**1. Introduction**

Edge-preserving smoothing is an image processing method that smooths away textures while preserving sharp edges. Most smoothing methods are generally linear low-pass filters that effectively reduce noise at the same time wipe out edges. Since the edges might concern important image information, they have to be protected in smoothing. Non-linear filters are employed for this purpose; however, most of these techniques focus on the problem of reducing additive noise from images, since it is by far the most popular type of corrupting multiplicative noise.

In the literature, there is various research on edge-preserving noise reduction algorithms. Chinrungrueng et al. have presented a study based on edge-preserving noise reduction on ultrasound images. They have introduced a modified 2D weighted Savitzky Golay filter based on the least-squares fitting in a polynomial function to image intensities [1]. Petryniak has described a dynamic image filter using both linear and non-linear image smoothing, based on the Gaussian function. Their filter removes noises in the graphic while preserving information on edges [2]. Yuan and Wang have suggested an edge-preserving and signal-preserving noise removing method based on a Bayesian framework. This filter reduces the number of noises and also adaptively protects edges on signals [3]. Hofheinz et al. have introduced a novel study, which is suitable for bilateral filtering for noise reduction and edge-preserving in the PET image dataset. Bilateral filtering exhibits a successful increase in

the smoothing of the PET images while preserving spatial resolution at edges in order to maintain the quantitative accuracy and obtain an acceptable signal-to-noise ratio (SNR) [4]. Pal et al. have presented a survey of benchmark edge-preserving smoothing methods, presented in the literature for computational photography. In their study, they have discussed various effects of the edge-preserving filters also within their optimized modifications and extensions according to their mathematical analysis [5]. Wang et al. have presented a study about a smoothing method with edge preservation for single-image de-hazing (removing haze from image). A novel variational model (VM) that optimizes the transmission in the dark channel has been proposed. This model has an effective linear time complexity in performing transmissions [6]. Storath et al. have introduced a reconstruction framework of edge-preserving and noise reducing for emerging medical imaging, magnetic particle imaging (MPI). Tikhonov regularization, a basic image reconstruction method, is used for MPIs to handle efficiently because of the high temporal resolution of 3D volumes. In their study, they improved an efficient noise removing and edge-preserving reconstruction technique for MPI, giving higher quality in reconstruction for the prototypical medical application of angioplasty [7]. A book chapter for edge-preserving smoothing filters has been written by Burger and Burge. In this detailed and extended study, they have presented noise reduction methods, adaptive smoothing filters for both color and grayscale images. They have especially stressed three conventional types of edge-preserving filters based on different strategies. These are the Kuwahara-type filters, the bilateral filters, and the anisotropic diffusion filters [8]. Additionally, Muhammad et al. have proposed a Bayesian method in which there is a hybrid filtering framework for images having more noises with an unknown variance. The framework, including an automatic parameter selection mechanism, removes noises by enabling an appropriate smoothing and feasible sharpening [9]. In another study, proposed by González-Hidalgo et al., a salt and pepper noise removal system is implemented by a special filter based on a fuzzy mathematical morphology [10]. Luengo et al. have studied noise removal differently by using a supervised learning approach. Specifically, their filter, named CNC-NOS (class noise cleaner with noise scoring), is designed on a noise scoring basis by using ensemble classifiers [11]. A noise-cleaning method for colorful images has been introduced by Pérez-Benito et al. A graph structure is constructed for each of the image pixels in the image by considering some constraints and criterions in order to characterize the pixels as the link cardinality of their connected components [12]. Tang et al. have prepared a detailed study of a smoothing method for edge-aware image manipulations by using a minimization formula of a convex objective function in order to regularize edge and texture pixels in the image [13]. Furthermore, Huang et al. have proposed a technique using an NP-hard method, rank minimization with matrix ranks for regularization in order to remove white Gaussian additive and Gamma multiplicative noises in an image [14].

Apart from those mentioned above, non-linear MCV (minimum coefficient of variation) and MLV (mean least variance) filters are proposed by Schulze et. al. [15] in which multiplicative noise is reduced while preserving the edge contours by employing sliding windows around the central pixel and selecting the pixel that has the minimum amount of coefficient of variation (MCV) and variance (MLV) in terms of intensity within its surrounding window to be low-pass filtered. By this approach, the varying contours, edge lines, and textures are preserved while multiplicative noise is reduced.

In this paper, an extended version of MCV and MLV filters are proposed by modifying its value, criterion, and selection functions to be better than MCV and MLV filters in terms of robustness in noise reduction and edge preservation.

This paper has five sections. There is an introduction with a literature review related to this proposed method in the first section above. Explanations and details of the suggested technique are placed in the second chapter. There are validations of experimental results using statistical metrics and discussion in the third section. Availability of the study is demonstrated in the fourth section. In the last summary section, future work is presented and the contributions are summarized.

