We want to integrate colourfulness in an image quality evaluation framework. This quality framework is meant to evaluate the perceptual impact of a compression algorithm or an error prone communication channel on the quality of an image. The image might go through various enhancement or compression algorithms, resulting in a different-but not necessarily worse-image. In other words, we will measure quality but not fidelity to the original picture.While modern colour appearance models are able to predict the perception of colourfulness of simple patches on uniform backgrounds, there is no agreement on how to measure the overall colourfulness of a picture of a natural scene. We try to quantify the colourfulness in natural images to perceptually qualify the effect that processing or coding has on colour. We set up a psychophysical category scaling experiment, and ask people to rate images using 7 categories of colourfulness. We then fit a metric to the results, and obtain a correlation of over 90% with the experimental data. The metric is meant to be used real time on video streams. We ignored any issues related to hue in this paper.

We give a new proof of a version of Klein's theorem on the existence of absolutely continuous spectrum for the Anderson model on the Bethe Lattice at weak disorder. Model and Statement of Main ResultsIt is widely believed that the Anderson model [An] should exhibit absolutely continuous spectrum at weak disorder in dimensions three and higher. But it is only for the Bethe lattice B,

Abstract. We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant λ. We show that the ground-state energy is an analytic function of λ and that the corresponding ground state can also be chosen to be an analytic function of λ. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground-state energy can be calculated using regular analytic perturbation theory.

Abstract.For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of α 3/2 , α being the fine structure constant. A suitably chosen ground state vector depends analytically on α 3/2 and it is twice continuously differentiable with respect to the nuclear coordinates.