The paper presents a unified approach to local likelihood estimation
for a broad class of nonparametric models, including e.g. the regression,
density, Poisson and binary response model. The method extends
the adaptive weights smoothing (AWS) procedure introduced in Polzehl
and Spokoiny (2000) in context of image denoising. Performance of the
proposed procedure is illustrated by a number of numerical examples
and applications to density or volatility estimation, classification and
estimation of the tail index parameter. We also establish a number of
important theoretical results on properties of the proposed procedure.
The adaptive weights smoothing (AWS) procedure was introduced in
Polzehl and Spokoiny (2000) in the context of image denoising. The
procedure has some remarkable properties like preservation of edges and
contrast, and (in some sense) optimal reduction of noise. The procedure
is fully adaptive and dimension free. Simulations with artificial images
show that AWS is superior to classical smoothing techniques especially
when the underlying image function is discontinuous and can be well
approximated by a piecewise constant function. However, the latter as-
sumption can be rather restrictive for a number of potential applications.
Here the AWS method is generalized to the case of an arbitrary local lin-
ear parametric structure. We also establish some important results about
properties of the AWS procedure including the so called "propagation
condition" and spatial adaptivity. The performance of the procedure is
illustrated by examples for local polynomial regression in univariate and
bivariate situations.
Data from functional magnetic resonance imaging (fMRI) consists of time series of brain images which are characterized
by a high noise level and a low signal-to-noise ratio. We provide a complete procedure for fMRI analysis. In order to reduce noise and to improve signal detection the fMRI data is spatially smoothed. However, the common application of a Gaussian filter does this at the cost of loss of information on spatial extend and shape of the activation area. We suggest to use the propagation-separation procedures introduced by Polzehl and Spokoiny (2005) instead. We show that this significantly improves the information on the spatial extend and shape of the activation region with similar results for the noise reduction. Signal detection is based on locally varying thresholds defined by random field theory. Effects of adaptive and non adaptive smoothing are illustrated by artificial examples and an analysis of real data.
An important problem of the analysis of fMRI experiments is to achieve some noise reduction of the data without blurring the shape of the activation areas. As a novel solution to this problem, the Propagation-Separation approach (PS), a structure adaptive smoothing method, has been proposed recently. PS adapts to different shapes of activation areas by generating a spatial structure corresponding to similarities and differences between time series in adjacent locations. In this paper we demonstrate how this method results in more accurate localization of brain activity. First, it is shown in numerical simulations that PS is superior over Gaussian smoothing with respect to the accurate description of the shape of activation clusters and and results in less false detections. Second, in a study of 37 presurgical planning cases we found that PS and Gaussian smoothing often yield different results, and we present examples showing aspects of the superiority of PS as applied to presurgical planning.
Diffusion Weighted Imaging has become and will certainly continue to be an important tool in medical research and diagnostics. Data obtained with Diffusion Weighted Imaging are characterized by a high noise level. Thus, estimation of quantities like anisotropy indices or the main diffusion direction may be significantly compromised by noise in clinical or neuroscience applications. Here, we present a new package dti for R, which provides functions for the analysis of diffusion weighted data within the diffusion tensor model. This includes smoothing by a recently proposed structural adaptive smoothing procedure based on the Propagation-Separation approach in the context of the widely used Diffusion Tensor Model. We extend the procedure and show, how a correction for Rician bias can be incorporated. We use a heteroscedastic nonlinear regression model to estimate the diffusion tensor. The smoothing procedure naturally adapts to different structures of different size and thus avoids oversmoothing edges and fine structures. We illustrate the usage and capabilities of the package through some examples.
Increasing the spatial resolution in functional Magnetic Resonance Imaging (fMRI) inherently lowers the signal-to-noise ratio (SNR). In order to still detect functionally significant activations in high-resolution images, spatial smoothing of the data is required. However, conventional non-adaptive smoothing comes with a reduced effective resolution, foiling the benefit of the higher acquisition resolution. We show how our recently proposed structural adaptive smoothing procedure for functional MRI data can improve signal detection of high-resolution fMRI experiments regardless of the lower SNR. The procedure is evaluated on human visual and sensory-motor mapping experiments. In these applications, the higher resolution could be fully utilized and high-resolution experiments were outperforming normal resolution experiments by means of both statistical significance and information content
Functional Magnetic Resonance Imaging inherently involves noisy measurements and a severe multiple test
problem. Smoothing is usually used to reduce the effective number of multiple
comparisons and to locally integrate the signal and hence increase the
signal-to-noise ratio. Here, we provide a new structural adaptive segmentation
algorithm (AS)
that naturally combines the signal detection with noise reduction in one procedure.
Moreover, the new method
is closely related to a recently proposed structural adaptive smoothing
algorithm and preserves shape and spatial extent of activation areas without
blurring the borders.
Modeling the orientation distribution function by mixtures of angular central Gaussian distributions
(2010)
In this paper we develop a tensor mixture model for diffusion weighted imaging
data using an automatic model selection criterion for the order of tensor
components in a voxel. We show that the weighted orientation distribution
function for this model can be expanded into a mixture of angular central
Gaussian distributions. We show properties of this model in extensive
simulations and in a high angular resolution experimental data set. The results
suggest that the model may improve imaging of cerebral fiber tracts. We
demonstrate how inference on canonical model parameters may give rise to new
clinical applications.
The package fmri is provided for analysis of single run functional
Magnetic Resonance Imaging data. It implements structural adaptive smoothing
methods with signal detection for adaptive noise reduction which avoids blurring
of edges of activation areas. fmri provides fmri analysis from time series
modeling to signal detection and publication-ready images.
Diffusion weighted imaging is a magnetic resonance based method to investigate
tissue micro-structure especially in the human brain via water diffusion.
Since the standard diffusion tensor model for the acquired data failes in
large portion of the brain voxel more sophisticated models have bee developed.
Here, we report on the package dti and how some of these models
can be used with the package.