Estudo do espectro Laplaciano na categorização de imagens

Uma imagem engloba informação que precisa ser organizada para interpretar e compreender seu conteúdo. Existem diversas técnicas computacionais para extrair a principal informação de uma imagem e podem ser divididas em três áreas: análise de cor, textura e forma. Uma das principais delas é a análise de forma, por descrever características de objetos baseadas em seus pontos fronteira. Propomos um método de caracterização de imagens, por meio da análise de forma, baseada nas propriedades espectrais do laplaciano em grafos. O procedimento construiu grafos G baseados nos pontos fronteira do objeto, cujas conexões entre vértices são determinadas por limiares T_l. A partir dos grafos obtêm-se a matriz de adjacência A e a matriz de graus D, as quais definem a matriz Laplaciana L=D -A. A decomposição espectral da matriz Laplaciana (autovalores) é investigada para descrever características das imagens. Duas abordagens são consideradas: a) Análise do vetor característico baseado em limiares e a histogramas, considera dois parâmetros o intervalo de classes IC_l e o limiar T_l; b) Análise do vetor característico baseado em vários limiares para autovalores fixos; os quais representam o segundo e último autovalor da matriz L. As técnicas foram testada em três coleções de imagens: sintéticas (Genéricas), parasitas intestinais (SADPI) e folhas de plantas (CNShape), cada uma destas com suas próprias características e desafios. Na avaliação dos resultados, empregamos o modelo de classificação support vector machine (SVM), o qual avalia nossas abordagens, determinando o índice de separação das categorias. A primeira abordagem obteve um acerto de 90 % com a coleção de imagens Genéricas, 88 % na coleção SADPI, e 72 % na coleção CNShape. Na segunda abordagem, obtém-se uma taxa de acerto de 97 % com a coleção de imagens Genéricas; 83 % para SADPI e 86 % no CNShape. Os resultados mostram que a classificação de imagens a partir do espectro do Laplaciano, consegue categorizá-las satisfatoriamente.

Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this article we present a computational framework for isolating spatial patterns arising in the steady states of reaction-diffusion systems. Such systems have been used to model many different phenomena in areas such as developmental and cancer biology, cell motility and material science. Often one is interested in identifying parameters which will lead to a particular pattern. To attempt to answer this, we compute eigenpairs of the Laplacian on a variety of domains and use linear stability analysis to determine parameter values for the system that will lead to spatially inhomogeneous steady states whose patterns correspond to particular eigenfunctions. This method has previously been used on domains and surfaces where the eigenvalues and eigenfunctions are found analytically in closed form. Our contribution to this methodology is that we numerically compute eigenpairs on arbitrary domains and surfaces. Here we present various examples and demonstrate that mode isolation is straightforward especially for low eigenvalues. Additionally we see that if two or more eigenvalues are in a permissible range then the inhomogeneous steady state can be a linear combination of the respective eigenfunctions. Finally we show an example which suggests that pattern formation is robust on similar surfaces in cases that the surface either has or does not have a boundary.

Existence and multiplicity results for quasilinear elliptic equations

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f (x, s), we show the following problem: -Delta(p)u = lambda f(x,u) in Omega, u/(partial derivative Omega) = 0, where Omega is a bounded open subset of R-N, N >= 2, with smooth boundary, lambda is a positive parameter and Delta(p) is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large lambda.

Asymptotic bifurcation results for quasilinear elliptic operators

We develop results for bifurcation from the principal eigenvalue for certain operators based on the p-Laplacian and containing a superlinear nonlinearity with a critical Sobolev exponent. The main result concerns an asymptotic estimate of the rate at which the solution branch departs from the eigenspace. The method can also be applied for nonpotential operators.

