Šiame darbe tiriamos naujo metodo, skirto funkcijų aproksimavimui baigtine eksponenčių suma galimybės, taikant šį metodą konkrečios diferencialinių lygčių sistemos, aprašančios mechaninius virpesius, sprendiniams. Viena iš galimų darbe pristatomos mechaninių virpesių sistemos taikymo sričių – jūros bangų arba vėjo sukeltus virpesius panaudoti kaip atsinaujinantį energijos šaltinį. Tokių mechanizmų veikimo principai prieš pradedant kurti realų veikiantį modelį analizuojami taikant matematinį modeliavimą. Sudėtingos lygčių sistemos sprendiniai, priklausomai nuo sprendimo metodo, gaunami laipsninių eilučių pavidale arba kaip taškų aibė, bet nei viena iš šių formų nėra patogi sprendinio kokybiniam tyrimui. Tačiau turint sprendinio išraišką eksponentinių funkcijų su kompleksiniais koeficientais suma, žinomi ir šį sprendinį sudarančių harmonikų dažniai – svarbi konkretaus virpesių sistemos režimo charakteristika. Atliekant skaitinius eksperimentus nustatyta, jog nusistovėjusį sistemos sprendinį galima įvertinti baigtine eksponenčių suma. Aproksimavimo paklaidos priklauso nuo žingsnio, aproksimuojamos funkcijos ir skaičiavimo paklaidos. / The aim of this work was to explore the possibilities of a new method, which gives an ability to approximate functions by a finite sum of exponential functions. This method was applied to the solutions of the concrete differential equations that describe the system of mechanical oscillations. One of the possible application areas of the system of oscillations presented in the paper is to use oscillations caused by the wind or water waves as a source of renewable energy. The action principles of such mechanisms are investigated using mathematical simulation before the real working model. The solutions of the sophisticated system of differential equations are obtained either in the form of power series or a set of points, depending of the solving method chosen. However, none of these forms is convenient for exploring properties of the solution. Therefore, we have a problem to approximate the solutions with linear formations of exponential functions. It is possible then to express the solutions as the linear formations of harmonics. It is demonstrated that a steady solution of the system can be expressed as a finite sum of exponential functions. Approximation errors vary depending on the distance between the points used, the function, which is being approximated, and the computation errors.

No / Abstract: We develop a novel approach to estimate the n unknown constituent frequencies of a sinusoidal signal that comprises of unknown number, n, of sinusoids of unknown phases and unknown amplitudes. The approach has been applied to multiple sinusoidal signals in the presence of white Gaussian noise with varying signal to noise ratio (SNR). The approach is based on eigenspace analysis of Hankel matrix formed with the samples from averaged frequency spectrum of the signal obtained through multiple measurements. The eigenspace analysis is based on the newly developed 3M relationship which reflects and exploits the relationship between the consecutive sets of Maximum, Middle and Minimum eigenvalues of square symmetric matrix of the Hankel matrix. The 3M relationship exhibits a pattern in line with the order of the Hankel matrix and leads to parametric estimation of the constituent sinusoids. This paper also presents the relationship equation between the size of 3M relationship pattern and the dimensions of the Hankel matrix. The performance of the developed approach has been tested to correctly estimate multiple constituent frequencies within a noisy signal.

