Superpixel-based image noise variance estimation with local statistical assessment
- Cheng-Ho Wu^{1} and
- Herng-Hua Chang^{2}Email author
https://doi.org/10.1186/s13640-015-0093-2
© Wu and Chang. 2015
Received: 11 December 2014
Accepted: 20 November 2015
Published: 28 November 2015
Abstract
Noise estimation is fundamental and essential in a wide variety of computer vision, image, and video processing applications. It provides an adaptive mechanism for many restoration algorithms instead of using fixed values for the setting of noise levels. This paper proposes a new superpixel-based framework associated with statistical analysis for estimating the variance of additive Gaussian noise in digital images. The proposed approach consists of three major phases: superpixel classification, local variance computation, and statistical determination. The normalized cut algorithm is first adopted to effectively divide the image into a set of superpixel regions, from which the noise variance is computed and estimated. Subsequently, the Jarque–Bera test is used to exclude regions that are not normally distributed. The smallest standard deviation in the remaining regions is finally selected as the estimation result. A wide variety of noisy images with various scenarios were used to evaluate this new noise estimation algorithm. Experimental results indicated that the proposed framework provides accurate estimations across various noise levels. Comparing with many state-of-the-art methods, our algorithm strikes a good compromise between low-level and high-level noise estimations. It is suggested that the proposed method is of potential in many computer vision, image, and video processing applications that require automation.
Keywords
Gaussian noise Noise estimation Image denoising Superpixel Jarque–Bera test1 Introduction
In the field of computer vision, signal, image, and video processing, noise is unfortunately inevitable during data acquisition and transmission. The accuracy of many algorithms significantly relies on well hand-tuned parameter adjustments to account for variations in noise [1–3]. To automate the process and achieve reliable procedures, the capability for accurate noise estimation is essential to motion estimation, edge detection, super-resolution, restoration, shape-from-shading, feature extraction, and object recognition [4–9]. In particular, image noise having a Gaussian-like distribution is quite often encountered, and it is characterized by adding to each pixel a random value obtained from a zero-mean Gaussian distribution, whose variance determines the magnitude of the corrupting noise. This zero-mean property enables such noise to be removed by locally averaging neighboring pixel values [10, 11].
Indeed, many noise reduction algorithms incorporate the knowledge of the noise level in the denoising process and assume that it is known a priori [12–15]. Accordingly, estimation for the amount of noise is critical in these methods, because it enables the process to adapt to the level of noise rather than using fixed values and thresholds. The challenge of noise estimation is to determine whether local image variations are due to color, texture, and lighting changes of images themselves, or caused by the noise. Nevertheless, existing noise estimation algorithms can be broadly classified into three major categories: filtered-based, block-based, and transform-based approaches [4, 5, 11, 16, 17].
In filtered-based methods, an input image is first filtered by a low-pass filter to smooth the structures and suppress the noise in the image [4]. The noise variance is then estimated from the difference between the noisy image and the filtered image. One fundamental problem of filtered-based methods is that the difference image is assumed to be the noise, but this assumption is not always true in general. This is because the low-pass filtered image is not equivalent to the original noise-free image, particularly when the image is with strong structures and complicated details. To minimize the influence and obtain a realistic basis for noise level estimation, Rank et al. [18] proposed to use the vertical and horizontal information of an image to extract the noise detail and histogram information in the corresponding components. However, it has a relatively higher computation load and many user-defined parameters to be set.
For block-based algorithms, an image is tessellated into a number of blocks followed by noise variance computation in a set of homogeneous blocks [5, 17, 19]. The philosophy underlying this approach is that a homogeneous block in an image is treated as a perfectly smooth image block with added noise, which has a relatively higher chance to contain useful visual activities. Consequently, the block with a smaller standard deviation has a weaker variation in intensity, leading to a smoother block. One main difficulty of block-based approaches is how to efficiently identify the homogeneous blocks. Lee and Hoppel [20] estimated noise level by assuming that the smallest standard deviation of a block is equivalent to additive white Gaussian noise. This method is simple but tends to produce overestimation results for small noise cases. Shin et al. [5] split an image into a number of blocks, which were further classified by the standard deviation in intensity. An adaptive Gaussian filtering process was then applied to relatively flat blocks, where the noise was estimated from the difference of the selected blocks between the noisy image and its filtered image.
