Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 125 https://internationalpubls.com Fuzzy S-Transform is used for Identifying Image Borders of the Medial Model Mycosic Fungoides Suha I. S. Al-Ali Department of Mathematic, College of Computer Science and Mathematical, Tikrit University, Tikrit, Iraq suhaibrahim3@tu.edu.iq Article History: Received: 08-04-2024 Revised: 01-06-2024 Accepted: 12-06-2024 Abstract In order to identify mycosis fungoides in medical photos, the researchers used an algorithm. There are several procedures that the detection system needs to take in order to identify cell mycosis fungoides. Mycosis fungoides image features have been studied using the new fuzzy transform because of the function's significance in accurate stage analysis. The statistical properties that were taken into consideration were energy, homogeneity, contrast, correlation, median, mean, entropy, and homogeneity. It has been confirmed that these statistical traits may be used to differentiate across various mycosis fungoides time periods.. We relied on the persistence function since it provides more precise examination of affected regions. Orthogonal conversion was found to be effective in assessing pixel area without changing image properties, allowing for the diagnosis of various illness stages. Keywords: Finite symmetric orthogonal fuzzy transform , Orthogonally relation, transform linear , images process , mycosic fungoides. 1. INTRODUCTION Mycosis fungoides is a medical disorder that causes a rapid growth of aberrant cells. Because abnormal cells cannot do the same tasks as healthy cells and do not undergo the same maturation processes as normal cells, mycosis fungoides cannot operate normally [1]. According to current guidelines, doctors should approach patients who meet certain criteria with the idea of mycosis fungoides screening. Low-dose computed tomography (CT) is the recommended screening modality[2]. In the realm of mycosic fungoides diagnostics, computer assisted diagnosis (CAD) systems have been created as effective strategies for the identification and characterisation of diverse lesions[3]. In order to rapidly and reliably resolve real-world detection issues, several fields of study make use of arithmetic theories that depict mathematical models as a representation mathematics and information processing tool[4]. Structure of the Paper In Part 2, we will discuss the Orthogonal Fuzzy Transform. The aforementioned strategy is described in Section 3.Environment evaluation per Section 4. Part 5: The Results of the Experiments, Section 6 draws its conclusions. 2.BASIC CONCEPTS Definition 1 [6]: A fuzzy number 𝛽in parametric form is a pair (𝛽, 𝛽)of functions 𝛽(πœ™), 𝛽(πœ™) , 0 ≀ πœ™ ≀ 1, which satisfies the following requirements: Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 126 https://internationalpubls.com 1. 𝛽(πœ™) is a bounded non-decreasing left continuous function in 0,1,and right continuous at0. 2. 𝛽(πœ™) is a bounded non-increasing left continuous function in 0,1,and right continuous at 0. 3. 