The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

The two-sided quaternionic Fourier transformation (QFT) was introduced in \cite{Ell:1993} for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical investigations \cite{10.1007/s00006-007-0037-8, EH:DirUP_QFT} a special split of quaternions was introduced, then called $\pm$split. In the current \change{chapter} we analyze this split further, interpret it geometrically as \change{an} \emph{orthogonal 2D planes split} (OPS), and generalize it to a freely steerable split of \H into two orthogonal 2D analysis planes. The new general form of the OPS split allows us to find new geometric interpretations for the action of the QFT on the signal. The second major result of this work is a variety of \emph{new steerable forms} of the QFT, their geometric interpretation, and for each form\change{,} OPS split theorems, which allow fast and efficient numerical implementation with standard FFT software.Comment: 25 pages, 5 figure

Factorization of Rational Curves in the Study Quadric and Revolute Linkages

Given a generic rational curve $C$ in the group of Euclidean displacements we construct a linkage such that the constrained motion of one of the links is exactly $C$. Our construction is based on the factorization of polynomials over dual quaternions. Low degree examples include the Bennett mechanisms and contain new types of overconstrained 6R-chains as sub-mechanisms.Comment: Changed arxiv abstract, corrected some type

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

Deep Quaternion Networks

The field of deep learning has seen significant advancement in recent years. However, much of the existing work has been focused on real-valued numbers. Recent work has shown that a deep learning system using the complex numbers can be deeper for a fixed parameter budget compared to its real-valued counterpart. In this work, we explore the benefits of generalizing one step further into the hyper-complex numbers, quaternions specifically, and provide the architecture components needed to build deep quaternion networks. We develop the theoretical basis by reviewing quaternion convolutions, developing a novel quaternion weight initialization scheme, and developing novel algorithms for quaternion batch-normalization. These pieces are tested in a classification model by end-to-end training on the CIFAR-10 and CIFAR-100 data sets and a segmentation model by end-to-end training on the KITTI Road Segmentation data set. These quaternion networks show improved convergence compared to real-valued and complex-valued networks, especially on the segmentation task, while having fewer parametersComment: IJCNN 2018, 8 pages, 1 figur