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.

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.

Computer and machine vision tasks can roughly be divided into a hierarchy of processing steps applied to input signals captured by a measuring device. In the case of image signals, the first stage in this hierarchy is also referred to as low-level vision or low-level image processing. The field of low-level image processing includes the mathematical description of signals in terms of certain local signal models. The choice of the signal model is often task dependent. A common task is the extraction of features from the signal. Since signals are subject to transformations, for example camera movements in the case of image signals, the features are supposed to fulfill the properties of invariance or equivariance with respect to these transformations. The chosen signal model should reflect these properties in terms of its parameters. This thesis contributes to the field of low-level vision. Local signal structures are represented by (sinusoidal) intrinsically one-dimensional signals and their superpositions. Each intrinsically one-dimensional signal consists of certain parameters such as orientation, amplitude, frequency and phase. If the affine group acts on these signals, the transformations induce a corresponding action in the parameter space of the signal model. Hence, it is reasonable, to estimate the model parameters in order to describe the invariant and equivariant features. The first and main contribution studies superpositions of intrinsically one-dimensional signals in the plane. The parameters of the signal are supposed to be extracted from the responses of linear shift invariant operators: the generalized Hilbert transform (Riesz transform) and its higher-order versions and the partial derivative operators. While well known signal representations, such as the monogenic signal, allow to obtain the local features amplitude, phase and orientation for a single intrinsically one-dimensional signal, there exists no general method to decompose superpositions of such signals into their corresponding features. A novel method for the decomposition of an arbitrary number of sinusoidal intrinsically one-dimensional signals in the plane is proposed. The responses of the higher-order generalized Hilbert transforms in the plane are interpreted as symmetric tensors, which allow to restate the decomposition problem as a symmetric tensor decomposition. Algorithms, examples and applications for the novel decomposition are provided. The second contribution studies curved intrinsically one-dimensional signals in the plane. This signal model introduces a new parameter, the curvature, and allows the representation of curved signal structures. Using the inverse stereographic projection to the sphere, these curved signals are locally identified with intrinsically one-dimensional signals in the three-dimensional Euclidean space and analyzed in terms of the generalized Hilbert transform and partial derivatives therein. The third contribution studies the generalized Hilbert transform in a non-Euclidean space, the two-sphere. The mathematical framework of Clifford analysis proposes a further generalization of the generalized Hilbert transform to the two-sphere in terms of the corresponding Cauchy kernel. Nonetheless, this transform lacks an intuitive interpretation in the frequency domain. A decomposition of the Cauchy kernel in terms of its spherical harmonics is provided. Its coefficients not only provide insights to the generalized Hilbert transform on the sphere, but also allow for fast implementations in terms of analogues of the convolution theorem on the sphere.

This work introduces a novel method to estimate the characteristic scale of low-level image structures, which can be modeled as superpositions of intrinsically one-dimensional signals. Rather than being a single scalar quantity, the characteristic scale of the superimposed signal model is an affine equivariant regional feature. The estimation of the characteristic scale is based on an accurate estimation scheme for the orientations of the intrinsically one-dimensional signals. Using the orientation estimations, the characteristic scales of the single intrinsically one-dimensional signals are obtained. The single orientations and scales are combined into a single affine equivariant regional feature describing the characteristic scale of the superimposed signal model. Being based on convolutions with linear shift invariant filters and one-dimensional extremum searches it yields an efficient implementation.

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.

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.

The local analysis of signals arising on the sphere is a common task in earth sciences. On the real line the analytic signal turned out to be an important representation in local one-dimensional signal processing. Its generalization to two dimensions is the monogenic signal, and the properties of the analytic and the monogenic signal in the Fourier domain are well known. A generalization to the sphere is given by the Hilbert transform on the sphere known from Clifford analysis. To obtain a spectral characterization, the transform has to be decomposed into spherical harmonic functions. In this paper, we derive the spherical harmonic coefficients of the Hilbert transform on the sphere and give a series expansion. This will show that it acts as a differential operator on the spherical harmonic basis functions of the Laplace equation solution, analogously to the Riesz transform in two dimensions. This allows an interpretation of the Hilbert transform suitable for signal processing of signals naturally arising on the two-sphere. We show that the scale space naturally arising is a Poisson scale space in the unit ball. In addition, the obtained interpretation of the Hilbert transform is used for orientation analysis of plane waves. This representation is justified as a novel signal model on the sphere which can be used to construct intensity and rotation-invariant operators for local signal analysis in a scale-space concept.

This work presents the isotropic and orthogonal decomposition of 2D signals into local geometrical and structural components. We will present the solution for 2D image signals in four steps: signal modeling in scale space, signal extension by higher order generalized Hilbert transforms, signal representation in classical matrix form, followed by the most important step, in which the matrix-valued signal will be mapped to a so called multivector. We will show that this novel multivector-valued signal representation is an interesting space for complete geometrical and structural signal analysis. In practical computer vision applications lines, edges, corners, and junctions as well as local texture patterns can be analyzed in one unified algebraic framework. Our novel approach will be applied to parameter-free multilayer decomposition.

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.

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.

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 out to be a gradient-like operator acting only on the angular parts of the signal. This leads to intensity and rotation invariant signal analysis techniques on the two-sphere in analogue to the Euclidean plane.

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 will consist of the generalization from 2D to multi-dimensional waves.

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.