Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

184 
https://internationalpubls.com 

Stability and Preservation of Fuzzy Membership Functions in Adaptive 

Gaussian Derivative Filters 

 

Aarthi D1, Panimalar A2, Santhosh Kumar S3, Anitha K4 
1,3 Sri Ramakrishna Mission Vidyalaya College of Arts and Science, Coimbatore, India  

2KGiSL Institute of Technology, Coimbatore, India 

4Amrita School of Engineering, Amrita Vishwa Vidyapeetham Chennai, India 

e-mail fuzzysansrmvcas@gmail.com 

 

Article History: 

Received: 10-07-2024 

Revised: 26-08-2024 

Accepted: 07-09-2024 

Abstract:  

This paper examines the stability and preservation of fuzzy membership functions in 

Gaussian filters, particularly focusing on the Adaptive Gaussian Derivative (AGD) filter. 

Gaussian filters are essential for smoothing and noise reduction in image processing. The 

AGD filter, which adapts based on local image statistics, outperforms traditional methods. 

Key concepts like fuzzy sets, Boundary Input Boundary Output (BIBO) stability, and 

convolution are defined, and the mathematical formulation of the Gaussian and its 

derivative is presented. The AGD filter's BIBO stability ensures bounded outputs for 

bounded inputs, guaranteeing consistent behavior. It also preserves fuzzy membership 

function properties, maintaining convexity and boundedness through linearity and 

continuity. Frequency response analysis using Fourier Transform confirms the AGD filter 

retains the Gaussian shape in the frequency domain, preserving image smoothness. 

Theorems and proofs validate the AGD filter's stability and its capability to preserve 

fuzzy membership functions, ensuring reliable processing in applications such as medical 

image analysis. These properties make the AGD filter a robust tool for advanced image 

processing tasks. 

Keywords: AGD filter, fuzzy, image, Gaussian. 

 

1. Introduction 

Fuzzy logic systems have garnered significant attention for their capability to handle uncertainty and 

imprecision in various computational tasks. These systems utilize fuzzy membership functions to map 

inputs to a degree of membership, which is crucial for decision-making processes. However, the 

stability and preservation of these membership functions are paramount for maintaining the accuracy 

and reliability of fuzzy logic systems [1]. In scenarios where these systems are applied, such as in 

image processing and pattern recognition, ensuring the robustness of membership functions is essential 

for consistent performance. 

The introduction of Gaussian derivative filtering [9] offers a promising approach to enhance the 

stability and preservation of fuzzy membership functions. Gaussian derivatives are widely recognized 

for their ability to capture fine details and subtle variations in data. By incorporating these derivatives 

into the filtering process, it is possible to maintain the integrity of the membership functions while also 

improving their resilience to noise and distortions. This integration can lead to more accurate and 

reliable outputs in applications that rely on fuzzy logic. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

185 
https://internationalpubls.com 

Adaptive filtering techniques [3] further enhance the potential of Gaussian derivative filtering. By 

adjusting the filter parameters dynamically based on the characteristics of the input data, adaptive 

filters can provide a more tailored and effective approach to preserving membership functions. This 

adaptability is particularly beneficial in complex and varying environments where static filter 

parameters may not suffice. The ability to adapt in real-time ensures that the membership functions 

remain stable and accurate under different conditions. 

Implementing adaptive Gaussian derivative filtering requires careful consideration of various factors, 

including the selection of appropriate derivative [5,7] orders and the design of adaptive mechanisms. 

The interplay between these elements determines the overall effectiveness of the filtering process. A 

thorough analysis and optimization of these parameters can lead to significant improvements in the 

stability [10,11] and preservation of fuzzy membership functions, thereby enhancing the performance 

of the associated fuzzy logic systems. 

This study explores the development and application of adaptive Gaussian derivative filtering for the 

stability and preservation of fuzzy membership functions. By evaluating different derivative orders 

and adaptive strategies, the research aims to identify the most effective configurations for maintaining 

the integrity of membership functions. The findings of this study hold promise for advancing the 

robustness and reliability of fuzzy logic systems, with potential implications for various fields where 

these systems are employed. 

1. Preliminaries 

2.1 Fuzzy Set: A fuzzy set A in a universe of discourse X is characterized by a membership function 

𝜇𝐴: 𝑋 ⟶ [0,1]. For each element 𝑥 ∈ 𝑋, the membership function 𝜇𝐴(𝑥) indicates the degree of 

membership of x in the fuzzy set A. 

2.2 Convolution: It is mathematical operation that combines two functions to produce a third function. 

In the context of image processing the convolution of an input image I with a filter G is given by 

(𝐼 ∗ 𝐺)(𝑥, 𝑦) = ∑ ∑ 𝐼(𝑥 − 𝑖, 𝑦 − 𝑗) ∙ 𝐺(𝑖, 𝑗)𝑘
𝑗=−𝑘

𝑘
𝑖=−𝑘  where k determines the size of the filter window. 

2.3 Bounded Input Bounded Output (BIBO) Stability: A system is BIBO stable [12,13] if every 

bounded input produces a bounded output. If the input image pixel intensities (or membership values 

in the case of fuzzy images) are bounded, the output image pixel intensities (or membership values) 

will also be bounded after processing with a filter. 

2.4 Membership Function: In a fuzzy set, the membership function 𝜇𝐴(𝑥) represent the degree of 

membership of an element x in the fuzzy set. It assigns a value between 0 and 1 to each element x , 

where 0 means no membership and 1 means full membership. 

2.5 Bounded function: A function 𝑓(𝑥) is bounded if there exist a real number M such that |𝑓(𝑥)| ≤

𝑀 for all x in the domain of 𝑓. 

2.6 Smooth function: A function 𝑓(𝑥) is smooth if it is continuously differentiable to a desired degree. 

In image processing, a smooth function typically refers to one that does not have abrupt changes in 

value. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

186 
https://internationalpubls.com 

2.7 Fourier Transform (FT): The Fourier Transform of a function 𝑓(𝑥, 𝑦) is a mathematically 

transformation used to analyze the frequencies present in the function. It is defined as:  

𝐹(𝑢, 𝑣) = ∫ ∫ 𝑓(𝑥, 𝑦)𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥 𝑑𝑦
∞

−∞

∞

−∞
  

2.8 Frequency Response: The frequency response of a filter describes how the filter affects the 

amplitude and phase of the input signals frequency components. For a Gaussian filter [14,15] the 

frequency response 𝐻(𝑢, 𝑣) is given by: 

𝐻(𝑢, 𝑣) = 𝑒−2𝜋2𝜎2(𝑢2+𝑣2) 

2.9 High Frequency components: It corresponds to rapid changes in intensity values, such as edges 

and noise. They are characterized by large values of u and v in the frequency domain. 

