An Introduction to Variational Calculus

This textbook introduces variational calculus and regularization methods for inverse problems, seamlessly blending classical concepts with contemporary computational applications, particularly in the field of image processing.

The classical perspective draws upon foundational topics explored by pioneers such as Euler and Lagrange, establishing a solid theoretical groundwork. In recent decades, the advent of disciplines such as computer vision has expanded the horizons of variational calculus, showcasing its effectiveness in addressing complex problems that necessitate computational solutions. Consequently, this book places a strong emphasis on the synergy between mathematical theory, practical applications, and algorithmic development.

To ensure the text is comprehensive and accessible, essential principles of functional analysis and Fourier analysis are incorporated, facilitating a deeper and more nuanced understanding of the applications presented. Covering both classic and more recent aspects of variational calculus, this book suggests that many topics of modern technology such as computer vision, robotics and especially digital image processing can be formulated in terms of variational problems.

State of the art problems of computer vision and data processing are presented in a variational approach Introduction to image processing problems from the point of view of variational calculus Includes applications to image processing, such as Denoising, Deblurring and Segmentation

Autorentext

Hebert Montegranario is an applied mathematician and computational scientist, currently serving as a professor at the Antioquia University, Medellín, Colombia. He holds a Ph.D. in Engineering from Universidad Nacional de Colombia, with a research focus on image processing, computer vision, and machine learning. Dr. Montegranario has extensive expertise in numerical analysis, scientific computing, and software design. He has authored several academic papers and a book on variational regularization of 3D data, showcasing his contributions to the field.

Inhalt

Introduction.- Mathematical Preliminaries.- Linear Operators and Functionals.- Minimization of Functionals.- Inverse Problems and Variational Regularization.- Variational Curve and Surface Interpolation.- Further Models and Applications.