While noise estimation methods in the first two categories work directly on the pixel intensity in the spatial domain, transform-based methods seek particular features in the transformed domain [21]. For example, the median absolute deviation method [22–24] used wavelet coefficients to estimate noise standard deviation based on the assumption that wavelet coefficients in the diagonal subband HH_{1} are dominated by noise. This approach provides good estimations for large noise cases, but it can overestimate the noise in small noise cases. The reason for overestimation is that wavelet coefficients in the diagonal subband contain not only added noise but also image details. Subsequently, Li et al. [25] proposed a modified noise estimation algorithm based on the wavelet coefficients in the HH_{1} subband. Better results were obtained by reducing the estimated original image contribution from HH_{1} comparing to the traditional methods. Liu and Lin [17] investigated the possibility to estimate noise in the singular value decomposition (SVD) domain. The authors used the tail of singular values to alleviate the influence of the signal in the noise estimation process and demonstrated the effectiveness of their method over wavelet-based approaches. However, due to the use of SVD twice in the estimation procedure, the computation time is more expensive.
Alternatively, there are other methods that estimate noise in various manners [26]. Immerkaer [27] proposed a Laplacian-based noise estimation algorithm, which computes the noise variance by convolving the image with a Laplacian-like mask with zero mean. This approach is fast and performs well on images that are corrupted by high level noise. However, for highly textured images, it perceives thin lines as noise, leading to overestimation. Tai and Yang [28] extended Immerkaer’s work by introducing the Sobel operator for edge detection to exclude the edge pixels. Salmeri et al. [29] introduced different weights to various subregions based on a similarity measure followed by a fuzzy procedure to estimate the variance of noise. Zoran and Weiss [30] proposed a statistical model to estimate the variance of noise and showed the effectiveness on images with low-level noise. Their assumption is that adding noise to images results in changes to kurtosis values throughout the scales. In their approach, the image was first convolved by the DCT filter to produce a response image, from which the variance and kurtosis were estimated. Aja-Fernández et al. [16] presented a noise estimation method based on the mode of local statistics (MLS). The authors demonstrated the efficiency of using the mode of the local sample statistical distribution for the variance estimation of additive noise provided that a great amount of low-variability areas exist in the image.
Among existing noise estimation methods, block-based algorithms are relatively simple and straightforward. Nonetheless, one main issue of this approach is how to effectively identify the homogeneous regions while alleviating the dependence of various noise levels. To address this major challenge and overcome the drawbacks in the existing methods, this paper proposes a new noise estimation algorithm that automatically and efficiently divides an image into a number of homogeneous subregions, which are called superpixels. To reduce noise influences, a statistical decision is then made to select the best superpixel, from which the noise variance is estimated. The ambition is to improve the estimation accuracy in low level noise while maintaining precision for higher level noise comparing to existing methods. The remainder of this paper is organized as follows. In Section 2, the noise model along with the probability density function of the Gaussian noise is described. Section 3 introduces the proposed algorithm that consists of three major phases: superpixel classification, local variance computation, and statistical determination. In Section 4, the performance of this new noise estimation method using a wide variety of images with various scenarios is evaluated and compared with many state-of-the-art methods. Finally, Section 5 discusses the results and Section 6 summarizes the contributions of the current work.
2 Noise model
3 Methods
3.1 Superpixel classification
The first step in our noise estimation framework is to divide a noisy image into several subregions. Unlike conventional block-based methods, each subregion is not necessary to be a rectangular block and it is usually not. In essence, each region is expected to have similar gray-level, color, and texture characteristics regardless of its geometry. To do this, the normalized cut algorithm [31] is adopted to achieve this goal. The basic idea is to use the theoretic criteria of graph to measure the goodness of an image partition. More specifically, it measures both the total dissimilarity between different groups as well as the total similarity within groups. The optimization of this criterion can be formulated as a generalized eigenvalue problem that can be efficiently solved. The concept of this perceptual grouping technique is briefly described as follows.
Given an image of N pixels, the set of pixels can be represented as a weighted undirected graph G = (V, E), where V represents nodes of the graph corresponding to the pixels in the feature space, E represents edges that are formed between every pair of nodes. A weight w(i, j) is assigned to each edge that captures the similarity between nodes i and j. In grouping, the goal is to partition the set of vertices into m disjoint sets V _{ 1 }, V _{ 2 },…, V _{ m }, where, by some measure, the similarity among the vertices in a set V _{ i } is high while it is low across different sets.
3.2 Local variance computation
3.3 Statistical determination
In other words, the JB value equals to 0 if the corresponding superpixel region is examined as a normal distribution; otherwise, it is set to 1.
- 1.
Sort R _{ i } based on the standard deviation \( {\sigma}_{R_i} \) in ascending order.
- 2.
Exclude the superpixel region whose JB value equals to 1.
- 3.