2.10 High Frequency components:  It corresponds to changes in intensity values, such as smooth 

regions. These are characterized by small values of u and v in the frequency domain. 

2.11 Gaussian Filter: A Gaussian filter [2] is a linear filter used in image processing to smooth images 

and reduce noise. It is characterized by a Gaussian function 𝐺𝜎(𝑥, 𝑦) which is defined as:  

𝐺𝜎(𝑥, 𝑦) =
1

2𝜋𝜎2
𝑒

−
𝑥2+𝑦2

2𝜎2  

Where 𝜎 is the standard deviation of the Gaussian distribution, controlling the width of the filter. 

2.12 Gaussian 2D Function: A two-dimensional Gaussian filter [4], 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(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦) =
1

2𝜋𝜎𝑥𝜎𝑦
𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2
−

𝑦2

2𝜎𝑦
2
) 

Where 𝐺(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦) 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. 

2.13 Fuzzy Gaussian 1 D and it’s Derivative: Let the function be represented as  

𝐹(𝑥, 𝜎,𝑚) =
1

√2𝜋𝜎
𝑒𝑥𝑝 (−

(𝑥 − 𝑚)2

2𝜎2
) 

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 (11) (i.e) fuzzy gaussian function with respect to x. The derivative 

is given as: 𝐹′(𝑥, 𝜎,𝑚) = −(
𝑥−𝑚

𝜎2
)

1

√2𝜋𝜎
𝑒𝑥𝑝 (−

(𝑥−𝑚)2

2𝜎2
)                                    

2.14 Fuzzy Gaussian 2 D and Derivative: Consider a 2D fuzzy Gaussian function 

𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥, 𝑚𝑦) where x and y are the variables 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦 are the standard deviation and 

mean along the respective axis [6]. The function is given by: 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

187 
https://internationalpubls.com 

 𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥,𝑚𝑦) =
1

2𝜋𝜎𝑥𝜎𝑦
𝑒𝑥𝑝 [−

(𝑥−𝑚𝑥)2

2𝜎𝑥
2 −

(𝑦−𝑚𝑦)
2

2𝜎𝑥
2 ]                                                             

The fuzzy partial derivative of this 2D fuzzy gaussian function with respect to x and y. the partial 

derivatives are given by: 

𝐹𝑥(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦) = −(
𝑥−𝑚𝑥

𝜎𝑥
2

)
1

2𝜋𝜎𝑥𝜎𝑦
𝑒𝑥𝑝 [−

(𝑥−𝑚𝑥)2

2𝜎𝑥
2

−
(𝑦−𝑚𝑦)

2

2𝜎𝑦
2

]                                                                        

 𝐹𝑦(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦) = −(
𝑦−𝑚𝑦

𝜎𝑦
2

)
1

2𝜋𝜎𝑥𝜎𝑦
𝑒𝑥𝑝 [−

(𝑥−𝑚𝑥)2

2𝜎𝑥
2 −

(𝑦−𝑚𝑦)
2

2𝜎𝑦
2 ]                           

2. Main Results  

Combining Gaussian filtering with derivatives to create an adaptive Gaussian filter offers significant 

advantages by enhancing edge detection and noise reduction capabilities. This combination is 

particularly effective for identifying subtle changes and edges, crucial for accurate feature extraction 

in image processing. Gaussian filters are renowned for their ability to smooth out noise while 

preserving essential signal characteristics, and when paired with derivatives, they further improve 

noise reduction while maintaining critical details. The adaptive nature of the filter allows it to 

dynamically adjust its parameters based on the specific characteristics of the input data, ensuring 

effectiveness across diverse conditions and applications. This approach optimizes [8] performance by 

balancing noise reduction and detail preservation, making it a versatile and powerful tool for various 

data processing and analysis tasks. 

Gaussian filters, both in 1D and 2D forms, are crucial for medical image processing, especially for 

MRI scans, as they effectively reduce noise while preserving key image details. By incorporating fuzzy 

logic to handle uncertainties in image data, fuzzy Gaussian filters further enhance this capability. Both 

standard and fuzzy Gaussian filters smooth images while maintaining edges, striking a balance in noise 

reduction.  

Adaptive filters excel in context-aware smoothing by distinguishing between different image regions 

and applying the appropriate level of smoothing to enhance image quality and reduce abnormalities. 

Adaptive Gaussian filters are particularly beneficial in scenarios where preserving fine structural 

details is essential, such as in medical imaging, which is vital for accurate diagnosis and analysis. Their 

adaptability and effectiveness make them highly suitable for these critical applications. The stability 

analysis of Adaptive filter is performed in this section. 

3.1 Adaptive Gaussian filter1D and Derivative of Gaussian 1D (AGD-1D) 

                                           𝐹(𝑥, 𝜎) =
1

√2𝜋𝜎
[1 − 𝛼.

𝑥

𝜎2
] 𝑒𝑥𝑝 (

−𝑥2

2𝜎2
)                                                              (1) 

 where 𝜎 = √
1

𝑚𝑛
∑ ∑ (𝐴(𝑖, 𝑗) − 𝑚)2𝑛

𝑗=1
𝑚
𝑖=1 ; ∝=

𝐴

max (|𝐴|)
 

Theorem 1: Boundary Input Boundary Output (BIBO) of the filter 

A filter 𝐹(𝑥, 𝜎) is BIBO stable if for every bounded input x, the output 𝑦 = 𝐹(𝑥, 𝜎) is also bounded. 

Proof: Suppose x is bounded then there exist a constant M such that |𝑥| ≤ 𝑀. Consider 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

188 
https://internationalpubls.com 

 𝐹(𝑥, 𝜎) =
1

√2𝜋𝜎
[1 − 𝛼.

