## Figures

## Abstract

In this paper we consider the problem of restoration of a image contaminated by a mixture of Gaussian and impulse noises. We propose a new statistic called ROADGI which improves the well-known Rank-Ordered Absolute Differences (ROAD) statistic for detecting points contaminated with the impulse noise in this context. Combining ROADGI statistic with the method of weights optimization we obtain a new algorithm called Optimal Weights Mixed Filter (OWMF) to deal with the mixed noise. Our simulation results show that the proposed filter is effective for mixed noises, as well as for single impulse noise and for single Gaussian noise.

**Citation: **Jin Q, Grama I, Liu Q (2017) Optimal Weights Mixed Filter for removing mixture of Gaussian and impulse noises. PLoS ONE 12(7):
e0179051.
https://doi.org/10.1371/journal.pone.0179051

**Editor: **Yuanquan Wang,
Beijing University of Technology, CHINA

**Received: **November 11, 2016; **Accepted: **May 23, 2017; **Published: ** July 10, 2017

**Copyright: ** © 2017 Jin et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Data Availability: **All relevant data are within the paper.

**Funding: **Jin and Liu have been partially supported by the National Natural Science Foundation of China (Grants No. 61661039, No. 11571052 and No. 11401590), the Natural Science Fund of Inner Mongolia Autonomous Region (Grant No. 2016MS0107), the Scientific Research Projection of Higher Schools of Inner Mongolia (Grant No. NJZY16017), and Hunan Provincial Natural Science Foundation (Grant no. 2017JJ2271).

**Competing interests: ** The authors have declared that no competing interests exist.

## 1 Introduction

Noise can be systematically introduced into digitized images during acquisition and transmission, which usually degrade the quality of digitized images. However, various image-related applications, such as aerospace, medical image analysis, object detection etc., generally require effective noise suppression to produce reliable results. The nature of the problem depends on the type of noise to the image. Generally, two noise models can adequately represent most noise added to images. Often in practice it is assumed that the noise has two components: an additive Gaussian noise and an impulse noise.

The additive Gaussian noise model is defined by
(1)
where *N* ∈ **N**, *Y* is the observed image brightness, *f*: **I** ↦ [*c*, *d*] is an unknown target regression function, and *ϵ*(*x*), *x* ∈ **I**, are independent and identically distributed (i.i.d.) Gaussian random variables with mean 0 and standard deviation *σ* > 0. The Model (1) adds to each digitized image pixel a value from a zero-mean Gaussian distribution. Such noise is usually introduced during image acquisition. The zero-mean property of this Gaussian distribution makes it possible to remove the Gaussian noise by Non-Local weighted averaging. Important denoising methods for the Gaussian noise model have been well developed in recent years, see for example [1–15]

The random impulse noise model is defined by
(2)
where **B** is the set of pixels contaminated by impulse noise, is the impulse probability (the proportion of the occurrence of the impulse noise), *n*(*x*) are independent random variables uniformly distributed on some interval [*c*, *d*]. The impulse noise is characterized by replacing a pixel value with a random one. Such a noise can be introduced due to transmission errors, malfunctioning pixel elements in the camera sensors, faulty memory locations, and timing errors in analog-to-digital conversion. Recently, some important methods have been proposed to remove the impulse noise, see for example: [16–23].

However, the above mentioned methods are not effective when we apply them to remove a mixture of the Gaussian and impulse noises defined by (3) The Gaussian noise removal methods cannot adequately remove impulse noise, for they interpret the impulse noise pixel as edges to be preserved; when impulse removal methods are applied to an image corrupted with the Gaussian noise, such filters, in practice, leave grainy, visually disappointing results. Garnett et al. [24] introduced a new local image statistic called Rank Ordered Absolute Difference (ROAD) to identify the impulse noisy pixels and incorporated it into a filter designed to remove the additive Gaussian noise. As a result they have obtained a trilateral filter capable to remove mixed Gaussian and impulse noise. This method also performs well for removing the single impulse noise. A variant of the ROAD statistic called ROLD was introduced in Dong et al. [22] which amplifies the differences between noisy and noise-free pixels, so that the noise detection becomes more accurate. An impulse detector and a filter which efficiently removes impulse/Gaussian mixed noise has been proposed in Xiong and Yin [25]. Lien et al [26] employed a decision-tree-based impulse noise detector and an edge-preserving filter to reconstruct the intensity values of noisy pixels, whose hardware cost was low. For other developpements in this direction we refer to [27–30]. Recently Delon and Desolneux [31, 32] and Hu et al. [33] introduced patch-based approaches to deal with the impulse noise and the mixture of Gaussian and impulse noises.

