A Novel Approach for Digital Image Compression in Some Fixed Point Results on Complete G − metric Space Using Comparison Function

. In the present time world, digital images are crucial for various applications, that includes the medical industry, aircraft and satellite imaging, underwater imaging and so on. For this huge quantities of digital images are produced and used by these applications. For a variety of reasons, these images also need to be transmitted and stored. Therefore, a technique known as compression is applied to resolve this storage issue while transmitting these images. In this article, by extending some unique ﬁxed point theorem results for comparison function on a complete symmetric G-metric space are used and it is a new approach. Moreover, this paper focuses on a compression method using the new structure of extended G-contraction mapping as it assists in compressing the size of the image. Thus, grayscale images are compressed using extended G-contraction mapping. And thus, grayscale images can be represented as matrices in this structure (pixel values). Also, similar images of reduced size can be obtained using an appropriate matrix G-metric and extended G-contraction mapping. The size of the matrix can be substantially reduced without losing any quality by controlling the order of sub matrices. These images are easy to store and transmit, with little variation between the original and contracted image.


Introduction
Fixed point theory in metric space is one of the primary research areas in applied and pure mathematics.Its applications are useful in most other areas in various parts of science and engineering.
The Banach contraction mapping concept [1], is an important tool in analysis, and also one of the fixed point theory's most basic and important results to be considered.Throughout the years, mathematicians have generalized this concept in various approaches and in different spaces.On the other hand, in metric spaces with partial ordering, fixed point theory has drawn a lot of research.(Ran and Reurings, 2004) provided the first finding in this direction and demonstrated applications of their discoveries to matrix equations [21].There was various generalization of a metric, notably a 2-metric, a D-metric, a G-metric, a cone metric, and a complex-valued metric.
In [2], Gähler first proposed the idea of a 2-metric.Consider a 2-metric is not a continuous function of its variables compared to a standard metric.The results obtained in 2-metric spaces and metric spaces could not then easily relate to one another.The fixed point theorems on 2-metric spaces and metric spaces may be readily unconnected.Due to this, Dhage proposed the idea of a D-metric in [3].
However, Mustafa and Sims demonstrated in [4] found that most of the topological characteristics of the D-metric were incorrect.To overcome the drawbacks of a D-metric, Mustafa and Sims established the idea of a G-metric in [5].The authors analyzed the topological characteristics of this space and demonstrated how G-metric spaces can use the analogue of the Banach contraction mapping concept and numerous fixed point theorems on G-metric spaces have been established since then.
The concept of a cone metric was defined by Huang and Zhang in [7].Subsequently, many authors developed numerous fixed point theorems from metric spaces to cone metric spaces.The fixed point results in cone metric spaces can also be obtained by reducing cone metric spaces to their standard metric counterparts.And, it was also demonstrated by various authors.Azam, Fisher, and Khan [8] demonstrated the concept of a complex-valued metric and stated some fixed point theorems.
A digital image is made up of a number of pixels, each of which has a unique position and value.Mathematical representations of these images are achievable.These are also referred to as raster images or bitmap images.Due to its numerous uses in various industries, digital imaging is in high demand.The drawbacks of digital images are that they require more memory space and consequently take a longer time to transfer from one device to another.Therefore, the size of the image is crucial in these situations in order to get better outcomes.A contraction is a tool that reduces the length of a distance, and it is at the centre of a wide range of image processing tools.Using general topology and functional analysis, digital topology is linked to the 2D and 3D features of digital images in [18,26].
In [9], Rosenfeld found that the field of digital topology has influenced a wide range of uses, including pattern recognition and image processing.The concept of digital continuity for 2D and 3D digital images is being further developed in [33,40].In [19][20][21], Boxer studied a variety of continuous digital functions as well as the digital version of several topological concepts.In [11,12], Ege et al.
explored the Banach fixed point theorem as it is pertinent to digital images.Therefore, introduced different outcomes and characteristics on 2D digital homology groups, proved the Lefschetz fixed point theorem for digital images in [11], gave some examples of the fixed point property, and illustrated that sphere-like digital images have the fixed point property.By implementing the Banach contraction principle to digital metric spaces at digital intervals, simply closed k-curves, and simply closed 18surfaces, in [10,13] Sang-Eon Han illustrated in the digital metric space which is complete.The need for storage space is growing in combination with the rapid demand for large amounts of image files.
There are different kinds of compression methods available to help in reducing the size of these data files.Hence, in this paper, an image contraction approach is used to decrease the dimension of image files in extended G-contractive mapping applications.Image resizing (decrease) can be accomplished by reducing the total number of pixels.

Basic Concepts
Definition 2.1.[5] Let X be a non empty set, and let G : X × X × X → R + be a function satisfying the following conditions: (2) G(x, x, y ) > 0; for all x, y ∈ X with x = y (3) G(x, x, y ) ≤ G(x, y , z) for all x, y , z ∈ X with y = z (4) G(x, y , z) = G(x, z , y ) = G(y , z, x) = ... (symmetry in all three variables) (5) G(x, y , z) ≤ G(x, a, a) + G(a, y , z) for all x, y , z, a ∈ X (rectangle inequality) Then the function G is called a generalized metric or more specially, a G-metric on X , and (X , G) is called a G-metric space.Definition 2.2.[5] Let (X , G) be a G-metric space.
(i ) Each iterate ϕ n is also comparison function.

