1,949 research outputs found
Wavelet Estimators in Nonparametric Regression: A Comparative Simulation Study
Wavelet analysis has been found to be a powerful tool for the nonparametric estimation of spatially-variable objects. We discuss in detail wavelet methods in nonparametric regression, where the data are modelled as observations of a signal contaminated with additive Gaussian noise, and provide an extensive review of the vast literature of wavelet shrinkage and wavelet thresholding estimators developed to denoise such data. These estimators arise from a wide range of classical and empirical Bayes methods treating either individual or blocks of wavelet coefficients. We compare various estimators in an extensive simulation study on a variety of sample sizes, test functions, signal-to-noise ratios and wavelet filters. Because there is no single criterion that can adequately summarise the behaviour of an estimator, we use various criteria to measure performance in finite sample situations. Insight into the performance of these estimators is obtained from graphical outputs and numerical tables. In order to provide some hints of how these estimators should be used to analyse real data sets, a detailed practical step-by-step illustration of a wavelet denoising analysis on electrical consumption is provided. Matlab codes are provided so that all figures and tables in this paper can be reproduced
Rotationally Invariant Image Representation for Viewing Direction Classification in Cryo-EM
We introduce a new rotationally invariant viewing angle classification method
for identifying, among a large number of Cryo-EM projection images, similar
views without prior knowledge of the molecule. Our rotationally invariant
features are based on the bispectrum. Each image is denoised and compressed
using steerable principal component analysis (PCA) such that rotating an image
is equivalent to phase shifting the expansion coefficients. Thus we are able to
extend the theory of bispectrum of 1D periodic signals to 2D images. The
randomized PCA algorithm is then used to efficiently reduce the dimensionality
of the bispectrum coefficients, enabling fast computation of the similarity
between any pair of images. The nearest neighbors provide an initial
classification of similar viewing angles. In this way, rotational alignment is
only performed for images with their nearest neighbors. The initial nearest
neighbor classification and alignment are further improved by a new
classification method called vector diffusion maps. Our pipeline for viewing
angle classification and alignment is experimentally shown to be faster and
more accurate than reference-free alignment with rotationally invariant K-means
clustering, MSA/MRA 2D classification, and their modern approximations
Estimation of Translation, Rotation, and Scaling between Noisy Images Using the Fourier–Mellin Transform
In this paper we focus on extended Euclidean registration of a set of noisy images. We provide an appropriate statistical model for this kind of registration problems, and a new criterion based on Fourier-type transforms is proposed to estimate the translation, rotation and scaling parameters to align a set of images. This criterion is a two step procedure which does not require the use of a reference template onto which aligning all the images. Our approach is based on M-estimation and we prove the consistency of the resulting estimators. A small scale simulation study and real examples are used to illustrate the numerical performances of our procedure
A Multiscale Approach for Statistical Characterization of Functional Images
Increasingly, scientific studies yield functional image data, in which the observed data consist of sets of curves recorded on the pixels of the image. Examples include temporal brain response intensities measured by fMRI and NMR frequency spectra measured at each pixel. This article presents a new methodology for improving the characterization of pixels in functional imaging, formulated as a spatial curve clustering problem. Our method operates on curves as a unit. It is nonparametric and involves multiple stages: (i) wavelet thresholding, aggregation, and Neyman truncation to effectively reduce dimensionality; (ii) clustering based on an extended EM algorithm; and (iii) multiscale penalized dyadic partitioning to create a spatial segmentation. We motivate the different stages with theoretical considerations and arguments, and illustrate the overall procedure on simulated and real datasets. Our method appears to offer substantial improvements over monoscale pixel-wise methods. An Appendix which gives some theoretical justifications of the methodology, computer code, documentation and dataset are available in the online supplements
Image Decomposition and Separation Using Sparse Representations: An Overview
This paper gives essential insights into the use of sparsity and morphological diversity in image decomposition and source separation by reviewing our recent work in this field. The idea to morphologically decompose a signal into its building blocks is an important problem in signal processing and has far-reaching applications in science and technology. Starck , proposed a novel decomposition method—morphological component analysis (MCA)—based on sparse representation of signals. MCA assumes that each (monochannel) signal is the linear mixture of several layers, the so-called morphological components, that are morphologically distinct, e.g., sines and bumps. The success of this method relies on two tenets: sparsity and morphological diversity. That is, each morphological component is sparsely represented in a specific transform domain, and the latter is highly inefficient in representing the other content in the mixture. Once such transforms are identified, MCA is an iterative thresholding algorithm that is capable of decoupling the signal content. Sparsity and morphological diversity have also been used as a novel and effective source of diversity for blind source separation (BSS), hence extending the MCA to multichannel data. Building on these ingredients, we will provide an overview the generalized MCA introduced by the authors in and as a fast and efficient BSS method. We will illustrate the application of these algorithms on several real examples. We conclude our tour by briefly describing our software toolboxes made available for download on the Internet for sparse signal and image decomposition and separation
A Hierarchical Bayesian Model for Frame Representation
In many signal processing problems, it may be fruitful to represent the
signal under study in a frame. If a probabilistic approach is adopted, it
becomes then necessary to estimate the hyper-parameters characterizing the
probability distribution of the frame coefficients. This problem is difficult
since in general the frame synthesis operator is not bijective. Consequently,
the frame coefficients are not directly observable. This paper introduces a
hierarchical Bayesian model for frame representation. The posterior
distribution of the frame coefficients and model hyper-parameters is derived.
Hybrid Markov Chain Monte Carlo algorithms are subsequently proposed to sample
from this posterior distribution. The generated samples are then exploited to
estimate the hyper-parameters and the frame coefficients of the target signal.
Validation experiments show that the proposed algorithms provide an accurate
estimation of the frame coefficients and hyper-parameters. Application to
practical problems of image denoising show the impact of the resulting Bayesian
estimation on the recovered signal quality
Representation Learning: A Review and New Perspectives
The success of machine learning algorithms generally depends on data
representation, and we hypothesize that this is because different
representations can entangle and hide more or less the different explanatory
factors of variation behind the data. Although specific domain knowledge can be
used to help design representations, learning with generic priors can also be
used, and the quest for AI is motivating the design of more powerful
representation-learning algorithms implementing such priors. This paper reviews
recent work in the area of unsupervised feature learning and deep learning,
covering advances in probabilistic models, auto-encoders, manifold learning,
and deep networks. This motivates longer-term unanswered questions about the
appropriate objectives for learning good representations, for computing
representations (i.e., inference), and the geometrical connections between
representation learning, density estimation and manifold learning
A Universal Scheme for Wyner–Ziv Coding of Discrete Sources
We consider the Wyner–Ziv (WZ) problem of lossy compression where the decompressor observes a noisy version of the source, whose statistics are unknown. A new family of WZ coding algorithms is proposed and their universal optimality is proven. Compression consists of sliding-window processing followed by Lempel–Ziv (LZ) compression, while the decompressor is based on a modification of the discrete universal denoiser (DUDE) algorithm to take advantage of side information. The new algorithms not only universally attain the fundamental limits, but also suggest a paradigm for practical WZ coding. The effectiveness of our approach is illustrated with experiments on binary images, and English text using a low complexity algorithm motivated by our class of universally optimal WZ codes
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