Showing 1 to 45 of 45 matching Articles
Results per page:
Export (CSV)
By
BayroCorrochano, Eduardo
In this paper the authors use the framework of geometric algebra for applications in computer vision, robotics and learning . 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. The authors show that framework of geometric algebra can be in general of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems. For learning the paper presents the Clifford Support Vector Machines as a generalization of the real and complexvalued Support Vector Machines.
more …
By
AltamiranoGómez, Gerardo E.; BayroCorrochano, Eduardo
3 Citations
In this paper, we introduce a novel geometric voting scheme that extends previous algorithms, like Hough transform and tensor voting, in order to tackle perceptual organization problems. Our approach is grounded in three methodologies: representation of information using Conformal Geometric Algebra, a local voting process, which introduce global perceptual considerations at low level, and a global voting process, which clusters salient geometric entities in the whole image. Since geometric algebra is the mathematical framework of our approach, our algorithm infers highlevel geometric representations from tokens that are perceptually salient in an image.
more …
By
CasarrubiasVargas, Heriberto; PetrilliBarceló, Alberto; BayroCorrochano, Eduardo
1 Citations
The understanding of scenes is a key aspect of computer vision. Edge detection helps us to understand more about the scene structure since the edges mark a clear distinction for a transition from one region with similar properties to another one. When the edges are obtained from changes in orientation, we can use them to find key planes and describe the scene. This paper describes a method for fast edge detection in RGBD images. The edge detection algorithm for depth images is based on the use of smooth constraints in orientation. We show experimental results that demonstrate the potential of the approach proposed for edge detection.
more …
By
BayroCorrochano, Eduardo
Abstract
In this paper the authors use the framework of geometric algebra for applications in computer vision, robotics and learning . 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. The authors show that framework of geometric algebra can be in general of great advantage for applications using stereo vision, range data, laser, omnidirectional and odometry based systems. For learning the paper presents the Clifford Support Vector Machines as a generalization of the real and complexvalued Support Vector Machines.
more …
By
BayroCorrochano, Eduardo; Torre Gomora, Miguel Angel
Abstract
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 multiresolution analysis using three phases. The paper can be very useful for researchers and practitioners interested in understanding and applications of the quaternion wavelet transform.
more …
By
GonzalezAguirre, David; Asfour, Tamim; BayroCorrochano, Eduardo; Dillmann, Ruediger
Show all (4)
3 Citations
A novel modelbased approach for global selflocalization using active stereo vision and density Gaussian spheres is presented. The proposed object recognition components deliver noisy percept subgraphs, which are filtered and fused into an egocentered reference frame. In subsequent stages, the required visiontomodel associations are extracted by selecting egopercept subsets in order to prune and match the corresponding worldmodel subgraph. Ideally, these coupled subgraphs hold necessary information to obtain the modeltoworld transformation, i.e., the pose of the robot. However, the estimation of the pose is not robust due to the uncertainties introduced when recovering Euclidean metric from images and during the mapping from the camera to the egocenter. The approach models the uncertainty of the percepts with a radial normal distribution. This formulation allows a closedform solution which not only derives the maximal density position depicting the optimal egocenter but also ensures the solution even in situations where pure geometric spheres might not intersect.
more …
By
BayroCorrochano, Eduardo; ZamoraEsquivel, Julio; LópezFranco, Carlos
In this paper the authors use the framework of conformal geometric algebra for the treatment of robot vision tasks. In this mathematical system we calculated projective invariants using omnidirectional vision for object recognition. We show the power of the mathematical system for handling differential kinematics in visual guided tracking.
more …
By
BayroCorrochano, Eduardo; Torre Gomora, Miguel Angel
3 Citations
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 multiresolution analysis using three phases. The paper can be very useful for researchers and practitioners interested in understanding and applications of the quaternion wavelet transform.
more …
By
LópezGonzález, G.; AranaDaniel, Nancy; BayroCorrochano, Eduardo
1 Citations
This paper presents the Quaternion Support Vector Machines for classification as a generalization of the real and complex valued Support Vector Machines. In this framework we handle the design of kernels involving the Clifford or quaternion product. The QSVM allows to change the metric involved in the quaternion product. The application section shows experiments in pattern recognition and colour image processing.