Application of Naïve Bayesian sequential analysis to primary care optometry

The objective of this study was to investigate the effects of circularity, comorbidity, prevalence and presentation variation on the accuracy of differential diagnoses made in optometric primary care using a modified form of naïve Bayesian sequential analysis. No such investigation has ever been reported before. Data were collected for 1422 cases seen over one year. Positive test outcomes were recorded for case history (ethnicity, age, symptoms and ocular and medical history) and clinical signs in relation to each diagnosis. For this reason only positive likelihood ratios were used for this modified form of Bayesian analysis that was carried out with Laplacian correction and Chi-square filtration. Accuracy was expressed as the percentage of cases for which the diagnoses made by the clinician appeared at the top of a list generated by Bayesian analysis. Preliminary analyses were carried out on 10 diagnoses and 15 test outcomes. Accuracy of 100% was achieved in the absence of presentation variation but dropped by 6% when variation existed. Circularity artificially elevated accuracy by 0.5%. Surprisingly, removal of Chi-square filtering increased accuracy by 0.4%. Decision tree analysis showed that accuracy was influenced primarily by prevalence followed by presentation variation and comorbidity. Analysis of 35 diagnoses and 105 test outcomes followed. This explored the use of positive likelihood ratios, derived from the case history, to recommend signs to look for. Accuracy of 72% was achieved when all clinical signs were entered. The drop in accuracy, compared to the preliminary analysis, was attributed to the fact that some diagnoses lacked strong diagnostic signs; the accuracy increased by 1% when only recommended signs were entered. Chi-square filtering improved recommended test selection. Decision tree analysis showed that accuracy again influenced primarily by prevalence, followed by comorbidity and presentation variation. Future work will explore the use of likelihood ratios based on positive and negative test findings prior to considering naïve Bayesian analysis as a form of artificial intelligence in optometric practice.

On Multidimensional Analogue of Marchaud Formula for Fractional Riesz-Type Derivatives in Domains in R^n

2000 Mathematics Subject Classification: 26A33, 42B20

Nonexistence Results of Solutions of Semilinear Differential Inequalities with Temperal Fractional Derivative on the Heinsenberg Group

2000 Mathematics Subject Classification: 26A33, 33C60, 44A15, 35K55

Generalized Fractional Evolution Equation

2000 Mathematics Subject Classification: Primary 46F25, 26A33; Secondary: 46G20

On a Stochastic Partial Differential Equation with a Noisy Term

2000 Mathematics Subject Classification: 60H15, 60H40

Spectral Shift Function for the Perturbations of Schrödinger Operators at High Energy

2000 Mathematics Subject Classification: 35P20, 35J10, 35Q40.

Probabilistic topographic information visualisation

The focus of this thesis is the extension of topographic visualisation mappings to allow for the incorporation of uncertainty. Few visualisation algorithms in the literature are capable of mapping uncertain data with fewer able to represent observation uncertainties in visualisations. As such, modifications are made to NeuroScale, Locally Linear Embedding, Isomap and Laplacian Eigenmaps to incorporate uncertainty in the observation and visualisation spaces. The proposed mappings are then called Normally-distributed NeuroScale (N-NS), T-distributed NeuroScale (T-NS), Probabilistic LLE (PLLE), Probabilistic Isomap (PIso) and Probabilistic Weighted Neighbourhood Mapping (PWNM). These algorithms generate a probabilistic visualisation space with each latent visualised point transformed to a multivariate Gaussian or T-distribution, using a feed-forward RBF network. Two types of uncertainty are then characterised dependent on the data and mapping procedure. Data dependent uncertainty is the inherent observation uncertainty. Whereas, mapping uncertainty is defined by the Fisher Information of a visualised distribution. This indicates how well the data has been interpolated, offering a level of ‘surprise’ for each observation. These new probabilistic mappings are tested on three datasets of vectorial observations and three datasets of real world time series observations for anomaly detection. In order to visualise the time series data, a method for analysing observed signals and noise distributions, Residual Modelling, is introduced. The performance of the new algorithms on the tested datasets is compared qualitatively with the latent space generated by the Gaussian Process Latent Variable Model (GPLVM). A quantitative comparison using existing evaluation measures from the literature allows performance of each mapping function to be compared. Finally, the mapping uncertainty measure is combined with NeuroScale to build a deep learning classifier, the Cascading RBF. This new structure is tested on the MNist dataset achieving world record performance whilst avoiding the flaws seen in other Deep Learning Machines.

