Showing papers on "Zernike polynomials published in 2004"

Shape retrieval using 3D Zernike descriptors

Abstract: We advocate the usage of 3D Zernike invariants as descriptors for 3D shape retrieval. The basis polynomials of this representation facilitate computation of invariants under rotation, translation and scaling. Some theoretical results have already been summarized in the past from the aspect of pattern recognition and shape analysis. We provide practical analysis of these invariants along with algorithms and computational details. Furthermore, we give a detailed discussion on influence of the algorithm parameters like the conversion into a volumetric function, number of utilized coefficients, etc. As is revealed by our study, the 3D Zernike descriptors are natural extensions of recently introduced spherical harmonics based descriptors. We conduct a comparison of 3D Zernike descriptors against these regarding computational aspects and shape retrieval performance using several quality measures and based on experiments on the Princeton Shape Benchmark.

Characterizing specimen induced aberrations for high NA adaptive optical microscopy

Abstract: Aberrations are known to severely compromise image quality in optical microscopy, especially when high numerical aperture (NA) lenses are used in confocal fluorescence microscopy (CFM) and two-photon microscopy (TPM). The method of adaptive optics may correct aberrations and restore diffraction limited operation. So far the problem of aberrations that occur in the imaging of biological specimens has not been quantified. However, this information is essential for the design of adaptive optics systems. We have therefore built an interferometer incorporating high NA objective lenses to measure the aberrations introduced by biological specimens. The measured wavefronts were decomposed into their Zernike mode content in order both to classify and quantify the aberrations. We calculated the potential benefit of correcting different numbers of Zernike modes using different NAs in an adaptive CFM by comparing the signal levels before and after correction. The results indicate that adaptive correction of low order Zernike modes can provide significant benefit for many specimens. The results also show that quantitative fluorescence microscopy may be strongly affected by specimen induced aberrations in non-adaptive systems.

Adaptive optics with a programmable phase modulator: applications in the human eye

Abstract: Adaptive optics for the human eye has two main applications: to obtain high-resolution images of the retina and to produce aberration-free retinal images to improve vision. Additionally, it can be used to modify the aberrations of the eye to perform experiments to study the visual function. We have developed an adaptive optics prototype by using a liquid crystal spatial light modulator (Hamamatsu Programmable Phase Modulator X8267). The performance of this device both as aberration generator and corrector has been evaluated. The system operated either with red (633nm) or infrared (780nm) illumination and used a real-time Hartmann-Shack wave-front sensor (25 Hz). The aberration generation capabilities of the modulator were checked by inducing different amounts of single Zernike terms. For a wide range of values, the aberration production process was found to be linear, with negligible cross-coupling between Zernike terms. Subsequently, the modulator was demonstrated to be able to correct the aberrations of an artificial eye in a single step. And finally, it was successfully operated in close-loop mode for aberration correction in living human eyes. Despite its slow temporal response, when compared to currently available deformable mirrors, this device presents advantages in terms of effective stroke and mode independence. Accordingly, the programmable phase modulator allows production and compensation of a wide range of aberrations, surpassing in this respect the performance of low-cost mirrors and standing comparison against more expensive devices.

Sketched symbol recognition using Zernike moments

Abstract: We present an on-line recognition method for hand-sketched symbols. The method is independent of stroke-order, -number, and -direction, as well as invariant to scaling, translation, rotation and reflection of symbols. Zernike moment descriptors are used to represent symbols and three different classification techniques are compared: support vector machines (SVM), minimum mean distance (MMD), and nearest neighbor (NN). We have obtained a 97% recognition accuracy rate on a dataset consisting of 7,410 sketched symbols using Zernike moment features and a SVM classifier.

Measurement of specimen-induced aberrations of biological samples using phase stepping interferometry.