Exclude the superpixel region whose pixel number is less than \( 10\times \min \left({\sigma}_{R_i}\right) \), where \( \min \left({\sigma}_{R_i}\right) \) is the smallest value of \( {\sigma}_{R_i} \) in region R.
- 4.
Choose the smallest value of \( {\sigma}_{R_i} \) from the remaining regions as the noise estimation result.
The reason for excluding the regions with a small pixel number in rule 3 is due to the fact that these regions may not have an enough sample quantity to reflect the real noise distribution, leading to poor estimations. As the region size is somewhat related to the value of \( {\sigma}_{R_i} \), the threshold is thus defined as it is and scaled by an experimental constant.
4 Experimental results
Quantitative analysis in noise level estimation with σ _{ a } = 1, 3, 5, 7, 10, 15, 20 on the Bird image using different methods
JI96 | Z&W09 | MLS09 | SVD13 | Ours | ||||||
---|---|---|---|---|---|---|---|---|---|---|
σ _{ a } | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) |
1 | 7.43 | 642.71 | 0.00 | 100.00 | 1.39 | 38.73 | 2.96 | 195.76 | 1.10 | 9.71 |
3 | 7.91 | 163.80 | 0.00 | 100.00 | 2.66 | 11.23 | 4.02 | 33.90 | 2.85 | 4.93 |
5 | 8.82 | 76.38 | 2.68 | 46.40 | 4.90 | 1.98 | 5.46 | 9.14 | 4.81 | 3.72 |
7 | 10.00 | 42.89 | 4.43 | 36.67 | 6.84 | 2.31 | 7.11 | 1.55 | 6.47 | 7.58 |
10 | 12.10 | 20.97 | 6.96 | 30.44 | 9.65 | 3.5 | 9.88 | 1.17 | 9.63 | 3.73 |
15 | 16.13 | 7.55 | 11.24 | 25.05 | 14.66 | 2.24 | 14.85 | 0.97 | 14.49 | 3.43 |
20 | 20.15 | 0.07 | 15.32 | 23.40 | 19.03 | 4.83 | 19.73 | 1.35 | 17.92 | 10.42 |
\( \mathrm{Average}\kern0.5em {\overline{\boldsymbol{\varepsilon}}}_{\boldsymbol{r}} \) | 136.34 | 51.71 | 9.26 | 34.83 | 6.22 |
Statistical comparison of noise level estimation on the six representative images in Fig. 4 between different methods
JI96 | Z&W09 | MLS09 | SVD13 | Ours | ||||||
---|---|---|---|---|---|---|---|---|---|---|
σ _{ a } | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) |
1 | 5.15 ± 2.15 | 415.23 | 0.92 ± 0.71 | 58.62 | 1.60 ± 0.36 | 59.91 | 1.75 ± 0.92 | 96.18 | 1.22 ± 0.48 | 38.12 |
3 | 5.97 ± 1.86 | 99.03 | 2.16 ± 1.34 | 35.92 | 3.18 ± 0.20 | 6.79 | 3.24 ± 0.61 | 17.53 | 3.20 ± 0.27 | 10.26 |
5 | 7.25 ± 1.56 | 44.98 | 4.3 ± 0.88 | 15.81 | 5.08 ± 0.22 | 3.64 | 5.08 ± 0.43 | 6.70 | 4.83 ± 0.37 | 6.27 |
7 | 8.77 ± 1.30 | 25.34 | 6.15 ± 0.84 | 12.25 | 7.04 ± 0.36 | 3.84 | 7.01 ± 0.36 | 3.76 | 7.34 ± 1.01 | 8.65 |
10 | 11.28 ± 1.00 | 12.80 | 8.89 ± 0.91 | 11.06 | 9.95 ± 0.46 | 3.34 | 9.97 ± 0.30 | 2.11 | 9.72 ± 0.57 | 5.85 |
15 | 15.79 ± 0.69 | 5.26 | 13.52 ± 1.06 | 9.84 | 14.63 ± 0.61 | 4.19 | 15.06 ± 0.34 | 1.49 | 14.05 ± 0.53 | 6.33 |
20 | 20.28 ± 0.67 | 2.28 | 17.99 ± 1.28 | 10.04 | 19.34 ± 0.53 | 3.29 | 20.16 ± 0.42 | 1.02 | 17.59 ± 1.43 | 12.03 |