In this paper, we propose a new patch-based filter that we call Optimal Weights Mixed Filter (OWMF), by improving the ROAD statistic of [24] and combining it with the Optimal Weights Filter in [11]. We introduce a new statistic called ROADGI (Rank-Ordered Absolute Differences for mixture of Gaussian and Impulse noises) which detects more effectively the impulse noise when it is mixed with Gaussian noise. The ROADGI statistic will give a weight for all pixels in the image, which take value in the interval (0, 1]. The weight will get low value (near to 0) when a pixel is contaminated by impulse noise; otherwise, it will carry a high value (near to 1). The ROADGI statistic is then combined with the Optimal Weights Filter (OWF) to deal with the mixed noise, by assigning nearly 0 weights for impulse noise points. The simulation results show that the proposed filter can effectively remove the mixture of impulse noise and the Gaussian noise. Moreover, when applied to either the single impulse noise or the single Gaussian noise it performs as good as the best filters specialized to single noises.

Let us point out the differences with the patch-based approaches in Delon and Desolneux [31, 32] and Hu et al. [33] which are all adapted for the mixed noise. The method in [31, 32] consists in finding the *n* most similar patches according to a suitably chosen distance between patches, with which one then constructs a maximum likelihood estimator. The filter in [33] is an extension of the Non-Local Means filter to the case of mixed noise, with weights depending on the ROAD statistic. In the present paper we use the optimal weights approach from [11, 34] and an improved version of ROAD statistic to appropriately measure the impact of the impulse noise pixels.

The rest of the paper is organized as follows. In Section 2 after a short recall of the Optimal Weights Filter and a brief presentation of the Trilateral Filter whose main ideas will be used in the definition of our new filter, we introduce our filter. In section 3, we provide visual examples and numerical results that demonstrate our method’s soundness. Section 4 is a brief conclusion.

## 2 Algorithms

### 2.1 Optimal Weights Filter

For any pixel *x*_{0} ∈ **I** and a given *h* > 0, the square window of pixels
(4)
will be called *search window* at *x*_{0}, where *h* is a positive integer. The size of the square search window **U**_{x0,h} is the positive integer number *M* = (2*h* + 1)^{2} = card **U**_{x0,h}. For any pixel *x* ∈ **U**_{x0,h} and a given integer *η* > 0 a second square window of pixels **V**_{x,η} = **U**_{x,η} will be called for short a *patch window* at *x* in order to be distinguished from the search window **U**_{x0,h}. The size of the patch window **V**_{x,η} is the positive integer *m* = (2*η* + 1)^{2} = card **V**_{x0,η}. The vector **Y**_{x,η} = (*Y* (*y*))_{y∈Vx,η} formed by the values of the observed noisy image *Y* at pixels in the patch **V**_{x,η} will be called simply *data patch* at *x* ∈ **U**_{x0,h}. For any *x*_{0} ∈ **I** and any *x* ∈ **U**_{x0,h}, a distance between the data patches **Y**_{x,η} = (*Y* (*y*))_{y∈Vx,η} and **Y**_{x0,η} = (*Y* (*y*))_{y∈Vx0,η} is defined by
(5)
where
and *T*_{x} is the translation mapping: *T*_{x} *y* = *x* + (*y* − *x*_{0}). If we use the approximation
and the law of large numbers, it seems reasonable that
(6)
For our filter, however, ne need an estimate for *ρ*_{f,x0}(*x*) without the square. As shown in [11], in practice, good denoising results are obtained by using the following approximation
(7)
rather than extracting the root in Eq (6). The fact that is a reasonable estimator of *ρ*_{f,x0} was justified by the convergence results in [11] (cf. Theorems 3 and 4 of [11]). The Optimal Weights Filter is defined by
(8)
where *κ*_{tr} is the usual triangular kernel:
(9)

The bandwidth is the solution of
and can be calculated as follows. We sort the set in the ascending order , where *M* = card **U**_{x0,h}. Let
(10)
and
(11)
with the convention that *a*_{k} = ∞ if and that . The solution can be expressed as ; moreover, *k** is the unique integer *k* ∈ {1, ⋯, *M*} such that and if *k* < *M*.