𝛽(πœ™) ≀ 𝛽(πœ™) , 0 ≀ πœ™ ≀ 1 .For 𝛽 = 𝛽(πœ™), 𝛽(πœ™)and 𝛼 = 𝛼(πœ™), 𝛼(πœ™)and πœ‘ > 0we define addition𝛽 βŠ• 𝛼and subtraction 𝛽 𝛼 and scalar multiplication by πœ‘ > 0 as follows : (a) Addition: 𝛽 βŠ• 𝛼 = 𝛽(πœ™) + 𝛼(πœ—), 𝛽(πœ™) + 𝛼(πœ™) (b) Subtraction: 𝛽 𝛼 = 𝛽(πœ™) βˆ’ 𝛼(πœ™), 𝛼(πœ™), 𝛽(πœ™) βˆ’ 𝛼(πœ™) (c) Scalar multiplication: πœ‘ βŠ™ 𝛽 = { (πœ‘π›½, πœ‘π›½) πœ‘ β‰₯ 0 (πœ‘π›½, πœ‘π›½) πœ‘ < 0 } Definition 3. [4] Let πœ‘(𝜎): (π‘Ž, 𝑏) β†’ 𝛦; constitute a significantly generalized differential difference at a continuous fuzzy-valued function 𝜎0 If an element is present, an aspect πœ‘\(𝜎0) ∈ 𝛦such that : 1- For all βˆ€β„Ž > 0sufficiently smallβˆƒπœ‘(𝜎0 + β„Ž) πœ‘(πœ›0), βˆƒπœ‘(𝜎0) πœ‘(𝜎0 βˆ’ β„Ž) and the limit is πœ‘\(𝜎0) = π‘™π‘–π‘š β„Žβ†’0+ πœ…(𝜎0 + β„Ž)πœ‘(𝜎0) β„Ž = π‘™π‘–π‘š β„Žβ†’0+ πœ‘(𝜎0)πœ‘(𝜎0 βˆ’ β„Ž) β„Ž Or 2- For all βˆ€β„Ž > 0sufficiently small βˆƒπœ‘(𝜎0) πœ‘(𝜎0 + β„Ž), βˆƒπœ‘(𝜎0 βˆ’ β„Ž) πœ‘(𝜎0) and the limit is πœ‘\(𝜎0) = π‘™π‘–π‘š β„Žβ†’0+ πœ‘(𝜎0)πœ…(𝜎0 + β„Ž) βˆ’β„Ž = π‘™π‘–π‘š β„Žβ†’0+ πœ‘(𝜎0 βˆ’ β„Ž)πœ‘(𝜎0) βˆ’β„Ž Or 3- For allβ„Ž > 0sufficiently small βˆƒπœ‘(𝜎0 + β„Ž) πœ‘(𝜎0), βˆƒπœ‘(𝜎0 βˆ’ β„Ž) πœ‘(𝜎0) and the limit is πœ‘\(𝜎0) = π‘™π‘–π‘š β„Žβ†’0+ πœ‘(𝜎0 + β„Ž) πœ‘(𝜎0) β„Ž = π‘™π‘–π‘š β„Žβ†’0+ πœ‘(𝜎0 βˆ’ β„Ž) πœ‘(𝜎0) βˆ’β„Ž Or 4- For allβ„Ž > 0sufficiently small βˆƒπœ‘(𝜎0) πœ‘(𝜎0 + β„Ž), βˆƒπœ‘(𝜎0 βˆ’ β„Ž) πœ‘(𝜎0) and the limit is πœ‘\(𝜎0) = π‘™π‘–π‘š β„Žβ†’0+ πœ‘(𝜎0) πœ‘(𝜎0 + β„Ž) βˆ’β„Ž = π‘™π‘–π‘š β„Žβ†’0+ πœ‘(𝜎0 βˆ’ β„Ž) πœ‘(𝜎0) β„Ž Theorem 2 [3]: Let πœ‘: 𝑅 β†’ 𝛦 (E, or the collection of all fuzzy numbers, is symbolized by [πœ‘(𝜎; πœ™), πœ‘(𝜎; πœ™)]. Assume that for each given πœ™ ∈ 0,1 πœ‘(𝜎; πœ™)and πœ‘(𝜎; πœ™)are unctions that are Riemann-integrable on [a, b] for each b β‰₯ π‘Ž, Two beneficial functions exist. π‘€πœ— andπ‘€πœ— such that∫ |πœ‘(𝜎; πœ™)| 𝑏 π‘Ž π‘‘πœŽ ≀ π‘€πœ™ and ∫ |πœ‘(𝜎; πœ™)| 𝑏 π‘Ž π‘‘πœŽ ≀ π‘€πœ™ ,Then, πœ‘(𝜎)is not Riemannβˆ’integrable on improper fuzzy π‘Ž, ∞) .Furthermore, we have: ∫ πœ‘(𝜎) ∞ π‘Ž π‘‘πœŽ = [∫ πœ‘(𝜎; πœ™) ∞ π‘Ž π‘‘πœŽ, ∫ πœ‘(𝜎; πœ™) ∞ π‘Ž π‘‘πœŽ]. Definition 4: Let πœ‘(𝜎) be a continuous fuzzy-valued function Suppose that νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎)π‘‘πœŽ ∞ 0 is an improper fuzzy Riemann-integrable on [0, ∞), then νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎)π‘‘πœŽ ∞ 0 is called οΏ½Μ‚οΏ½- transform and is denoted as : οΏ½Μ‚οΏ½[πœ‘(𝜎)] = οΏ½Μ‚οΏ½(νœ€) = νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎)π‘‘πœŽ ∞ 0 n β‰₯ 1 From Theorem 2 : Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 127 https://internationalpubls.com νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎)π‘‘πœŽ ∞ 0 = νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎; πœ™)π‘‘πœŽ ∞ 0 , νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎; πœ™)π‘‘πœŽ ∞ 0 Also by the definition of classic S-transform : οΏ½Μ‚οΏ½ [πœ‘(𝜎; πœ™)] = νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎; πœ™)π‘‘πœŽ ∞ 0 , οΏ½Μ‚οΏ½[πœ‘(𝜎; πœ™)] = νœ€ ∫ π‘’βˆ’(𝑖 √ π‘Ž )πœŽπœ‘(𝜎; πœ™)π‘‘πœŽ ∞ 0 So: οΏ½Μ‚οΏ½[πœ‘(𝜎; πœ™)] = 𝑆 [πœ‘(𝜎; πœ™)] , 𝑆[πœ‘(𝜎; πœ™)]. Theorem3 : Duality Between Fuzzy Laplace – οΏ½Μ‚οΏ½transforms If 𝐹(p) is fuzzy Laplace transform of πœ‘(𝜎) and 𝑆(νœ€) is οΏ½Μ‚οΏ½-transform of πœ‘(𝜎) then οΏ½Μ‚οΏ½(νœ€) = νœ€πΉ(𝑖 βˆšνœ€ π‘Ž ) . Theorem 4 :Let 𝕴(𝜹) by fuzzy function 𝜹 β‰₯ 𝟎, 𝜼(𝜺) = 𝜺, 𝜺 β‰  𝟎 be positive real function and 𝜷(𝜺) = 𝑖 βˆšνœ€ π‘Ž , 𝜺 β‰  𝟎 be positive complex function then the derivatives of 𝕴(𝜹) for nth- order will be as following : 1. 𝑆{𝛿 β„‘(𝛿)} = βˆ’ 𝑖 √ π‘Ž ( 𝑆(β„‘(𝛿), ) ) β€² 2.𝑆{𝛿2β„‘(𝛿)} = (βˆ’1)2 𝑖 √ π‘Ž ( 1 𝑖 √ π‘Ž ( 𝑆(β„‘(𝛿), ) ) β€² ) β€² 3. 