25 | 24.76 ± 0.83 | 3.06 | 22.51 ± 1.63 | 9.97 | 24.10 ± 0.67 | 3.88 | 25.27 ± 0.21 | 1.10 | 23.44 ± 1.31 | 6.94 |
\( \mathrm{Average}\kern0.5em {\overline{\boldsymbol{\varepsilon}}}_{\boldsymbol{r}} \) | 76.00 | 20.44 | 11.11 | 16.24 | 11.81 |
Statistical comparison of noise level estimation on 100 images obtained from the Berkeley image database between different methods
JI96 | Z&W09 | MLS09 | SVD13 | Ours | ||||||
---|---|---|---|---|---|---|---|---|---|---|
σ _{ a } | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) | σ _{ e } ± std. | ε _{ r } (%) |
1 | 5.68 ± 2.26 | 468.22 | 1.25 ± 1.61 | 114.72 | 2.51 ± 2.59 | 150.99 | 3.58 ± 3.35 | 302.81 | 1.87 ± 1.39 | 100.62 |
3 | 6.41 ± 2.05 | 113.78 | 2.75 ± 1.68 | 40.33 | 4.35 ± 2.67 | 46.09 | 4.40 ± 3.05 | 76.67 | 3.61 ± 1.31 | 32.50 |
5 | 7.59 ± 1.78 | 51.86 | 4.78 ± 1.59 | 22.60 | 6.22 ± 2.52 | 25.89 | 5.58 ± 2.76 | 38.80 | 5.42 ± 1.19 | 18.74 |
7 | 9.04 ± 1.55 | 29.14 | 6.62 ± 1.47 | 15.91 | 8.10 ± 2.25 | 17.43 | 6.98 ± 2.47 | 25.75 | 7.80 ± 1.49 | 17.37 |
10 | 11.44 ± 1.29 | 14.89 | 9.45 ± 1.37 | 12.55 | 10.91 ± 2.04 | 11.34 | 9.29 ± 2.10 | 18.07 | 10.57 ± 1.88 | 13.20 |
15 | 15.75 ± 1.09 | 6.61 | 14.12 ± 1.49 | 9.51 | 15.67 ± 1.62 | 6.66 | 13.49 ± 1.69 | 13.70 | 15.47 ± 1.83 | 8.76 |
20 | 20.20 ± 1.10 | 3.75 | 18.65 ± 1.69 | 8.95 | 20.36 ± 1.55 | 4.66 | 17.87 ± 1.43 | 12.01 | 19.71 ± 1.98 | 7.90 |
25 | 24.63 ± 1.28 | 3.25 | 23.04 ± 1.95 | 9.16 | 25.12 ± 1.47 | 3.93 | 22.26 ± 1.22 | 11.43 | 24.36 ± 1.92 | 6.15 |
30 | 29.02 ± 1.53 | 3.93 | N/A | N/A | 29.85 ± 1.47 | 3.46 | 26.78 ± 1.15 | 10.88 | 29.07 ± 1.86 | 5.87 |
35 | 33.30 ± 1.80 | 5.09 | N/A | N/A | 34.46 ± 1.54 | 3.73 | 31.15 ± 1.19 | 11.04 | 33.63 ± 2.24 | 5.49 |
40 | 37.47 ± 2.09 | 6.37 | N/A | N/A | 39.41 ± 1.52 | 3.22 | 35.65 ± 1.13 | 10.89 | 37.91 ± 3.13 | 6.95 |
\( \mathrm{Average}\kern0.5em {\overline{\boldsymbol{\varepsilon}}}_{\boldsymbol{r}} \) | 64.26 | N/A | 25.22 | 48.37 | 20.32 |
5 Discussion
A new superpixel-based algorithm was proposed to estimate the additive Gaussian noise level in images. The approach relied on the normalized cut algorithm to classify the image in order to obtain the superpixel map, from which the local variance was computed and selected. As illustrated in Fig. 6, each superpixel region had similar gray-level, color, and texture characteristics such that it provided great flexibility for subsequent statistical analysis. Since the Gaussian noise obeys a normal distribution, we proposed to use the Jarque–Bera test [33, 34] to separate those superpixel regions that follow a Gaussian distribution from all other regions based on the skewness and kurtosis. After excluding the superpixel regions that had a relatively small number of pixels, the remaining smallest local standard deviation was selected as the noise estimation result.
