Geometric analysis and computing of projective structure and motion

Real physical systems with reflective and rotational symmetries such as viruses, fullerenes and quasicrystals have recently been modeled successfully in terms of three-dimensional (affine) Coxeter groups. Motivated by this progress, we explore here the benefits of performing the relevant computations in a Geometric Algebra framework, which is particularly suited to describing reflections. Starting from the Coxeter generators of the reflections, we describe how the relevant chiral (rotational), full (Coxeter) and binary polyhedral groups can be easily generated and treated in a unified way in a versor formalism. In particular, this yields a simple construction of the binary polyhedral groups as discrete spinor groups. These in turn are known to generate Lie and Coxeter groups in dimension four, notably the exceptional groups D_4, F_4 and H_4. A Clifford algebra approach thus reveals an unexpected connection between Coxeter groups of ranks 3 and 4. We discuss how to extend these considerations and computations to the Conformal Geometric Algebra setup, in particular for the non-crystallographic groups, and construct root systems and quasicrystalline point arrays. We finally show how a Clifford versor framework sheds light on the geometry of the Coxeter element and the Coxeter plane for the examples of the two-dimensional non-crystallographic Coxeter groups I_2(n) and the three-dimensional groups A_3, B_3, as well as the icosahedral group H_3.

Under geometric algebra framework, generalized definition of instantaneous power quantity in the multiphase system without any co-ordinate transformation is possible. In the area of active power filtering, with an objective to reduce inverter capacity, the Series Hybrid Active Power Filter (SHAPF) has been taken into account increasingly. This paper proposes a novel concept of decomposing the multiphase voltage vector into different quantities that correspond to different components of the power quantity. The separated components of voltage vector are directly used to derive reference for SHAPF. The mathematical formulation of proposed control scheme with its applications to SHAPF is presented. The validity of the proposed control scheme is verified by the simulation study.

A method of incorporating the results of Grassmann calculus within the framework of geometric algebra is presented, and shown to lead to a new concept, the multivector Lagrangian. A general theory for multivector Lagrangians is outlined, and the crucial role of the multivector derivative is emphasised. A generalisation of Noether’s theorem is derived, from which conserved quantities can be found conjugate to discrete symmetries.

We present an autonomous visual landmark recognition and pose estimation algorithm designed for use in navigation of spacecraft around small asteroids. Landmarks are selected as generic points on the asteroid surface that produce strong Harris corners in an image under a wide range in viewing and illumination conditions; no particular type of morphological feature is required. The set of landmarks is triangulated to obtain a tightly fitting mesh representing an optimal low resolution model of the natural asteroid shape, which is used onboard to determine the visibility of each landmark and enables the algorithm to work with highly concave bodies. The shape model is also used to estimate the centre of brightness of the asteroid and eliminate large translation errors prior to the main landmark recognition stage. The algorithm works by refining an initial estimate of the spacecraft position and orientation. Tests with real and synthetic images show good performance under realistic noise conditions. Using simulated images, the median landmark recognition error is 2m, and the error on the spacecraft position in the asteroid body frame is reduced from 45m to 21m at a range of 2km from the surface. With real images the translation error at 8km to the surface increases from 107m to 119m, due mainly to the larger range and lack of sensitivity to translations along the camera boresight. The median number of landmarks detected in the simulated and real images is 59 and 44 respectively. This algorithm was partly developed and tested during industrial studies for the European Space Agency’s Marco Polo-R asteroid sample return mission.

In signal processing, the approach of the analytic signal is a capable and often used method. For signals of finite length, quadrature filters yield a bandpass filtered approximation of the analytic signal. In the case of multidimensional signals, the quadrature filters can only be applied with respect to a preference direction. Therefore, the orientation has to be sampled, steered or orientation adaptive filters have to be used. Up to now, there has been no linear approach to obtain an isotropic analytic signal which means that the amplitude is independent of the local orientation. In this paper, we present such an approach using the framework of geometric algebra. Our result is closely related to the Riesz transform and the structure tensor. It is seamless embedded in the framework of Clifford analysis. In a suitable coordinate system, the filter response contains information about local amplitude, local phase and local orientation of intrinsically one-dimensional signals. We have tested our filters on two- and three-dimensional signals.Keywordsquadrature filteranalytic signalRiesz transform

A position and attitude control law is developed using geometric algebra (GA). GA is a powerful representational and computational system for geometry. The rigid body motion can be represented by the versor product in GA framework. Using the kinematics of the motor (the versor which represents the rigid body motion in GA), the control law of the rigid body motion can be developed. This paper provides a GA-based position and attitude control law by using the negative feedback of the motor. The stability of the control law is validated by the Lyapunov theorem and the numerical simulation.

In this letter, a structure filling and matching method is proposed for the 3-D reconstruction of buildings, particularly those with combinational structures, from a single high-resolution synthetic aperture radar (SAR) image. This method consists of two stages. First, the structure model of the investigated building is constructed by filling the building structure elements, i.e., walls and roofs, into an initialized empty scene. Subsequently, a series of simulated SAR images is generated by this structure model and matched with the real SAR image of the building. By finding the maximal mutual information between the simulated and real images, the optimal geometric parameters of the model are obtained, which make the output more accurate. Compared with other methods, this method can reconstruct different kinds of practical buildings and retrieve their exact geometric parameters, using only a single SAR image. The performance of this method is evaluated on real TerraSAR-X images to show its effectiveness.

