Development of Adaptive Gaussian Filter Based Denoising as an Image Enhancement Technique

Medical imaging is indispensable in modern healthcare, offering non-invasive insights into internal structures for clinical analysis and intervention. It facilitates detailed visualization and monitoring of organs and tissues, aiding accurate diagnoses and treatment plans. The continual evolution of imaging technology significantly contributes to enhanced patient care and treatment outcomes, making medical imaging integral part of modern medicine.

The synergy between mathematics and the medical field is a powerful force, leveraging abstract analytical tools to address practical healthcare challenges. Quantitative methods, including statistical analysis and mathematical modeling, are crucial for understanding disease patterns, predicting outcomes, and optimizing healthcare processes. This symbiotic relationship not only deepens our understanding of biological systems but also drives innovation in diagnostic techniques, treatment strategies, and overall healthcare delivery.

Image processing, situated at the crossroads of computer science, mathematics, and engineering, plays a pivotal role in medical imaging by applying algorithms for tasks like reconstruction, segmentation, and feature extraction. Denoising [1], a critical technique within image processing, aims to reduce unwanted noise and enhance image quality. In medical imaging, denoising [2] directly influences diagnostic accuracy and treatment planning, contributing to more reliable clinical assessments. The continuous development of denoising methods ensures the production of high-fidelity images across various applications, from healthcare to computer vision and scientific research. Fuzzy logic [3] plays a significant role in image processing, providing a framework for handling uncertainty and imprecision inherent in image data. In image processing, fuzzy logic [4] allows for the development of algorithms that can effectively deal with vague or subjective information, such as edge detection, segmentation, and pattern recognition. By incorporating fuzzy sets, fuzzy inference systems, and fuzzy clustering techniques, image processing tasks can be performed with greater robustness and flexibility, enabling the extraction of meaningful information from complex and noisy image datasets. Fuzzy logic-based [5, 6] image processing techniques have been successfully applied in various fields, including medical imaging, remote sensing, object recognition, and computer vision, contributing to advancement in automated image analysis and decision-making systems.

An image [16] can be assumed to be an encoded form of matrix with grey-level or color pixel intensity values as its elements. In case of grey scale images, it can be referred to as pixel value in two-dimension (x, u(x)) where u(x) is the pixel intensity value at location x. An image noise model can roughly be approximated as:

$f(x)=u(x)+n(x), x \subset X, X \subset Z^2$

where, u(x) denotes the original pixel matrix and n(x) denotes noise at location x. The image used is an MRI brain image (Figure 1).

11.png

(a) Original image

12.png

(b) Salt and pepper

13.png

(c) Gaussian noise   

14.png

(d) Speckle noise

With Noise Density 0.06

Figure 1. MRI image displaying original image and noised images

Filtering techniques are essential tools in image processing that play a crucial role in enhancing image quality and extracting relevant information. These techniques involve the application of filters, which are mathematical operations applied to pixel values in an image. Filtering techniques [18, 19] find widespread use in various fields, including medical imaging, computer vision, and remote sensing. They are employed for tasks such as noise reduction [20], edge enhancement, and feature extraction. The choice of a filtering technique depends on the specific characteristics of the image and the objectives of the image processing task. While some filters are designed to smoothen the image and reduce noise, others are tailored for edge detection or sharpening details. The continuous evolution of filtering techniques contributes to advancements in image processing, facilitating improved analysis and interpretation of visual data in diverse applications.

Gaussian filters, both 1D and 2D, are vital for medical image processing, particularly MRI, since they effectively reduce noise while maintaining important imagine characteristics. By using fuzzy logic to address uncertainties in picture data, the fuzzy Gaussian filters improve this capability even more. Standard and fuzzy Gaussian filters smooth images while maintaining edges, providing a balanced approach to noise reduction. They are computationally efficient, can tolerate a wide range of noise kinds, and serve as the foundation for numerous sophisticated image processing methods. While traditional filters such as wiener and median filters have their advantages, Gaussian filters are often more versatile, computationally efficient, and able to handle a wider range of noise types.

When it comes to context-aware smoothing, adaptive filters are excellent at distinguishing between different kinds of image regions and applying the right amount of smoothing to improve overall image quality and minimize abnormalities. Adaptive Gaussian filters are particularly helpful in situations where preserving minute structural details is crucial, such medical imaging, is required for accurate diagnosis and analysis, due to their versatility and effectiveness.

