The vector quantization (VQ) technique uses a codebook containing block patterns with corresponding index on each of them. In this thesis, we simple TSP (traveling salesperson) scheme in the VQ (vector quantization) index compression. The goal of this method is to improve bit ratio scheme with the same image quality. We apply the TSP (traveling salesperson) scheme to reorder the codewords in the codebook such that the di erence between the indexes in neighboring blocks of the image becomes small. Then, the block indexes in the image are remapped according to the reordered codebook. Thus, the variation between two neighboring block indexes is reduced. Finally, we compress the block indexes of the image with some lossless compression methods. Adding our TSP scheme as a step in VQ (vector quantization) index compression really achieves signi cant reducxtion of bit rates. Our experiment results show that the bpp (bits per pixel) in our method is less than the bpp of those without the TSP scheme.

In this project we introduce the general idea of geometric quantization and demonstratehow to apply the process on a few examples. We discuss how to construct a line bundleover the symplectic manifold with Dirac’s quantization conditions and how to determine if we are able to quantize a system with the help of Weil’s integrability condition. To reducethe prequantum line bundle we employ real polarization such that the system does notbreak Heisenberg’s uncertainty principle anymore. From the prequantum bundle and thepolarization we construct the sought after Hilbert space.

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The aim of this thesis is to present and discuss some mathematical and physical problems in the theory of weak, electromagnetic and strong interactions. Our main concern is a parallel development of mathematical and physical concepts and when it is possible, an attempt to bridge the abstract mathematical formulations with physical ideas. A central role in this thesis is played by a general construction scheme, which enables us to calculate explicitly all the mathematical quantities like matrix elements, Clebsch-Gordan series, Clebsch-Gordan coefficients which are necessary for a Grand Unification model construction. In this content, we have followed two basic principles: simplicity and applicability. To meet the first principle, all the construction methods developed are based on first principles and basic concepts of the Lie algebras and its representation theory, like roots and weights. Moreover, the requirement of applicability is met with the implementation of all the algorithms into computer programs. In the physical area, we have concentrated on the problem of mass. The lepton mass spectrum us studied in a theory of weak and electromagnetic interactions, while the mass problem of the SO(10) Grand Unified theory is analysed as a direct application of our Lie group construction scheme.

Deep neural networks are proven to be highly effective tools in various domains, yet their computational and memory costs restrict them from being widely deployed on portable devices. The recent rapid increase of edge computing devices has led to an active search for techniques to address the above-mentioned limitations of machine learning frameworks. The quantization of artificial neural networks (ANNs), which converts the full-precision synaptic weights into low-bit versions, emerged as one of the solutions. At the same time, spiking neural networks (SNNs) have become an attractive alternative to conventional ANNs due to their temporal information processing capability, energy efficiency, and high biological plausibility. Despite being driven by the same motivation, the simultaneous utilization of both concepts has not been fully studied. Therefore, this thesis work aims to bridge the gap between recent progress in quantized neural networks and SNNs. It presents an extensive study on the performance of the quantization function, represented as a linear combination of sigmoid functions, exploited in low-bit weight quantization in SNNs. The given quantization function demonstrates the state-of-the-art performance on four popular benchmarks, CIFAR10-DVS, DVS128 Gesture, N-Caltech101, and N-MNIST, for binary networks (64.05%, 95.45%, 68.71%, and 99.365 respectively) with small accuracy drops (8.03%, 1.18%, 3.47%, and 0.17% respectively) and up to 32x memory savings, which outperforms the existing methods.

This thesis investigates the problem of optimal design of binary joint watermarking and scalar quantization (JWSQ) systems that are robust under additive Gaussian attacks. A binary JWSQ system consists of two quantizers with disjoint codebooks. The joint quantization and embedding are performed by choosing the quantizer corresponding to the embedded message. The optimal JWSQ design for both fixed-rate and variable-rate cases was considered in the past, but the solution approaches exhibited high computational complexity. In this thesis, we propose faster binary JWSQ design algorithms for both the fixed-rate and variable-rate scenarios. We achieve the speed up by mapping the corresponding optimization problem to a minimum weight path problem in a certain weighted directed acyclic graph (with a constraint on the length of the path in the fixed-rate case). For this mapping to be possible we discretize the quantizer space and use an approximation for the probability of decoding error. The proposed solution algorithms have $O(LN^3)$ and $O(N^4)$ time complexity in the two cases respectively, where $N$ is the size of discretized source alphabet, and in the fixed-rate scenario $L$ is the number of cells in each quantizer. The effectiveness of the proposed designs is assessed through extensive experiments on a Gaussian source. Our results show that our algorithms are able to achieve performance very close to the prior existing schemes, but only at a small fraction of their running time. / Thesis / Master of Applied Science (MASc)

Formality quasi-isomorphisms Cobar(C) -> O are a necessary component of the machinery used in deformation quantization to produce quantized algebras of observables, however they are often constructed via transcendental methods, resulting in computational difficulties and quasi-isomorphisms defined over extensions of Q We will show that these formality quasi-isomorphisms can be "demystified" for a large class of dg-operads, by showing that they can be constructed recursively via an algorithm that builds them from systems of linear equations over Q, given certain assumptions on H(O). / Mathematics

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