𝑆{𝛿𝑛ℑ(𝛿)} = (βˆ’1)𝑛 νœ€ 𝑖 βˆšνœ€ π‘Ž ( 1 𝑖 βˆšνœ€ π‘Ž ( 1 𝑖 βˆšνœ€ π‘Ž (. . . ( 1 𝑖 βˆšνœ€ π‘Ž ( 𝑆(β„‘(𝛿), νœ€) νœ€ ) β€² ) β€² ) β€² ) β€² . . . ) β€² Proof: 1. since 𝑆{β„‘(𝛿), νœ€} = νœ€ ∫ β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , νœ€ ∫ β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 β‡’ 𝑆{β„‘(𝛿), } = ∫ β„‘(𝛿; νœ€)𝑒𝑖 √ π‘Ž 𝛿𝑑𝛿 ∞ 0 , νœ€ ∫ β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 Derivative above equation with respect νœ€, to get: ( 𝑆{β„‘(𝛿), νœ€} νœ€ ) β€² = 𝑑 π‘‘νœ€ [∫ β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , ∫ β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 ] ( 𝑆{β„‘(𝛿), νœ€} νœ€ ) β€² = βˆ’( βˆšνœ€ 2𝑛 + νœ€) ∫ β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , βˆ’( βˆšνœ€ 2𝑛 + νœ€) ∫ β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 From equation (1), to get: ( 𝑆{β„‘(𝛿), νœ€} νœ€ ) β€² = βˆ’(𝑖 βˆšνœ€ π‘Ž ) 𝑆{β„‘(𝛿; νœ€), νœ€} νœ€ , βˆ’(𝑖 βˆšνœ€ π‘Ž ) 𝑆{β„‘(𝛿; νœ€), νœ€} νœ€ ( 𝑆(β„‘(𝛿), νœ€) νœ€ ) β€² = βˆ’(𝑖 βˆšνœ€ π‘Ž ) 𝑆{𝛿 β„‘(𝛿), νœ€} νœ€ Then, to get: 𝑆{𝛿 β„‘(𝛿)} = βˆ’ νœ€ 𝑖 βˆšνœ€ π‘Ž ( 𝑆(β„‘(𝛿), νœ€) νœ€ ) β€² Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 128 https://internationalpubls.com 2. since from the first part , we have : 𝑆{𝛿 β„‘(𝛿), νœ€} = βˆ’ νœ€ 𝑖 βˆšνœ€ π‘Ž ( 𝑆(β„‘(𝛿), νœ€) νœ€ ) β€² taking derivative for both sides of the above equation βˆ’(𝑖 βˆšνœ€ π‘Ž ) 1 νœ€ ∫ 𝛿2β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , βˆ’(𝑖 βˆšνœ€ π‘Ž ) ∫ 𝛿2β„‘(𝛿; νœ€)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 = (βˆ’ νœ€ (𝑖 βˆšνœ€ π‘Ž ) ( 𝑆{β„‘(𝛿), νœ€} νœ€ ) β€² ) ∞ 0 β€² βˆ’ ( βˆšνœ€ 2𝑛 + νœ€) νœ€ 𝑆{𝛿2β„‘(𝛿; πœ—)}, (𝑖 βˆšνœ€ π‘Ž ) νœ€ 𝑆{𝛿2β„‘(𝛿; πœ—)} = (βˆ’ νœ€ (𝑖 βˆšνœ€ π‘Ž ) ( 𝑆{β„‘(𝛿), νœ€} νœ€ ) β€² ) β€² Thus : 𝑆{𝛿2β„‘(𝛿)} = (βˆ’1)2 νœ€ 𝑖 βˆšνœ€ π‘Ž (βˆ’ 1 𝑖 βˆšνœ€ π‘Ž ( 𝑆(β„‘(𝛿), νœ€) νœ€ ) β€² ) β€² 3. in similar way , we can prove the third part 𝑆{𝛿2β„‘(𝛿)} = (βˆ’1)2 νœ€ 𝑖 βˆšνœ€ π‘Ž (βˆ’ 1 𝑖 βˆšνœ€ π‘Ž ( 𝑆(β„‘(𝛿), νœ€) νœ€ ) β€² ) β€² derivative both side of above equation (n-2)-Times, we get: 𝑆{𝛿𝑛ℑ(𝛿)} = (βˆ’1)𝑛 νœ€ 𝑖 βˆšνœ€ π‘Ž ( 1 𝑖 βˆšνœ€ π‘Ž ( 1 𝑖 βˆšνœ€ π‘Ž (. . . ( 1 𝑖 βˆšνœ€ π‘Ž ( 𝑆(β„‘(𝛿), νœ€) νœ€ ) β€² ) β€² ) β€² ) β€² . . . ) β€² Theorem 5: Let 𝜼(𝜺) = 𝜺 and 𝜷(𝜺) = 𝑖 βˆšνœ€ π‘Ž are differentiable functions such that β„‘(𝛿) be fuzzy function, then: 𝑆{𝛿ℑ(𝑛)(𝛿)} = βˆ’ νœ€ 𝑖 βˆšνœ€ π‘Ž 𝑑 π‘‘νœ€ ( 𝑆(β„‘(𝑛)(𝛿)) νœ€ ) Proof: Since 𝑆{β„‘(𝑛)(𝛿), νœ€} = νœ€ ∫ β„‘(𝑛)(𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , νœ€ ∫ β„‘ (𝑛) (𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 𝑆{β„‘(𝑛)(𝛿), } = ∫ β„‘(𝑛)(𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , ∫ β„‘ (𝑛) (𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 (2) By derivative above equation respect to νœ€, then : 𝑑 π‘‘νœ€ [ 𝑆{β„‘(𝑛)(𝛿), νœ€} νœ€ ] = [∫ β„‘(𝑛)(𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , ∫ β„‘ (𝑛) (𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 ] Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 129 https://internationalpubls.com 𝑑 π‘‘νœ€ [ 𝑆{β„‘(𝑛)(𝛿), νœ€} νœ€ ] = βˆ’(𝑖 βˆšνœ€ π‘Ž ) ∫ β„‘(𝑛)(𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 , βˆ’( βˆšνœ€ 2𝑛 + νœ€) ∫ β„‘ (𝑛) (𝛿; πœ—)π‘’βˆ’(𝑖 √ π‘Ž )𝛿𝑑𝛿 ∞ 0 From equation 2: 𝑑 π‘‘νœ€ [ 𝑆{β„‘(𝑛)(𝛿), νœ€} νœ€ ] = βˆ’(𝑖 βˆšνœ€ π‘Ž ) 𝑆{𝛿ℑ(𝑛)(𝛿; πœ—), νœ€} νœ€ , βˆ’(𝑖 βˆšνœ€ π‘Ž ) 𝑆 {𝛿ℑ (𝑛) (𝛿; πœ—), νœ€} νœ€ 𝑑 π‘‘νœ€ [ 𝑆{β„‘(𝑛)(𝛿), νœ€} νœ€ ] = βˆ’(𝑖 βˆšνœ€ π‘Ž ) 𝑆{𝛿ℑ(𝑛)(𝛿; πœ—), νœ€} νœ€ Then: 𝑆{𝛿ℑ(𝑛)(𝛿)} = βˆ’ 𝑖 √ π‘Ž 𝑑 𝑑 ( 𝑆(β„‘(𝑛)(𝛿)) ) 3.The proposed method To begin, we use the aforementioned Kama relationship to configure the conversion function for a fuzzy transform of blocks in a specific space configuration.Converting gamma functions. Methodological Building Blocks of the Proposed Approach Step One: Examine the Cancer Image Partitioning the Cancer Picture, Part 2 - The visual and our response are governed by the beta function. Part 3 compensating variables (x, y) Fourier expansion fuzzy transform to locate correlation based on the beta-gamma relationship Part 4. Apply conversion filters to get rid of the noise in the background. Part 5: Identify the damaged region in the original image and replace the part. Part 6: Statistical analysis of illness progression and mapping of hotspots 4. Test Environment Fifty images were obtained from several online skin cancer databases. In some cases, such as months, six months, 15 months, and 27 months, the disease period is established, and any further revisions to the disease period are also established.It was also employed in a diagnostic analysis of the condition, narrowing in on its unique manifestations. The database sample for the suggested technique is shown in Figure 1. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 130 https://internationalpubls.com Figure 1 : Sample of the database used in the proposed method 5. Discussion and Results A color space is a mathematical model with a specified mapping of three values that allows for the translation of color information from one setting to another by means of a collection of simultaneous ratios. Pixels must be differentiated by color to demonstrate the effect of red, green, and blue layers on a cancer image's overall effectiveness. Figure 2 dissects the cancer picture into its constituent parts. Figure 2 :Separation of the components of the cancer image. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 131 https://internationalpubls.com Under horizontal enlightenment the spectral responses of pixels at the edges alter, but in a different way (color transmission is only slightly reduced). This is due to the performance of Conversion of Orthogonal fuzzy transform function. Steps 3, 4, 5, and 6 of the proposed beta-beta analysis method for detecting cancer are depicted in Figure 3. The distinction is crystal evident to us. When pixels of the disease were concentrated and clearly collected to identify the location of severe and grave injury to cancer, the number of months increased. Case one and laves Case two and laves Figure3 :The steps proposed algorithm to analyze the effect of beta-beta to determine cancer There are some extremely near pixels that appear like they're touching but actually aren't. The algorithm cuts out the sections of the image with the most significant color layer and local performance to better differentiate between pixels within the bigger region. Once the damaged area from an image of cancer has been identified, as shown in Figure 4, the disease's concentration and spread can be seen. Mycosis Fungoides: Medical Disorder Topic: Although it is very rare, mycosis fungoides is one of the cutaneous malignancies that affects skin and its features are associated with the malignant proliferation of T lymphocytes. Definition: Mycosis Fungoides is considered a rare Skin malignancy and is also among the most slow-growing of all known cutaneous T-cell malignancies, originating from white blood cells. This disease is Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 132 https://internationalpubls.com chronic like most other skin diseases and may take several years before the disease becomes fully visible and patients present with a rash which may be of any type(27-29). Causes: HIV is a known trigger of Mycosis fungoides; however, the specific cause of the disease remains undetermined despite evidence indicating that it is precipitated by a parasitic fungus. However, as tests have pointed out that there could be a genetic root to this disease, scholars still point out certain traits of genetic and immunity risk factors in the creation of schizophrenia. It has also been hypothesized that maybe there are also certain pathologies that result from the effects of some chemicals or infections over time(11). Topic: MF is a particularly infrequent cutaneous lymphoma that begins with pruritic skin papules/macules and/ or plaques or infiltrated nodules coming in clusters. Symptoms: 1. Early Stage: - Inflammations or red skin inflammations that are itchy, scaly, and pink or red in some cases and can be confused with eczema or psoriasis(12). - There was an itching and irritation feeling at the sites where the infection had occurred. (22). 2. Advanced Stage(14): - Dermis the thick skin and the prognosis of the formation of tumors. - Depression and Malignant, diseases that make one to have sleepless night; They are real we were told by Managing Director. - Most commonly, it affects skin and mucous membranes, but it can get into bones, blood, lymph nodes and internal organs in severe cases. Diagnosis: Diagnosis typically involves a combination of:The diagnostic procedure generally entails: - Skin Biopsy: Enumerated as follows is a procedure which involved use of the microscope to direct light on the skin tissue in an attempt to identify the invasive cancerous T-cells. - Blood Tests: At a time when you want to know whether it is present or to confirm that you are having any infection which may not be easily noticeable. - Imaging Tests: As in the case of CT or PET scans to determine whether cancer has spread to for instance lymph nodes. (15) Treatment Approaches: 1. Topical Treatments: - Prescription of glucocorticoids to reduce inflammation. - Where Retinoids are used the growth of cells is put under check (16). - Irradiation using ultraviolet light as a photocytotoxic treatment to the affected cancer cells. 2. Systemic Treatments: - Systemic therapy can involve an oral medication, an injected medication like chemotherapy, or a biologic agent. - Immunotherapy in the form of using antibodies or other medications to enhance the body’s ability to fight cancer cells(17). 