In cardiac cine magnetic resonance imaging (MRI), the heart is repeatedly imaged at numerous time points during the cardiac cycle. Frequently, the temporal evolution of a certain region of interest such as the ventricles or the atria is highly relevant for clinical diagnosis. In this paper, we devise a novel approach that allows for an automatized propagation of an arbitrary region of interest (ROI) along the cardiac cycle from respective annotated ROIs provided by medical experts at two different points in time, most frequently at the end-systolic (ES) and the end-diastolic (ED) cardiac phases. At its core, a 3D TV-$\boldsymbol {L^{1}}$-based optical flow algorithm computes the apparent motion of consecutive MRI images in forward and backward directions. Subsequently, the given terminal annotated masks are propagated by this bidirectional optical flow in 3D, which results, however, in improper initial estimates of the segmentation masks due to numerical inaccuracies. These initially propagated segmentation masks are then refined by a 3D U-Net-based convolutional neural network (CNN), which was trained to enforce consistency with the forward and backward warped masks using a novel loss function. Moreover, a penalization term in the loss function controls large deviations from the initial segmentation masks. This method is benchmarked both on a new dataset with annotated single ventricles containing patients with severe heart diseases and on a publicly available dataset with different annotated ROIs. We emphasize that our novel loss function enables fine-tuning the CNN on a single patient, thereby yielding state-of-the-art results along the complete cardiac cycle.

Biological evidence shows that there are neural networks specialized for recognition of signals and patterns acting as associative memories. The spiking neural networks are another kind which receive input from a broad range of other brain areas to produce output that selects particular cognitive or motor actions to perform. An important contribution of this work is to consider the geometric processing in the modeling of feed-forward neural networks. Since quaternions are well suited to represent 3D rotations, it is then well justified to extend real-valued neural networks to quaternion-valued neural networks for task of perception and control of robot manipulators. This work presents the quaternion spiking neural networks which are able to control robots, where the examples confirm that these artificial neurons have the capacity to adapt on-line the robot to reach the desired position. Also, we present the quaternionic quantum neural networks for pattern recognition using just one quaternion neuron. In the experimental analysis, we show the excellent performance of both quaternion neural networks.

In this paper, a CNN based perception system for collision avoidance in mobile robots is presented. In the considered scenario, a mobile robot is ordered to reach a target on its workspace, where several types of objects influence a collision risk for a suitable movement to the desired goal. To ensure collision-free planning through the environment, a set of convolutional neural networks in parallel are employed to detect a set of static or dynamic objects of interest in the environment, as well as objects on the floor that could imply a collision risk during a movement execution. Afterward, stereo vision and filtering algorithms are employed to recover and track the spatial position of detections, in order to generate enough information to plan a collision-free trajectory. All the above steps are evaluated in real-time and real environments, proving to be enough robust and fast for a wide range of mobile robot applications.

In this paper we introduce a novel Depth-Aware Video Saliency approach to predict human focus of attention when viewing videos that contain a depth map (RGBD) on a 2D screen. Saliency estimation in this scenario is highly important since in the near future 3D video content will be easily acquired yet hard to display. Despite considerable progress in 3D display technologies, most are still expensive and require special glasses for viewing, so RGBD content is primarily viewed on 2D screens, removing the depth channel from the final viewing experience. We train a generative convolutional neural network that predicts the 2D viewing saliency map for a given frame using the RGBD pixel values and previous fixation estimates in the video. To evaluate the performance of our approach, we present a new comprehensive database of 2D viewing eye-fixation ground-truth for RGBD videos. Our experiments indicate that it is beneficial to integrate depth into video saliency estimates for content that is viewed on a 2D display. We demonstrate that our approach outperforms state-of-the-art methods for video saliency, achieving 15% relative improvement.

In this paper, we present a geometric algebra approach for detection of geometric entities in images. Our algorithm is grounded on two methodologies: representation of geometric entities and perceptual properties using Conformal Geometric Algebra, and a voting scheme which is implemented using a clustering algorithm. Our method is applied in a hierarchical way, so that, we extract local and global information from images. Experimental results show the application of our approach to detection of circles, lines, complex structures of shape, and symmetry axis. In addition, we show an FPGA implementation that speed-up the execution time of the algorithm.

In this work the authors 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 that are supported in the whole image. Geometric algebra provides a suitable mathematical framework, which allows our algorithm to infer high-level geometric representations of percepts in an image. Experiments show the capability of our algorithm for representing objects in images in terms of circles and lines, even though it contains a noisy input, incomplete data, illusory or non-linear contours.

This paper is concerned with 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 application section shows experimen ts in pattern recognition and colour image processing. We also present a way to expand the amount of classes without the need to increase the number of classifiers as in standar approaches.

This paper presents the design of radial basis function geometric bioinspired networks and their applications. Until now, the design of neural networks has been inspired by the biological models of neural networks but mostly using vector calculus and linear algebra. However, these designs have never shown the role of geometric computing. The question is how biological neural networks handle complex geometric representations involving Lie group operations like rotations. Even though the actual artificial neural networks are biologically inspired, they are just models which cannot reproduce a plausible biological process. Until now researchers have not shown how, using these models, one can incorporate them into the processing of geometric computing. Here, for the first time in the artificial neural networks domain, we address this issue by designing a kind of geometric RBF using the geometric algebra framework. As a result, using our artificial networks, we show how geometric computing can be carried out by the artificial neural networks. Such geometric neural networks have a great potential in robot vision. This is the most important aspect of this contribution to propose artificial geometric neural networks for challenging tasks in perception and action. In our experimental analysis, we show the applicability of our geometric designs, and present interesting experiments using 2-D data of real images and 3-D screw axis data. In general, our models should be used to process different types of inputs, such as visual cues, touch (texture, elasticity, temperature), taste, and sound. One important task of a perception-action system is to fuse a variety of cues coming from the environment and relate them via a sensor-motor manifold with motor modules to carry out diverse reasoned actions.

Geometric algebra (GA) is a powerful and versatile mathematical tool which helps to intuitively express and manipulate complex geometric relationships. It has recently been used in engineering problems such computer graphics, machine vision, robotics, among others. The problem with GA in its numeric version is that it requires many arithmetic operations, and the length of the input vectors is unknown until runtime in a generic architecture operating over homogeneous elements. Few works in hardware architectures for GA were developed to improve the performance in GA applications. In this work, a hardware architecture of a unit for GA operations (geometric product) for FPGA is presented. The main contribution of this work is the use of parallel memory arrays with access conflict avoidance for dealing with the issue of unknown length of input/output vectors, the intention is to reduce memory wasted when storing the input and output vectors. In this first stage of the project, we have implemented only a single access function (fixed-length) in the memory array in order to test the core of geometric product. In future works we will implement a full set of access functions with different lengths and shapes. In this work, only the simulations are presented; in the future, we will also present the experimental results

In this letter, we propose a global method for registering color images with respect to translation. Our approach is based on the idea of representing translations as convolutions with unknown shifted delta functions, and performing Wiener deconvolution in order to recover the shift between two images. We then derive a quaternionic version of the Wiener deconvolution filter in order to register color images. The use of Wiener filter also allows us to explicitly take into account the effect of noise. We prove that the well-known algorithm of phase correlation is a special case of our method, and we experimentally demonstrate the advantages of our approach by comparing it to other known generalizations of the phase correlation algorithm.

Controlling walking biped robots is a challenging problem due to its complex and uncertain dynamics. In order to tackle this, we propose a sliding mode controller based on a dynamic model which was obtained using the conformal geometric algebra approach (CGA). The CGA framework permits us to use lines, points, and other geometric entities, to obtain the Lagrange equations of the system. The references for the joints of the robot were bio-inspired in the kinematics of a walking human body. The first and second derivatives of the reference signal were obtained through an exact robust differentiator based on high order sliding modes. The performance of the proposed control scheme is illustrated through simulation.