On computer vision field Structure from Motion (SfM) algorithms offer good advantages for numerous applications (augmented reality, autonomous navigation, motion capture, remote sensing, object recognition, image-base 3D modeling, among others), nevertheless, these algorithms show some weakness; in the present paper we propose the use of Bio-inspired Aging Model-PSO (BAM-PSO) to improve the accuracy of SfM algorithms. The BAM-PSO algorithm is used over a Geometric Algebra (GA) framework in order to compute the rigid movement on images and this allows us to obtain a numerically stable algorithm.

Abstract. We propose a new adaptive strategy for text recognition that attempts to derive knowledge about the dominant font on a given page. The strategy uses a linguistic observation that over half of all words in a typical English passage are contained in a small set of less than 150 stop words. A small dictionary of such words is compiled from the Brown corpus. An arbitrary text page first goes through layout analysis that produces word segmentation. A fast procedure is then applied to locate the most likely candidates for those words, using only widths of the word images. The identity of each word is determined using a word shape classifier. Using the word images together with their identities, character prototypes can be extracted using a previously proposed method. We describe experiments using simulated and real images. In an experiment using 400 real page images, we show that on average, eight distinct characters can be learned from each page, and the method is successful on 90% of all the pages. These can serve as useful seeds to bootstrap font learning.

Hyperspectral image provides a way to analyze component and the fraction by spectral unmixing. The classical method is Least Squares algorithm. But it is defined without any constraint on the abundance. The Fully constrained Least Squares (FCLS) algorithm allows resulted abundances satisfying sum-to-one and non-negative, which is practical. It has been widely used till now because it may work quickly with a small unmxing error. But when the number of endmembers is large, it gets much slow. And unmxing error is high if the endmembers are not found out completely. As for improving the speed of the algorithm, there have been many research works. But fewer algorithms are developed for increasing accurateness. Here in this paper, a primal-dual interior-point method is tested for the fully constraint unmixing with the linear unmixing model being modified slightly. A dual problem is established firstly, and then the process of the algorithm is designed to decrease distance between original problem and the dual problem based on result of FCLS. In order to promise high unmxing accurateness, the unmxing error of resulted abundances is added into the optimizing goal. Several algorithms of fully constraint abundance estimation are tested on both simulation and real hyperspectral images. The experimental results show that the prompted algorithm may find the optimal abundance for existing endmembers even though the endmember set is not complete. Accurateness of unmxing is increased, especially on the real image.

A robust star identification algorithm with star shortlisting

This thesis presents a nonlinear method to estimate the fundamental matrix, a key problem arising in projective motion estimation and reconstruction, based on a general Errors-in-Variables (EV) model. In this model, the method considers that all the measurements are corrupted by noises, and minimizes a cost function derived from a nonlinear criterion to estimate both the fundamental matrix and corrupted data, involving the rank-2 constraint With reasonably adjusted data, this method turns out to significantly increase the accuracy and robustness with a simple form of computation. The performance of the proposed approach is justified by theory and assessed by several experiments on real images.

Body sensor calibration and construction of 3D maps for robot navigation using the framework of conformal geometric algebra

In recent years, automatic target recognition (ATR) based on deep learning has achieved great success in RGB field, which has huge data support. However, due to the confidentiality of military targets, weather constraints, and high shooting costs, it is difficult to obtain a large number of real IR images which leads to the performance degradation of deep learning algorithms in IR field. This paper discusses the method of using simulation IR images as training set to get rid of dependence on the real image. However, there are still great differences between the original simulated image and the real image, which leads to many defects when using the original simulated image for training. Therefore, in this paper, we use cycleGAN to convert the original simulation image into the intermediate image closer to the real image based on generative adversarial networks (GAN). Finally, the effectiveness of this method is proved by experiments.

Practical simulations of low-dose CT images have a possibility of being helpful means for optimization of the CT exposure dose. Because current methods reported by several researchers are limited to specific vendor platforms and generally rely on raw sinogram data that are difficult to access, we have developed a new computerized scheme for producing simulated low-dose CT images from real high-dose images without use of raw sinogram data or of a particular phantom. Our computerized scheme for low-dose CT simulation was based on the addition of a simulated noise image to a real high-dose CT image reconstructed by the filtered back-projection algorithm. First, a sinogram was generated from the forward projection of a high-dose CT image. Then, an additional noise sinogram resulting from use of a reduced exposure dose was estimated from a predetermined noise model. Finally, a noise CT image was reconstructed with a predetermined filter and was added to the real high-dose CT image to create a simulated low-dose CT image. The noise power spectrum and modulation transfer function of the simulated low-dose images were very close to those of the real low-dose images. In order to confirm the feasibility of our method, we applied this method to clinical cases which were examined with the high dose initially and then followed with a low-dose CT. In conclusion, our proposed method could simulate the low-dose CT images from their real high-dose images with sufficient accuracy and could be used for determining the optimal dose setting for various clinical CT examinations.