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The representer theorem from the reproducing kernel Hilbert space theory is the origin of many kernel-based machine learning and signal modelling techniques that are popular today. Most kernel functions used in practical applications behave in a homogeneous manner across the domain of the signal of interest, and they are called stationary kernels. One open problem in the literature is the specification of a non-stationary kernel that is computationally tractable. Some recent works solve large-scale optimization problems to obtain such kernels, and they often suffer from non-identifiability issues in their optimization problem formulation. Many practical problems can benefit from using application-specific prior knowledge on the signal of interest. For example, if one can adequately encode the prior assumption that edge contours are smooth, one does not need to learn a finite-dimensional dictionary from a database of sampled image patches that each contains a circular object in order to up-convert images that contain circular edges. In the first portion of this thesis, we present a novel method for constructing non-stationary kernels that incorporates prior knowledge. A theorem is presented that ensures the result of this construction yields a symmetric and positive-definite kernel function. This construction does not require one to solve any non-identifiable optimization problems. It does require one to manually design some portions of the kernel while deferring the specification of the remaining portions to when an observation of the signal is available. In this sense, the resultant kernel is adaptive to the data observed. We give two examples of this construction technique via the grayscale image up-conversion task where we chose to incorporate the prior assumption that edge contours are smooth. Both examples use a novel local analysis algorithm that summarizes the p-most dominant directions for a given grayscale image patch. The non-stationary properties of these two types of kernels are empirically demonstrated on the Kodak image database that is popular within the image processing research community. Tensors and tensor decomposition methods are gaining popularity in the signal processing and machine learning literature, and most of the recently proposed tensor decomposition methods are based on the tensor power and alternating least-squares algorithms, which were both originally devised over a decade ago. The algebraic approach for the canonical polyadic (CP) symmetric tensor decomposition problem is an exception. This approach exploits the bijective relationship between symmetric tensors and homogeneous polynomials. The solution of a CP symmetric tensor decomposition problem is a set of p rank-one tensors, where p is fixed. In this thesis, we refer to such a set of tensors as a rank-one decomposition with cardinality p. Existing works show that the CP symmetric tensor decomposition problem is non-unique in the general case, so there is no bijective mapping between a rank-one decomposition and a symmetric tensor. However, a proposition in this thesis shows that a particular space of rank-one decompositions, SE, is isomorphic to a space of moment matrices that are called quasi-Hankel matrices in the literature. Optimization over Riemannian manifolds is an area of optimization literature that is also gaining popularity within the signal processing and machine learning community. Under some settings, one can formulate optimization problems over differentiable manifolds where each point is an equivalence class. Such manifolds are called quotient manifolds. This type of formulation can reduce or eliminate some of the sources of non-identifiability issues for certain optimization problems. An example is the learning of a basis for a subspace by formulating the solution space as a type of quotient manifold called the Grassmann manifold, while the conventional formulation is to optimize over a space of full column rank matrices. The second portion of this thesis is about the development of a general-purpose numerical optimization framework over SE. A general-purpose numerical optimizer can solve different approximations or regularized versions of the CP decomposition problem, and they can be applied to tensor-related applications that do not use a tensor decomposition formulation. The proposed optimizer uses many concepts from the Riemannian optimization literature. We present a novel formulation of SE as an embedded differentiable submanifold of the space of real-valued matrices with full column rank, and as a quotient manifold. Riemannian manifold structures and tangent space projectors are derived as well. The CP symmetric tensor decomposition problem is used to empirically demonstrate that the proposed scheme is indeed a numerical optimization framework over SE. Future investigations will concentrate on extending the proposed optimization framework to handle decompositions that correspond to non-symmetric tensors.

Recognition of complex events in consumer uploaded Internet videos, captured under realworld settings, has emerged as a challenging area of research across both computer vision and multimedia community. In this dissertation, we present a systematic decomposition of complex events into hierarchical components and make an in-depth analysis of how existing research are being used to cater to various levels of this hierarchy and identify three key stages where we make novel contributions, keeping complex events in focus. These are listed as follows: (a) Extraction of novel semi-global features – firstly, we introduce a Lie-algebra based representation of dominant camera motion present while capturing videos and show how this can be used as a complementary feature for video analysis. Secondly, we propose compact clip level descriptors of a video based on covariance of appearance and motion features which we further use in a sparse coding framework to recognize realistic actions and gestures. (b) Construction of intermediate representations – We propose an efficient probabilistic representation from low-level features computed from videos, based on Maximum Likelihood Estimates which demonstrates state of the art performance in large scale visual concept detection, and finally, (c) Modeling temporal interactions between intermediate concepts – Using block Hankel matrices and harmonic analysis of slowly evolving Linear Dynamical Systems, we propose two new discriminative feature spaces for complex event recognition and demonstrate significantly improved recognition rates over previously proposed approaches.