Spectral one-homogeneous framework

This chapter introduces a main topic of this book, of viewing variational methods through a nonlinear spectral perspective. It is shown how all regularization methods—gradient flow, variational methods, and inverse scale space can be used to decompose the image in a new way that is similar in some sense to linear spectral or Fourier decompositions. We use the gradient descent as the canonical scale space and show how a nonlinear transform can be defined, based on its solution. This transform takes any nonlinear eigenfunction to appear in a singular time (scale) which is inverse proportional to its eigenvalue. Moreover, effective, contrast-preserving filtering can be applied by simple amplification, preservation, or attenuation of the different spectral components. We begin with a more informal presentation of the topic, where in the later part of this chapter more rigorous results are shown in the finite-dimensional case. A fundamental result is that in some settings we can show a precise decomposition of the input signal into eigenfunctions. In addition, the spectral components turn to be orthogonal to each other.