Abstract: Confocal or multiphoton microscopes, which deliver optical sections and three-dimensional (3D) images of thick specimens, are widely used in biology. These techniques, however, are sensitive to aberrations that may originate from the refractive index structure of the specimen itself. The aberrations cause reduced signal intensity and the 3D resolution of the instrument is compromised. It has been suggested to correct for aberrations in confocal microscopes using adaptive optics. In order to define the design specifications for such adaptive optics systems, one has to know the amount of aberrations present for typical applications such as with biological samples. We have built a phase stepping interferometer microscope that directly measures the aberration of the wavefront. The modal content of the wavefront is extracted by employing Zernike mode decomposition. Results for typical biological specimens are presented. It was found for all samples investigated that higher order Zernike modes give only a small contribution to the overall aberration. Therefore, these higher order modes can be neglected in future adaptive optics sensing and correction schemes implemented into confocal or multiphoton microscopes, leading to more efficient designs.

Advantages and disadvantages of the Zernike expansion for representing wave aberration of the normal and aberrated eye.

Abstract: PURPOSE: Zernike expansion has been selected for use in describing wavefront aberrations in the human eye. The advantages and limitations of this approach are assessed for eyes with varying degrees of aberration. METHODS:Corneal topography examinations were taken with the Nidek OPD-Scan topographer/aberrometer. These higher data density corneal topography examinations were converted to height data and subsequently to wavefront representations. System noise was evaluated with a 2D frequency analysis of 43-D test balls. Both Zernike polynomials and 2D Fourier transforms were used to evaluate fidelity in the presentation of the point spread function. A display format for potential clinical use was developed based upon Zernike decomposition. RESULTS: Systematic noise from the corneal topographer was found to be minimal and, when eliminated, produced small changes in the point spread function. Using Zernike decomposition up to the 30th order failed to preserve the higher frequency aberrations present in aberrated eyes. Use of a Zernike decomposition display with a fixed micron scale presented only clinically significant details of spherical aberration, coma, trefoil, irregular components above third order and total higher-order aberrations (above second order). CONCLUSIONS: Zernike polynomials excel in extracting the low-order optical characteristics of visual optics. Zernikes accurately represent both low- and high-order aberrations in normal eyes where high-order aberrations are clinically insignificant. For eyes after corneal surgery or eyes with corneal pathology such as keratoconus that have significant higher-order aberrations, the Zernike method fails to capture all clinically significant higher-order aberrations.

Variability of wavefront aberration measurements in small pupil sizes using a clinical Shack-Hartmann aberrometer

Abstract: Recently, instruments for the measurement of wavefront aberration in the living human eye have been widely available for clinical applications. Despite the extensive background experience on wavefront sensing for research purposes, the information derived from such instrumentation in a clinical setting should not be considered a priori precise. We report on the variability of such an instrument at two different pupil sizes. A clinical aberrometer (COAS Wavefront Scienses, Ltd) based on the Shack-Hartmann principle was employed in this study. Fifty consecutive measurements were perfomed on each right eye of four subjects. We compared the variance of individual Zernike expansion coefficients as determined by the aberrometer with the variance of coefficients calculated using a mathematical method for scaling the expansion coefficients to reconstruct wavefront aberration for a reduced-size pupil. Wavefront aberration exhibits a marked variance of the order of 0.45 microns near the edge of the pupil whereas the central part appears to be measured more consistently. Dispersion of Zernike expansion coefficients was lower when calculated by the scaling method for a pupil diameter of 3 mm as compared to the one introduced when only the central 3 mm of the Shack – Hartmann image was evaluated. Signal-to-noise ratio was lower for higher order aberrations than for low order coefficients corresponding to the sphero-cylindrical error. For each subject a number of Zernike expansion coefficients was below noise level and should not be considered trustworthy. Wavefront aberration data used in clinical care should not be extracted from a single measurement, which represents only a static snapshot of a dynamically changing aberration pattern. This observation must be taken into account in order to prevent ambiguous conclusions in clinical practice and especially in refractive surgery.