more …
By
MoyaSánchez, E. Ulises; BayroCorrochano, Eduardo
Atomic Functions are widely used in different applications in image processing, pattern recognition, computational physics and also in the digital interpretation of signal measurements. In 1D signals, is usual to compute the phase and the magnitude of a signal using the analytic signal (the signal and its Hilbert transform using complex numbers). However, for high dimensional signals the monogenic signal (the signal and its Riesz transform) has been used to obtain the local phase and orientation with good results. The main aim of this work is to present a new way to make the computation of the Hilbert transform using the atomic function. The computation of the Hilbert transform take relevance when the phase computation is required.
more …
By
BayroCorrochano, Eduardo; AranaDaniel, Nancy; VallejoGutiérres, J. Refugio
This paper introduces the Clifford Support Vector Machines as a generalization of the real and complex valued Support Vector Machines. The major advantage of this approach is that one requires only one CSVM which can admit multiple multivector inputs and it can carry multiclass classification. In contrast one would need many real valued SVMs for a multiclass problem which is time consuming.
more …
By
BayroCorrochano, Eduardo; Trujillo, Noel; Naranjo, Michel
32 Citations
This paper presents an application of the quaternion Fourier transform for the preprocessing for neuralcomputing. 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.
more …
By
RiveraRovelo, Jorge; Herold, Silena; BayroCorrochano, Eduardo
2 Citations
In this paper we present a method based on selforganizing neural networks to extract the shape of a 2D or 3D object using a set of transformations expressed as versors in the conformal geometric algebra framework. Such transformations, when applied to any geometric entity of this geometric algebra, define the shape of the object. This approach was tested with several images, but here we show its utility first using a 2D magnetic resonance image to segment the ventricle. Then we present some examples of an application for the case of 3D objects.
more …
By
Bayro Kaiser, Esteban Tobias; CorreaArameda, Eduardo; BayroCorrochano, Eduardo
This paper presents a gray scale image compression method using the Wavelet Transform and key feature detection for mobile phone video transmission. The major contribution of this work is to show the application of the wavelet transform in image compression and to add a new method to reduce redundant information in video transmission which is key feature detection. An algorithm is designed in Matlab to accomplish this task using a face to face video.
more …
By
CastilloMuñiz, Efrain; RiveraRovelo, Jorge; BayroCorrochano, Eduardo
1 Citations
The use of haptic interfaces in surgery could provide the surgeon useful sensing information about the patient tissues. Our goal in this work, is to use the haptic interface to obtain some sample points on the surface of an object or organ tissue in medical applications. This elasticity information feeds an artificial neural network. The output of the neural network is an approximation of the compliance of the object which is touched, as well as the coordinates of 3D surface points which in fact are used for the 3D surface reconstruction of the object. Experimental results show that the reconstruction of objects from a elasticity point of view is possible, and that the use of a haptic interface can improve the performance of 3D reconstruction algorithms.
more …
By
MoyaSánchez, E. Ulises; BayroCorrochano, Eduardo
5 Citations
Atomic Functions are widely used in different applications in image processing, pattern recognition, computational physics and also in the digital interpretation of signal measurements. The main contribution of this work is to develop a Quaternionic Atomic Function Wavelet as a new quaternionic image wavelet transform. This filter have a real part and three imaginary parts (i,j,k) of the Quaternion Atomic Function, as a result we can extract more information from the image by the three phases (φ,θ, ϕ) of the quaternion representation. The experimental part shows clearly that the phase information of the image is not afected by illumination changes.
more …
By
Reyes, Leo; BayroCorrochano, Eduardo
1 Citations
In this paper, we compare the various methods for the simultaneous and sequential reconstruction of points, lines, planes, quadrics, plane conics and degenerate quadrics using Bundle Adjustment, both in projective and metric space. In contrast, most existing work on projective reconstruction focuses mainly on one type of primitive. We also compare the simultaneous refinement of all primitives through Bundle Adjustment with various sequential methods were only certain primitives are refined together. We found that even though the sequential methods may seem somewhat arbitrary on the choice of which primitives are refined together, a higher precision and speed is achieved in most cases.
