Gerald Sommer

The monogenic signal

This paper introduces a two-dimensional (2-D) generalization of the analytic signal. This novel approach is based on the Riesz transform, which is used instead of the Hilbert transform. The combination of a 2-D signal with the Riesz transformed one yields a sophisticated 2-D analytic signal: the monogenic signal. The approach is derived analytically from irrotational and solenoidal vector fields. Based on local amplitude and local phase, an appropriate local signal representation that preserves the split of identity, i.e., the invariance-equivariance property of signal decomposition, is presented. This is one of the central properties of the one-dimensional (1-D) analytic signal that decomposes a signal into structural and energetic information. We show that further properties of the analytic signal concerning symmetry, energy, allpass transfer function, and orthogonality are also preserved, and we compare this with the behavior of other approaches for a 2-D analytic signal. As a central topic of this paper, a geometric phase interpretation that is based on the relation between the 1-D analytic signal and the 2-D monogenic signal established by the Radon (1986) transform is introduced. Possible applications of this relationship are sketched, and references to other applications of the monogenic signal are given.

Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case

The construction of Gabors (1946) complex signal-which is also known as the analytic signal-provides direct access to a real one-dimensional (1-D) signals local amplitude and phase. The complex signal is built from a real signal by adding its Hilbert transform-which is a phase-shifted version of the signal-as an imaginary part to the signal. Since its introduction, the complex signal has become an important tool in signal processing, with applications, for example, in narrowband communication. Different approaches to an n-D analytic or complex signal have been proposed in the past. We review these approaches and propose the hypercomplex signal as a novel extension of the complex signal to n-D. This extension leads to a new definition of local phase, which reveals information on the intrinsic dimensionality of the signal. The different approaches are unified by expressing all of them as combinations of the signal and its partial and total Hilbert transforms. Examples that clarify how the approaches differ in their definitions of local phase and amplitude are shown. An example is provided for the two-dimensional (2-D) hypercomplex signal, which shows how the novel phase concept can be used in texture segmentation.

The Monogenic Scale-Space: A Unifying Approach to Phase-Based Image Processing in Scale-Space

In this paper we address the topics of scale-space and phase-based image processing in a unifying framework. In contrast to the common opinion, the Gaussian kernel is not the unique choice for a linear scale-space. Instead, we chose the Poisson kernel since it is closely related to the monogenic signal, a 2D generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. The Riesz transform itself yields the flux of the Poisson scale-space and the combination of flux and scale-space, the monogenic scale-space, provides the local features phase-vector and attenuation in scale-space. Under certain assumptions, the latter two again form a monogenic scale-space which gives deeper insight to low-level image processing. In particular, we discuss edge detection by a new approach to phase congruency and its relation to amplitude based methods, reconstruction from local amplitude and local phase, and the evaluation of the local frequency.

Dynamic cell structure learns perfectly topology preserving map

Dynamic cell structures (DCS) represent a family of artificial neural architectures suited both for unsupervised and supervised learning. They belong to the recently (Martinetz 1994) introduced class of topology representing networks (TRN) that 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 (Fritzke 1993c) leads to an efficient and elegant algorithm that can beat conventional models on similar tasks.

Intrinsic dimensionality estimation with optimally topology preserving maps

A new method for analyzing the intrinsic dimensionality (ID) of low-dimensional manifolds in high-dimensional feature spaces is presented. Compared to a previous approach by Fukunaga and Olsen (1971), the method has only linear instead of cubic time complexity with respect to the dimensionality of the input space. Moreover, it is less sensitive to noise than the former approach. Experiments include ID estimation of synthetic data for comparison and illustration as well as ID estimation of an image sequence.

Gabor wavelet networks for efficient head pose estimation

In this paper, we first introduce the Gabor wavelet network (GWN) as a model-based approach for effective and efficient object representation. GWNs combine the advantages of the continuous wavelet transform with RBF networks. They have additional advantages such as invariance to some degree with respect to affine deformations. The use of Gabor filters enables the coding of geometrical and textural object features. Gabor filters as a model for local object features ensure considerable data reduction while at the same time allowing any desired precision of the object representation ranging from sparse to photo-realistic representation. As an application we present an approach for the estimation of head pose based on the GWNs. Feature information is encoded in the wavelet coefficients. An artificial neural network is then used to compute the head pose from the wavelet coefficients.

A New Extension of Linear Signal Processing for Estimating Local Properties and Detecting Features

The analytic signal is one of the most capable approaches in one-dimensional signal processing. Two-dimensional signal theory suffers from the absence of an isotropic extension of the analytic signal. Accepting the fact that there is no odd filter with isotropic energy in higher dimensions, one tried to circumvent this drawback using the one-dimensional quadrature Alters with respect to several preference directions. Disadvantages of these methods are an increased complexity, the loss of linearity and a lot of different heuristic approaches. In this paper we present a filter that is isotropic and odd, which means that the whole theory of local phase and amplitude can directly be applied to images. Additionally, a third local property is obtained which is the local orientation. The advantages of our approach are demonstrated by a stable orientation detection algorithm and an adaption of the phase congruency method which yields a superior edge detector with very low complexity.

Real-Time Tracking of Moving Objects with an Active Camera

This article is concerned with the design and implementation of a system for real-time monocular tracking of a moving object using the two degrees of freedom of a camera platform. Figure-ground segregation is based on motion without making anya prioriassumptions about the object form. Using only the first spatiotemporal image derivatives, substraction of the normal optical flow induced by camera motion yields the object image motion. Closed-loop control is achieved by combining a stationary Kalman estimator with an optimal Linear Quadratic Regulator. The implementation on a pipeline architecture enables a servo rate of 25 Hz. We study the effects of time-recursive filtering and fixed-point arithmetic in image processing and we test the performance of the control algorithm on controlled motion of objects.

A lie group approach to steerable filters

Recently, Freeman and Adelson (1991) and Simoncelli et al. (1992) published an approach to steer filters in their orientation and scale by Fourier decompositions. We present a generalization of their formalism based on Lie group theory. Within this framework we especially clarify the following points: (I) the possible scope of steerability by Fourier decompositions. (2) approximate steerability with a limited number of basis functions, (3) the singularity that occurs when steering the scale.

Affine real-time face tracking using a wavelet network

We present a method for visual face tracking that is based on a wavelet representation of a face template. The wavelet representation allows arbitrary affine variations of the facial image, it allows to generalize from an individual face template to a rather general face template and it allows to adapt the computational needs of the tracking algorithm to the computational resources available. The method presented runs in real-time (25 Hz) on a Linux Pentium 450 MHz and was tested on several common sequences including the salesman-sequence.