The need of finding a way to design a company trademark, without the worry of possible infringement on the intellectual property rights, has become exceedingly important as the economy and the accompanying intellectual property concerns advanced greatly in recent years. Traditionally, registered trademarks are stored in image databases and are categorized and retrieved by descriptions and keywords given by human workers. This is extremely time-consuming and considered by many as inappropriate. In this work we focus on image feature and content related techniques, or content-based image retrieval (CBIR) methods. Nevertheless, we still need human inputs since by law the most crucial basis for discerning the similarity or difference of two trademarks has to rely on human¡¦s naked eye. Therefore in this work we created a program which incorporates an man-machine interface allowing users to input various weighting factors each emphasizing a specific feature or shape of the trademark. The Zernike moments, and some new image compactness indices are used in the computations for image comparisons.

The performance of optical imaging systems relies on control of aberrations that can arise from limitations in the design, manufacture, or alignment. This dissertation addresses the form of aberrations that occur for misaligned reflective systems, such as telescopes. The relationship between a characteristic set of field-dependent aberrations and the misalignments that cause them is systematically explored. A comprehensive technique that quantifies field performance for a 5-mirror system is given, using Monte Carlo analysis to provide confidence levels of image quality as functions of manufacturing and alignment errors. This analysis is an example of the "forward problem"— determining optical performance of a system if the errors are assumed. The inverse problem — determining the state of alignment based on measurements of performance — is more difficult. The solution to the inverse problem for a multiple mirror system requires an understanding of the complex coupling between many degrees of freedom (tilt, decenter, despace, shape error) of the optical elements and field-dependent aberrations.This work builds on previous treatment of field dependent optical aberrations from Tessieres, Thompson, Shack, Buchroeder and others. A basis set of field-dependent aberrations orthogonal over both field and pupil are developed here and used to describe systems with misaligned and misshapen optics. This description allows complete representation of high order and non-linear effects. The functional form of aberrations that are characteristic of mirror tilt, shift, and deformation show some useful patterns that provide insight to the fundamental effects of misalignment.The use of singular value decomposition to create orthogonal combinations of the field dependent aberrations provides a powerful tool for evaluating a system and for estimating the state of alignment using wavefront measurements. The following optical systems are evaluated to investigate the linear coupling between misalignment and the resulting field dependent aberrations:* 2-mirror telescopes, evaluating well-understood effects for an axisymmetric system and developing the relationships for an unobscured system.* 4-mirror correctors for a spherical primary telescope.The tools and methods are applied to reflective optical systems for astronomical telescopes, but the methods are general and can be useful for any optical imaging system.

The term biometrics is used to describe the process of analyzing biological and behavioral traits that are unique to an individual in order to confirm or determine his or her identity. Many biometric modalities are currently being researched and implemented including, fingerprints, hand and facial geometry, iris recognition, vein structure recognition, gait, voice recognition, etc... This project explores the possibility of using corneal topography measurements as a trait for biometric identification. Two new corneal topographers were developed for this study. The first was designed to function as an operator-free device that will allow a user to approach the device and have his or her corneal topography measured. Human subject topography data were collected with this device and compared to measurements made with the commercially available Keratron Piccolo topographer (Optikon, Rome, Italy). A third topographer that departs from the standard Placido disk technology allows for arbitrary pattern illumination through the use of LCD monitors. This topographer was built and tested to be used in future research studies. Topography data was collected from 59 subjects and modeled using Zernike polynomials, which provide for a simple method of compressing topography data and comparing one topographical measurement with a database for biometric identification. The data were analyzed to determine the biometric error rates associated with corneal topography measurements. Reasonably accurate results, between three to eight percent simultaneous false match and false non-match rates, were achieved.