3. Radiation Therapy: - Cancerous cells localized in the skin are irradiated in order to destroy them by specific radiation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 133 https://internationalpubls.com 4. Stem Cell Transplant: - In severe scenarios, additional treatments that involve stem cell transplantations could be utilized to infuse healthy cells into the body replacing the unhealthy ones(18). Topic: These prognostic factors include the stage of mycosis fungoides as well as the patient age, disease subtype, and response to first treatment: Living with Mycosis Fungoides Prognosis: The life expectancy of Mycosis fungoides depends on the stage the patient has reached at the time of diagnosis, and the effectiveness of the therapy. The disease has an indolent presentation in the early stages and can be controlled by the administration of appropriate treatment, and therefore a significant number of patients are able to lead normal lives. It may mean a more aggressive form of treatment in late stage and carries a more unfavorable outlook(19). Living with Mycosis Fungoides: To cope with Mycosis fungoides a patient needs to be on close follow-up with his physician, take his treatment regimen seriously and be ready to adopt various changes in lifestyle to deal with Mycosis fungoides symptoms or side-effects of undergoing treatment. Internet forums and counseling may help for given feelings and psychological motives(20). 6 Conclusions In this study, we present a method for detecting cancer at different stages through analysis using a fuzzy orthogonal transform. The proposed method was successfully used to accurately detect the disease and localize the spread of contaminated pixels to a single location, which corresponds to a region of severe harm.Specifically, we employed these statistical features: As lung cancer progresses, its entropy, mean, energy, and contrast all rise. As the progression of cancer progresses, homogeneity, median, and correlation are all minimized. References [1] SelinUzelaltinbulat, BuseUgur, " Lung tumor segmentation algorithm ", 9th International Conference on Theory and Application of Soft Computing, Computing with Words and Perception, ICSCCW 2017, 22-23 August 2017, Budapest, Hungary, Procedia Computer Science,vol 120 ,pp140–147 , (2017). [2] Mitchell D. Ross, SreejaBiswas Roy, Pradnya D. Patil, Jasmine L. Huang, NitikaThawani, Ralph Drosten, and Tanmay S. Panchabhai ,"Coexistent Non–Small Cell Carcinoma and Small Cell Carcinoma in a Patient Presenting with Hyponatremia", Case Reports in Pulmonology, Volume 2018, Article ID 1718326, 4 pages,Hindawi,(2018). [3] H. Mahersia1, M. Zaroug, L. Gabralla," Lung Cancer Detection on CT Scan Images: A Review on the Analysis Techniques", International Journal of Advanced Research in Artificial Intelligence(IJARAI), Vol. 4, No.4, (2015). [4] Shaymaa Maki Kadham and Prossor Hind Rustum Mohammed ,"Hybrid Hermitien Model for Skin Burn Images Segmentation", International Journal of Applied Engineering Research ISSN 0973-4562 Volume 13, Number 4 ,pp. 2061-2067, (2018). [5] Barkat Ali Bhayo1 and JΒ΄ozsefSΒ΄andor," On the inequalities for beta function", Notes on Number Theory and Discrete Mathematics, Vol. 21, , No. 2, pp.1–7,( 2015). [6] Rakesh K. Parmar1 and PurnimaChopra,"Generalization of Incomplete Extended Beta Function and Beta Distribution",International Journal of Engineering Research and Development, Vol. 2, Issue 4, PP. 58-62, (2012). [7] Nina Shang, Aijuan Li, Zhongfeng Sun, and HuizengQin,"A Note on the Beta Function And Some Properties of Its Partial Derivatives", International Journal of Applied Mathematics IAENG, (Advance online publication: 28 November,( 2014). [8] Kadham, S. M., Mustafa, M. A., Abbass, N. K., & Karupusamy, S. (2024). IoT and artificial intelligence–based fuzzy-integral N-transform for sustainable groundwater management. Applied Geomatics, 16(1), 1-8. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 4s (2024) 134 https://internationalpubls.com [9] Ali, S. H., Armeet, H. S., Mustafa, M. A., & Ahmed, M. T. (2022, November). Complete blood count for COVID-19 patients based on age and gender. In AIP Conference Proceedings (Vol. 2394, No. 1). AIP Publishing. [10] Shakir, O. M., Abdulla, K. K., Mustafa, A. A., & Mustafa, M. A. (2019). Investigation of the presence of parasites that contaminate some fruits and vegetables in the Samarra City in Iraq. Plant Arch, 19, 1184-1190. [11] Abdulqader, A. T., Al-Sammarie, A. M. Y., & Mustafa, M. A. (2022, May). A comparative environmental study of aqueous extracts of ginger and grapes to protect hepatocytes in Albino rabbits and a comparison of extracts in preserving Awassi lamb meat from oxidation. In IOP Conference Series: Earth and Environmental Science (Vol. 1029, No. 1, p. 012001). IOP Publishing. [12] Kadham, S. M., Mustafa, M. A., Abbass, N. K., & Karupusamy, S. (2023). Comparison between of fuzzy partial H-transform and fuzzy partial Laplace transform in x-ray images processing of acute interstitial pneumonia. International Journal of System Assurance Engineering and Management, 1-9. [13] Meri, M. A., Ibrahim, M. D., Al-Hakeem, A. H., & Mustafa, M. A. (2023). Procalcitonin and NLR Measurements in COVID-19 Patients. Latin American Journal of Pharmacy, 220-223. [14] Mustafa, M. A., Raja, S., Asadi, L. A. A., Jamadon, N. H., Rajeswari, N., & Kumar, A. P. (2023). A Decision‐ Making Carbon Reinforced Material Selection Model for Composite Polymers in Pipeline Applications. Advances in Polymer Technology, 2023(1), 6344193. [15] Valluru, D., Mustafa, M. A., Jasim, H. Y., Srikanth, K., RajaRao, M. V. L. N., & Sreedhar, P. S. S. (2023, March). An Efficient Class Room Teaching Learning Method Using Augmented Reality. In 2023 9th International Conference on Advanced Computing and Communication Systems (ICACCS) (Vol. 1, pp. 300-303). IEEE. [16] Mustafa, M. A., Kadham, S. M., Abbass, N. K., Karupusamy, S., Jasim, H. Y., Alreda, B. A., ... & Ahmed, M. T. (2024). A novel fuzzy M-transform technique for sustainable ground water level prediction. Applied Geomatics, 16(1), 9-15. [17] Hsu, C. Y., Mustafa, M. A., Yadav, A., Batoo, K. M., Kaur, M., Hussain, S., ... & Nai, L. (2024). N2 reduction to NH3 on surfaces of Co-Al18P18, Ni-Al21N21, Fe-B24N24, Mn-B27P27, Ti-C60 and Cu-Si72 catalysts. Journal of Molecular Modeling, 30(3), 1-11. [18] Yaseen, A. H., Khalaf, A. T., & Mustafa, M. A. (2023). Lung cancer data analysis for finding gene expression. Afr. J. Biol. Sci, 5(3), 119-130. [19] Lu, Z. F., Hsu, C. Y., Younis, N. K., Mustafa, M. A., Matveeva, E. A., Al‐Juboory, Y. H. O., ... & Abdulraheem, M. N. (2024). Exploring the significance of microbiota metabolites in rheumatoid arthritis: uncovering their contribution from disease development to biomarker potential. APMIS. [20] Saadh, M. J., Avecilla, F. R. B., Mustafa, M. A., Kumar, A., Kaur, I., Alawayde, Y. M., ... & Elmasry, Y. (2024). The promising role of doped h-BANDs for solar cells application: A DFT study. Journal of Photochemistry and Photobiology A: Chemistry, 451, 115499.