<title>Line-photogrammetric mathematical model for the reconstruction of polyhedral objects</title>

The problem of estimating the motion and orientation parameters of a rigid object from two m - D point set patterns is of significant importance in medical imaging, electrocardiogram (ECG) alignment, and fingerprint matching. The rigid parameters can be defined by an m times m rotation matrix, a diagonal m times m scale matrix, and an m times 1 translation vector. All together, the total number of parameters to be found is m(m + 2). Several least squares based algorithms have recently appeared in the literature. These algorithms are all based on a singular value decomposition (SVD) of the m times m cross-covariance matrix between the two data sets. However, there are cases where the SVD based algorithms return a reflection matrix rather than a rotation matrix. Some authors have introduced a simple correction for guarding against such cases. Other types of algorithm are based on unit quaternions which guarantee obtaining a true rotation matrix. In this paper we introduce a principal component based registration algorithm which is solved in closed-form. By using matrix vectorization properties the problem can be cast as one of finding a rank-1 symmetric projection matrix. This is equivalent to solving a Sylvester equation with equality constraints. Once the solution is obtained, we apply the inverse vectorization operation to estimate the rotation and scale matrices, along with the translation vector. We apply the proposed algorithm to the alignment of ECG signals and compare the results to those obtained by the SVD and quaternion based algorithms

3D reconstruction from a single image using geometric constraints

The subject of this paper is the research that aims at efficiency improvement of acquisition of 3D building models from digital images for Computer Aided Architectural Design (CAAD). The results do not only apply to CAAD, but to all applications where polyhedral objects are involved. The research is concentrated on the integration of a priori geometric object information in the modeling process. Parallelism and perpendicularity are examples of the a priori information to be used. This information leads to geometric constraints in the mathematical model. This model can be formulated using condition equations with observations only. The advantage is that the adjustment does not include object parameters and the geometric constraints can be incorporated in the model sequentially. As with the use of observation equations statistical testing can be applied to verify the constraints. For the initial values of orientation parameters of the images we use a direct solution based on a priori object information as well. For this method only two sets of (coplanar) parallel lines in object space are required. The paper concentrates on the mathematical model with image lines as the main type of observations. Advantages as well as disadvantages of a mathematical model with only condition equations are discussed. The parametrization of the object model plays a major role in this discussion.

There are three main frameworks to describe the orientation and rotation of non-spherical particles: Euler angles, rotation matrices and unit Quaternions. Of these methods, the latter seems the most attractive for describing the behaviour of non-spherical particles. However, there are a number of drawbacks when using unit Quaternions: the necessity of applying rotation matrices in conjunction to facilitate the transformation from body space to world space, and the algorithm integrating the Quaternion should inherently conserve the length of the Quaternion. Both drawbacks are addressed in this paper. The present paper derives a new framework to transform vectors and tensors by unit Quaternions, and the requirement of explicitly using rotation matrices is removed altogether. This means that the algorithm derived in this paper can describe the rotation of a non-spherical particle with four parameters only. Moreover, this paper introduces a novel corrector-predictor method to integrate unit Quaternions, which inherently conserves the length of the Quaternion. The novel framework and method are compared to a number of other methods put forward in the literature. All the integration methods are discussed, scrutinised and compared to each other by comparing the results of four test cases, involving a single falling particle, nine falling and interacting particles, a 2D prescribed torque on a sphere and a 3D prescribed torque on a non-spherical particle. Moreover, a convergence study is presented, comparing the rate of convergence of the various methods. All the test cases show a significant improvement of the new framework put forward in this paper over existing algorithms. Moreover, the new method requires less computational memory and fewer operations, due to the complete omission of the rotation matrix in the algorithm.

Mathematical Modeling for Slide Tracheoplasty.

Background. The bulk of the work on dual quaternions is devoted to their application to describe helical motion. Little attention is paid to the representation of points, lines, and planes (primitives) using them. Purpose. It is necessary to consistently present the dual quaternion theory of the representation of primitives and refine the mathematical formalism. Method. It uses the algebra of dual numbers, quaternions and dual quaternions, as well as elements of the theory of screws and sliding vectors. Results. Formulas have been obtained and systematized that use exclusively dual quaternionic operations and notation to solve standard problems of three-dimensional geometry. Conclusions. Dual quaternions can serve as a full-fledged formalism for the algebraic representation of a three-dimensional projective space.