𝑥

𝜎2
] 𝑒𝑥𝑝 (

−𝑥2

2𝜎2
) where the term 

1

√2𝜋𝜎
 is a constant for a given 𝜎. 

 Also, [1 − 𝛼.
𝑥

𝜎2] is bounded since x is bounded, 

|1 − 𝛼.
𝑥

𝜎2
| ≤ 1 + |𝛼|

𝑀

𝜎2
 

Also, 𝑒𝑥𝑝 (
−𝑥2

2𝜎2
)is bounded for all x 

⇒ |𝐹(𝑥, 𝜎)| ≤
1

√2𝜋𝜎
(1 + |𝛼|

𝑀

𝜎2
) 

Since all the components are bounded, then 𝐹(𝑥, 𝜎) is also bounded. Hence, 𝐹(𝑥, 𝜎) is BIBO stable. 

Hence the proof. 

Theorem 2: Preservation of Fuzzy Membership Distribution Theorem 

If a fuzzy membership function 𝜇(𝑥) is processed through a filter 𝐹(𝑥, 𝜎), the resulting function 

𝜇′(𝑥) = 𝐹(𝜇(𝑥), 𝜎) preserves the properties of a fuzzy membership function. 

Proof: A fuzzy membership function 𝜇(𝑥) is normalized such that sup 𝜇(𝑥) = 1. For any 𝑥1, 𝑥2𝜖ℝ 

and 𝜆 ∈ [0,1] then 

𝜇(𝜆𝑥1 + (1 − 𝜆)𝑥2) ≥ min(𝜇(𝑥1), 𝜇(𝑥2)) 

which is bounded that is 0 ≤ 𝜇(𝑥) ≤ 1. Since 𝜇(𝑥) is normalized, assume,  

𝜇(𝑥0) = 1, 𝑓𝑜𝑟 𝜇′(𝑥) = 𝐹(𝜇(𝑥), 𝜎) 

Also, 𝜇′(𝑥0) = 𝐹(𝜇(𝑥0), 𝜎) = 𝐹(1, 𝜎) =
1

√2𝜋𝜎
(1 −

𝛼

𝜎2) exp(−
1

2𝜎2) 

This ensures that 𝜇′(𝑥0) is well defined and bounded. 

Given the linearity and continuity of the filter 𝐹(𝑥, 𝜎), it preserves the convex combinations such that 

𝜇′(𝜆𝑥1 + (1 − 𝜆)𝑥2) = 𝐹(𝜇(𝜆𝑥1 + (1 − 𝜆)𝑥2), 𝜎) ≥ min(𝐹(𝜇(𝑥1), 𝜎), 𝐹(𝜇(𝑥2), 𝜎)) 

Thus, convexity is preserved. Since 𝜇(𝑥) is bounded within [0,1], 𝜇′(𝑥) = 𝐹(𝜇(𝑥), 𝜎) remains 

bounded whose proof is similar to BIBO stability. 

Theorem 3: Frequency Response and Fuzzy membership Function Theorem 

When a fuzzy membership function 𝜇(𝑥) is processed through a filter 𝐹(𝑥, 𝜎) the output maintains 

properties of a fuzzy membership function. 

Proof: The frequency response of 𝐹(𝑥, 𝜎) can be analyzed using the Fourier Transform (FT): 

ℱ{𝐹(𝑥, 𝜎)} = ℱ {
1

√2𝜋𝜎
(1 − 𝛼.

𝑥

𝜎2
) exp (−

𝑥2

𝜎2
)} 

Using properties of FT and Linearity 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

189 
https://internationalpubls.com 

ℱ{𝐹(𝑥, 𝜎)} = ℱ {
1

√2𝜋𝜎
} ∗ ℱ {(1 − 𝛼.

𝑥

𝜎2
)exp (−

𝑥2

𝜎2
)} 

Also, the filtered output in the frequency domain will maintain the bounded and smooth nature due to 

the Gaussian properties and linear filtering. 

Hence the proof. 

3.2 Adaptive Gaussian 2D filter and Derivative of 2D filter (AGD-2D) 

                       𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦 , 𝛼, 𝛽) =
1

2𝜋𝜎𝑥𝜎𝑦
[1 − 𝛼.

𝑥

𝜎𝑥
2
− 𝛽.

𝑦

𝜎𝑦
2
] 𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2
−

𝑦2

2𝜎𝑦
2
)                             (2)            

where, 𝜎𝑥 = √
1

𝑛
∑ (𝐴(𝑖, 𝑗) − 𝑚𝑥)2𝑛

𝑗=1 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖;  𝜎𝑦 = √
1

𝑚
∑ (𝐴(𝑖, 𝑗) − 𝑚𝑦)2𝑚

𝑖=1 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑗           

∝ =
𝐴(𝑖,𝑗)

max(|𝐴(𝑖,𝑗)|)𝑟𝑜𝑤
 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖, 𝑗;     𝛽 =

𝐴(𝑖,𝑗)

max(|𝐴(𝑖,𝑗)|)𝑐𝑜𝑙𝑢𝑚𝑛
 𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖, 𝑗                

Theorem 4: BIBO Stability  

The adaptive Gaussian 2D filter given by the function 𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦, 𝛼, 𝛽) is BIBO stable, provided 

that the input image 𝐴(𝑖, 𝑗) is bounded. 

Proof: Let 𝐴(𝑖, 𝑗) be a bounded input image. Then there exist a constant M such that |𝐴(𝑖, 𝑗)| ≤

𝑀 ∀(𝑖, 𝑗). Since 𝐴(𝑖, 𝑗) is bounded, ∝, 𝛽 are also bounded. Also, |∝| ≤ 1 and |𝛽| ≤ 1 because they are 

normalized by the maximum absolute value in their respective rows and columns.  

The Gaussian filter component of 𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦, 𝛼, 𝛽) given by 𝑒𝑥𝑝 (−
𝑥2

2𝜎𝑥
2 −

𝑦2

2𝜎𝑦
2
) is always non-

negative and integrates to 1 over entire (x, y) plane. 

The output of the convolution of the input image 𝐴(𝑖, 𝑗) with the Gaussian filter 𝐹(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝛼, 𝛽) 

can be expressed as: 

𝑔(𝑥, 𝑦) = ∑∑𝐴(𝑖, 𝑗)

𝑗

𝐹(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥 , 𝜎𝑦 , 𝛼, 𝛽)

𝑖

 

Now to prove, g(x, y) is bounded. 

Since, 𝐴(𝑖, 𝑗) is bounded and the properties of Gaussian filter then, 

𝑔(𝑥, 𝑦) ≤ ∑∑|𝐴(𝑖, 𝑗)|

𝑗

|
1

2𝜋𝜎𝑥𝜎𝑦
[1 − 𝛼

𝑥

𝜎𝑥
2
− 𝛽

𝑦

𝜎𝑦
2
] 𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2
−

𝑦2

2𝜎𝑦
2
)|

𝑖

 

Given |𝐴(𝑖, 𝑗)| ≤ 𝑀, |∝| ≤ 1 and |𝛽| ≤ 1 then, 

𝑔(𝑥, 𝑦) ≤ ∑∑𝑀 

𝑗

|
1

2𝜋𝜎𝑥𝜎𝑦
[1 −

𝑥

𝜎𝑥
2
−

𝑦

𝜎𝑦
2
] 𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2
−

𝑦2

2𝜎𝑦
2
)|

𝑖

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

190 
https://internationalpubls.com 

Since, the Gaussian function 𝑒𝑥𝑝 (−
𝑥2

2𝜎𝑥
2 −

𝑦2

2𝜎𝑦
2) is bounded and the terms involving ∝ and 𝛽 do not 

cause the expression to diverge, the output g (x, y) remains bounded. 

Therefore, |𝑔(𝑥, 𝑦)| ≤ 𝑘 for some constant k, proving that adaptive Gaussian 2D filter is BIBO stable.  

Hence the Proof. 

Corollary 

If a bounded input image 𝐴(𝑖, 𝑗) is processed by the adaptive Gaussian 2D filter defined by,  

𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦 , 𝛼, 𝛽) =
1

2𝜋𝜎𝑥𝜎𝑦
[1 − 𝛼

𝑥

𝜎𝑥
2
− 𝛽

𝑦

𝜎𝑦
2
] 𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2
−

𝑦2

2𝜎𝑦
2
) 

The output image g (x, y) will also be bounded, preserving the stability of the filtering process. 

Theorem 5: Frequency Response 

The frequency response of a filter describes how the filter modifies the amplitude and phase of input 

signals at different frequencies. For the adaptive Gaussian 2D filter defined by  

𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦 , 𝛼, 𝛽) =
1

2𝜋𝜎𝑥𝜎𝑦
[1 − 𝛼

𝑥

𝜎𝑥
2
− 𝛽

𝑦

𝜎𝑦
2
] 𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2
−

𝑦2

2𝜎𝑦
2
) 

The frequency response can be found by taking the Fourier Transform of F (x, y). 

Proof: The 2D Fourier Transform of F(x .y) is given by  

ℱ{𝐹(𝑥, 𝑦)} = ∫ ∫ 𝐹(𝑥, 𝑦) 𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥𝑑𝑦
∞

−∞

∞

−∞
 where (u, v) are the frequency variables. 

The FT of the Gaussian component  𝑒𝑥𝑝 (−
𝑥2

2𝜎𝑥
2 −

𝑦2

2𝜎𝑦
2) is another Gaussian function: 

ℱ {𝑒𝑥𝑝 (−
𝑥2

2𝜎𝑥
2
−

𝑦2

2𝜎𝑦
2
)} = 2𝜋𝜎𝑥𝜎𝑦 exp(−2𝜋2𝜎𝑥

2𝑢2 − 2𝜋2𝜎𝑦
2𝑣2) 

The FT of linear terms 𝑥 . 𝑒𝑥𝑝 (−
𝑥2

2𝜎𝑥
2 −

𝑦2

2𝜎𝑦
2) and 𝑦. 𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2 −

𝑦2

2𝜎𝑦
2) introduce additional frequency- 

dependent terms. 

The overall frequency response H(u ,v) will be a combination of these terms: 

𝐻(𝑢, 𝑣) = (1 − 𝑗 2𝜋𝜎𝑥𝑢𝛼 − 𝑗2𝜋𝜎𝑦𝑣𝛽) exp(−2𝜋2𝜎𝑥
2𝑢2 − 2𝜋2𝜎𝑦

2𝑣2) 

Hence the proof. 

Theorem 6: Preservation of Fuzzy Membership Distribution  

If a fuzzy membership function 𝜇(𝑥, 𝑦) is processed by an adaptive Gaussian 2D filter defined by  

𝐹(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦, 𝛼, 𝛽) =
1

2𝜋𝜎𝑥𝜎𝑦
[1 − 𝛼

𝑥

𝜎𝑥
2 − 𝛽

𝑦

𝜎𝑦
2] 𝑒𝑥𝑝 (−

𝑥2

2𝜎𝑥
2 −

𝑦2

2𝜎𝑦
2) the resulting 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥. 𝑦) will 

also be a valid fuzzy membership function, preserving its essential properties. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

191 
https://internationalpubls.com 

Proof: The Fuzzy membership function  𝜇(𝑥, 𝑦) is typically bounded between 0 and 1. For any fuzzy 

membership function 𝜇(𝑥, 𝑦)it holds that 0 ≤ 𝜇(𝑥, 𝑦) ≤ 1.  

The filtered membership function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥. 𝑦) is obtained by convolving 𝜇(𝑥, 𝑦) with the adaptive 

Gaussian filter 𝐹(𝑥, 𝑦) : 

𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥. 𝑦) = ∫ ∫ 𝜇(𝑢, 𝑣)
∞

−∞

𝐹(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥 , 𝜎𝑦 , 𝛼, 𝛽)𝑑𝑢 𝑑𝑣
∞

−∞

 

Since 𝜇(𝑢, 𝑣) is bounded between 0 and 1 and 𝐹(𝑥, 𝑦) is non-negative and integrates to 1, then, 

0 ≤ 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥. 𝑦) ≤ ∫ ∫ 𝜇(𝑢, 𝑣)
∞

−∞

𝐹(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥, 𝜎𝑦 , 𝛼, 𝛽)𝑑𝑢 𝑑𝑣
∞

−∞

≤ 1 

The adaptive Gaussian filter preserves the smoothness and continuity of the membership function. 