more …
By
BayroCorrochano, Eduardo; LópezFranco, Carlos
9 Citations
It has been proven that a catadioptric projection can be modeled by an equivalent spherical projection. In this paper we present an extension and improvement of those ideas using the conformal geometric algebra, a modern framework for the projective space of hyperspheres. Using this mathematical system, the analysis of diverse catadioptric mirrors becomes transparent and computationally simpler. As a result, the algebraic burden is reduced, allowing the user to work in a much more effective framework for the development of algorithms for omnidirectional vision. This paper includes complementary experimental analysis related to omnidirectional vision guided robot navigation.
more …
By
AranaDaniel, Nancy; LópezFranco, Carlos; BayroCorrochano, Eduardo
1 Citations
This paper presents an improvement of a recurrent learning system called LSTMCSVM (introduced in [1]) for robot navigation applications, this approach is used to deal with some of the main issues addressed in the research area: the problem of navigation on large domains, partial observability, limited number of learning experiences and slow learning of optimal policies. The advantages of this new version of LSTMCSVM system, are that it can find optimal paths through mazes and it reduces the number of generations to evolve the system to find the optimal navigation policy, therefore either the training time of the system is reduced. This is done by adding an heuristic methodoly to find the optimal path from start state to the goal state.can contain information about the whole environment or just partial information about it.
more …
By
RiveraRovelo, Jorge; BayroCorrochano, Eduardo; Dillmann, Ruediger
1 Citations
In this work we present an algorithm to approximate the surface of 2D or 3D objects combining concepts from geometric algebra and artificial neural networks. Our approach is based on the selforganized neural network called Growing Neural Gas (GNG), incorporating versors of the geometric algebra in its neural units; such versors are the transformations that will be determined during the training stage and then applied to a point to approximate the surface of the object. We also incorporate the information given by the generalized gradient vector flow to select automatically the input patterns, and also in the learning stage in order to improve the performance of the net. Several examples using medical images are presented, as well as images of automatic visual inspection. We compared the results obtained using snakes against the GSOM incorporating the gradient information and using versors. Such results confirm that our approach is very promising. As a second application, a kind of morphing or registration procedure is shown; namely the algorithm can be used when transforming one model at time t_{1} into another at time t_{2}. We include also examples applying the same procedure, now extended to models based on spheres.
more …
By
RiveraRovelo, Jorge; BayroCorrochano, Eduardo
This paper presents three different tasks: segmentation of medical images, volume representation and nonrigid registration. The first task is a necessary step before volume representation ant it is done with a simple but effective strategy using tomographic images, combining texture and boundary information in a region growing strategy, obtaining good results. For the second task, we present a new approach to model 2D surfaces and 3D volumetric data based on marching cubes idea using however spheres (modeling the surface of an object using spheres allows us to reduce the number of primitives representing it and to benefit from such reduction the registration process of two objects). We compare our approach based on marching cubes idea with other one using Delaunay tetrahedrization, and the results show that our proposed approach reduces considerably the number of spheres. Finally, we show how to do nonrigid registration of two volumetric data represented as sets of spheres using 5dimensional vectors in conformal geometric algebra.
more …
By
Wörsdörfer, Florian; Stock, Florian; BayroCorrochano, Eduardo; Hildenbrand, Dietmar
Show all (4)
1 Citations
The usage of Conformal Geometric Algebra leads to algorithms that can be formulated in a very clear and easy to grasp way. But it can also increase the performance of an implementation because of its capabilities to be computed in parallel. In this paper we show how a grasping algorithm for a robotic arm is accelerated using a Conformal Geometric Algebra formulation. The optimized C code is produced by the CGA framework Gaalop automatically. We compare this implementation with a CUDA implementation and an implementation that uses standard vector algebra.
