Gradient extraction operators for discrete interval-valued data

Digital images are generally created as discrete measurements of light, as performed by dedicated sensors. Consequently, each pixel contains a discrete approximation of the light inciding in a sensor element. The nature of this measurement implies certain uncertainty due to discretization matters. In this work we propose to model such uncertainty using intervals, further leading to the generation of so-called interval-valued images. Then, we study the partial differentiation of such images, putting a spotlight on antisymmetric convolution operators for such task. Finally, we illustrate the utility of the interval-valued images by studying the behaviour of an extended version of the well-known Canny edges detection method

Gradient extraction operators for discrete interval-valued data

Digital images are generally created as discrete measurements of light, as performed by dedicated sensors. Consequently, each pixel contains a discrete approximation of the light inciding in a sensor element. The nature of this measurement implies certain uncertainty due to discretization matters. In this work we propose to model such uncertainty using intervals, further leading to the generation of so-called interval-valued images. Then, we study the partial differentiation of such images, putting a spotlight on antisymmetric convolution operators for such task. Finally, we illustrate the utility of the interval-valued images by studying the behaviour of an extended version of the well-known Canny edges detection method

A bilateral schema for interval-valued image differentiation

Differentiation of interval-valued functions is an intricate problem, since it cannot be defined as a direct generalization of differentiation of scalar ones. Literature on interval arithmetic contains proposals and definitions for differentiation, but their semantic is unclear for the cases in which intervals represent the ambiguity due to hesitancy or lack of knowledge. In this work we analyze the needs, tools and goals for interval-valued differentiation, focusing on the case of interval-valued images. This leads to the formulation of a differentiation schema inspired by bilateral filters, which allows for the accommodation of most of the methods for scalar image differentiation, but also takes support from interval-valued arithmetic. This schema can produce area-, segment-and vector-valued gradients, according to the needs of the image processing task it is applied to. Our developments are put to the test in the context of edge detection

Estimation de l'enveloppe et de la fréquence locales par les opérateurs de Teager-Kaiser en interférométrie en lumière blanche.

In this work, a new method for surface extraction in white light scanning interferometry (WLSI) is introduced. The proposed extraction scheme is based on the Teager-Kaiser energy operator and its extended versions. This non-linear class of operators is helpful to extract the local instantaneous envelope and frequency of any narrow band AM-FM signal. Namely, the combination of the envelope and frequency information, allows effective surface extraction by an iterative re-estimation of the phase in association with a new correlation technique, based on a recent TK crossenergy operator. Through the experiments, it is shown that the proposed method produces substantially effective results in term of surface extraction compared to the peak fringe scanning technique, the five step phase shifting algorithm and the continuous wavelet transform based method. In addition, the results obtained show the robustness of the proposed method to noise and to the fluctuations of the carrier frequency

Koopman analysis of the long-term evolution in a turbulent convection cell

We analyse the long-time evolution of the three-dimensional flow in a closed cubic turbulent Rayleigh-B\'{e}nard convection cell via a Koopman eigenfunction analysis. A data-driven basis derived from diffusion kernels known in machine learning is employed here to represent a regularized generator of the unitary Koopman group in the sense of a Galerkin approximation. The resulting Koopman eigenfunctions can be grouped into subsets in accordance with the discrete symmetries in a cubic box. In particular, a projection of the velocity field onto the first group of eigenfunctions reveals the four stable large-scale circulation (LSC) states in the convection cell. We recapture the preferential circulation rolls in diagonal corners and the short-term switching through roll states parallel to the side faces which have also been seen in other simulations and experiments. The diagonal macroscopic flow states can last as long as a thousand convective free-fall time units. In addition, we find that specific pairs of Koopman eigenfunctions in the secondary subset obey enhanced oscillatory fluctuations for particular stable diagonal states of the LSC. The corresponding velocity field structures, such as corner vortices and swirls in the midplane, are also discussed via spatiotemporal reconstructions.Comment: 32 pages, 9 figures, article in press at Journal of Fluid Mechanic