A new perceptual organization approach to 3D measuring system based on the fuzzy integral

Formulation of proper and efficient algorithms for robot kinematics is essential for the analysis and design of serial manipulators. Kinematic modeling of manipulators is most often performed in Cartesian space. However, due to disadvantages of most widely used mathematical constructs for description of orientation such as Euler angles and rotational matrices, a need for unambiguous, compact, singularity free, computationally efficient method for representing rotational information is imposed. As a solution, unit quaternions are proposed and kinematic modeling in dual quaternion space arose. In this paper, an overview of spatial descriptions and transformations that can be applied together within these spaces in order to solve kinematic problems is presented. Special emphasis is on a different mathematical formalisms used to represent attitude of a rigid body such as rotation matrix, Euler angles, axis-angle representation, unit quaternions, and their mutual relation. Benefits of kinematic modeling in quaternion space are presented. New direct kinematics algorithm in dual quaternion space pertaining to a particular manipulator is given. These constructs and algorithms are demonstrated on the human centrifuge as 3 DoF robot manipulator.

A laser Doppler vibrometer (LDV) combined with a pair of orthogonal scan mirrors is a scanning LDV (SLDV). A mathematical model based on the scan mirrors configuration for determining the orientation and position of a SLDV in a structure coordinate system (SCS) is presented in this work. Coordinates of a scan point in a scan mirrors coordinate system (SMCS) are derived from the mathematical model and the relation of coordinates of the same scan point in the SMCS and SCS is derived using the rigid transformation theory. A rotation matrix and a translation vector from the SMCS to SCS are obtained by applying the least squares method and singular value decomposition. An experiment to scan a 3D structure was performed with three SLDVs placed at three different locations. Rotation matrices obtained from two different sets of reference points are almost the same at the three locations, which validates that the proposed methodology is reliable. Another experiment to scan a 2D clamped plate was performed and the rotation matrices were calculated from the proposed methodology as a special case, which shows universal applicability of the proposed methodology and extends its application scope from 3D structures to 2D structures.

Chapter 2 - Attitude Representation

Automatic reasoning for geometric constraints in 3D city models with uncertain observations

Analysis and design of control systems via parameter‐based approach

Head pose estimation suffers from several problems, including low pose tolerance under different disturbances and ambiguity arising from common head pose representation. In this study, a robust three-branch model with triplet module and matrix Fisher distribution module is proposed to address these problems. Based on metric learning, the triplet module employs triplet architecture and triplet loss. It is implemented to maximize the distance between embeddings with different pose pairs and minimize the distance between embeddings with same pose pairs. It can learn a highly discriminate and robust embedding related to head pose. Moreover, the rotation matrix instead of Euler angle and unit quaternion is utilized to represent head pose. An exponential probability density model based on the rotation matrix (referred to as the matrix Fisher distribution) is developed to model head rotation uncertainty. The matrix Fisher distribution can further analyze the head pose, and its maximum likelihood obtained using singular value decomposition provides enhanced accuracy. Extensive experiments executed over AFLW2000 and BIWI datasets demonstrate that the proposed model achieves state-of-the-art performance in comparison with traditional methods.

Often one must find the quaternion corresponding to a given rotation matrix. Of themanymethods that have been proposed for performing this computation [3–9], Shepperd’s algorithm [5], which is singularity free and requires only one square root, has been the most widely applied. In this Note, we review Shepperd’s algorithm and present a variant that always produces a normalized quaternion even if numerical errors cause the matrix A to be only approximately orthogonal. This modification of Shepperd’s algorithm also provides a very efficient method for computing an exactly orthogonal matrix that is close to an approximately orthogonal matrix.

A geometric constraint solving method is presented that takes well-constrained mating conditions between a base and a mating component and directly transforms them into a 4 × 4 matrix that determines the relative orientation and location of the mating component with respect to the base component. In the proposed procedure the 4 × 4 transformation matrix is determined by directly computing a rotation matrix T R and a translation matrix T L that define the relative orientation and location of the mating component, respectively. Thus first, the rotation matrix is computed by solving a set of linear constraint equations associated with the orientation of two mating components After repositioning the mating component by applying the rotation matrix T R , the translation matrix is calculated by solving a set of linear constraint equations associated with location. This new method is computationally very effective, since the transformation matrix for relative orientation and location of the mating component is algebraically derived directly from the linear equations associated with the mating conditions.