168,834 research outputs found
Statistical deconvolution of the free Fokker-Planck equation at fixed time
We are interested in reconstructing the initial condition of a non-linear
partial differential equation (PDE), namely the Fokker-Planck equation, from
the observation of a Dyson Brownian motion at a given time . The
Fokker-Planck equation describes the evolution of electrostatic repulsive
particle systems, and can be seen as the large particle limit of correctly
renormalized Dyson Brownian motions. The solution of the Fokker-Planck equation
can be written as the free convolution of the initial condition and the
semi-circular distribution. We propose a nonparametric estimator for the
initial condition obtained by performing the free deconvolution via the
subordination functions method. This statistical estimator is original as it
involves the resolution of a fixed point equation, and a classical
deconvolution by a Cauchy distribution. This is due to the fact that, in free
probability, the analogue of the Fourier transform is the R-transform, related
to the Cauchy transform. In past literature, there has been a focus on the
estimation of the initial conditions of linear PDEs such as the heat equation,
but to the best of our knowledge, this is the first time that the problem is
tackled for a non-linear PDE. The convergence of the estimator is proved and
the integrated mean square error is computed, providing rates of convergence
similar to the ones known for non-parametric deconvolution methods. Finally, a
simulation study illustrates the good performances of our estimator
Probabilistic Motion Estimation Based on Temporal Coherence
We develop a theory for the temporal integration of visual motion motivated
by psychophysical experiments. The theory proposes that input data are
temporally grouped and used to predict and estimate the motion flows in the
image sequence. This temporal grouping can be considered a generalization of
the data association techniques used by engineers to study motion sequences.
Our temporal-grouping theory is expressed in terms of the Bayesian
generalization of standard Kalman filtering. To implement the theory we derive
a parallel network which shares some properties of cortical networks. Computer
simulations of this network demonstrate that our theory qualitatively accounts
for psychophysical experiments on motion occlusion and motion outliers.Comment: 40 pages, 7 figure
Recent Progress in Image Deblurring
This paper comprehensively reviews the recent development of image
deblurring, including non-blind/blind, spatially invariant/variant deblurring
techniques. Indeed, these techniques share the same objective of inferring a
latent sharp image from one or several corresponding blurry images, while the
blind deblurring techniques are also required to derive an accurate blur
kernel. Considering the critical role of image restoration in modern imaging
systems to provide high-quality images under complex environments such as
motion, undesirable lighting conditions, and imperfect system components, image
deblurring has attracted growing attention in recent years. From the viewpoint
of how to handle the ill-posedness which is a crucial issue in deblurring
tasks, existing methods can be grouped into five categories: Bayesian inference
framework, variational methods, sparse representation-based methods,
homography-based modeling, and region-based methods. In spite of achieving a
certain level of development, image deblurring, especially the blind case, is
limited in its success by complex application conditions which make the blur
kernel hard to obtain and be spatially variant. We provide a holistic
understanding and deep insight into image deblurring in this review. An
analysis of the empirical evidence for representative methods, practical
issues, as well as a discussion of promising future directions are also
presented.Comment: 53 pages, 17 figure
- …