Two optical design scenarios—imaging and illumination—were investigated for their use of Cartesian- and polar-based functions to generate freeform optical surfaces. The imaging scenario investigated a single-element, refracting freeform surface that converts an on-axis object field to an off-axis image point. XY polynomials (Cartesian but not orthogonal) and Zernike polynomials (Polar and orthogonal) were the two different function sets used to manipulate the surfaces to achieve the freeform imaging scenarios. The investigation discovered that the results between both function sets did not differ enough to single out a more effective surface type. However, the results did indicate that the Zernike function set typically required fewer coefficients to converge on an optimal imaging solution. The illumination scenario utilized an architectural lighting situation surrounding the Rothko exhibit for Green on Blue at the University of Arizona Museum of Art. The source location was fixed to the light track in the exhibit space and pointed in many different orientations towards the painting. For each orientation, a point cloud of a freeform optical surface was generated such that the painting surface was illuminated with uniform and low-level light. For each of these generated point clouds, a Legendre (Cartesian and orthogonal) and a Zernike (polar and orthogonal) fitting function was applied, and the convergence results were compared. In general, it was found that, after the 20th included fit term, the Legendre function resulted in a smaller RMS fit error than the Zernike function. However, if the light source was pointed near the center of the painting, the Zernike function converged on a solution with fewer fit terms than Legendre. Amidst the imaging scenario, a definition for the extent to which a surface was freeform, or the "freeformity", was given. This definition proved to be an effective solution when the image size was compared for an F/3.33, F/4, F/5, and F/6.67 system for a range of different image focusing heights: the image size trends for each F-number overlapped, indicating a universal freeform term. In addition, a recursive formula for Cartesian Zernike polynomials was defined, which was used to generate an infinite number of Zernike terms using one single recursive expression.

Zernike moments are sets of mathematical quantities that uniquely characterize an image. It is known that they are invariant under rotation and reflection and robust to noise. In this thesis several other algorithms have been used to calculate these moments. The intent of this thesis is: 1. to use the classical method and the algorithms to reconstruct an image using Zernike moments and study their accuracy and 2. to examine if the invariance and noise insensitivity property of the calculated Zernike moments are upheld by these procedures. It is found that the constructed images using these algorithms do not resemble the original image. This prevents us from carrying out further study of these algorithms. The classical method has been successfully used to reconstruct an image when the height and width are equal. The classical method is also shown to be invariant under rotation and reflection and robust to Poisson noise. xxxvii

Zernike moments are sets of mathematical quantities that uniquely characterize an image. It is known that they are invariant under rotation and reflection and robust to noise. In this thesis several other algorithms have been used to calculate these moments. The intent of this thesis is: 1. to use the classical method and the algorithms to reconstruct an image using Zernike moments and study their accuracy and 2. to examine if the invariance and noise insensitivity property of the calculated Zernike moments are upheld by these procedures. It is found that the constructed images using these algorithms do not resemble the original image. This prevents us from carrying out further study of these algorithms. The classical method has been successfully used to reconstruct an image when the height and width are equal. The classical method is also shown to be invariant under rotation and reflection and robust to Poisson noise. xxxvii

Dans cette thèse, l'holographie numérique dans l'axe (DIH) est la principale méthode optique utilisée pour analyser des inclusions dans une gouttelette. L'holographie numérique dans l'axe est utilisée pour caractériser des inclusions du point de vue de leur taille, leur position 3D et leur trajectoire à l'intérieur de la gouttelette. Comme les particules sont situées à l'intérieur d'une gouttelette, le front d'onde incident sur l'inclusion est modifié avant qu'il l'illumine. Le défi de ce travail est double : premièrement de prendre en compte la forme de la gouttelette dans le modèle d'holographie et deuxièmement d'étendre l'analyse aux inclusions transparentes (type objet de phase). Pour décrire l'hologramme enregistré par le capteur CCD, l'intégrale d'Huygens-Fresnel et le formalisme des matrices ABCD ont été utilisés. Dans ce modèle, nous introduisons les polynômes de Zernike pour décrire la fonction de transmission d'une particule. Pour l'analyse des hologrammes, l'outil mathématique de la transformation de Fourier fractionnaire 2D (2D-FRFT) est utilisé pour restituer l'image des inclusions et dans ce cas une mesure la taille de l'inclusion et de sa position 3D sont réalisées. Les trajectoires des inclusions dans la goutte est possible avec un long temps de pose de l'obturateur du capteur CCD. Nous avons également proposé un nouveau modèle pour décrire des objets de phases quelconque et des particules opaques. Pour ce nouveau modèle, les mêmes procédés ont été utilisés. Dans le cas d'inclusions filiformes à l'intérieur d'une géométrie cylindrique comme un canal, une méthode de simulation d'imagerie interférométrique multi-coeurs est proposée. Dans ce cas, une somme de distributions de Dirac, localisées le long d'une droite, introduite dans l'intégrale de Fresnel généralisée (c'est-à-dire le formalisme des matrices ABCD et l'intégrale de Fresnel) permet d'obtenir un bon degré de similitude entre l'expérience et la simulation. / In this thesis, the digital in-line holography (DIH) is the main optical method used to analyze inclusions in a droplet. The digital in-line holography is used to characterize the inclusions in terms of of their size, their 3D position, and their trajectories inside the droplet. Since the particles are located within a droplet, the incident wavefront is changed before it illuminates the inclusions. The challenge of this work has two points : first to take into account the shape of the droplet in the holographic model and secondly to extend the analysis to the transparent inclusions (phase object). To describe the hologram recorded by the CCD sensor, the Huygens-Fresnel integral and the ABCD matrix formalism were used. In this model, we introduce the Zernike polynomials to describe the transmission function of a particle. For the analysis of holograms, the2D fractional Fourier transformation (2D-FRFT) is used to reconstruct the image of inclusions and in this case the size and their 3D position of the inclusions are performed.The trajectories of the inclusions in the drop are possible tracked with a long exposure shutter speed of the CCD. We also proposed a new simulation to describe objects of any phases and opaque particles. For this simulation, the same methods of reconstruction were used. In the case of micro-channel inclusions inside a cylindrical geometry such as a pipe, the interferometric imaging of multi-core pipe is proposed. In this case, summation of Dirac delta distribution, located along a line, introduced into the generalized Fresnel integral allows us to get a good agreement between the experiment and the simulation.