This paper presents four types of adaptive gaussian filter [21, 22] and their performance analysis.

5.1 Adaptive gaussian filter 1D and derivative of gaussian 1D (AGD-1D)

It is a filter whose impulse response is gaussian function. A Gaussian filter [23] is a popular image processing technique used for smoothening and reducing noise in images. It is based on the Gaussian distribution [24, 25] and operates by convolving the image with a Gaussian kernel. The mathematical formula for a one-dimensional Gaussian function is given by:

$G(x)=\frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left(-\frac{x^2}{2 \sigma^2}\right)$                  (1)

where, G(x) is the value of the Gaussian function at position x, σ is the standard deviation, determining the width of the gaussian curve. The standard deviation and the constant value are given by:

$\sigma=\sqrt{\frac{1}{m n} \sum_{i=1}^m \sum_{j=1}^n(A(i, j)-m)^2} ; \propto=\frac{A}{\max (|A|)}$                    (2)

where, max(|A|) is the maximum absolute value of any element in the matrix. Derivative [26, 27] of Gaussian 1D filter can indirectly contribute to denoising by emphasizing edges and suppressing noise. The steep response of the filter to intensity changes can help highlight significant structures in the image while minimizing the influence of random noise. The one-dimensional mathematical formula for the derivative of Gaussian 1D filter is given by:

$\frac{\partial G(x)}{\partial x}=-\frac{x}{\sigma^3} \exp \left(-\frac{x^2}{2 \sigma^2}\right)$                       (3)

where, $\frac{\partial G(x)}{\partial x}$ is the derivative of the Gaussian function with respect to x, x is the spatial coordinate, σ is the standard deviation, controlling the width of the Gaussian distribution.

$F(x, \sigma)=G(x)+\alpha \cdot \frac{\partial G(x)}{\partial x}$              (4)

where, α is a constant multiplier for the derivative term. The combine equation represents a linear combination of a Gaussian Function and its derivative.

$\begin{aligned} F(x, \sigma) & =\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(\frac{-x^2}{2 \sigma^2}\right)+\alpha \cdot\left[-\frac{x}{\sigma^2} \frac{1}{\sqrt{2 \pi} \sigma} \exp \left(\frac{-x^2}{2 \sigma^2}\right)\right] \\ & =\frac{1}{\sqrt{2 \pi} \sigma}\left[1-\alpha \cdot \frac{x}{\sigma^2}\right] \exp \left(\frac{-x^2}{2 \sigma^2}\right)\end{aligned}$                (5)

5.2 Adaptive gaussian 2D filter and derivative of 2D filter (AGD-2D)

A two-dimensional Gaussian filter, the formula is the product of two one-dimensional Gaussians along the rows and columns, forming a 2D kernel. The filter effectively reduces high-frequency noise in the image while preserving its overall structure. It is based on the mathematical formulation of a two-dimensional Gaussian distribution and operates by convolving the image with a Gaussian kernel. The mathematical formula for a two-dimensional Gaussian function is given by:

$G\left(x, y, \sigma_x, \sigma_y\right)=\frac{1}{2 \pi \sigma_x \sigma_y} \exp \left(-\frac{x^2}{2 \sigma_x^2}-\frac{y^2}{2 \sigma_y^2}\right)$             (6)

where, G(x, y, σ_x, σ_y) is the value of the two-dimensional Gaussian function at positions x and y, σ is the standard deviation determining the width of the Gaussian distribution. Also, the two-dimensional Gaussian derivative filter is given by:

$G_x\left(x, y, \sigma_x, \sigma_y\right)=-\frac{x}{\sigma_x^2} \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left(-\frac{x^2}{2 \sigma_x^2}-\frac{y^2}{2 \sigma_y^2}\right)$                  (7)

$G_y\left(x, y, \sigma_x, \sigma_y\right)=-\frac{y}{\sigma_y^2} \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left(-\frac{x^2}{2 \sigma_x^2}-\frac{y^2}{2 \sigma_y^2}\right)$                  (8)

where, G_x(x, y, σ_x, σ_y) represents $\frac{\partial G(x, y)}{\partial x}$, G_y(x, y, σ_x, σ_y) represents $\frac{\partial G(x, y)}{\partial y}$ are the partial derivatives of the Gaussian function with respect to x and y, σ is the standard deviation controlling the width of the Gaussian distribution, x and y are the spatial coordinates.

