Eduardo Bayro-Corrochano

The Theory and Use of the Quaternion Wavelet Transform

This paper presents the theory and practicalities of the quaternion wavelet transform (QWT). The major contribution of this work is that it generalizes the real and complex wavelet transforms and derives a quaternionic wavelet pyramid for multi-resolution analysis using the quaternionic phase concept. As a illustration we present an application of the discrete QWT for optical flow estimation. For the estimation of motion through different resolution levels we use a similarity distance evaluated by means of the quaternionic phase concept and a confidence mask. We show that this linear approach is amenable to be extended to a kind of quadratic interpolation.

Motor Algebra for 3D Kinematics: The Case of the Hand-Eye Calibration

In this paper we apply the Clifford geometric algebra for solving problems of visually guided robotics. In particular, using the algebra of motors we model the 3D rigid motion transformation of points, lines and planes useful for computer vision and robotics. The effectiveness of the Clifford algebra representation is illustrated by the example of the hand-eye calibration. It is shown that the problem of the hand-eye calibration is equivalent to the estimation of motion of lines. The authors developed a new linear algorithm which estimates simultaneously translation and rotation as components of rigid motion.

Clifford Support Vector Machines for Classification, Regression, and Recurrence

This paper introduces the Clifford support vector machines (CSVM) as a generalization of the real and complex-valued support vector machines using the Clifford geometric algebra. In this framework, we handle the design of kernels involving the Clifford or geometric product. In this approach, one redefines the optimization variables as multivectors. This allows us to have a multivector as output. Therefore, we can represent multiple classes according to the dimension of the geometric algebra in which we work. We show that one can apply CSVM for classification and regression and also to build a recurrent CSVM. The CSVM is an attractive approach for the multiple input multiple output processing of high-dimensional geometric entities. We carried out comparisons between CSVM and the current approaches to solve multiclass classification and regression. We also study the performance of the recurrent CSVM with experiments involving time series. The authors believe that this paper can be of great use for researchers and practitioners interested in multiclass hypercomplex computing, particularly for applications in complex and quaternion signal and image processing, satellite control, neurocomputation, pattern recognition, computer vision, augmented virtual reality, robotics, and humanoids.

Multi-resolution image analysis using the quaternion wavelet transform

Abstract This paper presents the theory and practicalities of the quaternion wavelet transform. The contribution of this work is to generalize the real and complex wavelet transforms and to derive for the first time a quaternionic wavelet pyramid for multi-resolution analysis using the quaternion phase concept. The three quaternion phase components of the detail wavelet filters together with a confidence mask are used for the computation of a denser image velocity field which is updated through various levels of a multi-resolution pyramid. Our local model computes the motion by the linear evaluation of the disparity equations involving the three phases of the quaternion detail high-pass filters. A confidence measure singles out those regions where horizontal and vertical displacement can reliably be estimated simultaneously. The paper is useful for researchers and practitioners interested in the theory and applications of the quaternion wavelet transform.

The dual quaternion approach to hand-eye calibration

In order to relate measurements made by a sensor mounted on a mechanical link to the robots coordinate frame we must first estimate the transformation between the sensor and the link frame. In this paper we introduce the use of dual quaternions which are the algebraic counterpart of screws. We prove algebraically that if we consider the camera and motor transformations as screws, then only the line coefficients of the screw axes are relevant regarding the hand-eye calibration. This new parametrization enables us to simultaneously solve for the hand-eye rotation and translation using the singular value decomposition.

Geometric algebra: a framework for computing point and line correspondences and projective structure using n uncalibrated cameras

We present geometric algebra as system for analysing the geometry of multiple-view images. The power of this approach is illustrated by giving purely geometric derivations of the constraints, for point and line correspondences in n-views and via a discussion of projective structure.

Quaternion Fourier Descriptors for the Preprocessing and Recognition of Spoken Words Using Images of Spatiotemporal Representations

Abstract This paper presents an application of the quaternion Fourier transform for the preprocessing for neural-computing. In a new way the 1D acoustic signals of French spoken words are represented as 2D signals in the frequency and time domain. These kind of images are then convolved in the quaternion Fourier domain with a quaternion Gabor filter for the extraction of features. This approach allows to greatly reduce the dimension of the feature vector. Two methods of feature extraction are tested. The features vectors were used for the training of a simple MLP, a TDNN and a system of neural experts. The improvement in the classification rate of the neural network classifiers are very encouraging which amply justify the preprocessing in the quaternion frequency domain. This work also suggests the application of the quaternion Fourier transform for other image processing tasks.

Inverse kinematics, fixation and grasping using conformal geometric algebra

In this paper the authors introduce the conformal geometric algebra in the field of visually guided robotics. As opposite to the standard projective geometry, we can deal simultaneously with incidence algebra operations and conformal transformations. As a result this framework appears promising for dealing with kinematics, dynamics and projective geometry problems without the need to abandon the mathematical system (as current approaches). Using this framework the authors compute the inverse kinematics of a robot arm and a pan-tilt unit and solve a problem of visually guided grasping.

A new methodology for computing invariants in computer vision

We present geometric algebra, as a new, framework for the theory and computation of invariants in computer vision and compare it with the currently popular Grassmann-Cayley algebra. We also discuss the formation, of 3D projective invariants in terms of image coordinates.

Conformal Geometric Algebra for Robotic Vision

In this paper the authors introduce the conformal geometric algebra in the field of visually guided robotics. This mathematical system keeps our intuitions and insight of the geometry of the problem at hand and it helps us to reduce considerably the computational burden of the problems.As opposite to the standard projective geometry, in conformal geometric algebra we can deal simultaneously with incidence algebra operations (meet and join) and conformal transformations represented effectively using spinors. In this regard, this framework appears promising for dealing with kinematics, dynamics and projective geometry problems without the need to resort to different mathematical systems (as most current approaches do). This paper presents real tasks of perception and action, treated in a very elegant and efficient way: body–eye calibration, 3D reconstruction and robot navigation, the computation of 3D kinematics of a robot arm in terms of spheres, visually guided 3D object grasping making use of the directed distance and intersections of lines, planes and spheres both involving conformal transformations. We strongly believe that the framework of conformal geometric algebra can be, in general, of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems.