Round 1

Reviewer 1 Report

The Reviewer Report on

Reduce image noise using fuzzy metric filter

The reviewer observations and conclusion:

In this paper, the authors introduced a fuzzy filter by using the fuzzy metric notion. define fuzzy color pixels by using the mean of neighborhoods. Due to the noise in the image, they used the bootstrap resampling method to reduce the effect of outliers.

This work falls in the category of the research work on applications of fuzzy metric and image processing.

Observations:

1.      I could not see the use of notion of t-norm.

2.      In section 3, introduction to fuzzy metric space, the authors are not given full definition of fuzzy metric. Also, I could not see the difference between fuzzy metric given by A. George, P. Veeramani and Ghasemi (definition 2). In this regards-see Contributions to Fixed Point Theory of Fuzzy Contractive Mappings, Advances in Metric Fixed Point Theory and Applications, Springer, 2021, 241-282

Considering this fact, I recommend the paper for compulsory minor revision.

Author Response

Point 1: I could not see the use of notion of t-norm.

 Response 1: When we express the concept of fuzzy metric in lines 101 to 105, we use the concept of t-norm. Also, in the definition of fuzzy metric for fuzzy sets (Definition 2), the concept of t-norm has been used again.

Point 2: In section 3, introduction to fuzzy metric space, the authors are not given full definition of fuzzy metric.

 Response 2: The following description was added to section 3.

“Various definitions of fuzzy metric space have been proposed by authors \cite{DENG198274, KALEVA1984215, ERCEG1979205}. George and Viramani in \cite{GEORGE1994395} modified the concept of fuzzy metric space introduced by Kramosil and Michalek \cite{kramosil1975fuzzy} and defined a Hausdorff topology on this fuzzy metric space. In this definition, which we will refer to in the following, $M(x,y,t)$ can be thought of as the degree of nearness between $x$ and $y$ with respect to $t$.”

Also, I could not see the difference between fuzzy metric given by A. George, P. Veeramani and Ghasemi (definition 2). In this regards-see Contributions to Fixed Point Theory of Fuzzy Contractive Mappings, Advances in Metric Fixed Point Theory and Applications, Springer, 2021, 241-282

The following paragraph was added to section 3. (lines 120 to 124)

“Now, if we have a fuzzy view of colors and consider the value of each pixel as a fuzzy number, the above method cannot be effective. In fact, the method presented by Morrilas et al. \cite{morillas2007new}  is suitable for when the value of each pixel is a crisp number, and if the value of each pixel can be defined as a fuzzy number, this method cannot be used anymore. In this case, the fuzzy metric for fuzzy sets should be used.”

In fact, the key point in Ghasemi's definition is the ability to use the fuzzy metric for fuzzy numbers, while George's definition can only be used for crisp numbers.

Author Response File: Author Response.docx

Reviewer 2 Report

Dear Authors,

Very important research on the topic of noise reduction in color images. The methods presentation is clear and mathematically rigorous. The BCFM filter, a generalization of the CFM filter by one of the authors (Morrilas, 2007) showed better performance considering the examples in the article. The reasons of this is clearly explained.
Only a detail in line 56: to define SLLN (strong law of large numbers).

Author Response

Point 1: Very important research on the topic of noise reduction in color images. The methods presentation is clear and mathematically rigorous. The BCFM filter, a generalization of the CFM filter by one of the authors (Morrilas, 2007) showed better performance considering the examples in the article. The reasons of this is clearly explained.
Only a detail in line 56: to define SLLN (strong law of large numbers).

Response 1:  In line 56, " strong law of large numbers " was added to the text.

Author Response File: Author Response.docx