M\"obius Invariants of Shapes and Images

Identifying when different images are of the same object despite changes caused by imaging technologies, or processes such as growth, has many applications in fields such as computer vision and biological image analysis. One approach to this problem is to identify the group of possible transformations of the object and to find invariants to the action of that group, meaning that the object has the same values of the invariants despite the action of the group. In this paper we study the invariants of planar shapes and images under the M\"obius group $\mathrm{PSL}(2,\mathbb{C})$, which arises in the conformal camera model of vision and may also correspond to neurological aspects of vision, such as grouping of lines and circles. We survey properties of invariants that are important in applications, and the known M\"obius invariants, and then develop an algorithm by which shapes can be recognised that is M\"obius- and reparametrization-invariant, numerically stable, and robust to noise. We demonstrate the efficacy of this new invariant approach on sets of curves, and then develop a M\"obius-invariant signature of grey-scale images

Geometric and photometric affine invariant image registration

This thesis aims to present a solution to the correspondence problem for the registration of wide-baseline images taken from uncalibrated cameras. We propose an affine invariant descriptor that combines the geometry and photometry of the scene to find correspondences between both views. The geometric affine invariant component of the descriptor is based on the affine arc-length metric, whereas the photometry is analysed by invariant colour moments. A graph structure represents the spatial distribution of the primitive features; i.e. nodes correspond to detected high-curvature points, whereas arcs represent connectivities by extracted contours. After matching, we refine the search for correspondences by using a maximum likelihood robust algorithm. We have evaluated the system over synthetic and real data. The method is endemic to propagation of errors introduced by approximations in the system.BAE SystemsSelex Sensors and Airborne System

Robust and efficient Fourier-Mellin transform approximations for invariant grey-level image description and reconstruction

International audienceThis paper addresses the gray-level image representation ability of the Fourier-Mellin Transform (FMT) for pattern recognition, reconstruction and image database retrieval. The main practical di±culty of the FMT lies in the accuracy and e±ciency of its numerical approximation and we propose three estimations of its analytical extension. Comparison of these approximations is performed from discrete and ¯nite-extent sets of Fourier- Mellin harmonics by means of experiments in: (i) image reconstruction via both visual inspection and the computation of a reconstruction error; and (ii) pattern recognition and discrimination by using a complete and convergent set of features invariant under planar similarities. Experimental results on real gray-level images show that it is possible to recover an image to within a speci¯ed degree of accuracy and to classify objects reliably even when a large set of descriptors is used. Finally, an example will be given, illustrating both theoretical and numerical results in the context of content-based image retrieval

Monocular Pose Estimation Based on Global and Local Features

The presented thesis work deals with several mathematical and practical aspects of the monocular pose estimation problem. Pose estimation means to estimate the position and orientation of a model object with respect to a camera used as a sensor element. Three main aspects of the pose estimation problem are considered. These are the model representations, correspondence search and pose computation. Free-form contours and surfaces are considered for the approaches presented in this work. The pose estimation problem and the global representation of free-form contours and surfaces are defined in the mathematical framework of the conformal geometric algebra (CGA), which allows a compact and linear modeling of the monocular pose estimation scenario. Additionally, a new local representation of these entities is presented which is also defined in CGA. Furthermore, it allows the extraction of local feature information of these models in 3D space and in the image plane. This local information is combined with the global contour information obtained from the global representations in order to improve the pose estimation algorithms. The main contribution of this work is the introduction of new variants of the iterative closest point (ICP) algorithm based on the combination of local and global features. Sets of compatible model and image features are obtained from the proposed local model representation of free-form contours. This allows to translate the correspondence search problem onto the image plane and to use the feature information to develop new correspondence search criteria. The structural ICP algorithm is defined as a variant of the classical ICP algorithm with additional model and image structural constraints. Initially, this new variant is applied to planar 3D free-form contours. Then, the feature extraction process is adapted to the case of free-form surfaces. This allows to define the correlation ICP algorithm for free-form surfaces. In this case, the minimal Euclidean distance criterion is replaced by a feature correlation measure. The addition of structural information in the search process results in better conditioned correspondences and therefore in a better computed pose. Furthermore, global information (position and orientation) is used in combination with the correlation ICP to simplify and improve the pre-alignment approaches for the monocular pose estimation. Finally, all the presented approaches are combined to handle the pose estimation of surfaces when partial occlusions are present in the image. Experiments made on synthetic and real data are presented to demonstrate the robustness and behavior of the new ICP variants in comparison with standard approaches

A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group $SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group $SE(2,N)$, restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this semi-discrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.Comment: 123 pages, revised versio

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

Monocular pose estimation based on global and local features

The presented thesis work deals with several mathematical and practical aspects of the monocular pose estimation problem. Pose estimation means to estimate the position and orientation of a model object with respect to a camera used as a sensor element. Threemain aspects of the pose estimation problem are considered. These are themodel representations, correspondence search and pose computation. Free-form contours and surfaces are considered for the approaches presented in this work. The pose estimation problem and the global representation of free-form contours and surfaces are defined in the mathematical framework of the conformal geometric algebra (CGA), which allows a compact and linear modeling of the monocular pose estimation scenario. Additionally, a new local representation of these entities is presented which is also defined in CGA. Furthermore, it allows the extraction of local feature information of these models in 3D space and in the image plane. This local information is combined with the global contour information obtained from the global representations in order to improve the pose estimation algorithms. The main contribution of this work is the introduction of new variants of the iterative closest point (ICP) algorithm based on the combination of local and global features. Sets of compatible model and image features are obtained from the proposed local model representation of free-form contours. This allows to translate the correspondence search problem onto the image plane and to use the feature information to develop new correspondence search criteria. The structural ICP algorithm is defined as a variant of the classical ICP algorithm with additional model and image structural constraints. Initially, this new variant is applied to planar 3D free-form contours. Then, the feature extraction process is adapted to the case of free-form surfaces. This allows to define the correlation ICP algorithm for free-form surfaces. In this case, the minimal Euclidean distance criterion is replaced by a feature correlation measure. The addition of structural information in the search process results in better conditioned correspondences and therefore in a better computed pose. Furthermore, global information (position and orientation) is used in combination with the correlation ICP to simplify and improve the pre-alignment approaches for the monocular pose estimation. Finally, all the presented approaches are combined to handle the pose estimation of surfaces when partial occlusions are present in the image. Experiments made on synthetic and real data are presented to demonstrate the robustness and behavior of the new ICP variants in comparison with standard approaches

View generated database

This document represents the final report for the View Generated Database (VGD) project, NAS7-1066. It documents the work done on the project up to the point at which all project work was terminated due to lack of project funds. The VGD was to provide the capability to accurately represent any real-world object or scene as a computer model. Such models include both an accurate spatial/geometric representation of surfaces of the object or scene, as well as any surface detail present on the object. Applications of such models are numerous, including acquisition and maintenance of work models for tele-autonomous systems, generation of accurate 3-D geometric/photometric models for various 3-D vision systems, and graphical models for realistic rendering of 3-D scenes via computer graphics