Main Results
Theorem 3.1.In a complete symmetric G-metric space (X , G), a self mapping T : X → X has a unique fixed point if there exists a comparison function with for all x, y , z ∈ X , and ϕ(G(T x, T y , T z)) ≤ αϕ(G(x, y , z)) + βϕ (max {G(x, T x, T x), G(x, y , z)}) where α, β, γ ∈ [0, 1) and 0 Proof.Let u 0 ∈ X be a arbitrary point and the sequence {u n } in X , define as follows Continuing in the similar fashion, we have Taking limit n, m → ∞, we have To prove: Since, (X , G) be G-complete, there exists p ∈ X such that, Suppose that T p = p and T q = q then by (3.1), we have ϕ(G(p, q, q)) = ϕ(G(T p, T q, T q)) ≤ αϕ(G(p, q, q)) + βϕ (max{G(p, T p, T p), G(p, q, q)}) Since, ϕ(t) ≤ t, for all t ≥ 0.
G(p, q, q) ≤ LG(p, q, q) which is contradiction, since L < 1 p = q (3.9) Thus, T has a unique fixed point in X .
Proof.Put x = z, we get the conclusion by using Theorem 3.1.
Theorem 3.2.In a complete symmetric G-metric space (X , G), a self mapping T : X → X has a unique fixed point and also {u n } converging to a point p ∈ X if there exists a comparison function and {u n } has a subsequence converging to a point p ∈ X with for all x, y , z ∈ X , where α, β, γ ∈ [0, 1) and 0 ≤ L = α + β + γ < 1.
Proof.Let u 0 ∈ X be a arbitrary point and the sequence {u n } in X , define as follows Since, ϕ(t) ≤ t, for all t ≥ 0. To prove: {u n } converging to p in X . Consider, Thus, {u n } converging to p in X .
Corollary 3.2.In a complete symmetric G-metric space (X , G), a self mapping T : X → X has a unique fixed point and also {u n } converging to a point p ∈ X if there exists a comparison function and {u n } has a subsequence converging to a point p ∈ X with for all x, y ∈ X , ϕ(G(T x, T y , T x)) ≤ αϕ(G(x, y , x)) + βϕ (max {G(x, T x, T x), G(x, y , x)}) where α, β, γ ∈ [0, 1) and 0 ≤ L = α + β + γ < 1.

Proof.
Put x = z, we get the conclusion by using Theorem 3.2.

Application of Proposed G-metric Space Based Image Contraction Algorithm
Any grayscale digital image can be figured off as a matrix, with each element indicating the pixel's grey value in the corresponding index.Rectangular arrays of square pixels are used to represent digital images.These pixels have 256 shades [0, 255] in gray scale, 8 but, with 0 being black and 255 being white, and 254 shades of grey in between.As described in the following algorithm, the methodology involves parallel local operations on each element (pixel value) and immediate states.
Step 1: Place the grayscale image I into the programme.
Step 2: From I, get matrix A of order p × q.
Step 3: Divide the space into fixed-size blocks N = n × n(n ≥ 2).
Step 4: Using extend G-contraction condition, find the B 1 s that corresponds to each A n s .
Step 5: Obtain matrix B, which has been contracted.
Step 6: Get the image I that was contracted.
For the implementation of the above algorithm, the authors used matlab.The intensity of each pixel in the image is used as a starting point, and each element of I is transformed into a matrix representing that pixel's intensity.Divide I into several fixed-block-size segments, below are the pixel values for the illustrated box in Fig. 1, as well as the contracted region that was considered.As an outcome, the pixel values will differ according to the size of the contracted part.For the depicted region,    Fig. 8 shows that, the 16 × 16 region blown up to emphasize the independent pixels, whereas Fig. 9 and Fig. 10 show the subdivision corresponding to 64 non-overlapping A 2 s , each of which is a 2 × 2 block, and 64 non-overlapping B 1 s , each of which is a 1 × 1 block.Below are the pixel values for the highlighted box in Fig. 8 and the contracted region when considering A n s for n = 2.For n > 2, as a result of the varying pixel values, the contracted region's size will vary.In the region illustrated, where, ϕ is comparison function with α, β, γ ∈ [0, 1) and 0 ≤ L = α + β + γ < 1.

Conclusion
The notion of extended G-contraction mapping was implemented in this paper.The metric is appropriately formed in a nonempty space, and its application to use in a digital image is demonstrated.
The original image size has been reduced without significant loss of image quality.As a result, the contracted image that occupied less storage space, and it is also easier to send.The framework is implemented with a variety of block sizes to see how far an image can be contracted.Since the designed function iterates repeatedly on the input image, the concluding contracted images are of high quality when the input image is exceptionally large or has little color variation.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.

G
(p, T p, T p) ≤ βG(p, T p, T p) which is contradiction, since β < 1 Hence, T p = p (3.14) Using 3.1 and rectangular inequality in def.2.1, we obtain p is the unique fixed point of T by Theorem 3.1.

Figure 1 . 5 . 1 .
Figure 1.The above figure to show steps of scheme

Table 1 .
Summary of Original Image 2 After Contracted Chest Image: (720 × 820).The reference of chest image has been reduced in size to its original size, as shown in Fig.3, while the obtained contracted images are shown in Figs.15, 16, 17 and 18.

Table 2 .
Summary of Original Image 3 After Contracted Flower Image: (600×471).The reference of flower image has been reduced in size to its original size, as shown in Fig. 4, while the obtained contracted images are shown in Figs. 19, 20, 21 and 22.

Table 3 .
Summary of Original Image 4 After Contracted High Resolution Images.The reference of high resolution images has been reduced in size to its original size, as shown in Figs. 5, 6 and 7.

Table 4 .
Summary of Original Image 5 After Contracted

Table 5 .
Summary of Original Image 6 After Contracted Stadium Image: (1920 × 1200).The reference of stadium image has been reduced in size to its original size, as shown in Fig. 7, while the obtained contracted images are shown in Figs.31, 32, 33 and 34.

Table 6 .
Summary of Original Image 7 After Contracted