more …
By
OrtegónAguilar, Jaime; BayroCorrochano, Eduardo
1 Citations
This paper addresses the parameters’ estimation of 2D and 3D transformations. For the estimation we present a method based on system identification theory, we named it the “Amethod”. The transformations are considered as elements of the Lie group GL(n) or one of its subgroups. We represent the transformations in terms of their Lie Algebra elements. The Lie algebra approach assures to follow the shortest path or geodesic in the involved Lie group. To prove the potencial of our method, two experiments are presented. The first one is a monocular estimation of 3D rigid motion of an object in the visual space. With this aim, the six parameters of the rigid motion are estimated based on measurements of the six parameters of the affine transformation in the image. Secondly, we present the estimation of the affine or projective transformations involved in monocular region tracking.
more …
By
LópezGonzález, Gehová; AranaDaniel, Nancy; BayroCorrochano, Eduardo
This work presents a new method to apply the Hough Transform to 2D and 3D cloud points using the conformal geometric algebra framework. The objective is to detect geometric entities, with the use of simple parametric equations and the properties of the geometric algebra. We show with real images and RGBD data that this new method is very useful to detect lines and circles in 2D and planes and spheres in 3D.
more …
By
HeroldGarcía, Silena; RiveraRovelo, Jorge; BayroCorrochano, Eduardo
1 Citations
One necessary task in the operating room is to establish a common reference frame, in order to relate the information obtained from different sensors, and to combine both the preoperative with the intraoperative information. To estimate the transformations between different data, fiducial markers are typically used. In this paper we present a formulation of the known handeye calibration problem, to estimate the transformation between an endoscopic camera and the set of spherical markers placed on it, using the conformal geometric algebra framework. Such markers are tracked by an optical stereo tracking system, which help to relate the real world with the virtual model created before surgery. Experimental results shows that our method is reliable and useful for medical applications in real time like neurosurgery.
more …
By
BernalMarin, Miguel; BayroCorrochano, Eduardo
This paper describes a new approach for building 3D geometric maps using a laser rangefinder, a stereo camera system and a mathematical system the Conformal Geometric Algebra. The use of a known visual landmarks in the map helps to carry out a good localization of the robot. A machine learning technique is used for recognition of objects in the environment. These landmarks are found using the Viola and Jones algorithm and are represented with their position in the 3D virtual map.
more …
By
BayroCorrochano, Eduardo; RiveraRovelo, Jorge
14 Citations
We present a new approach to model 2D surfaces and 3D volumetric data, as well as an approach for nonrigid registration; both are developed in the geometric algebra framework. The approach for modeling is based on marching cubes idea using however spheres and their representation in the conformal geometric algebra; it will be called marching spheres. Note that before we can proceed with the modeling, it is needed to segment the object we are interested in; therefore, we include an approach for image segmentation, which is based on texture and border information, developed in a regiongrowing strategy. We compare the results obtained with our modeling approach against the results obtained with other approach using Delaunay tetrahedrization, and our proposed approach reduces considerably the number of spheres. Afterward, a method for nonrigid registration of models based on spheres is presented. Registration is done in an annealing scheme, as in ThinPlate Spline Robust Point Matching (TPSRPM) algorithm. As a final application of geometric algebra, we track in real time objects involved in surgical procedures.
more …
By
MachuchoCadena, Ruben; CruzRodríguez, Sergio; BayroCorrochano, Eduardo
2 Citations
We present an approach for reconstructing the 3D shape of brain tumors for applications in neurosurgery from 2D ultrasound (US) images. We record simultaneously endoscopic and ultrasonic images, and the pose of the endoscopic camera by an optical tracking system. The 3D pose of the ultrasound probe is determined by tracking the 2D position of the ultrasound probe in successive endoscopic images by image processing techniques (Hough Transform, Particle Filtering) and finally the 3D position is computed by the known camera geometry and multiple view geometry using conformal geometric algebra. When the 3D US probe position is calculated, we compound multiple 2D US images into a 3D volume.
more …
By
Rosenhahn, Bodo; BayroCorrochano, Eduardo
1 Citations
The authors of this paper adopted the characteristics of the image of the absolute conic in terms of Pascal's theorem to propose a new camera calibration method. Employing this theorem in the geometric algebra framework enables the authors to compute a projective invariant using the conics of only two images which expressed using brackets helps us to set enough equations to solve the calibration problem.