Full optical column characterization of DUV lithographic projection tools

Abstract: Advanced optical systems for low k1 lithography require accurate characterisation of various imaging parameters to insure that OPC strategies can be maintained. Among these parameters lens aberrations and illumination profiles are the most important optical column charcteristics. The phase measurement interferometer hardware (ILIAS: Integrated Lens Interferometer At Scanner) integrated into high-NA ArF lithographic projection tools opens novel pathways to measure and control tool critical performance parameters. In this presentation we address new extensions of this in-line tool that will allow the measurement of optical parameters of the full optical column. The primary functionality of the ILIAS system is to measure and analyse wavefront aberrations across the full image field with high accuracy and speed. In this paper performance data of the in-line wavefront sensor over multiple high-NA ArF lithographic systems is presented. In addition to the acquisition of wavefront aberrations in terms of Zernike polynomials, detailed measurements of high resolution wavefronts are now possible. Examples of such wavefronts and PSD analysis thereof are presented. Besides the projection lens properties, the detailed shape of the pupil distribution and transmission (apodisation) becomes critical for system optimization. The integrated ILIAS hardware can also be used to measure these parameters.

Wave aberrations of the isolated crystalline lens.

Abstract: A method to measure wave aberrations in the isolated crystalline lens is demonstrated. The method employs a laser scanning technique in which the trajectories of narrow refracted laser beams are measured for an array of sample positions incident on the lens. The local slope of the emerging wavefront is calculated for each sample position, and a least squares procedure is used to fit a Zernike polynomial function to define the wave aberration. Measurements of the aberrations of an isolated porcine lens and macaque lens undergoing changes in accommodative state with mechanical stretching are shown. Many aberrations were present, but negative spherical aberration dominated. In the macaque lens, many aberrations underwent systematic changes with accommodation, most notably the 4th order spherical aberration, which became more negative, and the 6th order spherical aberration, which progressed from negative to positive.

Theoretical modeling and evaluation of the axial resolution of the adaptive optics scanning laser ophthalmoscope.

Abstract: We present axial resolution calculated using a mathemati- cal model of the adaptive optics scanning laser ophthalmoscope (AOSLO). The peak intensity and the width of the axial intensity re- sponse are computed with the residual Zernike coefficients after the aberrations are corrected using adaptive optics for eight subjects and compared with the axial resolution of a diffraction-limited eye. The AOSLO currently uses a confocal pinhole that is 80 mm, or 3.48 times the width of the Airy disk radius of the collection optics, and projects to 7.41 mm on the retina. For this pinhole, the axial resolution of a diffraction-limited system is 114 mm and the computed axial resolu- tion varies between 120 and 146 mm for the human subjects included in this study. The results of this analysis indicate that to improve axial resolution, it is best to reduce the pinhole size. The resulting reduction in detected light may demand, however, a more sophisticated adap- tive optics system. The study also shows that imaging systems with large pinholes are relatively insensitive to misalignment in the lateral positioning of the confocal pinhole. However, when small pinholes are used to maximize resolution, alignment becomes critical. © 2004

System for manufacturing an optical lens

Abstract: A system for manufacturing an optical lens that is configured to correct optical aberrations, including, e.g., high order aberrations such as described by Zernike polynomials. The system can include a measurement system configured to measure optical aberrations in a patient's eye and to create measured optical aberration data. A calculation system is configured to receive the measured optical aberration data and to determine a lens definition based on the measured optical aberration data. A fabrication system is configured to produce a correcting lens based on the lens definition.

X-ray omni microscopy.

Abstract: Summary The science of wave-field phase retrieval and phase measurement is sufficiently mature to permit the routine reconstruction, over a given plane, of the complex wave-function associated with certain coherent forward-propagating scalar wave-fields. This reconstruction gives total knowledge of the information that has been encoded in the complex wave-field by passage through a sample of interest. Such total knowledge is powerful, because it permits the emulation in software of the subsequent action of an infinite variety of coherent imaging systems. Such ‘virtual optics’, in which software forms a natural extension of the ‘hardware optics’ in an imaging system, may be useful in contexts such as quantitative atom and X-ray imaging, in which optical elements such as beam-splitters and lenses can be realized in software rather than optical hardware. Here, we develop the requisite theory to describe such hybrid virtual-physical imaging systems, which we term ‘omni optics’ because of their infinite flexibility. We then give an experimental demonstration of these ideas by showing that a lensless X-ray point projection microscope can, when equipped with the appropriate software, emulate an infinite variety of optical imaging systems including those which yield interferograms, Zernike phase contrast, Schlieren imaging and diffraction-enhanced imaging.