The human eye suffers various optical aberrations that degrade the retinal image. These aberrations include defocus and astigmatism, as well as the higher order aberrations that also play an important role in our vision. The optics of the eye are not static, but are continuously fluctuating. The work reported in this thesis has studied the nature of the microfluctuations of the wavefront aberrations of the eye and has investigated factors that influence the microfluctuations. The fluctuations in the ocular surface of the eye were investigated using high speed videokeratoscopy which measures the dynamics of the ocular surface topography. Ocular surface height difference maps were computed to illustrate the changes in the tear film in the inter-blink interval. The videokeratoscopy data was used to derive the ocular surface wavefront aberrations up to the 4th radial order of the Zernike polynomial expension. We examined the ocular surface dynamics and temporal changes in the ocular surface wavefront aberrations in the inter-blink interval. During the first 0.5 sec following a blink, the tear thickness at the upper edge of the topography map appeared to thicken by about 2 microns. The influence of pulse and instantaneous pulse rate on the microfluctuations in the corneal wavefront aberrations was also investigated. The fluctuations in ocular surface wavefront aberrations were found to be uncorrelated with the pulse and instantaneous heart rates. In the clinical measurement of the ocular surface topography using videokeratoscopy, capturing images 2 to 3 seconds after a blink will result in more consistent results. To investigate fluctuations in the wavefront aberrations of the eye and their relation to pulse and respiration frequencies we used a wavefront sensor to measure the dynamics of the aberrations up to the Zernike polynomial 4th radial order. Simultaneously, the subject's pulse rate was measured, from which the instantaneous heart rate was derived. An auto-regressive process was used to derive the power spectra of the Zernike aberration signals, as well as pulse and instantaneous heart rate signals. Linear regression analysis was performed between the frequency components of Zernike aberrations and the pulse and instantaneous heart rate frequencies. Cross spectrum density and coherence analyses were also applied to investigate the relation between fluctuations of wavefront aberrations and pulse and instantaneous heart rate. The correlations between fluctuations of individual Zernike aberrations were also determined. A frequency component of all Zernike aberrations up to the 4th radial order was found to be significantly correlated with the pulse frequency (all > 2R0.51, p<0.02), and a frequency component of 9 out of 12 Zernike aberrations was also significantly correlated with instantaneous heart rate frequency (all>2R0.46, p<0.05). The major correlations among Zernike aberrations occurred between second order and fourth order aberrations with the same angular frequencies. Higher order aberrations appear to be related to the cardiopulmonary system in a similar way to that reported for the accommodation signal and pupil fluctuations. A wavefront sensor and high speed videokeratoscopy were used to investigate the contribution of the ocular surface, the effect of stimulus vergence, and refractive error on the microfluctuations of the wavefront aberrations of the eye. The fluctuations of the Zernike wavefront aberrations were quantified by their variations around the mean and using power spectrum analysis. Integrated power was determined in two regions: 0.1 Hz ─ 0.7 Hz (low frequencies) and 0.8 Hz ─ 1.8 Hz (high frequencies). Changes in the ocular surface topography were measured using high speed videokeratoscopy and variations in the ocular wavefront aberrations were calculated. The microfluctuations of wavefront aberrations in the ocular surface were found to be small compared with the microfluctuations of the wavefront aberrations in the total eye. The variations in defocus while viewing a closer target at 2 D and 4 D stimulus vergence were found to be significantly greater than variations in defocus when viewing a far target. This increase in defocus fluctuations occurred in both the low and high frequency regions (all p<0.001) of the power spectra. The microfluctuations in astigmatism and most of the 3rd order and 4th order Zernike wavefront aberrations of the total eye were found to significantly increase with the magnitude of myopia. The experiments reported in this thesis have demonstrated the characteristics of the microfluctuations of the wavefront aberrations of the eye and have shown some of the factors that can influence the fluctuations. Major fluctuation frequencies of the eye's wavefront aberrations were shown to be significantly correlated with the pulse and instantaneous heart rate frequencies. Fluctuations in the ocular surface wavefront aberrations made a small contribution to those of the total eye. Changing stimulus vergence primarily affected the fluctuations of defocus in both low and high frequency components. Variations in astigmatism and most 3rd and 4th order aberrations were associated with refractive error magnitude. These findings will aid our fundamental understanding of the complex visual optics of the human eye and may allow the opportunity for better dynamic correction of the aberrations with adaptive optics.

