Quantifying and containing the curse of high resolution coronal imaging

Future missions such as Solar Orbiter (SO), InterHelioprobe, or Solar Probe aim at approaching the Sun closer than ever before, with on board some high resolution imagers (HRI) having a subsecond cadence and a pixel area of about $(80km)^2$ at the Sun during perihelion. In order to guarantee their scientific success, it is necessary to evaluate if the photon counts available at these resolution and cadence will provide a sufficient signal-to-noise ratio (SNR). We perform a first step in this direction by analyzing and characterizing the spatial intermittency of Quiet Sun images thanks to a multifractal analysis. We identify the parameters that specify the scale-invariance behavior. This identification allows next to select a family of multifractal processes, namely the Compound Poisson Cascades, that can synthesize artificial images having some of the scale-invariance properties observed on the recorded images. The prevalence of self-similarity in Quiet Sun coronal images makes it relevant to study the ratio between the SNR present at SoHO/EIT images and in coarsened images. SoHO/EIT images thus play the role of 'high resolution' images, whereas the 'low-resolution' coarsened images are rebinned so as to simulate a smaller angular resolution and/or a larger distance to the Sun. For a fixed difference in angular resolution and in Spacecraft-Sun distance, we determine the proportion of pixels having a SNR preserved at high resolution given a particular increase in effective area. If scale-invariance continues to prevail at smaller scales, the conclusion reached with SoHO/EIT images can be transposed to the situation where the resolution is increased from SoHO/EIT to SO/HRI resolution at perihelion.Comment: 25 pages, 1 table, 7 figure

Multifractal stationary random measures and multifractal random walks with log-infinitely divisible scaling laws

We define a large class of continuous time multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal Multifractal Random Walk (MRW) [Bacry-Delour-Muzy] and the log-Poisson "product of cynlindrical pulses" [Barral-Mandelbrot]. Our construction is based on some ``continuous stochastic multiplication'' from coarse to fine scales that can be seen as a continuous interpolation of discrete multiplicative cascades. We prove the stochastic convergence of the defined processes and study their main statistical properties. The question of genericity (universality) of limit multifractal processes is addressed within this new framework. We finally provide some methods for numerical simulations and discuss some specific examples.Comment: 24 pages, 4 figure

Virtual Super Resolution of Scale Invariant Textured Images Using Multifractal Stochastic Processes

International audienceWe present a new method of magnification for textured images featuring scale invariance properties. This work is originally motivated by an application to astronomical images. One goal is to propose a method to quantitatively predict statistical and visual properties of images taken by a forthcoming higher resolution telescope from older images at lower resolution. This is done by performing a virtual super resolution using a family of scale invariant stochastic processes, namely compound Poisson cascades, and fractional integration. The procedure preserves the visual aspect as well as the statistical properties of the initial image. An augmentation of information is performed by locally adding random small scale details below the initial pixel size. This extrapolation procedure yields a potentially infinite number of magnified versions of an image. It allows for large magnification factors (virtually infinite) and is physically conservative: zooming out to the initial resolution yields the initial image back. The (virtually) super resolved images can be used to predict the quality of future observations as well as to develop and test compression or denoising techniques

Virtual resolution enhancement of scale invariant textured images using stochastic processes

4 pagesInternational audienceWe present a new method of magnification for textured im- ages featuring scale invariance properties. The procedure pre- serves the visual aspect as well as the statistical properties of the initial image. An augmentation of information is per- formed by locally adding small scale details below the ini- tial pixel size. This is made possible thanks to a family of scale invariant stochastic processes, namely compound Pois- son cascades. This extrapolating procedure yields a poten- tially infinite number of magnified versions of an image. It allows for large magnification factors (virtually infinite) and is physically conservative: zooming out to the initial reso- lution yields the initial image back. This work is motivated by an application to images of the quiet Sun to quantitatively predict statistical and visual properties of images taken by a forthcoming high resolution telescope

Wavelet and Multiscale Analysis of Network Traffic

The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behaviour in tele-traffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present many challenges, especially in non-stationary environments. In this paper we describe the state of the art in this area, focusing on the capabilities of the wavelet transform as a key tool for unravelling the mysteries of traffic statistics and dynamics

Synthèse de champs scalaires multifractals : application à la synthèse de texture

Nous étendons la définition des cascades infiniment divisibles de 1 à N 2 dimensions et résumons leurs principales propriétés (distributions, lois d'échelle...). Nous obtenons une méthode originale de synthèse de champs scalaires stochastiques multifractals dans un espace à N dimensions. L'implémentation de cette méthode s'avère très simple. En dimension N = 2, l'application à la synthèse de textures illustre la richesse des propriétés de ces processus. Nous obtenons une classe de textures possédant de nombreux degrés de liberté et permettant la synthèse directement sur une surface courbe, par exemple sur une sphère. D'autres applications sont envisagées en imagerie, en turbulence, en astrophysique..

Lp{L^p}-variations for multifractal fractional random walks

A multifractal random walk (MRW) is defined by a Brownian motion subordinated by a class of continuous multifractal random measures $M[0,t], 0\le t\le1$. In this paper we obtain an extension of this process, referred to as multifractal fractional random walk (MFRW), by considering the limit in distribution of a sequence of conditionally Gaussian processes. These conditional processes are defined as integrals with respect to fractional Brownian motion and convergence is seen to hold under certain conditions relating the self-similarity (Hurst) exponent of the fBm to the parameters defining the multifractal random measure $M$. As a result, a larger class of models is obtained, whose fine scale (scaling) structure is then analyzed in terms of the empirical structure functions. Implications for the analysis and inference of multifractal exponents from data, namely, confidence intervals, are also provided.Comment: Published in at http://dx.doi.org/10.1214/07-AAP483 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Random Curves by Conformal Welding

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally related to SLE$(\kappa)$ for $\kappa<4$.Comment: 5 page

On optimal wavelet bases for the realization of microcanonical cascade processes

International Journal of Wavelets, Multiresolution and Information ProcessingInternational audienceMultiplicative cascades are often used to represent the structure of turbulence. Under the action of a multiplicative cascade, the relevant variables of the system can be understood as the result of a successive transfer of information in cascade from large to small scales. However, to make this cascade transfer explicit (i.e, being able to decompose each variable as the product of larger scale contributions) is only achieved when signals are represented in an optimal wavelet basis. Finding such a basis is a data-demanding, highly-complex task. In this paper we propose a formalism that allows to find the optimal wavelet of a signal in an efficient, little data-demanding way. We confirm the appropriateness of this approach by analyzing the results on synthetic signals constructed with prescribed optimal bases. We show the validity of our approach constrained to given families of wavelets, though it can be generalized for a continuous unconstrained search scheme