$\begin{gathered}\quad F\left(x, y, \sigma_x, \sigma_y, \alpha, \beta\right)=G\left(x, y, \sigma_x, \sigma_y\right) \\ +\alpha \cdot G_x\left(x, y, \sigma_x, \sigma_y\right)+\beta \cdot G_y\left(x, y, \sigma_x, \sigma_y\right)\end{gathered}$               (9)

where, α, β are constant multiplier for the partial derivative in the x and y directions respectively. The final combined equation is:

$\begin{aligned} & F\left(x, y, \sigma_x, \sigma_y, \alpha, \beta\right)=\frac{1}{2 \pi \sigma_x \sigma_y} \exp \left(-\frac{x^2}{2 \sigma_x^2}-\frac{y^2}{2 \sigma_y^2}\right) \\ & +\alpha \cdot\left[-\frac{x}{\sigma_x^2} \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left(-\frac{x^2}{2 \sigma_x^2}-\frac{y^2}{2 \sigma_y^2}\right)\right] \\ & \quad+\beta\left[-\frac{y}{\sigma_y^2} \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left(-\frac{x^2}{2 \sigma_x^2}-\frac{y^2}{2 \sigma_y^2}\right)\right]\end{aligned}$

$\begin{gathered}F\left(x, y, \sigma_x, \sigma_y, \alpha, \beta\right) \\ =\frac{1}{2 \pi \sigma_x \sigma_y}\left[1-\alpha \cdot \frac{x}{\sigma_x^2}-\beta \cdot \frac{y}{\sigma_y^2}\right] \exp \left(-\frac{x^2}{2 \sigma_x^2}-\frac{y^2}{2 \sigma_y^2}\right)\end{gathered}$         (10)

where,

$\sigma_x=\sqrt{\frac{1}{n} \sum_{j=1}^n\left(A(i, j)-m_x\right)^2}$ for each i;

$\sigma_y=\sqrt{\frac{1}{m} \sum_{i=1}^m\left(A(i, j)-m_y\right)^2}$ for each j;

$\propto=\frac{A(i, j)}{\max (|A(i, j)|)_{\text {row }}}$ for each i, j;

$\beta=\frac{A(i, j)}{\max (|A(i, j)|)_{\text {column }}}$ for each i, j.

5.3 Adaptive fuzzy gaussian 1D filter and derivative fuzzy gaussian 1D filter (AFGD-1D)

Let the function be represented as:

$F(x, \sigma, m)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{(x-m)^2}{2 \sigma^2}\right)$               (11)

where, F(x, σ, m) is Fuzzy Gaussian 1D, x is a variable, σ is the standard deviation and m is a fuzziness parameter. Also, the derivative of Eq. (11) (i.e.) fuzzy gaussian function with respect to x. The derivative is given as:

$F^{\prime}(x, \sigma, m)=-\left(\frac{x-m}{\sigma^2}\right) \frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{(x-m)^2}{2 \sigma^2}\right)$        (12)

The final combination function is given by:

$G(x, \sigma, m, \alpha)=F(x, \sigma, m)+\alpha \cdot F^{\prime}(x, \sigma, m)$

where, σ is a constant multiplier and is calculated using (2):

$\begin{aligned} & G(x, \sigma, m, \alpha)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{(x-m)^2}{2 \sigma^2}\right) \\ & \quad+\alpha \cdot\left[-\left(\frac{x-m}{\sigma^2}\right) \frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{(x-m)^2}{2 \sigma^2}\right)\right]\end{aligned}$

$G(x, \sigma, m, \alpha)=\frac{1}{\sqrt{2 \pi} \sigma} \exp \left(-\frac{(x-m)^2}{2 \sigma^2}\right)\left[1-\alpha \cdot\left(\frac{x-m}{\sigma^2}\right)\right]$                 (13)

where, $m=\frac{1}{m n} \sum_{i=1}^m \sum_{j=1}^n A(i, j)$.