Since 𝐹(𝑥, 𝑦) is smooth and the convolution operation with 𝐹(𝑥, 𝑦) maintain this smoothness, the 

filtered membership function  𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥. 𝑦) will also be smooth. 

The filtered output 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥. 𝑦) remains a valid fuzzy membership function, preserving the essential 

properties such as boundedness, normalization and smoothness. 

Hence the proof. 

Corollary: 

If a fuzzy membership function 𝜇(𝑥, 𝑦) is processed by the adaptive Gaussian 2D filter, the resulting 

function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥. 𝑦) will not only be bounded between 0 and 1 but will also maintain the essential 

properties of a fuzzy membership function, ensuring that the concept represented by the membership 

function is preserved in the filtered output. 

3.3 Adaptive Fuzzy Gaussian 1D Filter and Derivative Fuzzy Gaussian 1D Filter (AFGD-1D) 

                                      𝐺(𝑥, 𝜎,𝑚, 𝛼) =
1

√2𝜋𝜎
𝑒𝑥𝑝 (−

(𝑥−𝑚)2

2𝜎2 ) [1 −  𝛼. (
𝑥−𝑚

𝜎2 )]                                         (3)  

where, 𝑚 =
1

𝑚𝑛
∑ ∑ 𝐴 (𝑖, 𝑗)𝑛

𝑗=1
𝑚
𝑖=1 . 

Theorem 7: BIBO Stability 

The adaptive Gaussian filter 𝐺(𝑥, 𝜎,𝑚, 𝛼) is BIBO stable. That is, for any bounded input image 

𝐴(𝑖, 𝑗) the output image will also be bounded. 

Proof: Let 𝐴(𝑖, 𝑗) be a bounded input image, then there exist a constant M such that | 𝐴(𝑖, 𝑗)| ≤

𝑀∀(𝑖, 𝑗) where 𝜎 is standard deviation, m is mean of the image which is bounded because 𝐴(𝑖, 𝑗) is 

bounded, 𝛼 is adaptive parameter which can be bounded based on the range of 𝐴(𝑖, 𝑗).  

The gaussian part 𝑒𝑥𝑝 (−
(𝑥−𝑚)2

2𝜎2
) is bounded by 1 and the term 1 −  𝛼. (

𝑥−𝑚

𝜎2
) is also bounded because 

𝛼 and (
𝑥−𝑚

𝜎2 ) are bounded. 

The output of the filter applied to 𝐴(𝑖, 𝑗) can be expressed as a convolution: 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

192 
https://internationalpubls.com 

𝑔(𝑥) = (𝐴 ∗ 𝐺)(𝑥) = ∫ 𝐴(𝑡) 𝐺(𝑥 − 𝑡, 𝜎,𝑚, 𝛼)𝑑𝑡
∞

−∞

 

Given the boundedness of 𝐴(𝑡) and 𝐺(𝑥, 𝜎,𝑚, 𝛼), then 

|𝑔(𝑥)| ≤ ∫ |𝐴(𝑡)||𝐺(𝑥 − 𝑡, 𝜎, 𝑚, 𝛼)| 𝑑𝑡
∞

−∞

≤ 𝑀 ∫ |𝐺(𝑥 − 𝑡, 𝜎,𝑚, 𝛼)| 𝑑𝑡
∞

−∞

 

Since, 𝐺(𝑥, 𝜎,𝑚, 𝛼) is a bounded function that integrates to a finite value, the output g(x) is also 

bounded. Therefore, the adaptive filter 𝐺(𝑥, 𝜎,𝑚, 𝛼) is BIBO stable. 

Corollary: Stability of the Adaptive Gaussian Filter 

If a bounded input image 𝐴(𝑖, 𝑗) is processed by the adaptive Gaussian filter 𝐺(𝑥, 𝜎,𝑚, 𝛼) the output 

will remain bounded, ensuring the stability of the filtering process. 

Proof: Same as BIBO stability. 

Theorem 8: Preservation of Fuzzy Membership Distribution 

If a fuzzy membership function 𝜇(𝑥) is processed by the adaptive Gaussian filter 𝐺(𝑥, 𝜎,𝑚, 𝛼) the 

resulting function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥) is also a valid fuzzy membership function, preserving its essential 

properties. 

Proof: the fuzzy membership function 𝜇(𝑥) is typically bounded between 0 and 1. For any fuzzy 

membership function 𝜇(𝑥), it holds that 0 ≤ 𝜇(𝑥) ≤ 1. 

The filtered membership function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥) is obtained by convolving 𝜇(𝑥) with the adaptive 

Gaussian filter 𝐺(𝑥, 𝜎,𝑚, 𝛼):  

𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥) = ∫ 𝜇(𝑡)𝐺(𝑥 − 𝑡, 𝜎,𝑚, 𝛼)𝑑𝑡
∞

−∞

 

Since, 𝜇(𝑡) is bounded between 0 and 1 and G(x) is non-negative and integrates to a finite value then,  

0 ≤ 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥) ≤ ∫ 𝜇(𝑡)𝐺(𝑥 − 𝑡, 𝜎,𝑚, 𝛼)𝑑𝑡
∞

−∞

≤ 1 

The adaptive Gaussian filter preserves the smoothness and continuity of the membership function. 

Since, G(x) is smooth and the convolution operation with G(x) maintains the smoothness, the filtered 

membership function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥) will also be smooth. 

Thus, 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥) remains a valid fuzzy membership function, preserving the essential properties such 

as boundedness, normalization and smoothness. 

Corollary: Preservation of Fuzzy Membership Properties 

If a fuzzy membership function 𝜇(𝑥) is processed by the adaptive Gaussian filter 𝐺(𝑥, 𝜎,𝑚, 𝛼) the 

resulting function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥) will not only be bounded between 0 and 1 but will also maintain the 

essential properties of a fuzzy membership function. 

Proof: Follows directly from the preservation of fuzzy membership distribution theorem. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

193 
https://internationalpubls.com 

Theorem 9: Frequency Response 

The frequency response of the adaptive Gaussian filter 𝐺(𝑥, 𝜎,𝑚, 𝛼)can be derived from its Fourier 

Transform. 

