Oliver Fleischmann

The geometry of 2D image signals

This paper covers a fundamental problem of local phase based signal processing: the isotropic generalization of the classical 1D analytic signal to two dimensions. The well known analytic signal enables the analysis of local phase and amplitude information of 1D signals. Local phase, amplitude and additional orientation information can be extracted by the 2D monogenic signal with the restriction 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 this work we present the 2D analytic signal as a novel generalization of both the analytic signal and the 2D monogenic signal. In case of 2D image signals the 2D analytic signal enables the isotropic analysis of lines, edges, corners and junctions in one unified framework. Furthermore, we show that 2D signals exist per se in a 3D projective subspace of the homogeneous conformal space which delivers a descriptive geometric interpretation of signals providing new insights on the relation of geometry and 2D signals.

2D Image Analysis by Generalized Hilbert Transforms in Conformal Space

This work presents a novel rotational invariant quadrature filter approach - called the conformal monogenic signal - for analyzing i(ntrinsic)1D and i2D local features of any curved 2D signal such as lines, edges, corners and junctions without the use of steering. The conformal monogenic signal contains the monogenic signal as a special case for i1D signals and combines monogenic scale space, phase, direction/orientation, energy and curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation of the 3D Radon transform and the generalized Hilbert transform on the sphere. The main idea is to lift up 2D signals to the higher dimensional conformal space where the signal features can be analyzed with more degrees of freedom. Results of this work are the low computational time complexity, the easy implementation into existing Computer Vision applications and the numerical robustness of determining curvature without the need of any derivatives.

Image Analysis by Conformal Embedding

This work presents new ideas in isotropic multi-dimensional phase based signal theory. The novel approach, called the conformal monogenic signal, is a rotational invariant quadrature filter for extracting local features of any curved signal without the use of any heuristics or steering techniques. The conformal monogenic signal contains the recently introduced monogenic signal as a special case and combines Poisson scale space, local amplitude, direction, phase and curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation between the Radon transform and the generalized Hilbert transform. The main idea of the conformal monogenic signal is to lift up n-dimensional signals by inverse stereographic projections to a n-dimensional sphere in ℝn+1 where the local signal features can be analyzed with more degrees of freedom compared to the flat n-dimensional space of the original signal domain. As result, it delivers a novel way of computing the isophote curvature of signals without partial derivatives. The philosophy of the conformal monogenic signal is based on the idea to use the direct relation between the original signal and geometric entities such as lines, circles, hyperplanes and hyperspheres. Furthermore, the 2D conformal monogenic signal can be extended to signals of any dimension. The main advantages of the conformal monogenic signal in practical applications are its compatibility with intrinsically one dimensional and special intrinsically two dimensional signals, the rotational invariance, the low computational time complexity, the easy implementation into existing software packages and the numerical robustness of calculating exact local curvature of signals without the need of any derivatives.

Lens-Based Depth Estimation for Multi-focus Plenoptic Cameras

Multi-focus portable plenoptic camera devices provide a reasonable tradeoff between spatial and angular resolution while enlarging the depth of field of a standard camera. Many applications using the data captured by these camera devices require or benefit from correspondences established between the single microlens images. In this work we propose a lens-based depth estimation scheme based on a novel adaptive lens selection strategy. Coarse depth estimates serve as indicators for suitable target lenses. The selection criterion accounts for lens overlap and the amount of defocus blur between the reference and possible target lenses. The depth maps are regularized using a semi-global strategy. For insufficiently textured scenes, we further incorporate a semi-global coarse regularization with respect to the lens-grid. In contrast to algorithms operating on the complete lightfield, our algorithm has a low memory footprint. The resulting per-lens dense depth maps are well suited for volumetric surface reconstruction techniques. We show that our selection strategy achieves similar error rates as selection strategies with a fixed number of lenses, while being computationally less time consuming. Results are presented for synthetic as well as real-world datasets.

A novel curvature estimator for digital curves and images

We propose a novel curvature estimation algorithm which is capable of estimating the curvature of digital curves and two-dimensional curved image structures. The algorithm is based on the conformal projection of the curve or image signal to the two-sphere. Due to the geometric structure of the embedded signal the curvature may be estimated in terms of first order partial derivatives in R3. This structure allows us to obtain the curvature by just convolving the projected signal with the appropriate kernels. We show that the method performs an implicit plane fitting by convolving the projected signals with the derivative kernels. Since the algorithm is based on convolutions its implementation is straightforward for digital curves as well as images. We compare the proposed method with differential geometric curvature estimators. It turns out that the novel estimator is as accurate as the standard differential geometric methods in synthetic as well as real and noise perturbed environments.