5.4 Adaptive fuzzy gaussian 2D filter and derivative of fuzzy gaussian 2D (AFGD-2D)

Consider a 2D fuzzy Gaussian function $F\left(x, y, \sigma_x, \sigma_y, m_x, m_y\right)$ where x and y are the variables $\sigma_x, \sigma_y, m_x, m_y$ are the standard deviation and mean along the respective axis. The function is given by:

$\begin{gathered}F\left(x, y, \sigma_x, \sigma_y, m_x, m_y\right) \\ =\frac{1}{2 \pi \sigma_x \sigma_y} \exp \left[-\frac{\left(x-m_x\right)^2}{2 \sigma_x^2}-\frac{\left(y-m_y\right)^2}{2 \sigma_x^2}\right]\end{gathered}$                    (14)

The fuzzy partial derivative of this 2D fuzzy gaussian function with respect to x and y. The partial derivatives are given by:

$\begin{gathered}F_x\left(x, y, \sigma_x, \sigma_y, m_x, m_y\right) \\ =-\left(\frac{x-m_x}{\sigma_x{ }^2}\right) \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left[-\frac{\left(x-m_x\right)^2}{2 \sigma_x{ }^2}-\frac{\left(y-m_y\right)^2}{2 \sigma_y{ }^2}\right]\end{gathered}$            (15)

$\begin{gathered}F_y\left(x, y, \sigma_x, \sigma_y, m_x, m_y\right) \\ =-\left(\frac{y-m_y}{\sigma_y^2}\right) \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left[-\frac{\left(x-m_x\right)^2}{2 \sigma_x^2}-\frac{\left(y-m_y\right)^2}{2 \sigma_y^2}\right]\end{gathered}$                  (16)

Combining all above equations, we get:

$\begin{gathered}G\left(x, y, \sigma_x, \sigma_y, m_x, m_y, \alpha, \beta\right) =F\left(x, y, \sigma_x, \sigma_y, m_x, m_y\right)+\alpha . F_x\left(x, y, \sigma_x, \sigma_y, m_x, m_y\right)+\beta . F_y\left(x, y, \sigma_x, \sigma_y, m_x, m_y\right)\end{gathered}$

$\begin{array}{r}G\left(x, y, \sigma_x, \sigma_y, m_x, m_y, \alpha, \beta\right)=\frac{1}{2 \pi \sigma_x \sigma_y} \exp \left[-\frac{\left(x-m_x\right)^2}{2 \sigma_x^2}-\frac{\left(y-m_y\right)^2}{2 \sigma_x^2}\right]+\alpha \cdot\left\{-\left(\frac{x-m_x}{\sigma_x^2}\right) \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left[-\frac{\left(x-m_x\right)^2}{2 \sigma_x{ }^2}-\frac{\left.\left(y-m_y\right)^2\right]}{2 \sigma_y{ }^2}\right]\right\}+\beta \cdot\left\{-\left(\frac{y-m_y}{\sigma_y^2}\right) \frac{1}{2 \pi \sigma_x \sigma_y} \exp \left[-\frac{\left(x-m_x\right)^2}{2 \sigma_x{ }^2}-\frac{\left(y-m_y\right)^2}{2 \sigma_y^2}\right]\right\}\end{array}$

$\begin{gathered}G\left(x, y, \sigma_x, \sigma_y, m_x, m_y, \alpha, \beta\right)=\frac{1}{2 \pi \sigma_x \sigma_y} \exp \left[-\frac{\left(x-m_x\right)^2}{2 \sigma_x^2}-\frac{\left(y-m_y\right)^2}{2 \sigma_y^2}\right]\left[1-\alpha \cdot\left(\frac{x-m_x}{\sigma_x^2}\right)-\right. \left.\beta \cdot\left(\frac{y-m_y}{\sigma_y^2}\right)\right]\end{gathered}$                        (17)

For 2D the constants are calculated using formulas for a given matrix A of size m x n: $m_x=\frac{1}{n} \sum_{j=1}^n A(i, j)$ for each i; $m_y=\frac{1}{m} \sum_{i=1}^m A(i, j)$ for each j.

5.5 Numerical example of proposed methodology

In order to display the numerical calculation of the above designed filters, consider a sample pixel 5×5 matrix from the image given in Figure 1.