The proof of Remark 2.1 can be found in [11].

### 2.2 ROAD statistic and trilateral filter

In [24], Garnett et al introduced the Rank-Ordered Absolute Differences (ROAD) statistic to detect points contaminated by impulse noise. For any pixel *x*_{0} ∈ **I** and a given *d* > 0, we define the square window of pixels
where *d* is a positive integer. The square window will be called deleted neighborhood at *x*_{0}. The ROAD statistic is defined by
(12)
where *r*_{i}(*x*_{0}) is the *i*-th smallest term in the set and . In [24] it is advised to use *d* = 1 and *K* = 4. Note that if *x*_{0} is an impulse noisy point, the value of *ROAD*(*x*_{0}) is large; otherwise it is small.

Following [24] and [28] the authors define the “joint impulsivity” *J*_{I} (*x*_{0}, *x*) between *x*_{0} and *x* as:
(13)
where the function *J*_{I} (*x*_{0}, *x*) assumes values in [0, 1] and the parameter *σ*_{J} controls the shape of the function *J*_{I} (*x*_{0}, *x*). If *x*_{0} or *x* is an impulse noisy point, then the value of *ROAD*(*x*_{0}) or *ROAD*(*x*) is large and *J*_{I} (*x*_{0}, *x*) ≈ 0; otherwise, the value of *ROAD*(*x*_{0}) and *ROAD*(*x*) are small and *J*_{I} (*x*_{0}, *x*) ≈ 1. The trilateral filter (cf. [24]) is given by
where
This filter has been shown to be very efficient in removing a mixed noise composed of a Gaussian and random impulse noise.

### 2.3 Optimal Weights Mixed Filter

The ROAD statistic (cf. [24]) provides a effective measure to detection the pixel contaminated by impulse. In this paper, we take into account the character of Gaussian noise and modify the ROAD statistic to better adapt to the mixture of impulse and Gaussian noises. Instead of the ROAD statistic Eq (12) we propose to use the statistic
(14)
where *σ* is the standard deviation of the added Gaussian noise, *r*_{i}(*x*_{0}) is the *i*-th smallest term in the set , and . An advantage of the ROADGI statistic, compared to the ROAD statistic, is that it is relatively stable with respect to size *d* of the detection window , and takes into account the Gaussian noise level *σ*. Let
(15)
be a weight to estimate whether the point is impulse one, where the parameter *H* controls the shape of the function. In the case when the pixel *x* is an impulse point then *ROADGI*(*x*) is large and *J*(*x*, *H*) ≈ 0; otherwise *ROADGI*(*x*) ≈ 0 and *J*(*x*, *H*) ≈ 1.

Now, we modify the Optimal Weights Filter [11] in order to treat the mixture of impulse and Gaussian noises. Similar to Eq (5), we define the impulse detection distance by
where
and *κ* are some weights defined on **V**_{x0,η}. The corresponding estimate of brightness variation *ρ*_{f,x0}(*x*) is given by
(16)
The best denoising results are obtained when the smoothing kernel *κ* is defined by
(17)
if ‖*y* − *x*_{0}‖_{∞} = *j* for some *j* ∈ {0, 1, ⋯, *η*} and *y* ∈ **U**_{x0,η}. It is possible to use as *k* the Gaussian kernel, but the results are a bit less precise.

Now, we define a new filter, called *Optimal Weights Mixed Filter* (OWMF), by
(18)
where the bandwidth can be calculated as in Remark 2.1 (with and replaced by and respectively) and *H*_{2} is a parameter. Notice that *H*_{1} and *H*_{2} may take different values. The flowchart and the pseudocode of algorithm of the OWMF are given by Fig 1 and Algorithm 1.