The proposed framework was applied on a wide variety of images and compared with four state-of-the-art methods, namely JI96 [27], Z&W09 [30], MLS09 [16], and SVD13 [17]. As illustrated in Fig. 7, our technique provided high accuracy for both simple (Countryroad) and complicated (Bird) texture images, particularly for low-level noise estimations. This great capability of excellent accuracy in low-level noise estimation can also be observed from Tables 1 and 2, which were experimented on the images shown in Fig. 4. In addition, the algorithm was extensively evaluated on 100 randomly selected images from the Berkeley database, which contained a wide diversity of photographic images. In comparison with the JI96, Z&W09, MLS09, and SVD13 methods, the proposed approach provided more accurate estimations for low-level noise images as well as smaller overall estimation errors as presented in Table 3. In general, the JI96 method performs better in images with high-level noise but inadequately for images with low-level noise. In contrast, the Z&W09 method performs better in images with lower level noise, but it fails to produce estimation results for larger noise levels with σ _{ a } > 25. Both MLS09 and SVD13 methods can provide satisfactory results in larger noise level estimation but they may notably overestimate small noise levels.
Performance analysis in noise level estimation on the image in Fig. 8 using different methods
JI96 | Z&W09 | MLS09 | SVD13 | Ours | ||||||
---|---|---|---|---|---|---|---|---|---|---|
σ _{ a } | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) | σ _{ e } | ε _{ r } (%) |
1 | 10.83 | 983.20 | 9.90 | 890.33 | 11.61 | 1061.12 | 14.23 | 1323.03 | 4.48 | 348.05 |
3 | 11.23 | 274.27 | 10.59 | 252.90 | 11.49 | 282.95 | 14.67 | 388.92 | 8.58 | 186.15 |
5 | 11.89 | 137.74 | 11.59 | 131.86 | 12.46 | 149.16 | 14.74 | 194.84 | 9.51 | 90.25 |
7 | 12.91 | 84.46 | 13.18 | 88.26 | 13.14 | 87.74 | 15.33 | 118.96 | 10.79 | 54.11 |
10 | 14.77 | 47.74 | 15.70 | 57.04 | 15.33 | 53.27 | 17.05 | 70.52 | 13.23 | 32.25 |
15 | 18.45 | 23.00 | 20.04 | 33.57 | 18.00 | 20.03 | 19.97 | 33.13 | 16.70 | 11.33 |
20 | 22.71 | 13.56 | 24.61 | 23.04 | 23.12 | 15.59 | 22.84 | 14.19 | 20.88 | 4.40 |
25 | 27.26 | 9.05 | 29.23 | 16.92 | 25.20 | 0.81 | 27.25 | 9.00 | 23.92 | 4.32 |
30 | 31.79 | 5.97 | N/A | N/A | 30.59 | 1.95 | 31.68 | 5.60 | 27.28 | 9.07 |
35 | 36.17 | 3.34 | N/A | N/A | 34.53 | 1.35 | 34.57 | 1.23 | 33.67 | 3.81 |
40 | 40.41 | 1.02 | N/A | N/A | 34.66 | 13.35 | 39.59 | 1.02 | 38.42 | 3.95 |
\( \mathrm{Average}\kern0.5em {\overline{\boldsymbol{\varepsilon}}}_{\boldsymbol{r}} \) | 142.17 | N/A | 153.39 | 196.40 | 67.97 |
Comparison of computation time in seconds between tested methods based on different dimensions of images
Size | JI96 | SVD13 | MLS09 | Z&W09 | Ours |
---|---|---|---|---|---|
100 × 100 | 0.02 | 0.16 | 0.32 | 1.2 | 3.2 |
200 × 200 | 0.02 | 0.20 | 0.33 | 1.2 | 13.6 |
300 × 300 | 0.02 | 0.26 | 0.33 | 1.8 | 34.8 |
400 × 400 | 0.02 | 0.33 | 0.35 | 2.4 | 85.2 |
500 × 500 | 0.02 | 0.44 | 0.39 | 3.6 | 152.0 |
6 Conclusions
In summary, a new algorithm for additive Gaussian noise level estimation is described, which consists of three major phases: superpixel classification, local variance computation, and statistical determination. Th e algorithm strikes a good compromise between low-level and high-level noise estimations. Hundreds of images with various subjects, scenes, textures, and structures were used to evaluate the proposed framework. Experimental results demonstrated the feasibility and effectiveness of the algorithm in providing accurate estimation results across a wide range of noise levels. This robust noise estimation framework is advantageous to automating denoising algorithms that require noise variance information. Moreover, the proposed noise estimation algorithm is of potential and promising in computer vision, image, and video processing applications. Further research is needed to more effectively divide the image into appropriate superpixels, to investigate the incorporation of filtered-based techniques, and to accelerate the computation for real-time applications.
Declarations
Acknowledgements
This work was supported in part by the National Science Council under research grant no. NSC 100-2320-B-002-073-MY3 and the Ministry of Science and Technology of Taiwan under research grant no. MOST 104-2221-E-002-095.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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