矩形孔径上折射率均匀性数据拟合研究

泽尼克圆多项式在圆形光瞳的正交性和能够代表经典像差而被广泛应用到波前分析中,用泽尼克圆多项式作为矩形光瞳基底函数,通过推导得到在矩形光瞳上正交的多项式.这个在矩形光瞳上正交的多项式不仅是唯一的,而且也能够表示经典像差,就像泽尼克圆多项式在表示圆形光瞳时具有这样的特性一样.矩形光瞳上正交多项式像泽尼克圆多项式一样即可以用...     

Full-aperture wavefront reconstruction from annular subaperture interferometric data by use of Zernike annular polynomials and a matrix method for testing large aspheric surfaces

We propose a more accurate and efficient reconstruction method used in testing large aspheric surfaces with annular subaperture interferometry. By the introduction of the Zernike annular polynomials that are orthogonal over the annular region, the method proposed here eliminates the coupling problem in the earlier reconstruction algorithm based on Zernike circle polynomials. Because of the complexity of recurrence definition of Zernike annular polynomials, a general symbol representation of that in a computing program is established. The program implementation for the method is provided in detail. The performance of the reconstruction algorithm is evaluated in some pertinent cases, such as different random noise levels, different subaperture configurations, and misalignments.     

Orthonormal polynomials in wavefront analysis: analytical solution

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. In recent papers, we derived closed-form polynomials that are orthonormal over a hexagonal pupil, such as the hexagonal segments of a large mirror. We extend our work to elliptical, rectangular, and square pupils. Using the circle polynomials as the basis functions for their orthogonalization over such pupils, we derive closed-form polynomials that are orthonormal over them. These polynomials are unique in that they are not only orthogonal across such pupils, but also represent balanced classical aberrations, just as the Zernike circle polynomials are unique in these respects for circular pupils. The polynomials are given in terms of the circle polynomials as well as in polar and Cartesian coordinates. Relationships between the orthonormal coefficients and the corresponding Zernike coefficients for a given pupil are also obtained. The orthonormal polynomials for a one-dimensional slit pupil are obtained as a limiting case of a rectangular pupil.     

Image description with generalized pseudo-Zernike moments

A new set, to our knowledge, of orthogonal moment functions for describing images is proposed. It is based on the generalized pseudo-Zernike polynomials that are orthogonal on the unit circle. The generalized pseudo-Zernike polynomials are scaled to ensure numerical stability, and some properties are discussed. The performance of the proposed moments is analyzed in terms of image reconstruction capability and invariant character recognition accuracy. Experimental results demonstrate the superiority of generalized pseudo-Zernike moments compared with pseudo-Zernike and Chebyshev-Fourier moments in both noise-free and noisy conditions.     

Comparison of annular wavefront interpretation with Zernike circle polynomials and annular polynomials

A general wavefront fitting procedure with Zernike annular polynomials for circular and annular pupils is proposed. For interferometric data of typical annular wavefronts with smaller and larger obscuration ratios, the results fitted with Zernike annular polynomials are compared with those of Zernike circle polynomials. Data are provided demonstrating that the annular wavefront expressed with Zernike annular polynomials is more accurate and meaningful for the decomposition of aberrations, the calculation of Seidel aberrations, and the removal of misalignments in interferometry. The primary limitations of current interferogram reduction software with Zernike circle polynomials in analyzing wavefronts of annular pupils are further illustrated, and some reasonable explanations are provided. It is suggested that the use of orthogonal basis functions on the pupils of the wavefronts analyzed is more appropriate.     

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.     

Zernike annular polynomials and optical aberrations of systems with annular pupils

Zernike annular polynomials that represent orthogonal andbalanced aberrations suitable for systems with annular pupilsare described. Their numbering scheme is the same asfor Zernike circle polynomials. Expressions for standard deviationof primary and balanced primary aberrations are given.     

Gram-Schmidt orthonormalization of Zernike polynomials for general aperture shapes

We present analytical derivations of aberration functions for annular sector apertures. We show that the Zernike functions for circular apertures can be generalized for any aperture shape. Interferogram reduction when Zernike functions were used as a basis set was performed on annular sectors. We have created a computer program to generate orthogonal aberration functions. Completely general aperture shapes and user-selected basis sets may be treated with a digital Gram-Schmidt orthonormalization approach.     

Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms

Four modal methods of reconstructing a wavefront from its difference fronts based on Zernike polynomials in lateral shearing interferometry are currently available, namely the Rimmer-Wyant method, elliptical orthogonal transformation, numerical orthogonal transformation, and difference Zernike polynomial fitting. The present study compared these four methods by theoretical analysis and numerical experiments. The results show that the difference Zernike polynomial fitting method is superior to the three other methods due to its high accuracy, easy implementation, easy extension to any high order, and applicability to the reconstruction of a wavefront on an aperture of arbitrary shape. Thus, this method is recommended for use in lateral shearing interferometry for wavefront reconstruction.     

两种正交多项式模拟湍流波前的比较

针对Zemike多项式仅在连续单位圆上正交,用于在离散点上构造光学波前必然会引起误差的问题,本文提出用能够在离散点上正交的多项式来模拟经过大气湍流的光学波前.该方法根据湍流的统计理论,采用Gram-Schmidt正交化方法,构造了Malacara多项式表示的湍流波前,并进行了数值模拟.将模拟结果与直接用Zemike多项式模拟的结果进行了比较分析,结果表明:在相同的条件下,该方法的模拟结果更接近统计理论值.     