This issue marks the first anniversary issue of IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS after it changed its name from IEEE TRANSACTIONS ON NEURAL NETWORKS. I am happy to report that we had a great year! The number of new submissions in a year exceeded 1,000 for the first time in the history of TNN/TNNLS. IEEE TNN had a very successful development for 22 years from 1990 to 2011, and we have good reasons to believe that IEEE TNNLS will have many more years of successful growth.

The main goal of this work is to develop a geometric neural network which can be used as an interface between sensors and robot mechanisms. For this goal we have developed two new geometric network called Spherical Radial Basis Function Network and Spherical General Regression Network using the conformal geometric algebra framework. The motivation to use circles or spheres as activation functions is due to the fact that the sphere is the computational unity of the conformal geometric algebra, as a result these Networks can be advantageously used as interface between the sensor domain and the robotic mechanism so that all the computing can be done in the same mathematical framework. In fact, there will be no need to abandon the system for the interpolation or reconstruction using this network. This article presents the design principles and a comparison with a standard Radial Basis Function Network and a standard General regression Neural Network. In the area of medical robotics the use of haptics is quite common. This is an interesting domain to apply our network for capturing data with a haptic device and using spheres reconstruct automatically the shape of a human organ. As we shown in other works, the differential robot kinematics can be formulated using lines, planes and spheres using geometric algebra, having the organ tissue modelled also with spheres with the networks, this helps greatly to related the perceptual and the mechanical devices and ensure their control. We show reconstruction results of an organ using both Networks.

In this paper we present a new hypercomplex-valued model of recurrent neural network which is based on the Geometric Radial Basis (RBF) and Elman Network Models. This model is useful to recognize temporal sequences of geometric entities using geometric algebra. Our model combines features from the Elman recurrent neural network and geometric RBF networks. This network constitutes a generalization of the standard real-valued recurrent models. The network fed with geometric entities can be used in real time to learn a sequence of entities determined using a geometric language. This approach calculates the temporal geometric transformation between each two entity orientations which are presented to the network in different times.

Humanoid robots should be able to grasp and handle objects in the environment, even if the objects are seen for the first time. A plausible solution to this problem is to categorize these objects into existing classes with associated actions and functional knowledge. So far, efforts on visual object categorization using humanoid robots have either been focused on appearance-based methods or have been restricted to object recognition without generalization capabilities. In this work, a shape model-based approach using stereo vision and machine learning for object categorization is introduced. The state-of-the-art features for shape matching and shape retrieval were evaluated and selectively transfered into the visual categorization. Visual sensing from different vantage points allows the reconstruction of 3D mesh models of the objects found in the scene by exploiting knowledge about the environment for model-based segmentation and registration. These reconstructed 3D mesh models were used for shape feature extraction for categorization and provide sufficient information for grasping and manipulation. Finally, the visual categorization was successfully performed with a variety of features and classifiers allowing proper categorization of unknown objects even when object appearance and shape substantially differ from the training set. Experimental evaluation with the humanoid robot ARMAR-IIIa is presented.

The kinematics of a robotic manipulator defines the geometric relationships between its elements. The synthesis of the kinematical model for this type of robots is presented in this work using the Conformal Geometric Algebra approach. Moreover, two error feedback controllers, one for the position tracking problem and another for the orientation tracking problem, are developed using the same framework. The stability analysis is obtained for both controllers, and their application to a robotic system, composed of a serial manipulator of 5 DOF and a robotic head of 2 DOF, is presented via simulation.

A controller, based on nested sliding modes and super-twisting algorithm, is proposed for n-link robotic manipulator tracking problem. This controller has the robustness of nested sliding modes against matched and no matched perturbations and the softness of control signals of super-twisting algorithm (STA). The Nested Super-Twisting Algorithm (NSTA) allows obtaining a control law without knowing the exact model of the robotic manipulator. The application of the proposed controller in a two-link planar robot manipulator and a comparison with the Nested Sliding Mode Control (NSMC) are presented via simulation.