Detection and Classification of Whale Acoustic Signals

This dissertation focuses on two vital challenges in relation to whale acoustic signals: detection and classification.

In detection, we evaluated the influence of the uncertain ocean environment on the spectrogram-based detector, and derived the likelihood ratio of the proposed Short Time Fourier Transform detector. Experimental results showed that the proposed detector outperforms detectors based on the spectrogram. The proposed detector is more sensitive to environmental changes because it includes phase information.

In classification, our focus is on finding a robust and sparse representation of whale vocalizations. Because whale vocalizations can be modeled as polynomial phase signals, we can represent the whale calls by their polynomial phase coefficients. In this dissertation, we used the Weyl transform to capture chirp rate information, and used a two dimensional feature set to represent whale vocalizations globally. Experimental results showed that our Weyl feature set outperforms chirplet coefficients and MFCC (Mel Frequency Cepstral Coefficients) when applied to our collected data.

Since whale vocalizations can be represented by polynomial phase coefficients, it is plausible that the signals lie on a manifold parameterized by these coefficients. We also studied the intrinsic structure of high dimensional whale data by exploiting its geometry. Experimental results showed that nonlinear mappings such as Laplacian Eigenmap and ISOMAP outperform linear mappings such as PCA and MDS, suggesting that the whale acoustic data is nonlinear.

We also explored deep learning algorithms on whale acoustic data. We built each layer as convolutions with either a PCA filter bank (PCANet) or a DCT filter bank (DCTNet). With the DCT filter bank, each layer has different a time-frequency scale representation, and from this, one can extract different physical information. Experimental results showed that our PCANet and DCTNet achieve high classification rate on the whale vocalization data set. The word error rate of the DCTNet feature is similar to the MFSC in speech recognition tasks, suggesting that the convolutional network is able to reveal acoustic content of speech signals.

Positive solutions for parametric Dirichlet problems with indefinite potential and superdiffusive reaction

We consider a parametric semilinear Dirichlet problem driven by the Laplacian plus an indefinite unbounded potential and with a reaction of superdifissive type. Using variational and truncation techniques, we show that there exists a critical parameter value λ_{∗}>0 such that for all λ> λ_{∗} the problem has least two positive solutions, for λ= λ_{∗} the problem has at least one positive solutions, and no positive solutions exist when λ∈(0,λ_{∗}). Also, we show that for λ≥ λ_{∗} the problem has a smallest positive solution.

Cornered Asymptotically Hyperbolic Metrics

Thesis (Ph.D.)--University of Washington, 2016-08

Expedition in Data and Harmonic Analysis on Graphs

The graph Laplacian operator is widely studied in spectral graph theory largely due to its importance in modern data analysis. Recently, the Fourier transform and other time-frequency operators have been defined on graphs using Laplacian eigenvalues and eigenvectors. We extend these results and prove that the translation operator to the i’th node is invertible if and only if all eigenvectors are nonzero on the i’th node. Because of this dependency on the support of eigenvectors we study the characteristic set of Laplacian eigenvectors. We prove that the Fiedler vector of a planar graph cannot vanish on large neighborhoods and then explicitly construct a family of non-planar graphs that do exhibit this property. We then prove original results in modern analysis on graphs. We extend results on spectral graph wavelets to create vertex-dyanamic spectral graph wavelets whose support depends on both scale and translation parameters. We prove that Spielman’s Twice-Ramanujan graph sparsifying algorithm cannot outperform his conjectured optimal sparsification constant. Finally, we present numerical results on graph conditioning, in which edges of a graph are rescaled to best approximate the complete graph and reduce average commute time.