This unique text/reference presents a fresh look at nonlinear processing
through nonlinear eigenvalue analysis, highlighting how one-homogeneous
convex functionals can induce nonlinear operators that can be analyzed
within an eigenvalue framework. The text opens with an introduction to
the mathematical background, together with a summary of classical
variational algorithms for vision. This is followed by a focus on the
foundations and applications of the new multi-scale representation based
on non-linear eigenproblems. The book then concludes with a discussion
of new numerical techniques for finding nonlinear eigenfunctions, and
promising research directions beyond the convex case.
Topics and features: introduces the classical Fourier transform and its
associated operator and energy, and asks how these concepts can be
generalized in the nonlinear case; reviews the basic mathematical
notion, briefly outlining the use of variational and flow-based methods
to solve image-processing and computer vision algorithms; describes the
properties of the total variation (TV) functional, and how the concept
of nonlinear eigenfunctions relate to convex functionals; provides a
spectral framework for one-homogeneous functionals, and applies this
framework for denoising, texture processing and image fusion; proposes
novel ways to solve the nonlinear eigenvalue problem using special flows
that converge to eigenfunctions; examines graph-based and nonlocal
methods, for which a TV eigenvalue analysis gives rise to strong
segmentation, clustering and classification algorithms; presents an
approach to generalizing the nonlinear spectral concept beyond the
convex case, based on pixel decay analysis; discusses relations to other
branches of image processing, such as wavelets and dictionary based
methods.
This original work offers fascinating new insights into established
signal processing techniques, integrating deep mathematical concepts
from a range of different fields, which will be of great interest to all
researchers involved with image processing and computer vision
applications, as well as computations for more general scientific
problems.