A multibit geometrically robust image watermark based on Zernike moments

Abstract: In image watermarking, the watermark robustness to geometric transformations is still an open problem. Using invariant image features to carry the watermark is an effective approach to addressing this problem. In this paper, a multibit geometrically robust image watermarking algorithm using Zernike moments is proposed. Some Zernike moments of an image are selected, and their magnitudes are dither-modulated to embed an array of bits. The watermarked image is obtained via reconstruction from the modified moments and those left intact. In watermark extraction, the embedded bits are estimated from the invariant magnitudes of the Zernike moments using a minimum distance decoder. Simulation results show that the hidden message can be decoded at low error rates, robust against image rotation, scaling and flipping, and as well, a variety of other distortions such as lossy compression.

On the computation of the Nijboer-Zernike aberration integrals at arbitrary defocus

Abstract: We present a new computation scheme for the integral expressions describing the contributions of single aberrations to the diffraction integral in the context of an extended Nijboer-Zernike approach. Such a scheme, in the form of a power series involving the defocus parameter with coefficients given explicitly in terms of Bessel functions and binomial coefficients, was presented recently by the authors with satisfactory results for small-to-medium-large defocus values. The new scheme amounts to systemizing the procedure proposed by Nijboer in which the appropriate linearization of products of Zernike polynomials is achieved by using certain results of the modern theory of orthogonal polynomials. It can be used to compute point-spread functions of general optical systems in the presence of arbitrary lens transmission and lens aberration functions and the scheme provides accurate data for any, small or large, defocus value and at any spatial point in one and the same format. The cases with high numerical aperture, requiring a vectorial approach, are equally well handled. The resulting infinite series expressions for these point-spread functions, involving products of Bessel functions, can be shown to be practically immune to loss of digits. In this respect, because of its virtually unlimited defocus range, the scheme is particularly valuable in replacing numerical Fourier transform methods when the defocused pupil functions require intolerably high sampling densities.

Geometrically robust image watermarking via pseudo-Zernike moments

Abstract: In image watermarking, the watermark robustness to geometric transformations is still an open problem. Using invariant image features to carry the watermark is an effective approach to addressing this problem. In this paper a geometrically robust image watermarking scheme using pseudo-Zernike moments is proposed. Some selected pseudo-Zernike moments of an image are computed, and their magnitudes are quantized by dither modulation to embed an array of bits. The watermarked image is obtained via reconstruction from the modified moments and those left intact. In watermark extraction, the embedded bits are estimated from the invariant magnitudes of the pseudo-Zernike moments using a minimum distance decoder. Simulation results show that the hidden message can be decoded at low error rates, robust against image rotation, scaling, flipping, and as well, a variety of other processes such as lossy compression.

Gram-Schmidt orthogonalization of the Zernike polynomials on apertures of arbitrary shape

Abstract: An orthonormal hexagonal Zernike basis set is generated from circular Zernike polynomials apodized by a hexagonal mask by use of the Gram-Schmidt orthogonalization technique. Results for the first 15 hexagonal Zernike polynomials are shown. The Gram-Schmidt orthogonalization technique presented can be extended to both apertures of arbitrary shape and other basis functions.

A New Class of Rotational Invariants Using Discrete Orthogonal Moments

Abstract: This paper presents a new class of Tchebichef moments in polar coordinate form, using which rotational invariants can be easily constructed. The structure of the invariants is very similar to that of Zernike and Pseudo-Zernike moments, and their computation does not involve discrete approximation of continuous integral terms. The invariants are thus very robust in the presence of image noise, and have far better recognition capabilities when compared with Zernike/Legendre moments. The new class of moment invariants presented in this paper can be used in pattern and character recognition tasks.