### **2. Methods**

In the literature, MCV and MLV filters are well-known filters, which eliminate noises in an image while preserving edges. All these methods similarly use a certain kernel (mask) size, which represents the size and shape of the neighborhood to be sampled while computing the corresponding value. The kernel is a *q* × *q* square matrix where *q* is a small odd number, generally 3, 5, or 7.

### *2.1. MCV and MLV Filters*

The MCV and MLV filters are edge-preserving noise removing filters based on the concepts of mathematical morphology [16]. They are value-and-criterion filters that are aimed to filter an image only over regions that are generally homogeneous, have low contrast and contain less amount of edges or textures [16]. The difference between MCV and MLV filters is that MCV filter uses the coefficient of variation as the criterion function whereas MLV employs the variance to perform better on multiplicative noise [16].

The idea is basically sliding windows around each central pixel and finding the sub-window, which has the minimum criterion function output, and apply the value function (mean value) of the window belonging to the central pixel [17]. The coefficient of variation over a sliding kernel is calculated by the ratio of the standard deviation to the mean over the sliding kernel. If the image is uniform within the kernel, the variation coefficient becomes very low [17].

On the other hand, if the image has high amount of edge and texture within the kernel, both the coefficient of variation and the standard deviation will return high values [18]. The selection function of MCV and MLV filters is designed as the minimum so that the filtering operation can completely be done over the kernel, which has the smallest output of the criterion function [18]. In other words, the noise smoothing function only acts over these kernels with the smallest coefficient of variation for MCV filter or the variance for the MLV filter [18].

As the value function, mean value is employed over the regions that have the minimum amount of criterion function: The coefficient of variation for the MCV and variance for the MLV filters [18]. The filter structure and detailed explanation of filter design are explained in Section 2.2 within the proposed method.

### *2.2. Proposed Method: Minimum Index of Dispersion (MID) Filter*

The MID filter uses the same morphological structure with MCV and MLV filters to direct low-pass filtering operation to only execute over regions decided to be most nearly constant by calculating the index of dispersion as the criterion function. As previously explained in Section 2.1, MCV and MLV filters employ the coefficient of variation and variance as the criterion function, MID filter employs the index of dispersion as the criterion function. The index of dispersion is the ratio of the variance of a random process to its mean and defined as in Equation (1).

$$D = \begin{cases} \frac{q^2}{\mu} \text{ if } \mu \neq 0\\ 0 \text{ otherwise} \end{cases} \tag{1}$$

where σ is the standard deviation and μ is the mean value of given elements of a set. For an image that is corrupted only by stationary multiplicative noise, the index of dispersion in terms of intensity and orientation in theory is constant at every point. Estimates of the index of dispersion show whether a region is nearly constant under the multiplicative noise or it includes important features. Regions that contain edges or other features generate higher estimates of the index of dispersion in terms of intensity and orientation than areas that are approximately constant. Value, criterion, and selection functions are defined as follows:

$$
\omega(\mathbf{x}) = \varphi(f(\mathbf{x}); \mathbf{N}) \tag{2}
$$

$$\gamma(\mathbf{x}) = \delta(f(\mathbf{x}); \mathbf{N}) \tag{3}$$

$$\varphi(\mathbf{x}) = \omega(\{\mathbf{x'} : \mathbf{x'} \in \mathcal{N}\iota\_{\mathbf{x'}} ; \gamma(\mathbf{x'}) = \beta[\gamma(\mathbf{x}); \mathcal{N}'] \}) \tag{4}$$

where ω is the value function, γ is the criterion function, and ϕ is the selection function. Additionally, *f*(*x*) denotes the intensity function that gets the intensity values of each pixel in each window that has *N* number of elements (pixels) within the window. β is another value function that gets the value of intensity with the minimum index of dispersion. *x* denotes the pixels within each sub-window around the central pixel of *x* and *N* is the number of pixels in each sub-window. This filter structure can be described as having a set of sub-windows within an overall filter window. The value-and-criterion filter operation at a point is equivalent to examining each sub-window within the overall window centered at that point and finding which sub-window has the output of optimal criterion function as described by the selection function. Then, the value function output over this sub-window becomes the final filter output for the current point. Value functions are interpreted in Equations (5) and (6) as follows:

$$\omega(\mathbf{x}) = \frac{1}{|\mathcal{N}|} \sum\_{y \in \mathcal{N}\_x} f(y) \tag{5}$$

$$\theta(x) = \frac{1}{|\mathcal{N}|} \sum\_{y \in \mathcal{N}\_x} g(y) \tag{6}$$

where ω represents the mean value of intensity and θ denotes the mean value of orientation. As the second metric, in addition to intensity value, normalized gradient orientation values are employed within their magnitudes defined as in Equation (7).

