods are overestimated. In addition, according to the law of large number, the estimation accuracy is naturally dependent on the size of sample. Due to the hard segmentation, the number of valid candidates on some intensity segments are low. Therefore, the noise variance estimates on these intensity segments does not converge towards the expected value.

In the last step the obtained noise variance – intensity pairs are fitted into a global parametric model. Foi et al.[1] proposed to fit the estimated data into Poissonian-Gaussian noise model using iterative maximum likelihood fitting. Liu et al.[9] extended the method by using generalized signal-dependent noise model. Li et al.[7] proposed an iterative re-weighted least squares method, by taking the texture strengths of patches into account. Although these methods improve the accuracy of the noise estimation, they are computationally complex due to their iterative framework.

In this paper, we propose a fast and accurate method for image signal-dependent noise estimation. We believe, that increasing the sample size for estimation, by using soft segmentation approach instead of conventional hard segmentation, would improve the accuracy of the estimation. Based on the noisy image histogram we initialize a pre-fixed number of intensity centroids, whose occurrence is large enough[7], and directly compute the noise variance for each intensity centroid via weighted summation. Specifically, we assign the pixels, which have similar intensity with the intensity centroid and correspond to the low gradient region, with high weights and vice versa. Unlike other conventional approaches, we estimate the noise variance directly in the image intensity domain.

To fit the estimated noise variance – intensity pairs into a noise model we utilize a least mean square error approach. According to our experiments on synthetic and real noisy images, the proposed method provides fast, and reliable noise estimates.

The rest of the paper is organized as follows. In Section 2 we briefly review the image sensor noise model. The proposed method is then presented in Section 3. The accuracy and applications of the proposed method is discussed in Section 4. Finally we draw conclusions in Section 5.

We consider generalized signal-dependent noise model[10] of the form

where x∈X is the pixel position, z(x) is the observed noisy pixel value, y(x) is the original noise-free pixel value, γ is a parameter that controls the dependence of noise on signal, and u(x) and w(x) are two mutually independent zero-mean Gaussian variables with variances σu2 and σw2 respectively. The noise level function σ(y(x)) of generalized signal-dependent noise model can be derived from (1), and has the affine form

The Poissonian-Gaussian noise model is a special type of generalized signal-dependent noise model with γ=0.5. In this paper, we consider this noise model as it is naturally suited for the raw-data of digital imaging sensors[1]. Our objective is to estimate the noise level function parameters (σu2 and σw2) from a single raw image.

Ideally, one can obtain true noise level function parameters if the noise variances corresponding to every pixel intensity are known. Since we have a single raw-image at our disposal, it is impossible to estimate the noise variance for every pixel intensity. In order to obtain robust estimate of the noise variance the size of the sample has to be large enough. Conventionally the image is segmented into collection of non-overlapping level sets, where each set is characterized by its mean value and allowed deviation. The noise variance on each segment is assumed constant, and is computed only on the homogeneous regions. Therefore, for highly textured images the sample size(set of homogeneous patches in a given intensity level) is small. Our approach focuses on increasing the sample size for estimation. We proceed with detailed explanation of our proposed approach.

To evaluate the accuracy of the estimation we test algorithms on the noise free image set shown in Fig. 1(a). The synthetic noisy images are generated by using three parameter settings(σ_u = 1.0, σ_w = 2.0; σ_u = 1.5, σ_w = 3.0; σ_u = 2.0, σ_w = 4.0) corresponding to low, medium and heavy contaminations, respectively. We measure the accuracy of the estimation in terms of root mean squared error(RMSE) against the theoretical curve (2).

Fig. 1. 
Image sets used in experiments

Table 1 shows the detailed comparison of estimation accuracy of algorithms. The presented RMSE results are the average of ten realizations. The average RMSE indicates that the proposed method outperforms the state-of-the-art for medium and heavy noise levels. In addition, while the estimation accuracy of all competing algorithms degrade with increasing the noise level, the proposed method is comparatively robust against noise level variations. The estimated noise level functions for ‘Traffic’ test image are depicted in Fig. 2.

Fig. 2. 
Comparison of noise level function estimation with Foi et al.[1], and Pyatykh et al.[12] against theoretical noise level function (2). Estimation is performed on ‘Traffic’ test image for (a) Heavy, and (b) Low noise levels

For high noise level, conventional approaches overestimate the noise standard deviation, while the estimated noise level function of proposed approach almost coincides with the theoretical curve. Foi et al. approach[1] fails when the noise level is low, while Pyatykh et al. approach[12] performs well(best RMSE).

In Table 2, we summarized the average RMSE results and the average runtimes of corresponding algorithms. As shown, the proposed method on average has the best accuracy and is 20 times faster than[1], and 700 times faster than[12].

We presented a fast and practical noise level function estimation method for Poissonian-Gaussian noise from a single image. The hard segmentation approach used in the conventional noise estimation schemes reduces the sample size of estimation, consequently degrading the estimation accuracy.

We showed that increasing the sample size for estimation, by using the proposed soft segmentation approach, is not only computationally efficient, but also improves the accuracy of the estimation. We initialize a pre-fixed number of intensity centroids based on the noisy image histogram, and compute the noise variances for each intensity centroid in the image intensity domain via weighted summation. The pixels, which have similar intensity with the intensity centroid and correspond to the low gradient region, are assigned with high weights and vice versa. According to our experiments on synthetic and real noisy images, our method provides fast and reliable noise estimates.