Let

$A=\left[\begin{array}{lllll}102 & 168 & 199 & 209 & 195 \\ 158 & 195 & 202 & 190 & 172 \\ 197 & 209 & 197 & 174 & 158 \\ 208 & 206 & 189 & 166 & 157 \\ 190 & 181 & 172 & 159 & 154\end{array}\right]$

be the 5×5 pixel matrix and the numerical calculation to obtain denoised matrix using the above-described formulas are demonstrated below. The calculation is done using MATLAB and the following values are obtained:

$$\begin{aligned}& \begin{array}{l}\text { Image }_{\text {filtered }}=\text { Image }_{\text {original }}+\text { Convolution }_{\text {matrix }} \\\alpha=\left[\begin{array}{ccccc}0.4880 & 0.8038 & 0.9522 & 1 & 0.9330 \\0.7560 & 0.9330 & 0.9665 & 0.9091 & 0.8230 \\0.9426 & 1 & 0.9426 & 0.8325 & 0.7560 \\0.9952 & 0.9856 & 0.9043 & 0.7943 & 0.7512 \\0.9091 & 0.8660 & 0.8230 & 0.7608 & 0.7368\end{array}\right] ; \\\beta=\left[\begin{array}{ccccc}0.4904 & 0.7596 & 0.9471 & 1 & 0.9135 \\0.8038 & 0.9330 & 1 & 0.9856 & 0.8660 \\0.9851 & 1 & 0.9752 & 0.9356 & 0.8515 \\1 & 0.9091 & 0.8325 & 0.7943 & 0.7608 \\0.9135 & 0.8821 & 0.8103 & 0.8051 & 0.7897\end{array}\right]\end{array} \\& m_x=\left[\begin{array}{c}171 \\191.8 \\191.8 \\179.6 \\167.2\end{array}\right] m_y=\left[\begin{array}{c}174.6 \\183.4 \\187 \\185.2 \\171.2\end{array}\right] ; \\& \sigma_x=\left[\begin{array}{l}38.9204 \\18.1769 \\19.0221 \\21.3897 \\13.9628\end{array}\right] ; \sigma_y=\left[\begin{array}{l}38.7536 \\16.1196 \\18.4065 \\20.6436 \\13.3776\end{array}\right] ; \\& \sigma=24.0807 ; m=180.28\end{aligned}$$

Using AGD-1D:

$\begin{gathered}\text { Convolution matrix }\left(C_x\right)= \\ =\left[\begin{array}{ccccc}25.83 & 34.54 & 42.28 & 34.73 & 25.67 \\ 35.37 & 46.56 & 56.59 & 45.71 & 33.43 \\ 43.97 & 57.55 & 69.84 & 55.91 & 40.78 \\ 36.50 & 46.87 & 56.23 & 44.08 & 31.65 \\ 27.70 & 35.17 & 42.01 & 32.46 & 23.11\end{array}\right] \\ A_{A G D-1 D} \text { Filter }=A+C_x \\ =\left[\begin{array}{llllll}127.83 & 202.54 & 241.28 & 243.73 & 220.67 \\ 193.37 & 241.56 & 258.59 & 235.71 & 205.43 \\ 240.97 & 266.55 & 266.84 & 229.91 & 198.78 \\ 244.50 & 252.87 & 245.23 & 210.08 & 188.65 \\ 217.70 & 216.17 & 214.01 & 191.46 & 177.11\end{array}\right] \\ \cong\left[\begin{array}{lllll}128 & 203 & 241 & 244 & 221 \\ 193 & 242 & 259 & 236 & 205 \\ 241 & 267 & 267 & 230 & 199 \\ 245 & 253 & 245 & 210 & 189 \\ 218 & 216 & 214 & 191 & 177\end{array}\right]\end{gathered}$

$=\left[\begin{array}{lllll}128 & 203 & 241 & 244 & 221 \\ 193 & 242 & 255 & 236 & 205 \\ 241 & 255 & 255 & 230 & 199 \\ 245 & 253 & 245 & 210 & 189 \\ 218 & 216 & 214 & 191 & 177\end{array}\right]$

Using AGD-2D:

$$\begin{aligned}& C_x=\left[\begin{array}{lllll}0.56 & 0.77 & 0.95 & 0.80 & 0.60 \\0.78 & 1.04 & 1.28 & 1.04 & 0.77 \\1.19 & 1.58 & 1.94 &1.59 & 1.17 \\1.18 & 1.53 & 1.85 & 1.46 & 1.06 \\0.90 & 1.15 & 1.38 & 1.07 & 0.77\end{array}\right] \\& A_{A G D-2 D} \text { Filter } \\& =\left[\begin{array}{lllll}102.56 & 168.77 & 199.95 & 209.80 & 195.60 \\158.78 & 196.04 & 203.28 & 191.04 & 172.77 \\198.19 & 210.58 & 198.94 & 175.59 & 159.17 \\209.18 & 207.53 & 190.85 & 167.46 & 158.06 \\190.90 & 182.15 & 173.38 & 160.07 & 154.77\end{array}\right] \\