**Algorithm 1:** Optimal Weights Mixed Filter

**Input:** Noisy image *Y*; The set of parameters {*d*, *K*, *M*, *m*, *H*_{1}, *H*_{2}}

**Output:** Denoised image

1 **forall** *x* ∈ **I** **do**

2 compute

3 compute

4 compute

5 **end**

6 **for** *each* *x*_{0} ∈ **I** **do**

7 give an initial value of (it can be an arbitrary positive number)

8 compute by Eq (16)

9 reorder as increasing sequence, say

10 **for** *k* = 1 *to* *M* **do**

11 **if** **then** computer ;

12 **else** quit loop;

13 **end**

14 **forall** *x* ∈ **U**_{x0,h} **do**

15

16 **end**

17 compute

18 **end**

19 To avoid the undesirable border effects, in our simulations we mirror the image outside the image limits symmetrically with respect to the border. At the corners, the image is extended symmetrically with respect to the corner pixels.

To explain the new algorithm Eq (18), note that the function *J*(*x*, *H*_{2}) acts as a filter of the points contaminated by the impulse noise. In fact, if *x* is an impulse noisy point, then *J*(*x*, *H*_{2}) ≈ 0. When the impulse noisy points are filtered, the remaining part of the image is treated as a image distorted by solely the Gaussian noise. So, in the new filter, the basic idea is to apply the OWF [11] by giving nearly 0 weights to impulse noisy points.

## 3 Simulation and comparisons

The performance of a filter is measured by the usual Peak Signal-to-Noise Ratio (PSNR) in decibels (db) defined by
where *f* is the original image.

In the simulations, to avoid the undesirable border effects in our simulations, we mirror the image outside the image limits. In more detail, we extend the image outside the image limits symmetrically with respect to the border. At the corners, the image is extended symmetrically with respect to the corner pixels.

In our simulations the parameters are chosen as follows:
In [24] it is suggested to take *d* = 1 and *K* = 4. In [24], for low and moderate levels of noise (*p* < 25%), one iteration is sufficient and usually provides the best results; for high levels of noise (*p* > 25%), applying two to five iterations provides better results. Only one iteration is required in our simulations. If we choose *d* = 1 and *K* = 4, as recommended in [24], we found that a few spots of unremoved impulses often remain. This happens because impulses sometimes “clump” together, and the 3 × 3 detection window is too small to identify all the impulse noise points. Consequently, we select parameters *d* = 2 and *K* = 12 of detection windows for all levels of impulse noise. Fig 2 shows the comparison results between the restored images, with detection window 3 × 3 and with detection window 5 × 5, which have been added an impulse noise with *p* = 20%, 30%, 40%, and 50% respectively. When *p* = 30%, 40% and 50%, we can see clearly some impulse spots in the restored images with detection window 3 × 3, while the visual quality of the restored images with detection window 5 × 5 is very good, without impulse spots. In the case where *p* = 20%, impulse spots of the restored image with detection window 3 × 3 are not obvious, and the PSNR value is a little better than that with detection window 5 × 5, whereas Fig 3 shows that the first image has two clumpy impulse spots and the visual quality is not good enough. Consequently, we prefer detection window 5 × 5 for all levels impulse noise.

The fist column corresponds to images restored with size 3 × 3 of the detection window. The second one is restored with size 5 × 5. The lines correspond to impulse noise proportions *p* = 20%, 30%, 40% and 50% respectively.

The first image is restored with the size 3 × 3 of detection window, the second one with the size 5 × 5. The original image has been contaminated by an impulse noise with *p* = 20%.

The parameters *m* and *M* have been fixed to *m* = 25 × 25 and *M* = 13 × 13. Figs 4(C) and 5(C) show that the noise is reduced in a natural manner and significant geometric features, fine textures, and original contrasts are visually well recovered with no undesirable artifacts. To better appreciate the accuracy of the restoration process, we zoom a part of the picture.

(A) the original image and its part; (B) the image restored by PARIGI and its part; (C) denoised image by our method and its part.

(A) the original image and its part; (B) the image restored by PARIGI and its part; (C) denoised image by our method and its part.

For comparison, we show the images denoised by PARIGI (see the left of Figs 4(B) and 5(B)) and their zoomed parts (see the right of Figs 4(B) and 5(B)). We can see clearly that the images denoised by our method are better than those denoised by PARIGI, so our method provides a significant improvement. The overall visual impression and the numerical results are improved using our algorithm.

For comparison, we consider the following three cases: pure Gaussian noise, pure impulse noise and the mixture of Gaussian and impulse noises.