more …
By
BayroCorrochano, Eduardo
67 Citations
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 multiresolution 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.
more …
By
Machucho, Ruben; RiveraRovelo, Jorge; BayroCorrochano, Eduardo
Several methods (discrete and continuous) for surface reconstruction have been proposed over the past years. Convex hull is one of them, which is the minimal convex envelope for a set of points X in a real vector space V. We present a method to 3D surface reconstruction which refines the convex hull by means of a peeling process with an adaptive radius. Tests with points of different objects, some of them from the AimatShape Project, were carried out, showing a better approximation than the one using traditional convex hull, and a little reduction in number of points used and computer time elapsed.
more …
By
MachuchoCadena, Ruben; MoyaSánchez, Eduardo; CruzRodríguez, Sergio; BayroCorrochano, Eduardo
Show all (4)
The main result of this work is an approach for reconstructing the 3D shape and pose of tumors for applications in laparoscopy from stereo endoscopic ultrasound images using Conformal Geometric Algebra. We record simultaneously stereo endoscopic and ultrasonic images and then the 3D pose of the ultrasound probe is calculated using conformal geometric algebra. When the position in 3D of the ultrasound probe is calculated, we compound multiple 2D ultrasound images into a 3D volume. To segment 2D ultrasound images we have used morphological operators and compared its performance versus the obtained with segmentation using level set methods.
more …
By
SernaMorales, Andrés F.; Prieto, Flavio; BayroCorrochano, Eduardo
2 Citations
On the process of render brain tumors from endoneurosonography, one of the most important steps consists in track the axis line of an ultrasound probe throughout successive endoscopic images. Recognizing of this line is important because it allows computing its 3D coordinates using the projection matrix of the endoscopic cameras. In this paper we present a method to track an ultrasound probe in successive endoscopic images without relying on any external tracking system. The probe is tracked using a spatiotemporal technique based on optical flow and clustering algorithm. Firstly, we compute the optical flow using the HornSchunck algorithm. Secondly, a feature space using the optical flow magnitude and luminance is defined. Thirdly, feature space is partitioned in two regions using the kmeans clustering algorithm. After this, we calculate the axis line of the ultrasound probe using Principal Component Analysis (PCA) over segmented region. Finally, a motion restriction is defined over consecutive frames in order to avoid tracking errors. We have used endoscopic images from brain phantoms to evaluate the performance of the proposed method, we compare our methodology against ground truth and a based–color particle filter, and our results show that it is robust and accurate.
more …
By
BayroCorrochano, Eduardo
13 Citations
No abstract available
By
BayroCorrochano, Eduardo; ReyesLozano, Leo; ZamoraEsquivel, Julio
24 Citations
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.
more …
By
González Jiménez, Luis Enrique; Loukianov, Alexander; BayroCorrochano, Eduardo
2 Citations
A Discrete Integral Sliding Mode algorithm is proposed to control a Stereo Vision System (SVS) and perform Visual Object Tracking. The kinematical model of the structure is obtained using Geometric Algebra (GA). The localizing part was done in a real SVS in order to obtain the reference for orientation vector and the application for a Pan Tilt Unit is presented. The algorithm presents a good and robust performance.
more …
By
LechugaGutiérrez, Luis; BayroCorrochano, Eduardo
This work presents a new type of Spike Neural Networks (SNN) developed in the quaternion algebra framework. This new neural structure based on SNN is developed using the quaternion algebra. The training algorithm was extended adjusting the weights according to the quaternion multiplication rule, which allows accurate results with a decreased network complexity with respect to the real SNN. The experimental part shows a good performance for robot manipulator control.