Erros de alinhamento em sistemas ópticos não criam novas aberrações, mas alteram a dependência com o campo das aberrações já conhecidas. Neste trabalho, a sensibilidade teórica ao alinhamento, de sistemas ópticos reflexivos compostos por dois elementos, foi avaliada em função das constantes cônicas dos espelhos. Dentre as diferentes configurações consideradas nesta análise, uma específica apresenta menor sensibilidade à descentralização do espelho secundário. A utilização da teoria de aberração de onda aplicável a sistemas plano-simétricos revelou que a escolha apropriada da constante cônica do espelho secundário faz com que coma uniforme de terceira ordem seja compensado quando esse elemento encontra-se descentralizado, fazendo com que esse sistema seja livre da aberração mais importante causada a ele por desalinhamentos, tornando-o menos sensível. Este trabalho apresenta uma metodologia de alinhamento baseada na análise da frente de onda transmitida por sistemas ópticos, que utiliza redes neurais artificiais para a estimativa dos erros de alinhamento. A frente de onda transmitida por um sistema óptico carrega informações das aberrações desse sistema, que podem ser descritas em termos dos polinômios de Zernike. Esses polinômios podem ser usados para a análise dos efeitos de erros de alinhamento nas aberrações do sistema. Redes neurais artificiais são empregadas na análise dos coeficientes dos polinômios de Zernike visando avaliar o tipo de desalinhamento e a sua magnitude. As estimativas teóricas dos desalinhamentos tanto em sistemas reflexivos como em sistemas refrativos são satisfatórias quando o sistema é considerado perfeito, ou seja, as superfícies ópticas de seus elementos não apresentam erros de forma e não há ruído nos dados avaliados. Na presença de defeitos de fabricação ocorre degradação no desempenho do estimador. Além de descentralização e inclinação, redes neurais artificiais são capazes de fornecer uma estimativa de erros de posicionamento axial dos elementos do sistema. Com base nos estudos realizados, acredita-se que redes neurais artificiais constituem uma alternativa promissora no alinhamento de sistemas ópticos complexos. / Although misalignments in optical systems do not generate new aberration forms, they change the field-dependence of the known ones. In this research, the sensitivity of two-mirror optical systems due to misalignments is evaluated in function of the conic constants of the mirrors. Among the different configurations considered in this study, a specific one has shown low sensitivity due to decenter misalignments. The application of the wave aberration theory for plane-symmetric optical systems has revealed that the proper choice of the secondary mirror conic constant allows third-order uniform coma to be compensated, leading to a less sensitive system, free from the most important misalignment-induced aberration. This thesis also presents an alignment methodology based on the analysis of the transmitted wavefront utilizing artificial neural networks to estimate alignment errors in the components of the system. The transmitted wavefront carries information about the aberrations in the optical system, which can be described in terms of Zernike polynomials. Such polynomials are used for the analysis of the effects of misalignments on the aberrations of the system. Artificial neural networks are employed in the analysis of the coefficients of Zernike polynomials and used to evaluate both type and magnitude of the misalignments. Theoretical misalignments estimated in reflexive and refractive optical systems are satisfactory for perfect systems, i.e., systems with no surface errors, and noiseless data. When surface imperfections are considered, the performance of the estimator is reduced. Besides decenter and tilt misalignments, artificial neural networks can estimate axial positioning errors of the elements in the system, therefore they are believed to be a promising alternative for the alignment of complex optical systems.