A methodology for 3D modeling of virtualized reality objects using neural computing is presented. In this paper the objects are represented in virtualized reality and their 3D data are acquired by one of three acquisition systems: endoneurosonographic equipment (ENS), stereo vision system and non-contact 3D digitizer. These objects are modeled by one of three neural architectures: Multilayer Feed-Forward Neural Network (MLFFNN), Self-Organizing Maps (SOM) and Neural Gas Network (NGN). The 3D virtualized representations correspond to several objects as phantom brain tumors, faces, archaeological items, fruits, among others. We carry out comparisons in terms of computational cost, architectural complexity, training method, training epochs and performance. Finally, we present the modeling results and conclude that SOM and NGN models achieve the best performances and the lowest displaying times, while MLFFNN models have the lowest memory requirements and acceptable training times.

In this paper we present the application of a new hypercomplex-valued Radial Basis Network (RBF) to estimate unknown geometric transformations such as in the case of the Hand-Eye Calibration problem. This network constitutes a generalization of the standard real-valued RBF. The network fed with geometric entities can be used in real time to estimate changes in the linear transformation between the coordinate system of the camera and the coordinate system of the end-effector. This approach is more efficient than standard batch methods particularly because our method works in real time, estimating the rigid transformation under temporal perturbation. In contrast, the standard methods need to recalibrate each time first by collecting data and then by computing a batch procedure often using SVD or optimization techniques.

In this work we propose the use of machine learning techniques to improve Simultaneous Localization and Mapping (SLAM) using an extended Kalman filter (EKF) and visual information for robot navigation. We are using the Viola and Jones approach for looking specific visual landmarks in environment. The landmarks are used to improve the robot localization in the EKF-SLAM system. Our experiments validate the efficiency of our algorithm.

This paper presents a method for 3D reconstruction of tumors for applications in laparoscopy. This uses stereo endoscopic ultrasound images, which are simultaneously recorded. To do this, the ultrasound probe is tracked throughout the stereo endoscopic images using a particle filter and an auxiliary method based on thresholding in the HSV-space is used in order to improve the tracking. Then, the 3D pose of the ultrasound probe is calculated using conformal geometric algebra. The 2D ultrasound images have been segmented using two methods: the level sets method and morphological operators, and a comparison between their performances has been done. Finally, the processed ultrasound images are compounded into a 3D volume, using the calculated ultrasound pose.

This paper presents a new hyper complex valued Radial Basis Network. This network constitutes a generalization of the standard real valued RBF. This geometric RBF can be used in real time to estimate changes in linear transformations between sets of geometric entities. Experiments using stereo image sequences validate this proposal. We propose a Geometric RBF Network (GRBF-N) designed in the geometric algebra framework. We present an application to estimate linear transformations between sets of geometric entities. Our experiments validate our proposal.

Traditional cameras have a narrow field of view, to enlarge the field of view omnidirectional cameras can be used. In this work, we propose, a simple an elegant solution to the image formation model for omnidirectional cameras with parabolic mirrors. We propose the use of conformal geometric algebra (CGA), since the involved transformation operations in the model can be represented as an special group of multivectors. This representation is advantageous since the inversions are linearized, furthermore the transformation can be applied to all the geometric objects of the CGA. In consequence, the paracatadioptric image formation can be simplified, since the procedure is the same for points, point-pairs, lines, or circles. As an application example the control of a nonholonomic mobile robot using paracatadioptric line images and the proposed framework is described.

This paper presents a new hypercomplex valued Radial Basis Network. This network constitutes a generalization of the standard real valued RBF. This geometric RBF can be used in real time to estimate changes in linear transformations in 3D space between sets of geometric entities. Experiments using stereo image sequences validate this proposal. We show the tracking of changes in the orientation between sets of lines and planes. This is a promising approach for neural computing applications in visual guided robotics.