Proof: The Fourier Transform of 𝐺(𝑥, 𝜎,𝑚, 𝛼) is given: 

ℱ{𝐺(𝑥, 𝜎,𝑚, 𝛼)} = ∫ 𝐺(𝑥, 𝜎,𝑚, 𝛼)𝑒−𝑗2𝜋𝑢𝑥 𝑑𝑥
∞

−∞

 

The Fourier Transform of the Gaussian component 
1

√2𝜋𝜎
𝑒𝑥𝑝 (−

(𝑥−𝑚)2

2𝜎2
) is : 

ℱ {
1

√2𝜋𝜎
𝑒𝑥𝑝 (−

(𝑥−𝑚)2

2𝜎2
)} = exp(−2𝜋2𝜎2𝑢2)𝑒−𝑗2𝜋𝑚𝑢  

The linear term [1 −  𝛼. (
𝑥−𝑚

𝜎2 )] introduces additional frequency components. The overall frequency 

response H(u) is a combination of these terms:  

𝐻(𝑢) = (1 − 𝑗2𝜋𝛼𝜎𝑢) exp(−2𝜋2𝜎2𝑢2)𝑒−𝑗2𝜋𝑚𝑢 

Hence the Proof. 

Corollary:  

The frequency response of the adaptive Gaussian filter shows that it acts as a low-pass filter, 

attenuating high-frequency components and preserving low-frequency components of the input signal. 

Proof: Follows directly from the frequency response derivation, where the term exp(−2𝜋2𝜎2𝑢2) 

indicates attenuation of higher frequencies. 

3.4 Adaptive Fuzzy Gaussian 2D filter and Derivative of Fuzzy Gaussian 2D (AFGD-2D) 

𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) =
1

2𝜋𝜎𝑥𝜎𝑦
𝑒𝑥𝑝 [−

(𝑥−𝑚𝑥)2

2𝜎𝑥
2 −

(𝑦−𝑚𝑦)
2

2𝜎𝑦
2 ] [1 − 𝛼. (

𝑥−𝑚𝑥

𝜎𝑥
2 ) −  𝛽. (

𝑦−𝑚𝑦

𝜎𝑦
2

)]             

(4)                                

For 2D the constants are calculated using formulas for a given matrix A of size m x n 

𝑚𝑥 =
1

𝑛
∑ 𝐴(𝑖, 𝑗)𝑛

𝑗=1  𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑖  ;  𝑚𝑦 =
1

𝑚
∑ 𝐴(𝑖, 𝑗)𝑚

𝑖=1  𝑓𝑜𝑟 𝑒𝑎𝑐ℎ 𝑗 

Theorem 10: BIBO Stability 

The adaptive 2D Gaussian filter 𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦, 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) is Bounded Input Bounded Output 

(BIBO) Stable. 

Proof: Let 𝐴(𝑖, 𝑗) be a bounded input matrix. Then there exist a constant M such that |𝐴(𝑖, 𝑗)| ≤

𝑀 ∀(𝑖, 𝑗). Also, 𝜎𝑥and 𝜎𝑦 are standard deviations, 𝑚𝑥 and 𝑚𝑦 are the means of the image along rows 

and columns respectively which are bounded because 𝐴(𝑖, 𝑗) is bounded, 𝛼 and 𝛽 are adaptive 

parameters which can be bounded based on the range of 𝐴(𝑖, 𝑗) the Gaussian part 𝑒𝑥𝑝 [−
(𝑥−𝑚𝑥)2

2𝜎𝑥
2 −



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

194 
https://internationalpubls.com 

(𝑦−𝑚𝑦)
2

2𝜎𝑦
2 ] is bounded by 1, and the term [1 − 𝛼. (

𝑥−𝑚𝑥

𝜎𝑥
2 ) −  𝛽. (

𝑦−𝑚𝑦

𝜎𝑦
2

)] is also bounded because 

𝛼, 𝛽, (
𝑥−𝑚𝑥

𝜎𝑥
2 ) and (

𝑦−𝑚𝑦

𝜎𝑦
2

) are bounded. 

The output of the filter applied to 𝐴(𝑖, 𝑗) can be expressed as a convolution, 

𝑔(𝑥, 𝑦) = (𝐴 ∗ 𝐺)(𝑥, 𝑦) 

= ∫ ∫ 𝐴(𝑢, 𝑣)𝐺(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥,𝑚𝑦, 𝛼, 𝛽)𝑑𝑢 𝑑𝑣
∞

−∞

∞

−∞

 

Given the boundedness of A (u, v) and  𝐺(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥,𝑚𝑦, 𝛼, 𝛽) then, 

|𝑔(𝑥, 𝑦)| ≤  ∫ ∫ |𝐴(𝑢, 𝑣)||𝐺(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽)|𝑑𝑢 𝑑𝑣
∞

−∞

∞

−∞

 

≤ 𝑀 ∫ ∫ |𝐺(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽)|𝑑𝑢 𝑑𝑣
∞

−∞

∞

−∞

 

Since, 𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) is a bounded function that integrates to a finite value, the output 

g(x,y) is also bounded.  

Therefore, the adaptive 2D Gaussian filter 𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) is BIBO stable. 

Corollary: 

If a bounded input matrix 𝐴(𝑖, 𝑗) is processed by the adaptive 2D Gaussian filter 

𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) the output will remain bounded ensuring the stability of the filtering 

process. 

Proof: Follows directly from BIBO stability proof. 

Theorem 11: Preservation of Fuzzy Membership Distribution 

If the fuzzy membership function 𝜇(𝑥, 𝑦) is processed by the adaptive 2D Gaussian filter 

𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) the resulting function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥, 𝑦)  is also a valid fuzzy membership 

function, preserving its essential properties. 

Proof: The fuzzy membership function 𝜇(𝑥, 𝑦) is typically bounded between 0 and 1. For any fuzzy 

membership function 𝜇(𝑥, 𝑦)it holds that 0 ≤ 𝜇(𝑥, 𝑦) ≤ 1. 

