Nowadays, still images are used everywhere in the digital world. The shortages of storage capacity and transmission bandwidth make efficient compression solutions essential. A revolutionary mathematics tool, wavelet transform, has already shown its power in image processing. Minimage, the major topic of this paper, is an application that compresses still images by wavelets. Minimage is used to compress grayscale images and true color images. It implements the wavelet transform to code standard BMP image files to LET wavelet image files, which is defined in Minimage. The code is written in C++ on the Microsoft Windows NT platform. This paper illustrates the design and implementation details in MinImage according to the image compression stages. First, the preprocessor generates the wavelet transform blocks. Second, the basic wavelet decomposition is applied to transform the image data to the wavelet coefficients. The discrete wavelet transforms are the kernel component of MinImage and are discussed in detail. The different wavelet transforms can be plugged in to extend the functionality of MinImage. The third step is the quantization. The standard scalar quantization algorithm and the optimized quantization algorithm, as well as the dequantization, are described. The last part of MinImage is the entropy-coding schema. The reordering of the coefficients based on the Peano Curve and the different entropy coding methods are discussed. This paper also gives the specification of the wavelet compression parameters adjusted by the end user. The interface, parameter specification, and analysis of MinImage are shown in the final appendix.

Operators on a manifold with (geometric) singularities are degenerate in a natural way. They have a principal symbolic structure with contributions from the different strata of the configuration. We study the calculus of such operators on the level of edge symbols of second generation, based on specific quantizations of the corner-degenerate interior symbols, and show that this structure is preserved under compositions.

Electrical scooters have gained a lot of attention and popularity among commuters all around the world since they entered the market. After all, electrical scooters have shown to be efficient and cost-effective mode of transportation for commuters and travelers. As of today electrical scooters have firmly established themselves in the micromobility industry, with an increasing global demand.  Although, as the industry is booming so are the accidents as well as getting into dangerous situations of riding electrical scooters. There is a growing concern regarding the safety of the scooters where more and more people are getting injured.   This research focuses on training a computer vision model using semi-supervised learning to help detect traffic rule violations and also prevent collisions for people using electrical scooters. However, applying a computer vision model on an embedded system can be challenging due to the limited capabilities of the hardware. This is where the model can enable post-training quantizations. This thesis examines which post-training quantization has the best performance and if it can perform better compared to the non-quantized model. There are three post-training quantizations that are being applied to the model, dynamic range, full integer and float16 post-training quantizations. The results showed that the non-quantized model achieved a mean average precision (mAP) of 0.03894 with a mean average training and validation loss of 22.10 and 28.11. The non-quantized model was compared with the three post-training quantizations in terms of mAP where the dynamic range post-training quantization achieve the best performance with a mAP of 0.03933.

[ES] En esta tesis estudiamos operadores pseudodiferenciales, que son operadores integrales de la forma f 7→ ∫ Rd (∫ Rd ei(x−y)·ξa(x,y,ξ)f(y)dy)dξ, en las clases globales de funciones ultradiferenciables de tipo Beurling Sω(Rd) introducidas por Björck, cuando la función peso ω viene dada en el sentido de Braun, Meise y Taylor. Desarrollamos el cálculo simbólico para estos operadores, tratando además el cambio de cuantización, la existencia de paramétrix pseudodiferencial y aplicaciones al frente de ondas global. La tesis consta de cuatro capítulos. En el Capítulo 1 introducimos los símbolos y amplitudes globales, y demostramos que los correspondientes operadores pseudodiferenciales están bien definidos y son continuos en en Sω(Rd). Estos resultados son extendidos en el Capítulo 2 para cuantizaciones arbitrarias, lo que conduce al estudio del traspuesto de cualquier cuantización de un operador pseudodiferencial y a la composición de dos cuantizaciones distintas de operadores pseudodiferenciales. En el Capítulo 3, desarrollamos el método de la paramétrix, dando condiciones suficientes para la existencia de paramétrix por la izquierda de un operador pseudodiferencial, que motiva en el Capítulo 4 la definición de un nuevo frente de ondas global para ultradistribuciones en S′ω(Rd) dada en términos de cuantizaciones de Weyl. Comparamos este frente de ondas con el frente de ondas de Gabor definido mediante la STFT y damos aplicaciones a la regularidad de las cuantizaciones de Weyl. / [CAT] En aquesta tesi estudiem operadors pseudodiferencials, que són operadors integrals de la forma f 7→ ∫ Rd (∫ Rd ei(x−y)·ξa(x,y,ξ)f(y)dy)dξ, en les classes globals de funcions ultradiferenciables de tipus Beurling Sω(Rd) introduïdes per Björck, quan la funció pes ω ve donada en el sentit de Braun, Meise i Taylor. Desenvolupem el càlcul simbòlic per aquestos operadors, tractant, a més a més, el canvi de quantització, l'existència de paramètrix pseudodiferencial i aplicacions al front d'ones global. La tesi consisteix de quatre capítols. Al Capítol 1 introduïm els símbols i amplituds globals, i demostrem que els corresponents operadors pseudodiferencials estan ben definits i són continus en Sω(Rd). Aquestos resultats són estesos al Capítol 2 per a quantitzacions arbitràries, que condueix a l'estudi del transposat de qualsevol quantització d'un operador pseudodiferencial i a la composició de dues quantitzacions distintes d'operadors pseudodiferencials. Al Capítol 3 desenvolupem el mètode de la paramètrix, donant condicions suficients per a l'existència de paramètrix per l'esquerra d'un operador pseudodiferencial donat, que motiva al Capítol 4 la definició d'un nou front d'ones global per a ultradistribucions en S′ω(Rd) mitjançant quantitzacions de Weyl. Comparem aquest front d'ones amb el front d'ones de Gabor definit mitjançant la STFT i donem aplicacions a la regularitat de les quantitzacions de Weyl. / [EN] In this thesis we study pseudodifferential operators, which are integral operators of the form f 7→ ∫ Rd (∫ Rd ei(x−y)·ξa(x,y,ξ)f(y)dy)dξ, in the global class of ultradifferentiable functions of Beurling type Sω(Rd) as introduced by Björck, when the weight function ω is given in the sense of Braun, Meise, and Taylor. We develop a symbolic calculus for these operators, treating also the change of quantization, the existence of pseudodifferential parametrices and applications to global wave front sets. The thesis consists of four chapters. In Chapter 1 we introduce global symbols and amplitudes and show that the corresponding pseudodifferential operators are well defined and continuous in Sω(Rd). These results are extended in Chapter 2 for arbitrary quantizations, which leads to the study of the transpose of any quantization of a pseudodifferential operator, and the composition of two different quantizations of pseudodifferential operators. In Chapter 3 we develop the method of the parametrix, providing sufficient conditions for the existence of left parametrices of a pseudodifferential operator, which motivates in Chapter 4 the definition of a new global wave front set for ultradistributions in S′ω(Rd) given in terms of Weyl quantizations. Then, we compare this wave front set with the Gabor wave front set defined by the STFT and give applications to the regularity of Weyl quantizations. / Asensio López, V. (2021). Global pseudodifferential operators in spaces of ultradifferentiable functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/174847 / TESIS