2,395 research outputs found
A real algebra perspective on multivariate tight wavelet frames
Recent results from real algebraic geometry and the theory of polynomial
optimization are related in a new framework to the existence question of
multivariate tight wavelet frames whose generators have at least one vanishing
moment. Namely, several equivalent formulations of the so-called Unitary
Extension Principle by Ron and Shen are interpreted in terms of hermitian sums
of squares of certain nonnegative trigonometric polynomials and in terms of
semi-definite programming. The latter together with the recent results in
algebraic geometry and semi-definite programming allow us to answer
affirmatively the long standing open question of the existence of such tight
wavelet frames in dimension ; we also provide numerically efficient
methods for checking their existence and actual construction in any dimension.
We exhibit a class of counterexamples in dimension showing that, in
general, the UEP property is not sufficient for the existence of tight wavelet
frames. On the other hand we provide stronger sufficient conditions for the
existence of tight wavelet frames in dimension and illustrate our
results by several examples
A real algebra perspective on multivariate tight wavelet frames
Recent results from real algebraic geometry and the theory of polynomial optimization
are related in a new framework to the existence question of multivariate tight wavelet
frames whose generators have at least one vanishing moment. Namely, several equivalent
formulations of the so-called Unitary Extension Principle (UEP) from [33] are interpreted
in terms of hermitian sums of squares of certain nongenative trigonometric polynomials
and in terms of semi-definite programming. The latter together with the results in
[31, 35] answer affirmatively the long stading open question of the existence of such tight
wavelet frames in dimesion d = 2; we also provide numerically efficient methods for
checking their existence and actual construction in any dimension. We exhibit a class
of counterexamples in dimension d = 3 showing that, in general, the UEP property is
not sufficient for the existence of tight wavelet frames. On the other hand we provide
stronger sufficient conditions for the existence of tight wavelet frames in dimension d ≥ 3
and illustrate our results by several examples
Wavelet methods in speech recognition
In this thesis, novel wavelet techniques are developed to improve parametrization of
speech signals prior to classification. It is shown that non-linear operations carried out
in the wavelet domain improve the performance of a speech classifier and consistently
outperform classical Fourier methods. This is because of the localised nature of the
wavelet, which captures correspondingly well-localised time-frequency features
within the speech signal. Furthermore, by taking advantage of the approximation
ability of wavelets, efficient representation of the non-stationarity inherent in speech
can be achieved in a relatively small number of expansion coefficients. This is an
attractive option when faced with the so-called 'Curse of Dimensionality' problem of
multivariate classifiers such as Linear Discriminant Analysis (LDA) or Artificial
Neural Networks (ANNs). Conventional time-frequency analysis methods such as the
Discrete Fourier Transform either miss irregular signal structures and transients due to
spectral smearing or require a large number of coefficients to represent such
characteristics efficiently. Wavelet theory offers an alternative insight in the
representation of these types of signals.
As an extension to the standard wavelet transform, adaptive libraries of wavelet and
cosine packets are introduced which increase the flexibility of the transform. This
approach is observed to be yet more suitable for the highly variable nature of speech
signals in that it results in a time-frequency sampled grid that is well adapted to
irregularities and transients. They result in a corresponding reduction in the
misclassification rate of the recognition system. However, this is necessarily at the
expense of added computing time.
Finally, a framework based on adaptive time-frequency libraries is developed which
invokes the final classifier to choose the nature of the resolution for a given
classification problem. The classifier then performs dimensionaIity reduction on the
transformed signal by choosing the top few features based on their discriminant power. This approach is compared and contrasted to an existing discriminant wavelet
feature extractor.
The overall conclusions of the thesis are that wavelets and their relatives are capable
of extracting useful features for speech classification problems. The use of adaptive
wavelet transforms provides the flexibility within which powerful feature extractors
can be designed for these types of application
Dynamic Steerable Blocks in Deep Residual Networks
Filters in convolutional networks are typically parameterized in a pixel
basis, that does not take prior knowledge about the visual world into account.
We investigate the generalized notion of frames designed with image properties
in mind, as alternatives to this parametrization. We show that frame-based
ResNets and Densenets can improve performance on Cifar-10+ consistently, while
having additional pleasant properties like steerability. By exploiting these
transformation properties explicitly, we arrive at dynamic steerable blocks.
They are an extension of residual blocks, that are able to seamlessly transform
filters under pre-defined transformations, conditioned on the input at training
and inference time. Dynamic steerable blocks learn the degree of invariance
from data and locally adapt filters, allowing them to apply a different
geometrical variant of the same filter to each location of the feature map.
When evaluated on the Berkeley Segmentation contour detection dataset, our
approach outperforms all competing approaches that do not utilize pre-training.
Our results highlight the benefits of image-based regularization to deep
networks
Directional edge and texture representations for image processing
An efficient representation for natural images is of fundamental importance in image processing and analysis. The commonly used separable transforms such as wavelets axe not best suited for images due to their inability to exploit directional regularities such as edges and oriented textural patterns; while most of the recently proposed directional schemes cannot represent these two types of features in a unified transform. This thesis focuses on the development of directional representations for images which can capture both edges and textures in a multiresolution manner. The thesis first considers the problem of extracting linear features with the multiresolution Fourier transform (MFT). Based on a previous MFT-based linear feature model, the work extends the extraction method into the situation when the image is corrupted by noise. The problem is tackled by the combination of a "Signal+Noise" frequency model, a refinement stage and a robust classification scheme. As a result, the MFT is able to perform linear feature analysis on noisy images on which previous methods failed. A new set of transforms called the multiscale polar cosine transforms (MPCT) are also proposed in order to represent textures. The MPCT can be regarded as real-valued MFT with similar basis functions of oriented sinusoids. It is shown that the transform can represent textural patches more efficiently than the conventional Fourier basis. With a directional best cosine basis, the MPCT packet (MPCPT) is shown to be an efficient representation for edges and textures, despite its high computational burden. The problem of representing edges and textures in a fixed transform with less complexity is then considered. This is achieved by applying a Gaussian frequency filter, which matches the disperson of the magnitude spectrum, on the local MFT coefficients. This is particularly effective in denoising natural images, due to its ability to preserve both types of feature. Further improvements can be made by employing the information given by the linear feature extraction process in the filter's configuration. The denoising results compare favourably against other state-of-the-art directional representations
Koopman Neural Forecaster for Time Series with Temporal Distribution Shifts
Temporal distributional shifts, with underlying dynamics changing over time,
frequently occur in real-world time series, and pose a fundamental challenge
for deep neural networks (DNNs). In this paper, we propose a novel deep
sequence model based on the Koopman theory for time series forecasting: Koopman
Neural Forecaster (KNF) that leverages DNNs to learn the linear Koopman space
and the coefficients of chosen measurement functions. KNF imposes appropriate
inductive biases for improved robustness against distributional shifts,
employing both a global operator to learn shared characteristics, and a local
operator to capture changing dynamics, as well as a specially-designed feedback
loop to continuously update the learnt operators over time for rapidly varying
behaviors. To the best of our knowledge, this is the first time that Koopman
theory is applied to real-world chaotic time series without known governing
laws. We demonstrate that KNF achieves the superior performance compared to the
alternatives, on multiple time series datasets that are shown to suffer from
distribution shifts
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