This paper describes a new approach for building 3D geometric maps using a geometric line and plane detector together with laser range finder and a stereo camera system. This detector is developed in such a way that one can related easily it with the conformal geometric algebra framework, thus detected lines and planes can be used for algebra of incidence computation to find geometric constraints useful to perceive special configurations in the 3D visual space for exploration, navigation and obstacle avoidance.

In this work we show the use of potential fields in conformal geometric algebra to assist the humanoid manipulation planning. Since the computational unit in conformal geometry algebra is the sphere, it appears natural to use the potential fields for computing navigation trajectories avoiding obstacles. Furthermore, by covering obstacles with convex hulls, one can obtain certain shapes which are amenable of being parameterized as spheres, thus this equivalent potential fields can be used so that the robot device can move around the irregular shapes of obstacle without increasing the computation complexity. In this paper we present this theory and show with simulations the effectiveness of the approach.

A controller, based on integral nested sliding modes and super-twisting algorithm, is proposed for n-link robotic manipulator tracking problem. This controller has the robustness of nested sliding modes against matched and no matched perturbations, the capability of integral sliding modes to reduce the sliding functions gains, and the softness of control signals of super-twisting algorithm. The performance of the proposed algorithm is compared with integral nested sliding mode control via simulation. For this purpose, the application of both controllers in a two-link planar robot manipulator is presented.

The authors present a new algorithm to compute the forward Dynamics of n degrees of freedom serial kinematic chains, which is faster than the classical approaches. This algorithm was created rewriting the Lagrange equation in terms of lines and points in the framework of conformal geometric algebra, which allows having a new equation to compute the dynamics with less number of products. This algorithm not only performs less computations but it also takes the advantages of the newest multi core architectures by computing the dynamics in parallel.

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.

This paper presents a discrete-time decentralized control scheme for identification and trajectory tracking of a five degrees of freedom (DOF) robot manipulator. A recurrent high order neural network (RHONN) structure is used to identify the robot model, and based on this model a discrete-time control law is derived, which combines discrete-time block control and sliding modes techniques. The neural network learning is performed online using Kalman filtering. A controller is designed for each joint, using only local angular position and velocity measurements. These simple local joint controllers allow trajectory tracking with reduced computations. The applicability of the proposed scheme is illustrated via simulations.

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. These landmarks are found using the Viola and Jones algorithm and are represented with their position in the 3D virtual map. This landmarks help in the relocalization of a robot in a previously captured environment. This machine learning technique is used for recognition of objects in the environment.

This paper presents a discrete-time decentralized control scheme for identification and trajectory tracking of a two degrees of freedom (DOF) robot manipulator. A recurrent high order neural network (RHONN) structure is used to identify the plant model and based on this model, a discrete-time control law is derived, which combines discrete-time block control and sliding modes techniques. The neural network learning is performed online by Kalman filtering. A controller is designed for each joint, using only local angular position and velocity measurements. These simple local joint controllers allow trajectory tracking with reduced computations. The proposed scheme is implemented in real-time to control a two DOF robot manipulator.

In this paper we apply the classic theory of harmonic analysis and the conformal geometric algebra (CGA) to evaluate the Fourier transform on the unit sphere S2and on the rotation group SO(3). Since the images taken by omnidirectional sensors can be mapped to the sphere, the problem of attitude estimation of a 3D camera rotation can be treated as a problem of estimating rotations between spherical images. Usually, this rotation estimation problem has been solved using the Radon transform with point correlation or with line correlation. Using a catadioptric system with a parabolic mirror, 3D lines will be projected as great circles on S2and then projected as circles on the image plane. CGA is a mathematical framework where its basic entities, spheres, circles, lines and planes can be used with incidence algebra operations to improve the line correlation in the Radon transform as a dual-circle correlation. Thus, harmonic analysis theory and conformal geometric algebra will be joined in this paper to improve the search of a 3D rotation correspondence between two omnidirectional images.