Palmprint verification with moments

Abstract: Palmprint verification is an approach for verifying a palmprint input by matching the input to the claimed identity template stored in a database. If the dissimilarity measure between the input and the claimed template is below the predefined threshold value, the palmprint input is verified possessing same identity as the claimed identity template. This paper introduces an experimental evaluation of the effectiveness of utilizing three well known orthogonal moments, namely Zernike moments, pseudo Zernike moments and Legendre moments, in the application of palmprint verification. Moments are the most commonly used technique in character feature extraction. The idea of implementing orthogonal moments as palmprint feature extractors is prompted by the fact that principal features of both character and palmprint are based on line structure. These orthogonal moments are able to define statistical and geometrical features containing line structure information about palmprint. An experimental study about verification rate of the palmprint authentication system using these three orthogonal moments as feature descriptors has been discussed here. Experimental results show that the performance of the system is dependent on the moment order as well as the type of moments. The orthogonal property of these moments is able to characterize independent features of the palmprint image and thus have minimum information redundancy in a moment set. Pseudo Zernike moments of order of 15 has the best performance among all the moments. Its verification rate is 95.75%, which also represents the overall performance of this palmprint verification system.

An efficient method for human face recognition using wavelet transform and Zernike moments

Abstract: This paper presents a method of combining wavelet transforms (WT) and Zernike moments (ZM) as a feature vector for face recognition. Wavelet transform, with its approximate decomposition is used to reduce the noise and produce a representation in the low frequency domain, and hence making the facial images insensitive to facial expression and small occlusion. The Zernike moments, on the other hand, is selected as feature extractor due to its robustness to image noise, geometrical invariants property and orthogonal property. The simulation results on Essex database indicates that higher order degree of WT combine with ZM achieve better performance with respect to recognition rate rather than using WT or ZM alone. The optimum result is obtained for ZM of order 10 with Daubechies orthonormal wavelet filter of order 7 in the first decomposition level. It can achieve the verification of 94.26%.

Beam control system with extended beacon and method

Abstract: A beam control system and method: The inventive system includes, an arrangement for receiving a first beam of electromagnetic energy; measuring wavefront aberrations in the first beam with a wavefront sensor; and removing global tilt from the measured wavefront aberrations to provide higher order aberrations for beam control. In the illustrative embodiment, the invention uses a traditional (quad-cell) Shack-Hartmann wavefront sensor to measure wavefront aberrations. An adaptive optics processor electronically removes the global tilt (angular jitter) from this measurement leaving only the higher-order Zernike components. These higher-order aberrations are then applied to wavefront control elements, such as deformable mirrors or spatial light modulators that correct the tracker image and apply a conjugate distortion to the wavefront of the outgoing HEL beam. A track error (angular jitter) component is supplied by a separate fine track sensor. This jitter error is then applied by the adaptive optics processor to a fast steering mirror, which corrects jitter in the tracker image and applies a compensating distortion to the LOS of the HEL beam.

Measurement of lens focal length using multicurvature analysis of Shack-Hartmann wavefront data

Abstract: The lens is one of the most commonly used optical elements. Yet it is sometimes difficult to make accurate effective focal length and pupil position measurements, especially for long focal length lenses. Many measurement methods rely on a mechanical measurement to determine the back focal length, or may require careful operator discrimination in determining the best focus position. Aberrations may confuse an automatic focal length measurement system. However, an accurate determination of the optical properties of a lens is often critical for building an accurate system model. We have developed a method for measurement of the focal length, pupil plane and collimation positions of positive lenses using a Shack-Hartmann wavefront sensor. The SHWFS uses a micro-optic lens array to separate the incoming wavefront into a pattern of focal spots. The position of these focal spots is related to the local wavefront slope. Wavefront reconstruction allows the complete incident wavefront to be retrieved. A Zernike decomposition reconstructor is used to separate the effects of lens focal power from other aberrations. The lens under test is illuminated by a point source on a computer-controlled stage. The transmitted wavefront was recorded by the SHWFS while the source was translated over a few mm range. By analyzing the Zernike coefficient associated with defocus, we were able to extract the focal length, pupil plane and collimation positions using a least squares fitting procedure. This procedure was tested for a variety of lenses of varying focal lengths, from 10 to 1000 mm focal length, and showed excellent repeatability and accuracy. These measurements were compared to knife-edge, manufacturer’s specification, and ray-tracing analysis for verification testing.