Signal Analysis by Generalized Hilbert Transforms on the Unit Sphere

In 1D signal processing local energy and phase can be determined by the analytic signal. Local energy, phase and orientation of 2D signals can be analyzed by the monogenic signal for all i(ntrinsic)1D signals in an rotational invariant way by the generalized Hilbert transform. In order to analyze both i1D and i2D signals in one framework the main idea of this contribution is to lift up 2D signals to the higher dimensional conformal space in which the original signal can be analyzed with more degrees of freedom by the generalized Hilbert transform on the unit sphere. An appropriate embedding of 2D signals on the unit sphere results in an extended feature space spanned by local energy, phase, orientation/direction and curvature. In contrast to classical differential geometry, local curvature can now be determined by the generalized Hilbert transform in monogenic scale space without any derivatives.

Fast projector-camera calibration for interactive projection mapping

We propose a fast calibration method for projector-camera pairs which does not require any special calibration objects or initial estimates of the calibration parameters. Our method is based on a structured light approach to establish correspondences between the camera and the projector view. Using the vanishing points in the camera and the projector view the internal as well as the external calibration parameters are estimated. In addition, we propose an interactive projection mapping scheme which allows the user to directly place two-dimensional media elements in the tangent planes of the target surface without any manual perspective corrections.

The Conformal Monogenic Signal of Image Sequences

Based on the research results of the Kiel University Cognitive Systems Group in the field of multidimensional signal processing and Computer Vision, this book chapter presents new ideas in 2D/3D and multidimensional signal theory. The novel approach, called the conformal monogenic signal, is a rotationally invariant quadrature filter for extracting i(ntrinsic)1D and i2D local features of any curved 2D signal - such as lines, edges, corners and circles - without the use of any heuristics or steering techniques. The conformal monogenic signal contains the monogenic signal as a special case for i1D signals - such as lines and edges - and combines monogenic scale space, local energy, direction/orientation, both i1D and i2D phase and curvature in one unified algebraic framework. The conformal monogenic signal will be theoretically illustrated and motivated in detail by the relation of the 3D Radon transform and the generalized Hilbert transform on the sphere. The main idea of the conformal monogenic signal is to lift up 2D signals by stereographic projection to a higher dimensional conformal space where the local signal features can be analyzed with more degrees of freedom compared to the flat two-dimensional space of the original signal domain. The philosophy of the conformal monogenic signal is based on the idea to make use of the direct relation of the original two-dimensional signal and abstract geometric entities such as lines, circles, planes and spheres. Furthermore, the conformal monogenic signal can not only be extended to 3D signals (image sequences) but also to signals of any dimension. The main advantages of the conformal monogenic signal in practical applications are the completeness with respect to the intrinsic dimension of the signal, the rotational invariance, the low computational time complexity, the easy implementation into existing Computer Vision software packages and the numerical robustness of calculating exact local curvature of signals without the need of any derivatives.

The Solution of the Interference‐Problem of 2D Waves by Means of Clifford Structures

We will present a fundamental solution of the constructive interference‐problem of waves in two dimensions (also known as fringe patterns). Such problems are known from quantum physics and optics. In case of two one‐dimensional waves with same frequency but different phases and different amplitudes, the solution of their resulting superposition or interference is well known. We will generalize this solution to two dimensions. In case of two dimensions the waves can not only be described by their phases, amplitudes and frequencies, also geometric properties pop up since in two dimensions an infinite number of additional degrees of freedom exists. The wave equations will be given in a traditional Clifford‐valued tensor form. We will solve this problem in a hybrid matrix geometric algebra setting by mapping the traditional tensor expressions to Clifford numbers in conformal space. This Clifford number representation of two‐dimensional waves can be used to solve the interference‐problem linearly. Future work ...

A Spherical Harmonic Expansion of the Hilbert Transform on the Two‐Sphere

The classical Hilbert transform on the real line is a valuable tool in signal processing. It constitutes the analytic signal which allows the determination of the instantaneous phase and amplitude of a one dimensional signal. For signals in in the Euclidean plane its analogue is the monogenic signal based on the Riesz transform, a generalization of the Hilbert transform to the plane. In addition to the instantaneous phase and amplitude, the orientation of intrinsically one dimensional structures in the plane can be determined. Various disciplines like geosciences, omnidirectional vision or astrophysics have to deal with signals arising on the two‐sphere. A Hilbert transform on the two‐sphere is well known from Clifford analysis. Yet it lacks a suitable interpretation from a signal processing viewpoint, especially in the frequency domain. In this paper we derive a series expansion of the Hilbert transform on the two‐sphere in terms of spherical harmonics. It provides an intuitive interpretation and turns o...