In this paper, a geometric approach for global self-localization based on a world-model and active stereo vision is introduced. The method uses class specific object recognition algorithms to obtain the location of entities within the surroundings. The perceived entities in recognition trials are simultaneously filtered and fused to provide a robust set of class features. These classified perceptions which simultaneously satisfy geometric and topological constraints are employed for pruning purposes upon the world-model generating the location hypotheses set. Finally, the hypotheses are validated and disambiguated by applying visual recognition algorithms to selected entities of the world-model. The proposed approach has been successfully used with a humanoid robot.

We have developed a method to render brain tumors from endoneurosonography. We propose to track an ultrasound probe in successive endoscopic images without relying on an external optic or magnetic tracking system. The probe is tracked using two different methods, one of them based on generalized Hough transform and the other one on particle filters. With the estimation of the pose of the ultrasound probe in several contiguous images we use conformal geometric algebra to compute the geometric transformations that yield the 3D position of the tumor, which was segmented in the ultrasound image using morphological operators. We have used images from brain phantoms to evaluate the performance of the proposed methods and our results show that they are robust.

We present an algorithm in the Conformal Geometric Algebra framework which computes the transformation (rotation and translation) relating the coordinate system of an endoscopic camera, with the coordinate system of the markers placed on it. Such markers are placed in order to track the camera’s position in real time by using an optical tracking system. The problem is an adaptation of the so-called hand-eye calibration, well known in robotics; however we name it the endoscope-tracking system calibration. By this way, we can relate the preoperative data (3D model and planing of the surgery), with the data acquired in real time during the surgery by means of the optical tracking system and the endoscope. Results obtained with our approach are compared with other approach which computes separately (in a two-step algorithm) the rotation and translation, and they are promising.

In this paper the authors propose geometric techniques to deal with the problem of grasping of the objects relaying on their mathematical models. For that we use the geometric algebra framework to formulate the kinematics of a three linger robotic hand. Our main objective is by formulating a kinematic control law to close the loop between perception and actions. This allows us to perform a smooth visually guided object grasping.

This paper introduces the recurrent Clifford support vector machines (RCSVM). First we explain the 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 and 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 to build a recurrent CSVM.We study the performance of the recurrent CSVM with experiments using time series and tasks of visually guided robotics.

This paper presents the properties and the definition of Hopfield neural networks as a natural extension to complex Hopfield neural networks and quaternionic Hopfield neural networks. This extension allows us to generalize the concept of Hopfield neural networks to all type of algebras and also to describe the main characteristics of this networks.

Because of their large field of view, omnidirectional images can assist with localization tasks to robotic navigation algorithms. Since the images taken by omnidirectional sensors can be mapped to the sphere, the problem of attitude estimation of a 3D camera rotation can be treated as a problem of estimating rotations between spherical images. Usually, this rotation estimation problem has been solved using point correspondences or gradient information of the points of the images, with the respective computational time consuming of those point algorithms. We present an effective solution to the attitude estimation problem using line information of the images through the Radon transform, because line algorithms are less time consuming that those using points. In the formulation of the Radon transform we include a similarity function on the cross product of two images which assigns a weight to all lines pairs, and where this similarity function is integrated over all lines pairs that satisfy a constraint for lines. That is, we formulate the problem to obtain the Euler angles of the 3D camera rotation as a correlation of functions defined on the product of spheres S2times S2which are acted upon by elements of the direct product groups of rotations SO(3) timesSO(3). Because of the spherical treatment of the data, our approach utilizes the spherical fourier transform and spherical harmonics to produce a solution in the Fourier domain.

