No reference noise estimation in digital images using random conditional selection and sampling theory

Abstract

An accurate quantitative noise estimate is required in many image/video processing applications like denoising, computer vision, pattern recognition and tracking. But blind and accurate estimation of noise in an unknown image is a challenging task and hence is an open area of research. We propose the first elegant and novel blind noise estimation method based on random image tile selection and statistical sampling theory for estimating standard deviation of zero mean Gaussian and speckle noise in digital images. Randomly selected samples, i.e., pixels with $3\times 3$ neighborhood, are checked for availability of edges in the tile. If there is an edge in the tile at more than one neighboring pixel, the tile is excluded. Only non-edge tiles are used for estimation of noise in the tile and subsequently in the image using the concepts of statistical sampling theory. Finally, we propose a supervised curve fitting approach using the proposed noise estimation model for more accurate estimation of standard deviation of the two types of noise. The proposed technique is computationally efficient as it is a selective random sample-based spatial domain technique. Benchmarking with other contemporary techniques published so far shows that the proposed technique clearly outperforms the others by at least 5% improved noise estimates, over a very wide range of noise.

Appendix A: How many samples should be taken?

An image I of size $M\times N$ has approximately the same number of $3\times 3$ tiles neglecting the boundary pixel positions. In this work, we consider a small square tile as a sample of the complete image for noise estimation. For example, an image of size 1K $\times$ 1K = 1M pixels will have exactly (1M-4K-4) $3\times 3$ tiles or approximately equal to 1M (with accuracy of more than 99.5%) tiles. Thus out of the available 1M tiles, how many tiles should be randomly taken to represent the complete image population accurately is an issue of concern. Obviously, for smaller size images, the same number of samples will be more than enough.

Let us assume that an image I is added a zero mean Gaussian noise of standard deviation $\sigma$. The noisy image $I_{N }$ modeled using a Gaussian variable has a measured mean $\mu$ and standard deviation $\sigma$$_{n}$. Let the unknown number ‘n’ of tile samples taken randomly has complete true mean $\bar{{X}}$, while each sample mean is $\hat{{\bar{{X}}}}$ and the variance of sample mean is Var($\hat{{\bar{{X}}}})$. Then, the minimum number of samples n for truly representing the complete population ($I_{N}$) for estimating the mean of the population to p% accuracy of the true mean value $\mu$ is given by (14) [31].

$$\begin{aligned} \mathrm{Var}(\hat{{\bar{{X}}}})=\frac{\sigma _n^2 }{n}=\left( {\frac{p}{100}\mu } \right) ^{2} \end{aligned}$$

Leading to,

$$\begin{aligned} n=\left( {\frac{100\times \sigma _n }{p\mu }} \right) ^{2} \end{aligned}$$

(14)

Thus, using (14), for around 2% estimation error in estimation of the population mean that overall represents the complete population; the number of required samples are 1200. Further, for 5% error in the estimation of the true mean $\mu$, it comes down to around 100. In the experiment section, we experimentally establish that the number of samples above 100 (0.0004%) and up to 5000 does not yield much improvement in the accuracy of the estimate over a wide range of the added noise standard deviation for image Lena of size $512\times 512$.

The theoretical error in computation of noise standard deviation estimation can be obtained using (6). A simple re-arrangement of terms in (6) leads to (15).

$$\begin{aligned} e=\frac{\left| {\sigma _{\mathrm{est}} -\sigma _{\mathrm{added}} } \right| }{\sigma _{\mathrm{added}} }\times 100=\left| {1-\sqrt{1-\frac{1}{n}}} \right| \times 100 \end{aligned}$$

(15)

where $\sigma _{\mathrm{est}}$ is the estimated standard deviation estimated using (6) and $\sigma _{\mathrm{added}}$ is the true noise standard deviation of the complete population, i.e., added noise in our experiments.

It is clear from (15) that, as ntends to $\infty$, i.e., the complete population, error e tends to 0. This is obviously true. In case of practical images, object edges and segment texture region make the distribution of image non-Gaussian. Hence, selecting a large sample size will invalidate the basis of sampling theory due to increased probability of image edges and textures being included in the sample tiles. If we consider the smallest symmetric sample tile size $3\times 3$, it yields sample size of 9. Using (15) yields the percent error e as 5.71%. For 7% error in estimation, we require $\sim$7.4 samples (i.e., 8 samples) leading to unsymmetrical tile size. For a sample size 100, the percentage error yield using (15) is 0.5%. The error in estimation will go on reducing for larger sample size for a truly random distribution. However, practical noisy images are not truly random variables. Also bigger tile sizes lead to inaccurate estimates as already discussed.

Thus, we take number of samples $n = 100$ and the smallest symmetric and most probably smooth 3$\times 3 = 9$ sample (tile) size for our experiments.