$$g(\mathbf{x}) = \arctan\left(\frac{G\_y}{G\_\mathbf{x}}\right) \times \sqrt{G\_y^{-2} + G\_\mathbf{x}^{-2}}\tag{7}$$

where *Gy* is the vertical gradient vector's normalized magnitude and *Gx* is the horizontal one. A 3 × 3 Sobel operator is employed to find the gradient vectors for each pixel of the input image. The regions that have a chaotic distribution of orientations are also defined as noise since regular patterns of orientation distributions land on the regions without noise. Therefore, orientation dispersion is also employed as the second metric in the proposed filter. Less dispersion in the orientation will have more impact on the intensity distribution. Thus, the criterion function is re-modeled as follows:

$$\gamma(\mathbf{x}) = \frac{\frac{1}{|\mathbb{N}|} \sum\_{\mathbf{y} \in N\_x} [f(\mathbf{y}) - \omega(\mathbf{x})]^2}{\omega(\mathbf{x})} \times \left(1 - \frac{\frac{1}{|\mathbb{N}|} \sum\_{\mathbf{y} \in N\_x} [g(\mathbf{y}) - \theta(\mathbf{x})]^2}{\theta(\mathbf{x})}\right) \tag{8}$$

Elimination of multiplicative noise from images is commonly more difficult than additive noise since the noise intensity varies with the signal intensity. In order to avoid this, selection function is re-modeled by adding an alpha parameter, which adds a low-pass value to the output of selection function. Alpha is a normalized parameter and it transforms the filter into a fully mean filter when it is set to 1. The proposed selection function is defined as follows:

$$\forall (\mathbf{x}) = \omega (\{ \mathbf{x'} : \mathbf{x'} \in N'\_{\mathbf{x'}}; \gamma(\mathbf{x'}) = \min \{ \gamma(\mathbf{y}); \, \mathbf{y} \in N'\_{\mathbf{x}} \} \}) \tag{9}$$

$$\text{MID}\{f(\mathbf{x});\mathbf{N}\} = \,\,\forall (\mathbf{x}) \times (1 - a) + w(\mathbf{x}) \times a \tag{10}$$

The sample mean is accepted as the value function for the MID filter. Thus, the MID filter uses the sample mean for a value function, the index of dispersion as a criterion function, and the minimum as a selection function. This value-and-criterion filter is particularly designed to remove multiplicative noise. Theoretically, index of dispersion in the images corrupted by noises is minimum in structuring elements where there is the constant signal. The MID filter in images both preserves sharp edges between flat areas and enhances the edges, which are not perfect step edges.

This study especially indicates a structural design of a special filter, which is the extended version of MCV, and MLV architectures. This filtering method is an edge-preserving noise reduction technique designed for reducing multiplicative noise by using the structure of the value-and-criterion filter. The theoretical mechanism of this method stands on the well-known fundamentals of geometrical properties. The advantage of this method is that it successfully preserves edges in the regions corrupted by the multiplicative noise and enhances them while preserving the morphological structures of the image.

### **3. Experimental Results and Analysis**

Mainly, all of the experiments have been performed on the Computational and Subjective Image Quality (CSIQ) benchmark dataset [19] by using a normal computer. MID filtering is implemented using Processing in Java and testing is performed in the MATLAB environment.

### *3.1. CSIQ Image Quality Database Specifications*

The CSIQ database is a popular image quality-benchmark test set in order to evaluate algorithms. The database includes 30 original images at the resolution of 512×512 pixels. The set is distorted using one of six distortions with four to five different distortion levels. CSIQ images have been tested based on linear image displacements on four calibrated LCD screens placed side by side with equal viewing distance. This database contains 5000 subjective evaluations from 35 different observers and the assessment are presented in the form of difference mean opinion scores (DMOS) in which a larger one indicates greater visual impairments compared to the corresponding reference image.

### *3.2. Performance Measurement Criterions*

De-noising a picture requires a successful method providing that edges are to be preserved. In order to evaluate the performances of the methods, some quality metrics are preferred. Evaluation of de-noising quality is performed five fundamental metrics. These are mean squared error (MSE), peak signal-to-noise ratio (PSNR), the structural similarity index (SSIM), contrast, and standard deviation [20]. Additionally, *F* Score, which is an original hybrid metric for comparison, is proposed in this study by using the combinations of the basic measurements.

### 3.2.1. Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR)

Both MSE and PSNR are used to evaluate the performance for image manipulation algorithms. They are similar to each other and derived from signal processing. Implementation and calculation are straightforward, but the results are not always considered reliable as they show aspects in various situations. Nevertheless, they have a great role in the performance evaluation domain.

