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267
A Tensor Framework for Multidimensional Signal Processing
 Linkoping University, Sweden
, 1994
"... ii About the cover The figure on the cover shows a visualization of a symmetric tensor in three dimensions, G = λ1ê1ê T 1 + λ2ê2ê T 2 + λ3ê3ê T 3 The object in the figure is the sum of a spear, a plate and a sphere. The spear describes the principal direction of the tensor λ1ê1ê T 1, where the lengt ..."
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Cited by 66 (8 self)
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ii About the cover The figure on the cover shows a visualization of a symmetric tensor in three dimensions, G = λ1ê1ê T 1 + λ2ê2ê T 2 + λ3ê3ê T 3 The object in the figure is the sum of a spear, a plate and a sphere. The spear describes the principal direction of the tensor λ1ê1ê T 1, where the length is proportional to the largest eigenvalue, λ1. The plate describes the plane spanned by the eigenvectors corresponding to the two largest eigenvalues, λ2(ê1ê T 1 + ê2ê T 2). The sphere, with a radius proportional to the smallest eigenvalue, shows how isotropic the tensor is, λ3(ê1ê T 1 + ê2ê T 2 + ê3ê T 3). The visualization is done using AVS [WWW94]. I am very grateful to Johan Wiklund for implementing the tensor viewer module used. This thesis deals with filtering of multidimensional signals. A large part of the thesis is devoted to a novel filtering method termed “Normalized convolution”. The method performs local expansion of a signal in a chosen filter basis which
On Instantaneous Amplitude and Phase of Signals
 IEEE Trans. Signal Processing
, 1997
"... Abstract—In many questions of signal processing, it is important to use the concepts of instantaneous amplitude or phase of signals. This is especially the case in communication systems with amplitude or frequency modulation. These concepts are often introduced empirically. However, it is well known ..."
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Cited by 65 (0 self)
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Abstract—In many questions of signal processing, it is important to use the concepts of instantaneous amplitude or phase of signals. This is especially the case in communication systems with amplitude or frequency modulation. These concepts are often introduced empirically. However, it is well known that the correct approach for this purpose is to use the concept of analytic signal. Starting from this point, we show some examples of contradictions appearing when using other definitions of instantaneous amplitude or frequency that are commonly admitted. This introduces the problem of characterizing pure amplitudemodulated or pure phasemodulated signals. It is especially shown that whereas amplitude modulated signals can be characterized by spectral considerations, this is no longer the case for phasemodulated signals. Furthermore, signals with constant amplitude have very specific properties, which are analyzed in detail. Some consequences and extensions to random signals are finally discussed. I.
Characterization Of Signals By The Ridges Of Their Wavelet Transforms
 IEEE Trans. on Signal Processing
, 1994
"... We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of onedimensional signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter ..."
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Cited by 42 (5 self)
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We present a couple of new algorithmic procedures for the detection of ridges in the modulus of the (continuous) wavelet transform of onedimensional signals. These detection procedures are shown to be robust to additive white noise. We also derive and test a new reconstruction procedure. The latter uses only information from the restriction of the wavelet transform to a sample of points from the ridge. This provides with a very efficient way to code the information contained in the signal. Partially supported by ONR N00014911010 y Supported by NSF IBN 9405146 1 Introduction The characterization and the separation of amplitude and frequency modulated signals is a classical problem of signal analysis and signal processing. Applications can be found in many situations, such as for instance radar/sonar detection and speech processing [9]. Many methods have been proposed in the past few years to analyze the timefrequency localization of signals. The most noticeable are the family...
Computational Analysis Of NonFourier Motion
 Vision Research
, 1995
"... NonFourier motion is now commonplace in research on visual motion perception, yet lacks a computational framework. This paper examines this issue based on the observation that many nonFourier motion stimuli have a simple characterization in the frequency domain, in terms of oriented power distribu ..."
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Cited by 38 (5 self)
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NonFourier motion is now commonplace in research on visual motion perception, yet lacks a computational framework. This paper examines this issue based on the observation that many nonFourier motion stimuli have a simple characterization in the frequency domain, in terms of oriented power distributions that lie along lines (or planes) that do not pass through the origin. This provides a unifying theoretical framework for a very diverse class of nonFourier phenomena. It also allows us to examine some central issues concerning the computational nature of nonFourier models, and naturally occurring sources of nonFourier motion. For example, it is shown that the orientation of power in frequency domain corresponds to the velocity of a multiplicative envelope, and may arise as a restricted form of lighting effects, translucency or occlusion. We also show that both the location and orientation of spectral power may be extracted from the phase and amplitude output of bandpass filters, co...
