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Concept# Duality (projective geometry)

Summary

In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of duality, one through language () and the other a more functional approach through special mappings. These are completely equivalent and either treatment has as its starting point the axiomatic version of the geometries under consideration. In the functional approach there is a map between related geometries that is called a duality. Such a map can be constructed in many ways. The concept of plane duality readily extends to space duality and beyond that to duality in any finite-dimensional projective geometry.
Principle of duality
A projective plane C may be defined axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines. Thes

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Though deep learning (DL) algorithms are very powerful for image processing tasks, they generally require a lot of data to reach their full potential. Furthermore, there is no straightforward way to impose various properties, given by the prior knowledge about the target task, on the features extracted by a DL model. Therefore, in this thesis we propose several techniques that rely on the power of graph representations to embed prior knowledge inside the learning process. This allows to reduce the solution space and leads to faster optimization convergence and higher accuracy in the representation learning.
In our first work, inspired by the ability of a human to correctly classify rotated, shifted or flipped objects, we propose an algorithm that permits to inherently encode invariance to isometric transformations of objects in an image. Our DL architecture is based on graph representations and consists of three novel layers, which we refer to as graph convolutional, dynamic pooling and statistical layers. Our experiments on the image classification tasks show that our network correctly recognizes isometrically transformed objects even though such types of transformation are not seen by the network at training time. Standard DL techniques are typically not able to succeed in solving such a problem without extensive data augmentation.
Then, we propose to exploit the properties of graph-based approaches to efficiently process images with various types of projective geometry. In particular, we are interested in increasingly popular omnidirectional cameras, which have a 360 degree field of view. Despite their effectiveness, such cameras create images with specific geometric properties, which require special techniques for efficient processing. We propose an efficient way of adjusting the weights of the graph edges to adapt the filter responses to the geometric image properties introduced by omnidirectional cameras. Our experiments prove that using the proposed graph with properly adjusted edge weights permits to reach better performance as compared to using regular grid graph with equal weights.
Finally, the approach described above relies on the isotropic filters, which work well within our transformation invariant architecture for image classification. However, for other problems (e.g. image compression) or even when used without dynamic pooling and statistical layers that are defined within the proposed architecture, these filters are unable to efficiently encode the information about the object. Thus, we introduce a different technique based on anisotropic filters that adapt their shape and size according to the omnidirectional image geometry. The main advantage of this approach compared to the previous one is the ability to encode the orientation of an image pattern, which is important for various tasks such as image compression. Our experiments show that our approach adapts to different image projective geometries and achieves state-of-the-art performance on image classification and compression tasks.
Overall we propose several methods, which combine the power of DL and graph signal processing towards incorporating prior information about the target task inside the optimization procedure. We hope that the research efforts presented in this thesis will help the development of efficient DL algorithms that can use various types of prior knowledge to make them efficient even when the available training data is scarce.

This work is dedicated to developing algebraic methods for channel coding. Its goal is to show that in different contexts, namely single-antenna Rayleigh fading channels, coherent and non-coherent MIMO channels, algebraic techniques can provide useful tools for building efficient coding schemes. Rotated lattice signal constellations have been proposed as an alternative for transmission over the single-antenna Rayleigh fading channel. It has been shown that the performance of such modulation schemes essentially depends on two design parameters: the modulation diversity and the minimum product distance. Algebraic lattices, i.e., lattices constructed by the canonical embedding of an algebraic number field, or more precisely ideal lattices, provide an efficient tool for designing such codes, since the design criteria are related to properties of the underlying number field: the maximal diversity is guaranteed when using totally real number fields and the minimum product distance is optimized by considering fields with small discriminant. Furthermore, both shaping and labelling constraints are taken care of by constructing Zn-lattices. We present here the construction of several families of such n-dimensional lattices for any n, and compute their performance. We then give an upper bound on their minimal product distance, and show that with respect to this bound, existing lattice codes are optimal in the sense that no further significant coding gain could be reached. Cyclic division algebras have been introduced recently in the context of coherent Space-Time coding. These are non-commutative algebras which naturally yield families of invertible matrices, or in other words, linear codes that fulfill the rank criterion. In this work, we further exploit the algebraic structures of cyclic algebras to build Space-Time Block codes (STBCs) that satisfy the following properties: they have full rate, full diversity, non-vanishing constant minimum determinant for increasing spectral efficiency, uniform average transmitted energy per antenna and good shaping. We give algebraic constructions of such STBCs for 2, 3, 4 and 6 antennas and show that these are the only cases where they exist. We finally consider the problem of designing Space-Time codes in the noncoherent case. The goal is to construct maximal diversity Space-Time codewords, subject to a fixed constellation constraint. Using an interpretation of the noncoherent coding problem in terms of packing subspaces according to a given metric, we consider the construction of non-intersecting subspaces on finite alphabets. Techniques used here mainly derive from finite projective geometry.

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Ce cours traite des 3 sujets suivants : la perspective, la géométrie descriptive, et une initiation à la géométrie projective.

Computer Vision aims at modeling the world from digital images acquired using video or infrared cameras, and other imaging sensors.
We will focus on images acquired using digital cameras. We will introduce basic processing techniques and discuss their field of applicability.

Luigi Bagnato, Eugenio De Vito, Laurent Jacques, Pierre Vandergheynst

In this paper, we describe a method to infer 3-D scene information using a single view captured by an omnidirectional camera. The proposed technique is inscribed in the so called ``Shape from Texture'' problem : if the textures hold by 3-D surfaces respect some a priori models, the deformation due to its projection in the image contains both local information about surface depth and orientation. To estimate this deformation, we adapt the work of Garding and Lindeberg to the case of spherical images processing. The planar multiscale procedure allowing the definition of precise texture descriptor is here replaced by a multiscale representation compatible with the compactness of the sphere. More precisely, the multiscale representation is obtained by filtering the data by dilated copies of a mother function. The spherical dilation introduced is the emph{gnomonic} dilation, a simple variation of the stereographic dilation due to Antoine and Vandergheynst. This dilation has a simple interpretation in terms of projective geometry. It fits precisely the transformation that the apparent omnidirectional image of an object follows when the distance of this object to the sensor changes. A spherical texture descriptor, close to a deformation tensor, is then defined thanks to the use of simple filters that act as smoothed differential operators on the data. Results are provided in the analysis of a synthetic example to illustrate the capacity of the proposed method.

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