The filtered membership function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥, 𝑦)  is obtained by convolving 𝜇(𝑥, 𝑦) with the adaptive 

Gaussian filter 𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) : 

𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥, 𝑦) = ∫ ∫ 𝜇(𝑢, 𝑣)𝐺(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽)𝑑𝑢 𝑑𝑣
∞

−∞

∞

−∞

 

Since, 𝜇(𝑢, 𝑣) is bounded between 0 and 1 and 𝐺(𝑥, 𝑦)is non-negative and integrates to a finite value, 

then, 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

195 
https://internationalpubls.com 

0 ≤ 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥, 𝑦) ≤ ∫ ∫ 𝜇(𝑢, 𝑣)𝐺(𝑥 − 𝑢, 𝑦 − 𝑣, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽)𝑑𝑢 𝑑𝑣
∞

−∞

≤ 1
∞

−∞

 

The adaptive Gaussian filter preserves the smoothness and continuity of the membership function. 

Since 𝐺(𝑥, 𝑦) is smooth and the convolution operation with 𝐺(𝑥, 𝑦) maintains this smoothness, the 

filtered membership function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥, 𝑦)  will also be smooth. 

Thus, 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥, 𝑦)  remains a valid fuzzy membership function, preserving the essential properties 

such as boundedness, normalization and smoothness. 

Corollary: 

If a fuzzy membership function 𝜇(𝑥, 𝑦) is processed by the adaptive 2D Gaussian filter 

𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) the resulting function 𝜇𝑓𝑖𝑙𝑡𝑒𝑟𝑒𝑑(𝑥, 𝑦) will not only be bounded between 0 

and 1 but will also maintain the essential properties of a fuzzy membership function.  

Proof: Follows directly from the preservation of fuzzy membership distribution theorem. 

Theorem 12: Frequency Response 

The frequency response of the adaptive 2D Gaussian filter 𝐺(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦, 𝑚𝑥, 𝑚𝑦 , 𝛼, 𝛽) can be derived 

from its Fourier Transform. 

Proof: The Fourier Transform of 𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦, 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽) is:  

ℱ{𝐺(𝑥, 𝑦, 𝜎𝑥 , 𝜎𝑦 , 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽)} = ∫ ∫ 𝐺(𝑥, 𝑦, 𝜎𝑥, 𝜎𝑦, 𝑚𝑥, 𝑚𝑦, 𝛼, 𝛽)𝑒−𝑗2𝜋(𝑢𝑥+𝑣𝑦)𝑑𝑥 𝑑𝑦
∞

−∞

∞

−∞

 

Where (u, v) are the frequency variables. 

The Fourier Transform of the Gaussian component 
1

2𝜋𝜎𝑥𝜎𝑦
𝑒𝑥𝑝 [−

(𝑥−𝑚𝑥)2

2𝜎𝑥
2 −

(𝑦−𝑚𝑦)
2

2𝜎𝑦
2 ] is  

ℱ {
1

2𝜋𝜎𝑥𝜎𝑦
𝑒𝑥𝑝 [−

(𝑥 − 𝑚𝑥)
2

2𝜎𝑥
2

−
(𝑦 − 𝑚𝑦)

2

2𝜎𝑦
2

]} = exp [−2𝜋2(𝜎𝑥
2𝑢2 + 𝜎𝑦

2𝑣2)]𝑒−𝑗2𝜋(𝑚𝑥𝑢+𝑚𝑣𝑣) 

The linear terms [1 − 𝛼. (
𝑥−𝑚𝑥

𝜎𝑥
2 ) −  𝛽. (

𝑦−𝑚𝑦

𝜎𝑦
2

)] introduce additional frequency components. The 

overall frequency response H (u, v) is a combination of these terms: 

𝐻(𝑢, 𝑣) = (1 − 𝑗2𝜋𝛼𝜎𝑥𝑢 − 𝑗2𝜋𝛽𝜎𝑦𝑣)exp [−2𝜋2(𝜎𝑥
2𝑢2 + 𝜎𝑦

2𝑣2)]𝑒−𝑗2𝜋(𝑚𝑥𝑢+𝑚𝑦𝑣) 

Hence the proof. 

Corollary: Frequency Response Properties 

The frequency response of the adaptive 2D Gaussian filter shows that it acts as a low-pass filter, 

attenuating high-frequency components and preserving low-frequency components of the input signal. 

Proof: Follows directly from the frequency response deviation, where the term  exp[−2𝜋2(𝜎𝑥
2𝑢2 +

𝜎𝑦
2𝑣2)] indicates attenuation of higher frequencies. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

196 
https://internationalpubls.com 

3. Numerical Example 

Let us consider the 5x5 matrix sample from a image pixel matrix. Three different sample pixel matrix 

is considered which is given below: 

𝐴 =

[
 
 
 
 
1 2 3 4 5
5 4 3 2 1
1
4
2

3
2
5

5
1
4

2
5
3

4
3
1]
 
 
 
 

;  𝐵 =

[
 
 
 
 
255 255 249 255 252
251 253 255 248 255
247
255
251

255
255
252

254
255
255

255
252
255

247
255
251]

 
 
 
 

; 

 

                          𝐶 =

[
 
 
 
 
11 5 1 2 1
23 17 7 2 3
34
33
25

28
24
12

15
15
9

4
5
3

6
7
6]
 
 
 
 

 ; 𝐷 =

[
 
 
 
 
102 168 199 209 195
158 195 202 190 172
197
208
190

209
206
181

197
189
172

174
166
159

158
157
154]

 
 
 
 

; 

 

The function graph is shown in following fig.1. From the graph its concluded that as the 𝜎 value 

increases the function graph changes that it smoothens the intensity value accordingly. When the 

standard deviation value is less the image intensity value is enhanced. Using MATLAB 2017b, the 

results were obtained and the standard deviation of the matrices A, B, C, D are 𝜎𝐴 = 1.4142, 𝜎𝐵 =

2.6880, 𝜎𝐶 = 2.6880, 𝜎𝐷 = 10.1111 respectively. 

Pseudo Code: 

1. Initialize the pixel matrix 

2. Calculate the parameters of matrices:  

             Compute mean, standard deviation, normalized Matrix etc. 

