A Novel Representation for Two-dimensional Image Structures

This paper presents a novel approach towards two-dimensional (2D) image structures modeling. To obtain more degrees of freedom, a 2D image signal is embedded into a certain geometric algebra. Coupling methods of differential geometry, tensor algebra, monogenic signal and quadrature filter, we can design a general model for 2D structures as the monogenic extension of a curvature tensor. Based on it, a local representation for the intrinsically two-dimensional (i2D) structure is derived as the monogenic curvature signal. From it, independent features of local amplitude, phase and orientation are simultaneously extracted. Besides, a monogenic curvature scale-space can be built by applying a Poisson kernel to the monogenic curvature signal. Compared with the other related work, the remarkable advantage of our approach lies in the rotationally invariant phase evaluation of 2D structures in a multi-scale framework, which delivers access to phase-based processing in many computer vision tasks

Learning and Adaption: A Comparison of Methods in Case of Navigation in an Artificial Robot World

Neural networks, reinforcement learning systems and evolutionary algorithms are widely used to solve problems in real-world robotics. We investigate learning and adaptation capabilities of agents and show that the learning time required in continual learning is shorter than that of learning from scratch under various learning conditions. We argue that agents using appropriate hybridization of learning and evolutionary algorithms show better learning and adaptation capability as compared to agents using learning algorithms only. We support our argument with experiments, where agents learn optimal policies in an artificial robot worl

The Poisson Scale-Space: A Unified Approach to Phase-Based Image Processing in Scale-Space

In this paper we address the topics of scale-space and phase-based signal processing in a common framework. The involved linear scale-space is no longer based on the Gaussian kernel but on the Poisson kernel. The resulting scale-space representation is directly related to the monogenic signal, a 2D generalization of the analytic signal. Hence, the local phase arises as a natural concept in this framework which results in several advanced relationships that can be used in image processing

Dynamic Cell Structures: Radial Basis Function Networks with Perfect Topology Preservation

Dynamic Cell Structures (DCS) represent a family of artificial neural architectures suited both for unsupervised and supervised learning. They belong to the recently [Martinetz94] introduced class of Topology Representing Networks (TRN) which build perfectly topology preserving feature maps. DCS employ a modified Kohonen learning rule in conjunction with competitive Hebbian learning. The Kohonen type learning rule serves to adjust the synaptic weight vectors while Hebbian learning establishes a dynamic lateral connection structure between the units reflecting the topology of the feature manifold. In case of supervised learning, i.e. function approximation, each neural unit implements a Radial Basis Function, and an additional layer of linear output units adjusts according to a delta-rule. DCS is the first RBF-based approximation scheme attempting to concurrently learn and utilize a perfectly topology preserving map for improved performance. Simulations on a selection of CMU-Benchmarks indicate that the DCS idea applied to the Growing Cell Structure algorithm [Fritzke93b] leads to an efficient and elegant algorithm that can beat conventional models on similar tasks

Basic functions for early vision

It is commonly agreed on that the first step in early vision consists of projections of the image to a set of basis functions. Usually the spatial distribution of the basis functions is homogeneous and the projection is a convolution but in general this will not be the case. In the literature there is a wealth of different basis functions, each of them optimal with respect to certain criteria. On the other hand, there seems to be a convergence towards derivatives of Gaussians or harmonic modulations of Gaussians (Gabor functions). In this report we discuss the principles and analysing methods underlying the choice of these functions. One of these methods that recently became of exceptional importance is the energy/phase representation. We investigate in detail the quality of succesive orders of derivatives of Gaussians as odd/even pairs for the energy/phase concept. In addition we work out to which extent derivatives of Gaussians can be approximated by Gabor functions