& \cong\left[\begin{array}{lllll}103 & 169 & 200 & 210 & 196 \\159 & 196 & 203 & 191 & 173 \\198 & 211 & 199 & 176 & 159 \\209 & 208 & 191 & 167 & 158 \\191 & 182 & 173 & 160 & 155\end{array}\right]\end{aligned}$$

Using AFGD-1D:

$$\begin{aligned}& C_x=\left[\begin{array}{lllll}52.94 & 37.06 & 38.10 & 25.80 & 31.08 \\38.56 & 24.69 & 10.14 & 30.13 & 27.11 \\27.07 & 37.11 & 23.97 & 19.16 & 40.83 \\29.06 & 19.09 & 45.54 & 23.82 & 56.50 \\13.75 & 49.57 & 20.61 & 20.49 & 41.07\end{array}\right] \\& A_{A F G D-1 D F i l t e r} \\& =\left[\begin{array}{lllll}154.94 & 168.06 & 199.10 & 209.10 & 195.08 \\158.56 & 195.09 & 202.14 & 190.13 & 172.11 \\198.07 & 209.11 & 197.17 & 174.16 & 158.13 \\208.06 & 206.09 & 189.14 & 166.12 & 157.10 \\190.75 & 181.07 & 172.11 & 159.09 &154.07\end{array}\right] \\& \cong\left[\begin{array}{lllll}155 & 168 & 199 & 209 & 195 \\159 & 195 & 202 & 190 & 172 \\198 & 209 & 197 & 174 & 158 \\208 & 106 & 189 & 166 & 157 \\191 & 181 & 172 & 159 & 154\end{array}\right]\end{aligned}$$

Using AFGD-2D:

$$\begin{aligned}& C_x=\left[\begin{array}{lllll}0.26 & 0.48 & 1.48 & 0.98 & 0.83 \\0.26 & 058 & 1.47 & 0.29 & 0.92 \\0.24 & 0.33 & 0.44 & 0.93 & 0.27 \\0.92 & 0.57 & 0.08 & 0.63 & 1.98 \\0.85 & 1.93 & 0.81 & 1.02 & 1.05\end{array}\right] \\& A_{A F G D-2 D} \text { Filter } \\

& =\left[\begin{array}{lllll}102.26 & 168.48 & 200.48 & 209.98 & 195.83 \\158.26 & 195.58 & 203.47 & 190.29 & 172.92 \\197.24 & 209.33 & 197.44 & 174.93 & 158.27 \\208.92 & 206.57 & 189.08 & 166.63 & 158.98 \\190.85 & 182.93 & 172.81 & 160.02 & 155.05\end{array}\right] \\

& \cong\left[\begin{array}{lllll}102 & 168 & 200 & 210 & 196 \\158 & 196 & 203 & 190 & 173 \\197 & 209 & 197 & 175 & 158 \\209 & 207 & 189 & 167 & 159 \\191 & 182 & 173 & 160 & 155\end{array}\right]\end{aligned}$$

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Figure 3. Filtered image after using adaptive filters for noise density =0.06

Table 1. RMSE value for speckle noise

Noise Density	1D Gaussian	2D Gaussian	1D Derivative	2D Derivative	AGD-1D	AGD-2 D	AFGD-1D	AFGD-2D
0.01	7.109147881	9.51227493	48.35614443	46.49438233	4.4315	7.0499	3.6751	13.0718
0.02	7.671172547	9.835913216	48.22640418	46.40444893	4.7316	7.1425	3.9752	13.096
0.03	8.210896484	10.14732279	48.0989115	46.28819597	5.0585	7.2675	4.3021	13.1154
0.04	8.707606813	10.42382294	48.02646318	46.22959589	5.3075	7.3423	4.5511	13.1333
0.05	9.080211021	10.61466951	47.94066323	46.15142981	5.5821	7.4365	4.8257	13.1506
0.06	9.60487158	10.90986684	47.8783036	46.11420211	5.8534	7.5641	5.097	13.1673
0.07	9.990753023	11.14869392	47.79922848	46.0557318	6.1168	7.6674	5.3604	13.1814
0.08	10.34709934	11.3494867	47.7208218	45.96744576	6.3838	7.7944	5.6274	13.1929
0.09	10.66651743	11.49004254	47.67081004	45.93751735	6.5836	7.8929	5.8272	13.208
0.1	11.08357009	11.77125603	47.60464728	45.87470046	6.8535	8.0119	6.0971	13.2192