3. Apply the calculated parameters to the matrices 

4. Define 1D functions: 

             𝐹(𝜎, 𝛼, 𝑥), G(𝜎,𝑚, 𝛼, 𝑥),   

5. Define range of x 

6. Compute Values for F, G and plot the results. 

7. Define 2D functions: 

             𝐹(𝜎𝑥, 𝜎𝑦 , 𝛼, 𝛽, 𝑥, 𝑦), G(𝜎𝑥, 𝜎𝑦, 𝑚𝑦, 𝑚𝑦𝛼, 𝛽, 𝑥, 𝑦) 

8. Compute values for F and G of 2D functions. 

9. Plot the results 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

197 
https://internationalpubls.com 

  
(a) AGD -1D (b) AFGD -1D 

Fig.1. Functional graph of 1D filters where Blue –value of A, Green- value of B, Black –value of C, 

Red- value of D 

 
 

(a) AGD-2D (b)AFGD-2D 

Fig.2. Functional graph of 2D filters where Blue –value of A, Green- value of B, Black –value of C, 

Red- value of D 

4. Conclusion 

The integration of adaptive Gaussian derivative filtering into fuzzy logic systems demonstrates a 

significant advancement in maintaining the stability and accuracy of fuzzy membership functions. This 

approach leverages the precision of Gaussian derivatives to effectively capture subtle variations in 

data, while adaptive filtering techniques ensure that the filtering process dynamically adjusts to the 

specific characteristics of the input data. This combination results in a robust filtering mechanism 

capable of preserving the integrity of membership functions even in the presence of noise and complex 

data patterns. 

The theorem and corollaries stated above establish that the adaptive 1D and 2D Gaussian filters is 

BIBO stable, preserves the essential properties of fuzzy membership function and acts as a low pass 

filter in the frequency domain. the mathematical proof demonstrates the robustness and reliability of 

the filter in various applications, ensuring the integrity and stability of the processed outputs. 

The research underscores the critical importance of selecting appropriate derivative orders and 

designing adaptive mechanisms that are both robust and responsive. The careful balance between these 

elements is crucial to optimizing the performance of the filtering process. The findings indicate that 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

198 
https://internationalpubls.com 

adaptive Gaussian derivative filtering can greatly enhance the reliability and performance of fuzzy 

logic systems, making them more adept at handling real-world data that is often noisy and irregular. 

These advancements hold considerable promise for a wide range of applications, from image 

processing to pattern recognition and beyond. By ensuring the stability and accuracy of fuzzy 

membership functions, this approach can contribute to more reliable and precise computational 

models, ultimately enhancing the effectiveness of systems that rely on fuzzy logic for decision-making 

and analysis. The potential for improved robustness in fuzzy logic systems marks a significant step 

forward in the field, paving the way for future research and development. 

References 

[1] Akula Suneetha and E. Srinivasa Reddy: Robust Gaussian Noise Detection and Removal in Color Images Using 

Modified Fuzzy Set Filter, Journal of Intelligence System, Vol.30, pp. 240-257 (2021) 

[2] Alex A, Gorodetsky, Sertac Karaman, Youssef M. Marzouk, Low-rank Tensor Integration for Gaussian Filtering of 

Continuous Time Nonlinear Systems, Institute of Electrical and Electronics Engineers (IEEE), (2017). 

[3] Alex Keilmann, Michael Godehardt, Ali Moghiseh, Claudia Redenbach, Katja Schladitz : Improved Anisotropic 

Gaussian Filters, Computer Vision and Pattern Recognition, (2023) 

[4] Animikh Biswas, Michal Branicki, A Unified Framework for the Analysis of Accuracy and Stability of a Class of 

Approximate Gaussian Filters For The Navier-Stokes Equations, Feb (2024). 

[5] Chigansky P, Liptser R , Van Handel R, Intrinsic Methods in Filter Stability, The Oxford Handbook of Non-Linear 

Filtering, pp. 319-351, (2011). 

[6] Christoph D. Mathys, Ekaterina I.Lomakina, Jean Daunizeau, Sandra Iglesias, Kay H. Broderson, Karl J. Friston and 

Klaas E. Stephan, Uncertainty in Perception and the Hierarchical Gaussian Filter, Frontiers in Human Neuroscience, 

Vol. 8, Nov. (2014).  Doi:10.3389/fnhum.2014.00825 

[7] Harishvijey A, Benadict Raja J: Automated Technique for EEG Signal Processing to Detect Seizure with Optimized 

Variable Gaussian Filter and Fuzzy RBFELM Classifier, Biomedical Signal Processing and Control, Vol. 74, (2022). 

10.1016/j.bspc.2021.103450 

[8] Jin Won Kim, Anant A. Joshi, Prashant G. Mehta, Backward Map for Filter Stability Analysis, Optimization and 

Control,(2024) 

[9] Junfeng Zhang, Xiao He, Donghua Zhou: Generalised Proportional – Integral – Derivative Filter, IET Control Theory 

and Applications, Vol. 10, Issue 17, pp. 2339-2347, (2016). 

[10] Koichiro Watanabe, Yoshihiro Maeda, Norishige Fukushima, Stability of Recursive Gaussian Filtering for Piecewise 

Linear Bilateral Filtering, International Conference on Frontiers of Computer Vision, (2018). 

[11] Manuel Wuthrich, Sebastian Trimpe, Crisina Garcia Cifuentes, Daniel Kappler, Stefan Schaal, A New Perspective 

and Extension of Gaussian Filter, The International Journal of Robotics Research, Vol.35, Issue 14, pp. 1731-1749, 

(2016). 10.1177/0278364916684019 

[12] Qichun Zhang, Yuyang Zhou, Recent Advances in Non-Gaussian Stochastic Systems Control Theory and its 

Applications, International Journal of Network Dynamics and Intelligence, Vol. 1(1), pp 111-119, (2022). 

[13]  Toni Karvonen, Silvere Bonnabel, Eric Moulines, Simo Sarkka, On Stability of a Class od Filters Non-Linear 

Stochastic Systems, SIAM Journal of Control and Optimization, (2020). 

[14] Wernerson D. Parreira, Jose Carlos M. Bermudez, Cedric Richard, Jean – Yves Tourneret, Stability Behavior Analysis 

of the Gaussian Kernel-Least-Mean-Square Algorithm, IEEE Transactions on Signal Processing, Vol. 60, pp 2208-

2222, (2012). 

[15] Zhao, Zheng, Sarkka Simo, Non-Linear Gaussian Smoothing with Taylor Moment Expansion, IEEE Signal 

Processing Letters, vol. 29, pp 80-84, https://doi.org/10.1109/LSP.2021.3125831