Diffusion-Driven X-Ray Two-Dimensional Patterns Denoising

The use of a mathematical model is proposed in order to denoise X-ray two-dimensional patterns. The method relies on a generalized diffusion equation whose diffusion constant depends on the image gradients. The numerical solution of the diffusion equation provides an efficient reduction of pattern noise as witnessed by the computed peak of signal-to-noise ratio. The use of experimental data with different inherent levels of noise allows us to show the success of the method even in the case, experimentally relevant, when patterns are blurred by Poissonian noise. The corresponding MatLab code for the numerical method is made available.

Shot noise in multitracer constraints on fNL and relativistic projections: Power spectrum

ABSTRACT Multiple tracers of the same surveyed volume can enhance the signal-to-noise on a measurement of local primordial non-Gaussianity and the relativistic projections. Increasing the number of tracers comparably increases the number of shot noise terms required to describe the stochasticity of the data. Although the shot noise is white on large scales, it is desirable to investigate the extent to which it can degrade constraints on the parameters of interest. In a multitracer analysis of the power spectrum, a marginalization over shot noise does not degrade the constraints on fNL by more than ∼30 per cent so long as haloes of mass $M\lesssim 10^{12}\, \mathrm{M}_\odot$ are resolved. However, ignoring cross shot noise terms induces large systematics on a measurement of fNL at redshift z < 1 when small mass haloes are resolved. These effects are less severe for the relativistic projections, especially for the dipole term. In the case of a low and high mass tracer, the optimal sample division maximizes the signal-to-noise on fNL and the projection effects simultaneously, reducing the errors to the level of ∼10 consecutive mass bins of equal number density. We also emphasize that the non-Poissonian noise corrections that arise from small-scale clustering effects cannot be measured with random dilutions of the data. Therefore, they must either be properly modelled or marginalized over.

Ptychography of pure quantum states

Ptychography is an imaging technique in which a localized illumination scans overlapping regions of an object and generates a set of diffraction intensities used to computationally reconstruct its complex-valued transmission function. We propose a quantum analogue of this technique designed to reconstruct d-dimensional pure states. A set of n rank-r projectors “scans” overlapping parts of an input state and the moduli of the d Fourier amplitudes of each part are measured. These nd outcomes are fed into an iterative phase retrieval algorithm that estimates the state. Using d up to 100 and r around d / 2, we performed numerical simulations for single systems in an economic (n = 4) and a costly (n = d) scenario, as well as for multiqubit systems (n = 6logd). This numeric study included realistic amounts of depolarization and poissonian noise, and all scenarios yielded, in general, reconstructions with infidelities below 10−2. The method is shown, therefore, to be resilient to noise and, for any d, requires a simple and fast postprocessing algorithm. We show that the algorithm is equivalent to an alternating gradient search, which ensures that it does not suffer from local-minima stagnation. Unlike traditional approaches to state reconstruction, the ptychographic scheme uses a single measurement basis; the diversity and redundancy in the measured data—key for its success—are provided by the overlapping projections. We illustrate the simplicity of this scheme with the paradigmatic multiport interferometer.

On the long-range dependence of fractional Poisson and negative binomial processes

We discuss the short-range dependence (SRD) property of the increments of the fractional Poisson process, called the fractional Poissonian noise. We also establish that the fractional negative binomial process (FNBP) has the long-range dependence (LRD) property, while the increments of the FNBP have the SRD property. Our definitions of the SRD/LRD properties are similar to those for a stationary process and different from those recently used in Biard and Saussereau (2014).

Can irregularities of solar proxies help understand quasi-biennial solar variations?

We define, calculate and analyze irregularity indices λISSN of daily series of the International Sunspot Number ISSN as a function of increasing smoothing from N = 162 to 648 days. The irregularity indices λ are computed within 4-year sliding windows, with embedding dimensions m = 1 and 2. λISSN displays Schwabe cycles with ~5.5-year variations ("half Schwabe variations" HSV). The mean of λISSN undergoes a downward step and the amplitude of its variations strongly decreases around 1930. We observe changes in the ratio R of the mean amplitude of λ peaks at solar cycle minima with respect to peaks at solar maxima as a function of date, embedding dimension and, importantly, smoothing parameter N. We identify two distinct regimes, called Q1 and Q2, defined mainly by the evolution of R as a function of N: Q1, with increasing HSV behavior and R value as N is increased, occurs before 1915–1930; and Q2, with decreasing HSV behavior and R value as N is increased, occurs after ~1975. We attempt to account for these observations with an autoregressive (order 1) model with Poissonian noise and a mean modulated by two sine waves of periods T1 and T2 (T1 = 11 years, and intermediate T2 is tuned to mimic quasi-biennial oscillations QBO). The model can generate both Q1 and Q2 regimes. When m = 1, HSV appears in the absence of T2 variations. When m = 2, Q1 occurs when T2 variations are present, whereas Q2 occurs when T2 variations are suppressed. We propose that the HSV behavior of the irregularity index of ISSN may be linked to the presence of strong QBO before 1915–1930, a transition and their disappearance around 1975, corresponding to a change in regime of solar activity.

Can irregularities of solar proxies help understand quasi-biennial solar variations?

We define, calculate and analyze irregularity indices λWN and λaa of daily series of sunspot number WN and geomagnetic index aa as a function of increasing smoothing from N = 162 to 648 days. The irregularity indices λ are computed within 4 year sliding windows, with embedding dimensions m = 1 and 2. λWN and λaa display Schwabe cycles with sharp peaks not only at cycle maxima but also at minima: we call the resulting ~5.5 year variations "half Schwabe variations" (HSV). The mean of λWN undergoes a downward step and the amplitude of its variations strongly decreases around 1930. We observe changes in the ratio R of the mean amplitude of λ peaks at solar cycle minima with respect to peaks at solar maxima as a function of date, embedding dimension and importantly smoothing parameter N. We identify two distinct regimes, called Q1 and Q2, defined mainly by the evolution of R as a function of N: Q1, with increasing HSV behavior and R value as N is increased, occurs before 1915–1930 and Q2, with decreasing HSV behavior and R value as N is increased, occurs after ~1975. We attempt to account for these observations with an autoregressive (order 1) model with Poissonian noise and a mean modulated by two sine waves of periods T1 and T2 (T1 = 11 years, and intermediate T2 is tuned to mimic quasi-biennial oscillations QBO). The model can generate both Q1 and Q2 regimes. When m = 1, HSV appears in the absence of T2 variations. When m = 2, Q1 occurs when T2 variations are present, whereas Q2 occurs when T2 variations are suppressed. We propose that the HSV behavior of the irregularity index of WN may be linked to the presence of strong QBO before 1915–1930, a transition and their disappearance around 1975, corresponding to a change in regime of solar activity.