In this paper we introduce a conformal model for stereoscopic perception systems. This model fuses stereoscopic views using an extended 3D horopter concept. This model is suitable for building a powerful binocular head for humanoids, intuitive man-machine interfaces and advanced 3D automated visual inspection

We present an approach to ruled surfaces and complex 3D-curves using conformal geometric algebra. As robots and mechanisms move, any line attached to them will trace out a ruled surface. In robotics, the arm of a robot usually needs to follow a curve over a ruled surface or over the intersection of some of them. In this paper we present the way to model and to resolve these problems using algebra of incidence and rigid-motion transformations in the context of the conformal geometric algebra

The automatic landmark identification is very important in autonomous robot navigation tasks. In this work we use a monocular omnidirectional vision system to extract images features, and with help of the conformal geometric algebra, we show how these features can be used to calculate projective and permutation p2-invariants. These p2-invariants represent scene sub-landmarks and a set of them characterize a landmark. The advantage of this representation is that the landmarks are more robust than the single cross-ratio

This paper presents an analysis of the kinematics of a binocular head, and a new formulation of its differential kinematics using the conformal geometric algebra framework. The experimental part includes the smooth visual tracking of 3D objects

This paper presents the detailed theory of the Clifford support vector machines (CSVM). Remarkably enough, the CSVM's can be used for multiclass classification. In the experiment part we show interesting applications such as automatic learning in robot vision.

In this paper we propose a conformal model for human like vision. In our model the two views are fused in an extended 3D horopter concept. The inverse transformation of the horopter spheres and Poncelet points leads remarkably to a 3D log-polar representation of the visual space. This representation is quite effective for processing the visual space information. We believe that this model is very suitable for building a powerful binocular head for humanoids.

In this paper we present a method based on self-organizing neural networks to extract the shape of an object which is useful for segmentation tasks. For that, the method uses 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. The utility of this approach is showed with one synthetic and several medical images (computer tomography and magnetic resonance images), where the object of interest is well segmented even if there is no well defined contours in the original image. In fact, the segmentation results obtained are better than the results using the ggvf-snake, no matter if the initialization of the snake is given inside, outside or over the (blurred) contour of the object.

This paper presents an approach to deal with standard tasks of computer animations and robotics based on Conformal Geometric Algebra. We will show that this algebra is very well suitable for applications of all kind of robot manipulator kinematics, representation and visualization and object robot manipulation. Due to its geometric intuitiveness and simplicity Conformal Geometric Algebra appears to be a promising mathematical tool for building intelligent man-machine interfaces.

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 for linear and nonlinear classification and regression. The major advantage of our approach is that one requires only one CSVM with one kernel (involving the Clifford product) which can admit multiple multivector inputs and it can carry out multi-class classification and regression. In contrast one would need many real valued SVMs for a multi-class problem which is time consuming.

In this paper we present an object recognition technique based on neural computing and projective invariants but now using an omnidirectional vision system and conformal geometric algebra. We also show how to recover the projective invariants from a catadioptric image, where the projective invariants do not hold. With these invariants we train a multilayer perceptron (MLP) neural network to recognize objects.

This paper introduces the use of projective invariants for object recognition using omnidirectional vision. The catadioptric image formation is modeled with a two-step mapping via the sphere using the conformal geometric algebra. Also, the projective invariants are calculated using the conformal geometric algebra. Furthermore, we show that the projective invariants in a plane are equivalent to the invariants of circles in the unit sphere. This equivalence induces us to project features from the catadioptric image to the sphere, calculate their invariants and use these invariants to recognize objects.

We have developed a novel system for template tracking. Our system takes advantage of the theory of system identification to identify using minimal information the best model estimator of the affine and projective transformation. Lie algebras are used to characterize these transformations. This allows us to move along geodesics assuring the best approximation to the orbit of the Lie group involved.

The catadioptric projection can be modeled by an equivalent spherical projection. This idea is more versatile in the conformal geometric algebra, a modern framework for the projective space of hyper-spheres. In this framework we analyze diverse catadioptric mirrors, as a result the algebraic burden is reduced allowing the user to work in a more effective framework for the development of algorithms for omnidirectional vision. We present preliminary experiments useful for omnidirectional visually guided robotics.

This work design support vector machines using the conformal Clifford geometric algebra framework. In this study we map the feature space into hyperspheres in order to get a uniformly distribution data. In this domain we apply as classifier a support conformal vector machines. In this context the optimal hyperplane found by the support conformal vector machine will approach to the optimal Bayes point. An experimental analysis clarifies our approach.