Erros de alinhamento em sistemas ópticos não criam novas aberrações, mas alteram a dependência com o campo das aberrações já conhecidas. Neste trabalho, a sensibilidade teórica ao alinhamento, de sistemas ópticos reflexivos compostos por dois elementos, foi avaliada em função das constantes cônicas dos espelhos. Dentre as diferentes configurações consideradas nesta análise, uma específica apresenta menor sensibilidade à descentralização do espelho secundário. A utilização da teoria de aberração de onda aplicável a sistemas plano-simétricos revelou que a escolha apropriada da constante cônica do espelho secundário faz com que coma uniforme de terceira ordem seja compensado quando esse elemento encontra-se descentralizado, fazendo com que esse sistema seja livre da aberração mais importante causada a ele por desalinhamentos, tornando-o menos sensível. Este trabalho apresenta uma metodologia de alinhamento baseada na análise da frente de onda transmitida por sistemas ópticos, que utiliza redes neurais artificiais para a estimativa dos erros de alinhamento. A frente de onda transmitida por um sistema óptico carrega informações das aberrações desse sistema, que podem ser descritas em termos dos polinômios de Zernike. Esses polinômios podem ser usados para a análise dos efeitos de erros de alinhamento nas aberrações do sistema. Redes neurais artificiais são empregadas na análise dos coeficientes dos polinômios de Zernike visando avaliar o tipo de desalinhamento e a sua magnitude. As estimativas teóricas dos desalinhamentos tanto em sistemas reflexivos como em sistemas refrativos são satisfatórias quando o sistema é considerado perfeito, ou seja, as superfícies ópticas de seus elementos não apresentam erros de forma e não há ruído nos dados avaliados. Na presença de defeitos de fabricação ocorre degradação no desempenho do estimador. Além de descentralização e inclinação, redes neurais artificiais são capazes de fornecer uma estimativa de erros de posicionamento axial dos elementos do sistema. Com base nos estudos realizados, acredita-se que redes neurais artificiais constituem uma alternativa promissora no alinhamento de sistemas ópticos complexos. / Although misalignments in optical systems do not generate new aberration forms, they change the field-dependence of the known ones. In this research, the sensitivity of two-mirror optical systems due to misalignments is evaluated in function of the conic constants of the mirrors. Among the different configurations considered in this study, a specific one has shown low sensitivity due to decenter misalignments. The application of the wave aberration theory for plane-symmetric optical systems has revealed that the proper choice of the secondary mirror conic constant allows third-order uniform coma to be compensated, leading to a less sensitive system, free from the most important misalignment-induced aberration. This thesis also presents an alignment methodology based on the analysis of the transmitted wavefront utilizing artificial neural networks to estimate alignment errors in the components of the system. The transmitted wavefront carries information about the aberrations in the optical system, which can be described in terms of Zernike polynomials. Such polynomials are used for the analysis of the effects of misalignments on the aberrations of the system. Artificial neural networks are employed in the analysis of the coefficients of Zernike polynomials and used to evaluate both type and magnitude of the misalignments. Theoretical misalignments estimated in reflexive and refractive optical systems are satisfactory for perfect systems, i.e., systems with no surface errors, and noiseless data. When surface imperfections are considered, the performance of the estimator is reduced. Besides decenter and tilt misalignments, artificial neural networks can estimate axial positioning errors of the elements in the system, therefore they are believed to be a promising alternative for the alignment of complex optical systems.