This book presents a mathematically rigorous approach to high-precision image processing, where the accurate representation of pixels — even at fine detail — is crucial. It develops a novel methodology that treats discrete pixel data using continuous mathematics and numerical approximation, allowing image greyness values to be modeled as integrals of piecewise-continuous functions.The text introduces new discrete algorithms for 2D and 3D images with multiple grey levels, applicable under a wide range of linear and nonlinear geometric transformations, including affine, perspective, Coons, and PDE-based models. By integrating numerical analysis, geometry, probability, and statistics, the book provides a unified framework for image transformations with provable accuracy and high computational efficiency.Ideal for advanced study, the work bridges mathematics and computer science, offering powerful tools for image processing, computer vision, computer graphics, geometric computation, and pattern recognition.