# Normalized Weighting Schemes for Image Interpolation Algorithms
## 1 Introduction and Background  
Commonly used normalization methods are based on interval arithmetic and fuzzy arithmetic [1]. Usually, normalization means rescaling variables in the range between zero and one [2]. While this meaning holds in this work, it can vary from problem to problem in other works [3, 4]. Here, the news is the normalization of areas whose sum exceeds a unit square size, especially when those areas are based on the Pythagorean theorem equation. In mathematics, the Pythagorean theorem is a fundamental relation in Euclidean geometry among the three sides of a right triangle [5]. Here, it can be written as Equation (1) or Equation (2) relating the lengths of the sides \(a\), \(b\), and \(c\), of a right-angled triangle [5]. The Pythagorean equation has been widely used in many computer sciences and engineering techniques, including in [6, 7], but very few times for digital image interpolation purposes [8]. In mathematics, interpolation is an estimation method used to construct a new data value within the range of a set of known data [9]. In digital image processing, it is regarded as the process of generating estimates at each grid point which must preserve original pixels in the enlarged grid, because the image interpolation operation, by definition, must preserve the input pixel values of the smaller image. In real life, interpolation operation is routinely used for digital zooming [10]. Here, it is important to remember that digital zoom is one of the digital image processing techniques used to get a closer view of image objects or details of interest. Even if it is well known that digital zooming produces more visual artefacts than optical zooming in images, digital zooming is still widely used, for example, in modern digital cameras, imaging software (for processing and analysis applications in medicine, military, astronomy, etc. ), as well as in many other electronic devices because, unlike optical zooming, digital zooming does notrequire the mechanical device of lens elements such as the one used in optical zoom [10]. To date, artefact-free digital image zooming remains very challenging to achieve due to unrealistic estimates by traditional interpolation algorithms--in addition to the current digital format requirements. Therefore, more research is still needed in this direction. Here, it is important to remember that interpolation pervades or penetrates many image processing and analysis applications. It is also important to remember that, in this work, developing novel and efficient weighting schemes was the goal because they could affect the performance of linear image resizing algorithms or desired results or the ways to obtain them [11]. To the best of the author's knowledge, there are currently many approaches to interpolation problems, particularly techniques developed to efficiently reduce visual artefacts in interpolated images or contribute to other image processing tasks [12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43]. Among them, some use deep learning means to reduce visual artefacts in the output of interpolated images [44; 45], which are beyond the scope of image interpolation methods presented, mainly because they do not respect the image interpolation definition [23], unlike those presented in this paper. One of my recent works, presented in [46], examined the origin of image pixels and divided image interpolation approaches into two major categories of non-extra-pixel and extra-pixel interpolation. Unlike the extra-pixel approach, the non-extra-pixel approach only depends on original or source image pixels [46]. Another of my recent works, presented in [10], examined the effects of rounding functions in one of the fastest image interpolation algorithms--the nearest neighbor interpolation algorithm.