This chapter focuses on geometric operations in images, which modify the spatial arrangement of pixels rather than their intensity values. It introduces the mathematical foundations of spatial transformations—functions that map pixel coordinates from an original image to new positions through translation, scaling, rotation, and inclination. The chapter discusses how these transformations can be expressed using matrix formulations, emphasizing the use of homogeneous coordinates to simplify combined operations. It also presents the concept of affine transformations, which preserve linearity and parallelism while allowing complex manipulations such as skewing and resizing. Detailed derivations are provided for calculating transformation parameters and their inverses using point correspondences. Finally, practical Python implementations are included for each transformation type, demonstrating their application through code examples that manipulate images using libraries like OpenCV and NumPy. Overall, the chapter provides a comprehensive theoretical and practical foundation for performing geometric transformations in digital image processing and computer vision.

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Geometric Operations in Images

  • Erik Cuevas,
  • Alma Nayeli Rodriguez-Vazquez,
  • Beatriz A. Rivera-Aguilar,
  • Jesús A. López-Luquín,
  • Carlos Guzmán-Rosales

摘要

This chapter focuses on geometric operations in images, which modify the spatial arrangement of pixels rather than their intensity values. It introduces the mathematical foundations of spatial transformations—functions that map pixel coordinates from an original image to new positions through translation, scaling, rotation, and inclination. The chapter discusses how these transformations can be expressed using matrix formulations, emphasizing the use of homogeneous coordinates to simplify combined operations. It also presents the concept of affine transformations, which preserve linearity and parallelism while allowing complex manipulations such as skewing and resizing. Detailed derivations are provided for calculating transformation parameters and their inverses using point correspondences. Finally, practical Python implementations are included for each transformation type, demonstrating their application through code examples that manipulate images using libraries like OpenCV and NumPy. Overall, the chapter provides a comprehensive theoretical and practical foundation for performing geometric transformations in digital image processing and computer vision.