Orthogonal Aberration Functions for Microlithographic Optics

Abstract: Newly developed orthogonal aberration functions are reviewed. The new functions can be utilized to express aberrations of a high NA and wide field optical system like a microlithographic projection lens. The new functions are orthogonal to each other and expressed by a simple combination of Zernike function(s) of pupil coordinates and Zernike function(s) of field coordinates.

Corneal surface reconstruction algorithm that uses Zernike polynomial representation.

Abstract: We developed an algorithm that directly determines Zernike coefficients for the corneal anterior surface derived from the reflection image of a stimulus with pseudorandom encoding. This algorithm does not need to include calculation of corneal height maps. The numerical performance of the algorithm is good. It has the potential of determining corneal shape with submicrometer accuracy in obtaining Zernike coefficients. When applied to real eye measurements the accuracy of the procedure will be limited by the topographer that is used.

Determination of resist parameters using the extended Nijboer-Zernike theory

Abstract: This study presents an experimental method to determine the resist parameters that are at the origin of a general blurring of the projected aerial image. The resist model includes the effects of diffusion in the horizontal plane and a second cause for image blur that originates from a stochastic variation of the focus parameter. The used mathematical framework is the so-called Extended Nijboer-Zernike (ENZ) theory. The experimental procedure to extract the model parameters is demonstrated for several 193 nm resists under various conditions of post exposure baking temperature and baking time. The advantage of our approach is a clear separation between the optical parameters, such as feature size, projection lens aberrations and the illuminator setting on the one hand and process parameters introducing blur on the other.

Computer-aided alignment for a reference transmission sphere of an interferometer

Abstract: A reference transmission sphere is an important device to measure the spherical surfaces, and it offers a high-quality spherical wave and a reference spherical surface with peak-to-valley less than l/20 (l50.6328 mm). For this value, only the manual alignment is difficult to manage it. Thus, we study the computer-aided alignment (CAA), which can provide a guide to align the individual lens. We describe the following works concerning the CAA with no iteration for the transmission spheres: (1) by analysis of the relationship between the Zernike polyno- mial coefficients and the sensitive variables, for example, the air thick- ness error, the tilt, the decenter of the transmission sphere with an analy- sis program written in Zemax program language, only several Zernike coefficients that change linearly with these sensitive variables are cho- sen; (2) the magnitude and direction of the correction are found using the Moore-Penrose generalized inverse matrix; and (3) a numerical simula- tion and successful alignment are processed for a 4-in.-diam, f/5 trans- mission sphere. © 2004 Society of Photo-Optical Instrumentation Engineers.

Simulation of specimen-induced aberrations for objects with spherical and cylindrical symmetry.

Abstract: Wavefront aberrations caused by the refractive index structure of the specimen are known to compromise signal intensity and three-dimensional resolution in confocal and multiphoton microscopy. However, adaptive optics can measure and correct specimen-induced aberrations. For the design of an adaptive optics system, information on the type and amount of the aberration is required. We have previously described an interferometric set-up capable of measuring specimen-induced aberrations and a method for the extraction of the Zernike mode content. In this paper we have modelled specimen-induced aberrations caused by spherical and cylindrical objects using a ray tracing method. The Zernike mode content of the wavefronts was then extracted from the simulated wavefronts and compared with experimental results. Aberrations for a simple model of an oocyte cell consisting of two spherical regions and for a model of a well-characterized optical fibre are calculated. This simple model gave Zernike mode data that are in good agreement with experimental results.