Table 2. PSNR values of speckle noise

Noise Density	1D Gaussian	2D Gaussian	1D Derivative	2D Derivative	AGD-1D	AGD-2 D	AFGD-1D	AFGD-2D
0.01	31.09445264	28.56511573	14.44177029	14.78279396	35.2	31.1673	56.6179	36.0645
0.02	30.43356858	28.27450984	14.46510598	14.79961122	34.6307	31.0539	56.4605	35.4952
0.03	29.84299207	28.00377409	14.48809865	14.82139851	34.0504	30.9033	56.3343	34.9149
0.04	29.3328274	27.77026309	14.50119151	14.83240167	33.6331	30.8143	56.2182	34.4976
0.05	28.96888478	27.61267407	14.51672285	14.8471004	33.195	30.7035	56.1065	34.0595
0.06	28.48097236	27.37441462	14.52802852	14.85410964	32.7828	30.5557	55.9989	33.6473
0.07	28.13883915	27.18632376	14.54238587	14.86512986	32.4004	30.438	55.9082	33.2649
0.08	27.83443124	27.0312792	14.55664533	14.88179616	32.0294	30.2953	55.8337	32.8939
0.09	27.57035066	26.92437088	14.56575297	14.8874532	31.7617	30.1861	55.7366	32.6262
0.1	27.23721017	26.71434749	14.57781657	14.89933878	31.4126	30.0562	55.665	32.2771

Table 3. RMSE value for salt and pepper noise

Noise Density	1D Gaussian	2D Gaussian	1D Derivative	2D Derivative	AGD-1D	AGD-2 D	AFGD-1D	AFGD-2D
0.01	11.11702861	10.21946771	48.61942732	46.54211052	6.285	7.7701	4.1974	13.0604
0.02	14.44717898	11.5344193	48.63665398	46.43533838	7.9623	8.5877	5.8747	13.0778
0.03	17.50551308	12.93244987	48.67265104	46.3265297	9.3603	9.3822	7.2727	13.0931
0.04	19.63295042	14.04488319	48.69226808	46.26227233	10.7906	10.2824	8.703	13.1083
0.05	22.05272295	15.36976576	48.7141882	46.17514423	12.0355	11.1279	9.9479	13.1243
0.06	23.9057489	16.46080945	48.72247308	46.10760899	13.1609	11.9145	11.0733	13.1369
0.07	25.82604815	17.64797129	48.74933894	46.06733781	14.4544	12.9025	12.3668	13.149
0.08	27.51779109	18.68339019	48.7829182	46.06149024	15.4735	13.6981	13.3859	13.1661
0.09	29.32424472	19.93126336	48.80346442	46.0127109	16.5842	14.6072	14.4966	13.1775
0.1	31.16517892	21.20821542	48.87401754	46.02980714	17.7571	15.555	14.6695	13.1899

Table 4. PSNR value of salt and pepper noise

Noise Density	1D Gaussian	2D Gaussian	1D Derivative	2D Derivative	AGD-1D	AGD-2 D	AFGD-1D	AFGD-2D
0.01	27.21102914	27.9422381	14.39460683	14.77388215	32.1649	30.3224	56.6922	33.0625
0.02	24.93514255	26.89088891	14.39152982	14.79383132	30.1101	29.4534	56.5788	31.0077
0.03	23.26730672	25.89718754	14.38510358	14.81420824	28.7051	28.6848	56.4793	29.6027
0.04	22.27109221	25.18044098	14.38160352	14.82626439	27.47	27.889	56.3803	28.3676
0.05	21.26155918	24.39745863	14.37769422	14.8426384	26.5216	27.2026	56.2767	27.4192
0.06	20.56075654	23.80177986	14.37621712	14.85535158	25.7452	26.6094	56.1949	26.6428
0.07	19.88964448	23.19690784	14.37142899	14.8629413	24.9309	25.9174	56.1168	25.8285
0.08	19.33853222	22.70168993	14.36544808	14.86404392	24.3391	25.3977	56.0063	25.2367
0.09	18.7862669	22.14010706	14.36179056	14.87324719	23.737	24.8395	55.9329	24.6346
0.1	18.25741112	21.60072109	14.3492428	14.8700205	23.1435	24.2935	55.8533	24.0411

