1,611 research outputs found
The geometry of proper quaternion random variables
Second order circularity, also called properness, for complex random
variables is a well known and studied concept. In the case of quaternion random
variables, some extensions have been proposed, leading to applications in
quaternion signal processing (detection, filtering, estimation). Just like in
the complex case, circularity for a quaternion-valued random variable is
related to the symmetries of its probability density function. As a
consequence, properness of quaternion random variables should be defined with
respect to the most general isometries in , i.e. rotations from .
Based on this idea, we propose a new definition of properness, namely the
-properness, for quaternion random variables using invariance
property under the action of the rotation group . This new definition
generalizes previously introduced properness concepts for quaternion random
variables. A second order study is conducted and symmetry properties of the
covariance matrix of -proper quaternion random variables are
presented. Comparisons with previous definitions are given and simulations
illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure
Wing and body motion during flight initiation in Drosophila revealed by automated visual tracking
The fruit fly Drosophila melanogaster is a widely used model organism in studies of genetics, developmental biology and biomechanics. One limitation for exploiting Drosophila as a model system for behavioral neurobiology is that measuring body kinematics during behavior is labor intensive and subjective. In order to quantify flight kinematics during different types of maneuvers, we have developed a visual tracking system that estimates the posture of the fly from multiple calibrated cameras. An accurate geometric fly model is designed using unit quaternions to capture complex body and wing rotations, which are automatically fitted to the images in each time frame. Our approach works across a range of flight behaviors, while also being robust to common environmental clutter. The tracking system is used in this paper to compare wing and body motion during both voluntary and escape take-offs. Using our automated algorithms, we are able to measure stroke amplitude, geometric angle of attack and other parameters important to a mechanistic understanding of flapping flight. When compared with manual tracking methods, the algorithm estimates body position within 4.4±1.3% of the body length, while body orientation is measured within 6.5±1.9 deg. (roll), 3.2±1.3 deg. (pitch) and 3.4±1.6 deg. (yaw) on average across six videos. Similarly, stroke amplitude and deviation are estimated within 3.3 deg. and 2.1 deg., while angle of attack is typically measured within 8.8 deg. comparing against a human digitizer. Using our automated tracker, we analyzed a total of eight voluntary and two escape take-offs. These sequences show that Drosophila melanogaster do not utilize clap and fling during take-off and are able to modify their wing kinematics from one wingstroke to the next. Our approach should enable biomechanists and ethologists to process much larger datasets than possible at present and, therefore, accelerate insight into the mechanisms of free-flight maneuvers of flying insects
Quaternion Convolutional Neural Networks for End-to-End Automatic Speech Recognition
Recently, the connectionist temporal classification (CTC) model coupled with
recurrent (RNN) or convolutional neural networks (CNN), made it easier to train
speech recognition systems in an end-to-end fashion. However in real-valued
models, time frame components such as mel-filter-bank energies and the cepstral
coefficients obtained from them, together with their first and second order
derivatives, are processed as individual elements, while a natural alternative
is to process such components as composed entities. We propose to group such
elements in the form of quaternions and to process these quaternions using the
established quaternion algebra. Quaternion numbers and quaternion neural
networks have shown their efficiency to process multidimensional inputs as
entities, to encode internal dependencies, and to solve many tasks with less
learning parameters than real-valued models. This paper proposes to integrate
multiple feature views in quaternion-valued convolutional neural network
(QCNN), to be used for sequence-to-sequence mapping with the CTC model.
Promising results are reported using simple QCNNs in phoneme recognition
experiments with the TIMIT corpus. More precisely, QCNNs obtain a lower phoneme
error rate (PER) with less learning parameters than a competing model based on
real-valued CNNs.Comment: Accepted at INTERSPEECH 201
Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets
Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components.
For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper.
We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles.
Quaternions are isomorphic to an algebra of structured 4x4 real matrices.
This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms.
A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance.
Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs.
Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces
Radio astronomical polarimetry and phase-coherent matrix convolution
A new phase-coherent technique for the calibration of polarimetric data is
presented. Similar to the one-dimensional form of convolution, data are
multiplied by the response function in the frequency domain. Therefore, the
system response may be corrected with arbitrarily high spectral resolution,
effectively treating the problem of bandwidth depolarization. As well, the
original temporal resolution of the data is retained. The method is therefore
particularly useful in the study of radio pulsars, where high time resolution
and polarization purity are essential requirements of high-precision timing. As
a demonstration of the technique, it is applied to full-polarization baseband
recordings of the nearby millisecond pulsar, PSR J0437-4715.Comment: 8 pages, 4 figures, accepted for publication in Ap
Tracking 3-D Rotations with the Quaternion Bingham Filter
A deterministic method for sequential estimation of 3-D rotations is presented. The Bingham distribution is used to represent uncertainty directly on the unit quaternion hypersphere. Quaternions avoid the degeneracies of other 3-D orientation representations, while the Bingham distribution allows tracking of large-error (high-entropy) rotational distributions. Experimental comparison to a leading EKF-based filtering approach on both synthetic signals and a ball-tracking dataset shows that the Quaternion Bingham Filter (QBF) has lower tracking error than the EKF, particularly when the state is highly dynamic. We present two versions of the QBF, suitable for tracking the state of first- and second-order rotating dynamical systems
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