Multidimensional Wavelets and Computer Vision

This report deals with the construction and the mathematical analysis of multidimensional nonseparable wavelets and their efficient application in computer vision. In the first part, the fundamental principles and ideas of multidimensional wavelet filter design such as the question for the existence of good scaling matrices and sensible design criteria are presented and extended in various directions. Afterwards, the analytical properties of these wavelets are investigated in some detail. It will turn out that they are especially well-suited to represent (discretized) data as well as large classes of operators in a sparse form - a property that directly yields efficient numerical algorithms. The final part of this work is dedicated to the application of the developed methods to the typical computer vision problems of nonlinear image regularization and the computation of optical flow in image sequences. It is demonstrated how the wavelet framework leads to stable and reliable results for these problems of generally ill-posed nature. Furthermore, all the algorithms are of order O(n) leading to fast processing

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Nondyadic and nonlinear multiresolution image approximations

This thesis focuses on the development of novel multiresolution image approximations. Specifically, we present two kinds of generalization of multiresolution techniques: image reduction for arbitrary scales, and nonlinear approximations using other metrics than the standard Euclidean one. Traditional multiresolution decompositions are restricted to dyadic scales. As first contribution of this thesis, we develop a method that goes beyond this restriction and that is well suited to arbitrary scale-change computations. The key component is a new and numerically exact algorithm for computing inner products between a continuously defined signal and B-splines of any order and of arbitrary sizes. The technique can also be applied for non-uniform to uniform grid conversion, which is another approximation problem where our method excels. Main applications are resampling and signal reconstruction. Although simple to implement, least-squares approximations lead to artifacts that could be reduced if nonlinear methods would be used instead. The second contribution of the thesis is the development of nonlinear spline pyramids that are optimal for lp-norms. First, we introduce a Banach-space formulation of the problem and show that the solution is well defined. Second, we compute the lp-approximation thanks to an iterative optimization algorithm based on digital filtering. We conclude that l1-approximations reduce the artifacts that are inherent to least-squares methods; in particular, edge blurring and ringing. In addition, we observe that the error of l1-approximations is sparser. Finally, we derive an exact formula for the asymptotic Lp-error; this result justifies using the least-squares approximation as initial solution for the iterative optimization algorithm when the degree of the spline is even; otherwise, one has to include an appropriate correction term. The theoretical background of the thesis includes the modelisation of images in a continuous/discrete formalism and takes advantage of the approximation theory of linear shift-invariant operators. We have chosen B-splines as basis functions because of their nice properties. We also propose a new graphical formalism that links B-splines, finite differences, differential operators, and arbitrary scale changes

Preservation of 2-D signal symmetries in quincunx filter-banks

An important problem in the analysis of symmetric extension methods is to determine the conditions under which signal symmetries are preserved by the filtering (convolution) and downsampling operations. In this letter, we provide a complete characterization of four-fold two-dimensional signal symmetries viz. quadrantal symmetry, diagonal symmetry, and 90deg rotational symmetry. We then consider Quincunx filter-banks and determine the conditions under which the four-fold signal symmetries are preserved by the filtering and downsampling operations.IEE

Painting in Stone: The Symbolism of Colored Marbles in the Visual Arts and Literature from Antiquity until the Enlightenment

Colored marble has been used throughout the Mediterranean as a building material, architectural veneer, sculptural material, even a support for painting since at least the second century BC. This thesis examines the poetics and symbolism of marbles, as a medium more than a material, over many centuries along three predominant lines: as images of substance according to a pre-modern concept of matter and pre-modern notions of geology; marble's apparent ability to bear light due to its polish and occasional translucency; and the longue durée that colored marbles constituted a form of natural (hence divine) painting. The use of marble in architecture and sculpture, as well as its depiction in painting and its description in literature, is examined from the Augustan era up untnil the close of the seventeenth century. Examples range from Durham to Samarra, from Ottoman folklore to popular piety in Florida, from Etruscan tomb painting to installation art, but key monuments like Hagia Sophia and the Cornaro Chapel offer case studies for in-depth analysis

BUILT UTOPIAS IN THE COUNTRYSIDE: THE RURAL AND THE MODERN IN FRANCO’S SPAIN

Anchored by Hüppauf and Umbach’s notion of Vernacular Modernism and focusing on architecture and urbanism during Franco’s dictatorship from 1939 to 1975, this thesis challenges the hegemonic and Northern-oriented narrative of urban modernity. It develops arguments about the reciprocal influences between the urban and the rural that characterize Spanish modernity, and analyzes the intense architectural and urban debates that resulted from the crisis of 1898, as they focused on the importance of vernacular architecture, in particular the Mediterranean one, in the definition of an “other modernity.†This search culminated before 1936 with the “Lessons of Ibiza,†and was revived at the beginning of the 1950s, when architects like Coderch, Fisac, Bohigas, and the cosigners of the Manifiesto de la Alhambra brought back the discourse of the modern vernacular as a politically acceptable form of Spanish modernity, and extended its field of application from the individual house and the rural architecture to the urban conditions, including social and middle-class housing. The core of the dissertation addresses the 20th century phenomenon of the modern agricultural village as built emergence of a rural paradigm of modernity in parallel or alternative to the metropolitan condition. In doing so, it interrogates the question of tradition, modernity, and national identity in urban form between the 1920s and the 1960s. Regarding Spain, it studies the actuation of the two Institutes that were created to implement the Francoist policy of post-war reconstruction and interior colonization—the Dirección General de Regiones Devastadas, and the Instituto Nacional de Colonización. It examines the ideological, political, urban, and architectural principles of Franco’s reconstruction of the devastated countryside, as well as his grand “hydro-social dream†of modernization of the countryside. It analyzes their role in national-building policies in liaison with the early 20th-century Regenerationist Movement of Joaquín Costa, the first works of hydraulic infrastructure under Primo de Rivera, and the aborted agrarian reform of the Second Republic. Inspired by the Zionist colonization of Palestine and Mussolini’s reclaiming of the Pontine Marshes, Falangist planners developed a national strategy of “interior colonization†that, along with the reclamation and irrigation of extensive and unproductive river basins, entailed the construction of three hundred modern villages or pueblos between 1940 and 1971. Each village was designed as a “rural utopia,†centered on a plaza mayor and the church, which embodied the political ideal of civil life under the nationalcatholic regime and evolved from a traditional town design in the 1940s to an increasingly abstract and modern vision, anchored on the concept of the “Heart of the City†after 1952. The program was an important catalyst for the development of Spanish modern architecture after the first period of autarchy and an effective incubator for a new generation of architects, including Alejandro de la Sota, José Luis Fernández del Amo, and others. Between tradition and modernity, these architects reinvented the pueblos as platforms of urban and architectonic experimentation in their search for a depurated rural vernacular and a modern urban form. Whereas abstraction was the primary design tool that Fernández del Amo deployed to the limits of the continuity of urban form, de la Sota reversed the fundamental reference to the countryside that characterizes Spanish surrealism to bring surrealism within the process of rural modernization in Franco’s Spain