Table 5. RMSE values of Gaussian noise

Noise Density	1D Gaussian	2D Gaussian	1D Derivative	2D Derivative	AGD-1D	AGD-2 D	AFGD-1D	AFGD-2D
0.01	14.45374646	13.49420509	47.75976741	45.82554865	9.2092	9.8082	6.3197	13.2281
0.02	19.34256992	16.83674224	47.47599651	45.51995165	12.4663	12.2058	9.5768	13.3152
0.03	22.99063698	19.4753713	47.34050229	45.36223612	15.0327	14.2439	12.1432	13.3733
0.04	26.17116478	21.83692253	47.2459628	45.23099926	17.1824	16.9282	14.2929	13.4173
0.05	28.94662331	23.98696408	47.19347487	45.1444431	19.0999	17.6862	15.2104	13.4539
0.06	31.55328362	26.06022579	47.13545949	45.02103329	21.082	19.4344	16.1925	13.4786
0.07	33.91014375	27.91011346	47.13329082	45.01382967	22.7703	20.9124	19.8808	13.5032
0.08	36.08758531	29.63753666	47.1157946	44.99458047	24.3868	22.389	21.4973	13.5209
0.09	38.07428237	31.28258924	47.10518741	44.93996311	25.8149	23.6889	22.9254	13.5402
0.1	40.03755472	32.91757745	47.1501727	44.96015887	27.3009	25.0351	24.4114	13.5549

Table 6. PSNR value of gaussian noise

Noise Density	1D Gaussian	2D Gaussian	1D Derivative	2D Derivative	AGD-1D	AGD-2 D	AFGD-1D	AFGD-2D
0.01	24.93119496	25.52785748	14.54955955	14.90865014	28.8464	28.2991	55.608	30.6094
0.02	22.4005201	23.60564234	14.60132182	14.96676777	26.2161	26.3995	55.0532	27.9791
0.03	20.89978353	22.34108869	14.62614638	14.99691451	24.5901	25.0583	54.6862	26.3531
0.04	19.77434257	21.34697494	14.64350954	15.02207996	23.4291	24.0331	54.41	25.1921
0.05	18.89884542	20.53129791	14.65316449	15.03871761	22.5102	23.1781	54.1808	24.2732
0.06	18.14991238	19.81124012	14.66384871	15.06249444	21.6526	22.3594	54.0272	23.4156
0.07	17.52421099	19.21557157	14.66424835	15.06388434	20.9835	21.7227	53.8746	22.7465
0.08	16.98364714	18.69396153	14.66747321	15.06759947	20.3877	21.1301	53.7645	22.1507
0.09	16.51816908	18.22474976	14.66942889	15.07814938	19.8934	20.6399	53.6455	21.6564
0.1	16.0814527	17.78224629	14.66113785	15.07424686	19.4073	20.1598	53.5545	21.1703

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Figure 4. RMSE of different filters adding speckle noise

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Figure 5. PSNR of different filters added speckle noise

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Figure 6. RMSE of different filters added salt and pepper noise

7.png

Figure 7 PSNR of different filters added salt and pepper noise

8.png

Figure 8. RMSE of different filters added gaussian noise

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Figure 9. PSNR of different filters added salt and gaussian noise

An MRI image was taken and different noises were added and the RMSE and PSNR value where calculated and the results are shown in Tables 1-6 and their corresponding graphical representation is shown in Figures 4-9. The MATLAB software was used for the execution of algorithm. In this, four Adaptive gaussian filter is proposed namely AGD-1D, AGD-2D, AFGD-1D, AFGD-2D. The performance of the filter is calculated using the statistical method PSNR and RMSE. The calculated value is compared with some basic gaussian filter namely Gaussian 1D filter, Gaussian 2D filter. Derivative of Gaussian 1D filter and Derivative of Gaussian 2D filter. The results are shown in the Tables 1-6. From the tabulation it can be concluded that the adaptive filter works effectively. Also, the performance of AFGD-1D gives the best results at different level of noises. It can also be noticed that the AFGD-1D filter works better in different noises.