The MSE between the two signals is described as seen in Equation (11).

$$MSE = \frac{1}{N \times M} \sum\_{i=0}^{N-1} \sum\_{j=0}^{M-1} \left[ X(i,j) - \left| Y(i,j) \right|^2 \right] \tag{11}$$

where *X* and *Y* are two arrays of size *N* × *M*. The closer *Y* is to *X*, the smaller MSE will be. When the MSE is equal to zero, apparently, the maximum similarity is achieved.

The PSNR (in dB) accordingly is defined as follows:

$$PSNR = 10\log\_{10}\frac{L^2}{MSE} \tag{12}$$

In Equation (12) above, *L* is the maximum fluctuation in the data type of the input image. For instance, if the input image has a double-precision floating-point data type, then *L* is defined as 1. Similarly, if the input image has an 8-bit unsigned integer data type, *L* is defined as 255. Logarithm

transforms the ratio into a decibel (dB) scale, which is a common scale operation in signal processing. PSNR in decibels units calculates the PSNR between original and filtered images. The lower the PSNR value, the worse the quality of de-noised image. MSE and PSNR are the two-error measurement metrics used to compare the image de-noising quality.

MSE shows the cumulative squared error between filtered and original images. PSNR displays the measure of the peak error. In a little while, the higher the MSE value, the higher the error. If there are two identical images (in the absence of artificial noise), the MSE value becomes 0 and the PSNR value becomes infinite [21].

### 3.2.2. The Structural Similarity Index Measurement (SSIM)

The SSIM measurement is a common and well-known quality criterion to determine the similarity between two images. The SSIM index gives a similarity percentage in the interval of [0, 1].

This measurement style compares two images in the same size, the de-noised picture, and the original picture. The original picture is assumed as it has perfect quality. The de-noised one is for test and the original is for verification. SSIM index is defined as follows:

$$SSIM(\mathbf{x}, y) = \frac{\left(2\mu\_x \mu\_y + \mathcal{C}\_1\right)\left(2\sigma\_{xy} + \mathcal{C}\_2\right)}{\left(\mu\_x + \mu\_y + \mathcal{C}\_1\right)\left(\sigma\_x + \sigma\_y + \mathcal{C}\_2\right)}\tag{13}$$

where *x* and *y* are the two different images with μ*<sup>x</sup>* and μ*<sup>y</sup>* mean values of intensity and standard deviations of σ*<sup>x</sup>* and σ*<sup>y</sup>* with contrast values *C*<sup>1</sup> and *C*<sup>2</sup> for the two images separately. When comparing two images, MSE does not indicate highly perceived similarity while implementation is simple. Structural similarity is aimed at addressing this hardship.

### 3.2.3. Contrast

Contrast of an image might be simply explained as the difference between the minimum and maximum pixel intensity. Shortly, it is the difference in color or luminance for a group of objects. In this project, edge-based contrast measure (EBCM) for image quality evaluation is selected as a performance metric [22]. This metric is based on the fact that an enhanced image normally has more edge pixels than the original image. The EBCM metric calculates the intensity of edge pixels in small windows of the image.

### 3.2.4. Standard Deviation

The standard deviation of the pixel intensity values is used to quantify the amount of variation or dispersion of a grayscale image. It is calculated by Equation (14).

$$\sigma = \sqrt{\frac{1}{N \times M} \sum\_{i=0}^{N-1} \sum\_{j=0}^{M-1} \left( x\_{ij} - \mu \right)^2} \tag{14}$$

where σ is the standard deviation of matrix elements. *N* and *M* are the vertical and horizontal sizes of the image. *xij* is the pixel of the *i*th line and *j*th column. μ is the arithmetic mean. A low standard deviation value displays that the pixels tend to be close to the mean of the image, while a high value shows that the pixels are spread out over a wider range of values.

### 3.2.5. A Hybrid Assessment Metric: F Score

Handling each of the metrics separately in image quality assessment might be difficult. A hybrid approach is proposed to evaluate each of the filter methods as shown in Equation (15).

$$F\,\text{Score} = 100 \times \frac{PSNR \times SSIM \times \text{Contrast}}{Std.Dev \times MSE} \tag{15}$$

An optimal edge-preserving and noise-reducing filter should increase the PSNR, SSIM, and contrast values while reducing the standard deviation and MSE values. Therefore, a compact formula of *F* is generated in order to benchmark the filters. Higher values of PSNR, SSIM, and Contrast values indicate that there is a successful smoothing operation.

In contrast, higher values of standard deviation and MSE shows poor smoothing results. In other words, PSNR, SSIM, and contrast have a positive effect on image quality, whereas the others have a negative impact.

In the experiments, *F* scores result in very tiny values, even very close to zero. Hence, the *F* score results are multiplied by a constant value of 100 so as to optimize the outputs. A regularly higher *F* score rate indicates the successful filtering performance.