The 2D Analytic Signal

This technical report covers a fundamental problem of 2D local phase based signal processing: the isotropic generalization of the analytic signal (D. Gabor) for two dimensional signals. The analytic signal extends a real valued 1D signal to a complex valued signal by means of the classical 1D Hilbert transform. This enables the complete analysis of local phase and amplitude information. Local phase, amplitude and additional orientation information can be extracted by the monogenic signal (M. Felsberg and G. Sommer) which is always restricted to the subclass of intrinsically one dimensional signals. In case of 2D image signals the monogenic signal enables the rotationally invariant analysis of lines and edges. In contrast to the 1D analytic signal the monogenic signal extends all real valued signals of dimension n to a (n + 1) - dimensional vector valued monogenic signal by means of the generalized first order Hilbert transform (Riesz transform). In this technical report we present the 2D analytic signal as a novel generalization of the 2D monogenic signal which now extends the original 2D signal to a multivector valued signal in conformal space by means of higher order Hilbert transforms and by means of a hybrid matrix geometric algebra representation. The 2D analytic signal can be interpreted in conformal space which delivers a descriptive geometric interpretation of 2D signals. One of the main results of this work is, that all 2D signals exist per se in a 3D projective subspace of the conformal space and can be analyzed by means of geometric algebra. In case of 2D image signals the 2D analytic signal enables now the rotational invariant analysis of lines, edges, corners and junctions

Evolution of Neural Networks Through Incremental Acquisition of Neural Structures

In this contribution we present a novel method, called Evolutionary Acquisition of Neural Topologies (EANT), of evolving the structures and weights of neural networks. The method introduces an efficient and compact genetic encoding of a neural network onto a linear genome that enables one to evaluate the network without decoding it. The method uses a meta-level evolutionary process where new structures are explored at larger time-scale and the existing structures are exploited at lower time-scale. This enables it to find minimal neural structures for solving a given learning task

Pose Estimation in Conformal Geometric Algebra

2D-3D pose estimation means to estimate the relative position and orientation of a 3D object with respect to a reference camera system. This work has its main focus on the theoretical foundations of the 2D-3D pose estimation problem: We discuss the involved mathematical spaces and their interaction within higher order entities. To cope with the pose problem (how to compare 2D projective image features with 3D Euclidean object features), the principle we propose is to reconstruct image features (e.g. points or lines) to one dimensional higher entities (e.g. 3D projection rays or 3D reconstructed planes) and express constraints in the 3D space. It turns out that the stratification hierarchy \cite{faugerasstrat} introduced by Faugeras is involved in the scenario. But since the stratification hierarchy by Faugeras is based on pure point concepts a new algebraic embedding is required when dealing with higher order entities. The conformal geometric algebra (CGA) \cite{hli1} is well suited to solve this problem, since it subsumes the involved mathematical spaces. Operators are defined to switch entities between the algebras of the conformal space and its Euclidean and projective subspace. This leads to another interpretation of the stratification hierarchy, which is not restricted to be based solely on point concepts. This work summarizes the theoretical foundations needed to deal with the pose problem. Therefore it contains mainly basics of Euclidean, projective and conformal geometry. Since especially conformal geometry is not well known in computer science, we recapitulate the mathematical concepts in some detail. We believe that this geometric model is useful also for many other computer vision tasks and has been ignored so far. Applications of these foundations are presented in part II. Part II: Part II uses the foundations of part I to define constraint equations for 2D-3D pose estimation of different corresponding entities. Most articles on pose estimation concentrate on specific types of correspondences, mostly between point, and only rarely line correspondences. The first aim of this part is to extend pose estimation scenarios to correspondences of an extended set of geometric entities. In this context we are interested to relate the following (2D) image and (3D) model types: 2D point/3D point, 2D line/3D point, 2D line/3D line, 2D conic/3D circle, 2D circle/3D sphere. Furthermore, to handle articulated objects, we describe kinematic chains in this context in a similar manner. We ensure that all constraint equations end up in a distance measure in the Euclidean space, which is well posed in the context of noisy data. We also discuss the numerical estimation of the pose. We propose to use linearized twist transformations which result in well conditioned and fast solvable systems of equations. The key idea is not to search for the representation of the Lie group, describing the rigid body motion, but for the representation of their generating Lie algebra. This leads to real-time capable algorithms