more …
By
OrozcoAguirre, Rafael; RiveraRovelo, Jorge; BayroCorrochano, Eduardo
This paper presents a method for segmentation of medical images and the application of the so called geometric or Clifford algebras for volume representation, nonrigid registration of volumes and object tracking. Segmentation is done combining texture and boundary information in a region growing strategy obtaining good results. To model 2D surfaces and 3D volumetric data we present a new approach based on marching cubes idea however using spheres. We compare our approach with other method based on the delaunay tetrahedrization. The results show that our proposed approach reduces considerably the number of spheres. Also we show how to do nonrigid registration of two volumetric data represented as sets of spheres using 5dimensional vectors in conformal geometric algebra. Finally we show the application of geometric algebras to track surgical devices in real time.
more …
By
BayroCorrochano, Eduardo
29 Citations
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 multiresolution 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 multiresolution pyramid. Our local model computes the motion by the linear evaluation of the disparity equations involving the three phases of the quaternion detail highpass 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.
more …
By
BayroCorrochano, Eduardo; MoyaSánchez, Eduardo Ulises
In this work we introduce a new kernel for image processing called the atomic function. This kernel is compact in the spatial domain, and it can be adapted to the behavior of the input signal by broadening or narrowing its band ensuring a maximum signaltonoise ratio. It can be used for smooth differentiation of images in the quaternion algebra framework. We discuss the role of the quaternion atomic function with respect to monogenic signals. We then propose a steerable quaternion wavelet scheme for image structure and contour detection. Making use of the generalized Radon transform and images processed with the quaternion wavelet atomic function transform, we detect shape contours in color images. We believe that the atomic function is a promising kernel for image processing and scene analysis.
more …
By
BayroCorrochano, Eduardo; Banarer, Vladimir
18 Citations
A central task of computer vision is to automatically recognize objects in realworld scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing position. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes in the observation parameters. The study of such geometric invariance is a field of active research. This paper presents the theory and computation of projective invariants formed from points and lines using the geometric algebra framework. This work shows that geometric algebra is a very elegant language for expressing projective invariants using n views. The paper compares projective invariants involving two and three cameras using simulated and real images. Illustrations of the application of such projective invariants in visual guided grasping, camera selflocalization and reconstruction of shape and motion complement the experimental part.
more …
By
BayroCorrochano, Eduardo; AranaDaniel, Nancy
This paper introduces the Clifford Support Vector Machines (CSVM) as a generalization of the real and complexvalued Support Vector Machines using the Clifford geometric algebra. In this framework we handle the design of kernels involving the Clifford or geometric product for linear and nonlinear classification and regression. The major advantage of our approach is that we redefine 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 conduct comparisons between CSVM and the most used approaches to solve multiclass classification to show that our approach is more suitable for practical use on certain type of multiclass classification problems.
more …
By
BayroCorrochano, Eduardo; Daniilidis, Kostas; Sommer, Gerald
42 Citations
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 handeye calibration. It is shown that the problem of the handeye 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.
more …
By
BayroCorrochano, Eduardo; Zhang, Yiwen
19 Citations
In this paper the motor algebra for linearizing the 3D Euclidean motion of lines is used as the oretical basis for the development of a novel extended Kalman filter called the motor extended Kalman filter (MEKF). Due to its nature the MEKF can be used as online approach as opposed to batch SVD methods. The MEKF does not encounter singularities when computing the Kalman gain and it can estimate simultaneously the translation and rotation transformations. Many algorithms in the literature compute the translation and rotation transformations separately. The experimental part demonstrates that the motor extended Kalman filter is an useful approach for estimation of dynamic motion problems. We compare the MEKF with an analytical method using simulated data. We present also an application using real images of a visual guided robot manipulator; the aim of this experiment is to demonstrate how we can use the online MEKF algorithm. After the system has been calibrated, the MEKF estimates accurately the relative position of the endeffector and a 3D reference line. We believe that future vision systems being reliably calibrated will certainly make great use of the MEKF algorithm.
more …
By
Raviv, Dan; BayroCorrochano, Eduardo; Raskar, Ramesh
We present a novel algorithm for generating the mean structure of nonrigid stretchable shapes. Following an alignment process, which supports local affine deformations, we translate the search of the mean shape into a diagonalization problem where the structure is hidden within the kernel of a matrix. This is the first step required in many practical applications, where one needs to model bendable and stretchable shapes from multiple observations.
more …