Phase synchronization for the recognition of mental tasks in a braincomputer interface
 IEEE Trans Neural Syst Rehabil Eng
"... Abstract—Brain–computer interfaces (BCIs) may be a future communication channel for motordisabled people. In surface electroencephalogram (EEG)based BCIs, the extracted features are often derived from spectral estimates and autoregressive models. We examined the usefulness of synchronization betwe ..."
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Cited by 33 (1 self)
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Abstract—Brain–computer interfaces (BCIs) may be a future communication channel for motordisabled people. In surface electroencephalogram (EEG)based BCIs, the extracted features are often derived from spectral estimates and autoregressive models. We examined the usefulness of synchronization between EEG signals for classifying mental tasks. To this end, we investigated the performance of features derived from the phase locking value (PLV) and from the spectral coherence and compared them to the classification rates resulting from the power densities in,
A Comparison of the Energy Operator and the Hilbert Transform Approach to Signal and Speech Demodulation
, 1994
"... The Hilbert transform together with Gabor's analytic signal provides a standard linear integral approach to estimate the amplitude envelope and instantaneous frequency of signals with a combined amplitude modulation (AM) and frequency modulation (FM) structure. An alternative recent approach ..."
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Cited by 28 (8 self)
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The Hilbert transform together with Gabor's analytic signal provides a standard linear integral approach to estimate the amplitude envelope and instantaneous frequency of signals with a combined amplitude modulation (AM) and frequency modulation (FM) structure. An alternative recent approach uses a nonlinear differential `energy' operator to track the energy required to generate an AMFM signal and separate it into amplitude and frequency components. In this paper, we compare these two fundamentally different approaches for demodulation of arbitrary signals and of speech resonances modeled by AMFM signals. The comparison is done from several viewpoints: magnitude of estimation errors, computational complexity, and adaptability to instantaneous signal changes. We also propose a refinement of the energy operator approach that uses simple binomial convolutions to smooth the energy signals. This smoothed energy operator is compared to the Hilbert transform on tracking modul...
MultiRidge Detection and TimeFrequency Reconstruction
 IEEE Transactions on Signal Processing
, 1996
"... The ridges of the wavelet transform, the Gabor transform or any timefrequency representation of a signal contain crucial information on the characteristics of the signal. Indeed they mark the regions of the timefrequency plane where the signal concentrates most of its energy. We introduce a new ..."
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Cited by 25 (9 self)
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The ridges of the wavelet transform, the Gabor transform or any timefrequency representation of a signal contain crucial information on the characteristics of the signal. Indeed they mark the regions of the timefrequency plane where the signal concentrates most of its energy. We introduce a new algorithm to detect and identify these ridges. The procedure is based on an original penalization of the transitions of the random walk in a bounded domain of the plane. We show that this detection algorithm is especially useful for noisy signals with multiridge transforms. It is a common practice among practitioners to reconstruct a signal from the skeleton of a transform of the signal (i.e. the restriction of the transform to the ridges). After reviewing several known procedures we introduce a new reconstruction algorithm and we illustrate its usefulness on speech signals. Partially supported by ONR N00014911010 y Supported by NSF IBN 9405146 1 1 Introduction and Notations ...
Hypercomplex SignalsA Novel Extension of the Analytic Signal to the Multidimensional Case
 IEEE trans. on Signal Processing
, 2001
"... Abstract—The construction of Gabor’s complex signal—which is also known as the analytic signal—provides direct access to a real one–dimensional (1D) signal’s local amplitude and phase. The complex signal is built from a real signal by adding its Hilbert transform—which is a phaseshifted version of ..."
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Cited by 22 (1 self)
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Abstract—The construction of Gabor’s complex signal—which is also known as the analytic signal—provides direct access to a real one–dimensional (1D) signal’s local amplitude and phase. The complex signal is built from a real signal by adding its Hilbert transform—which is a phaseshifted version of the signal—as an imaginary part to the signal. Since its introduction, the complex signal has become an important tool in signal processing, with applications, for example, in narrowband communication. Different approaches to anD analytic or complex signal have been proposed in the past. We review these approaches and propose the hypercomplex signal as a novel extension of the complex signal toD. This extension leads to a new definition of local phase, which reveals information on the intrinsic dimensionality of the signal. The different approaches are unified by expressing all of them as combinations of the signal and its partial and total Hilbert transforms. Examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown. An example is provided for the two–dimensional (2D) hypercomplex signal, which shows how the novel phase concept can be used in texture segmentation. Index Terms—Analytic signal, Clifford Fourier transform, complex signal, Hilbert transform, hypercomplex Fourier transform,