This paper introduces the Clifford support vector machines 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 for linear and nonlinear classification. We present an interesting application to classify mechanical tools using the concept of Clifford moments which is invariant under affine transformations.

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.

This paper presents the design of kernels for nonlinear support vector machines using the Clifford geometric algebra framework. In this study we present the design of kernels involving the Clifford or geometric product making use of nonlinear mappings which map multi-vectors into higher dimensional geometric algebra. We introduce also the conformal geometric neuron for geometric classification. Experiments are given to demonstrate the usefulness of the approach.

This paper presents a preprocessing for neural computing using the quaternion Fourier transform. The 1D signals of French vocables are represented as 2D signals in the frequency and time domain. The images are convolved in the quaternion Fourier domain with a quaternion Gabor filter for the extraction of features. Two methods of feature extraction are tested. The features vectors are then used for the training of two neural network architectures. The results are very encouraging which justify plenty the preprocessing in the quaternion frequency domain.

This paper shows the analysis and design of feedforward neural networks using the coordinate-free system of Clifford or geometric algebra. It is shown that real-, complex-, and quaternion-valued neural networks are simply particular cases of the geometric algebra multidimensional neural networks and that some of them can also be generated using support multivector machines (SMVMs). Particularly, the generation of radial basis function for neurocomputing in geometric algebra is easier using the SMVM, which allows one to find automatically the optimal parameters. The use of support vector machines in the geometric algebra framework expands its sphere of applicability for multidimensional learning. Interesting examples of nonlinear problems show the effect of the use of an adequate Clifford geometric algebra which alleviate the training of neural networks and that of SMVMs.

The paper presents a geometric approach for the reconstruction of shape and motion using projective invariants as function of the trifocal tensor. We compute first the projective depth using groups of three uncalibrated cameras of an image sequence and then initialize the join image matrix which can be then decomposed into a 3D shape and motion by means of SVD. Experiments using real images show the the value of the approach. Using the algebra of incidence relations of the geometric algebra the 3D reconstruction was further improved.

The representation of the external world in biological creatures appears to be defined in terms of geometry. In this regard the author uses Clifford geometric algebra for the development of geometric type neural networks. The contribution of the paper is the extension of our past work including the use of support vector machines (SV machines) in the Clifford algebra framework. Thus geometric MLPs and RBF networks can be generated using SV machines straightforwardly. In this way we expanded the sphere of applicability of the SV machines by the treatment of multi-vectors which encode the geometry of the data manifold in a rich manner.

A central task of computer vision is to automatically recognize objects in real-world scenes. The parameters defining image and object spaces can vary due to lighting conditions, camera calibration and viewing positions. It is therefore desirable to look for geometric properties of the object which remain invariant under such changes. In this paper we present geometric algebra as a complete framework for the theory and computation of projective invariants formed from points and lines in computer vision. We will look at the formation of 3D projective invariants from multiple images, show how they can be formed from image coordinates and estimated tensors (F, fundamental matrix and T, trilinear tensor) and give results on simulated and real data.

We present geometric algebra as a new framework for the theory and computation of invariants and multiple-view constraints in computer vision. We discuss the formation of 3D projective invariants and a wholly geometric formulation of constraints from a number of images.

This paper presents a new self-organizing type RBF neural network and introduces the geometric algebra framework in the neurocomputing field. Real valued neural nets for function approximation require feature enhancement, dilation and rotation operations and are limited by the Euclidean metric. The authors believe that more general and flexible neural networks should be designed in order to capture important geometric characteristics of the manifolds. This is an important goal overlooked ever since. Geometric algebra is a system which allows the design of neural networks in a coordinate-free frame work to process patterns between layers using different dimensions and desired metric. The potential of such nets working in a Clifford algebra C(V/sub p,q/) is shown by a simple application of frame coordination in robotics.

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.

In order to relate measurements made by a sensor mounted on a mechanical link to the robot's 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.

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.