In the case of pure Gaussian white noise, we have done simulation on a commonly-used set of images (“Lena”, “Barbara”, “Boat” and “House”) available at http://decsai.ugr.es/javier/denoise/test_images/ and the comparison with several filters is given in Table 1. The PSNR values show that our approach work as well as relatively sophisticated methods, like Hirakawa and Parks [35], Kervrann and Boulanger [36], Hammond and Simoncelli [7] and Aharon et al. [4], and is better than the filters proposed in Buades et al. [2], Katkovnik et al. [37], Foi et al. [38], Roth and Black [39], Hu et al. [33] and Delon et al. [32]. Exept [33] and [32], these methods can only deal with pure Gaussian noise, while our method can cope not only with the Gaussian noise, but also with the impulse noise and the mixture of Gaussian and pure impulse noises. The proposed approach gives a quality of denoising which is competitive with one of the state-of-the art methods, BM3D (see [5]).

For the pure impulse noise, our method is also competitive. We choose a commonly used set of images “Baboon”, “Bridge”, “Lena” and “Pentagon”(where “Baboon”, “Bridge”, “Lena” and “Pentagon” available at http://www.math.cuhk.edu.hk/rchan/paper/dcx/), which is considered in Delon et al. [22]. Table 2 lists the restoration results using various known algorithms. It is clear that our method provides a significant improvement over Sun and Neuvo [40], Abreu et al. [17], Wang and Zhang [41], Chen et al. [42], Chen and Wu [18, 43], Crnojevic et al. [44], Wenbin [21], etc. Our approach works as well as Dong et al. [22], Yu et al. [23], Hu et al. [33] and Delon et al. [32]. It produces the best PSNR values in the cases of “Baboon” (40%) and “Pentagon” (40%), while Yu et al. [23] has the best results in the case of “Baboon” (20%) and “Bridge” (40%), and Dong et al. [22](ROLD-EPR) wins in the case of “Lena” (20% and 40%). Finally, in Table 3 we compare Garnett et al. [24], Hu et al. [33], Delon et al. [32] and our filter (OWMF) on the set of images “Lena”, “Bridge”, “Boat” and “Barbara”; from Table 3, it is clear that our method performs better. in most cases, especially when *σ* > 10.

For a mixture of Gaussian and impulse noises simulation results show that the new proposed filter OWMF is competitive with PARIGI from [31, 32] and the filter in [33]. Table 3 shows that the results of denoising using our filter are generally better than those of PARIGI and [33] in the cases of “Lena”, “Bridge” and “Boat” when *σ* > 10. For *σ* = 10 our filter gives the results close to the best, which are sometimes the best. When considering the pure impulse noise, our method improves PARIGI and [33] in most cases; for pure Gaussian noise, our method is better than PARIGI and [33] with a larger margin.

## 4 Conclusion

A new image denoising filter to deal with the mixture of Gaussian and impulse noises, based on weights optimization and the modified Rank-Ordered Absolute Differences statistic, is proposed. The implementation of the filter is straightforward. Our work leads to the following conclusions.

- The improved Rank-Ordered Absolute Differences statistic, used in the new filter, detects effectively the impulse noise in the case of mixture of Gaussian and impulse noises. This statistic is well adapted for use with the Weights Optimization Filter of [11].
- It is shown by simulations that the proposed filter is very efficient for removing both a mixture of impulse and Gaussian noises, and the pure impulse or pure Gaussian noise.
- Our numerical results demonstrate that the new filter is competitive with the known filters.

## Acknowledgments

Jin and Liu have been partially supported by the National Natural Science Foundation of China (Grants No. 61661039, No. 11571052 and No. 11401590), the Natural Science Fund of Inner Mongolia Autonomous Region (Grant No. 2016MS0107), the Scientific Research Projection of Higher Schools of Inner Mongolia (Grant No. NJZY16017), Hunan Provincial Natural Science Foundation (Grant no. 2017JJ2271).

## Author Contributions

**Conceptualization:**QJ.**Data curation:**QJ.**Formal analysis:**IG QL.**Funding acquisition:**QJ QL.**Investigation:**QJ.**Methodology:**QJ.**Project administration:**QJ.**Resources:**QJ.**Software:**QJ.**Supervision:**IG QL.**Validation:**QJ.**Visualization:**QJ.**Writing – original draft:**QJ IG QL.**Writing – review & editing:**IG QL.

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