报告题目：  Wavelet phase harmonic covariance models of stationary processes

报告人：    章斯鑫, 北京大学数据科学中心

时间：      2018年12月28日，上午10:00-11:00

地点：      数学系格物楼503

摘要：

Many sequential and image data such as textures and homogenous turbulence can be modeled as the stationary processes because of their translational invariant statistical properties in the time or the spatial domain. The spectral representation of a discrete stationary process decomposes each realization of the random process into a Fourier series. Using the building blocks of the sine and cosine functions is suitable to model Gaussian stationary processes, but when there are transient/intermittent and multi-scale geometric structures in the long-memory processes such as Fractional Brownian noise and Turbulence, it is more efficient to use the wavelet representation. This talk focuses on modeling the finite discrete image data with such structures. 

We introduce a class of statistical models of non-Gaussian long-range stationary processes from the covariance of wavelet phase harmonic coefficients. These coefficients are computed with the phase harmonic transform, which multiplied the phase of the complex wavelet coefficients by the integers. The covariance between these coefficients can be estimated from the observed data samples to build a maximum entropy micro-canonical model for the stationary process. The samples from the model are able to capture the intermittent and multi-scale geometric structures visible from the two-dimensional image data. We evaluate quantitatively the model error in terms of the bias and variance, and show their ability to approximate well the stationary processes as complex as the ones encountered in turbulent fluids, textures and many other natural signals.

This is a joint work with Stéphane Mallat at École normale supérieure Paris.

简历：

2018-present:  Research Associate

Center for Data Science, Peking University

2016-2018:    Postdoctoral Fellow, Advisor: Stéphane Mallat

Ecole Normale Supérieur (Paris), France.

2010-2016:    PhD in Computer Science, Advisor: Yann LeCun

Courant Institute, New York University, United States.

2008-2010:    Master in Mathematics, Computer Vision, and Machine Learning

Université Pierre et Marie Curie (Paris VI)

& Ecole Normale Supérieur (Cachan), France.

2004-2008:    Bachelor in Mathematics and Information Science, Shanghai University