基于Zernike矩的快速图像配准算法

图像配准是机器视觉中的重要研究课题.本文针对任意角度图像配准问题,提出了一种基于Zemike矩的互相关算法.首先根据三角函数的对称性和反对称性,仅基于八分之一单位圆信息,快速计算Zemike矩;然后通过求取图像多个矩的互相关性,确定配准点位置;进而通过最小二乘拟合,得到亚像素配准位置;最终利用Zemike相位信息,估算旋转角度.同时,在线阶段采用径向多项式查表法加速配准过程.实验结果表明,该算法在保障配准精度的前提下显著提高了配准速度.     

Variational solution for modal wave-front projection functions of minimum-error norm.

Common wave-front sensors such as the Hartmann or curvature sensor provide measurements of the local gradient or Laplacian of the wave front. The expression of wave fronts in terms of a set of orthogonal basis functions thus generally leads to a linear wave-front-estimation problem in which modal cross coupling occurs. Auxiliary vector functions may be derived that effectively restore the orthogonality of the problem and enable the modes of a wave front to be independently and directly projected from slope measurements. By using variational methods, we derive the necessary and sufficient condition for these auxiliary vector functions to have minimum-error norm. For the specific case of a slope-based sensor and a basis set comprising the Zernike circular polynomials, these functions are precisely the Gavrielides functions.     

Matrix method to find a new set of Zernike coefficients from an original set when the aperture radius is changed

A matrix method is developed that allows a new set of Zernike coefficients that describe a surface or wave front appropriate for a new aperture size to be found from an original set of Zernike coefficients that describe the same surface or wave front but use a different aperture size. The new set of coefficients, arranged as elements of a vector, is formed by multiplying the original set of coefficients, also arranged as elements of a vector, by a conversion matrix formed from powers of the ratio of the new to the original aperture and elements of a matrix that forms the weighting coefficients of the radial Zernike polynomial functions. In developing the method, a new matrix method for expressing Zernike polynomial functions is introduced and used. An algorithm is given for creating the conversion matrix along with computer code to implement the algorithm.     

Topography retrieval using different solutions of the transport intensity equation

The topography of a phase plate is recovered from the phase reconstruction by solving the transport intensity equation (TIE). The TIE is solved using two different approaches: (a) the classical solution of solving the Poisson differential equation and (b) an algebraic approach with Zernike functions. In this paper we present and compare the topography reconstruction of a phase plate with these solution methods and justify why one solution is preferable over the other.     

Zernike-Bessel representation and its application to Hankel transforms

The duality between the well-known Zernike polynomial basis set and the Fourier-Bessel expansion of suitable functions on the radial unit interval is exploited to calculate Hankel transforms. In particular, the Hankel transform of simple truncated radial functions is observed to be exact, whereas more complicated functions may be evaluated with high numerical accuracy. The formulation also provides some general insight into the limitations of the Fourier-Bessel representation, especially for infinite-range Hankel transform pairs.     

AUTOMATED COMPACT PARAMETRIC REPRESENTATION OF IMPACT CRATERS

We have developed a procedure for the measurement and automated parametric representation of 3-D impact craters. The 3-D stucture of the crater is measured using a lowcoherence optical interference technique and the parametric representation achieved in a two-step procedure. Sobel edge detection and morphological operations are used to define rigorously the impact region. The parameter set is defined with respect to a coordinate system whose origin and orientation are uniquely determined from the data. Subjective, arbitrary choices for this position and angle are thus eliminated and the coefficients of the Zernike polynomials are calculated by direct integration over this region to produce a parameter set. This set allows the representation of the three-dimensional geometry by a small set of numbers. Results are presented for impact craters generated at DERA Fort Halstead by the penetration of shaped charge jet particles. It is shown that the agreement between the crater profile generated by the Zernike parameter set and the real crater is very good.     

Pseudo Zernike Moment and Deep Stacked Sparse Autoencoder for COVID-19 Diagnosis

(Aim) COVID-19 is an ongoing infectious disease. It has caused more than 107.45 m confirmed cases and 2.35 m deaths till 11/Feb/2021. Traditional computer vision methods have achieved promising results on the automatic smart diagnosis. (Method) This study aims to propose a novel deep learning method that can obtain better performance. We use the pseudo-Zernike moment (PZM), derived from Zernike moment, as the extracted features. Two settings are introducing: (i) image plane over unit circle; and (ii) image plane inside the unit circle. Afterward, we use a deep-stacked sparse autoencoder (DSSAE) as the classifier. Besides, multiple-way data augmentation is chosen to overcome overfitting. The multiple-way data augmentation is based on Gaussian noise, salt-and-pepper noise, speckle noise, horizontal and vertical shear, rotation, Gamma correction, random translation and scaling. (Results) 10 runs of 10-fold cross validation shows that our PZM-DSSAE method achieves a sensitivity of 92.06% ± 1.54%, a specificity of 92.56% ± 1.06%, a precision of 92.53% ± 1.03%, and an accuracy of 92.31% ± 1.08%. Its F1 score, MCC, and FMI arrive at 92.29% ±1.10%, 84.64% ± 2.15%, and 92.29% ± 1.10%, respectively. The AUC of our model is 0.9576. (Conclusion) We demonstrate “image plane over unit circle” can get better results than “image plane inside a unit circle.” Besides, this proposed PZM-DSSAE model is better than eight state-of-the-art approaches.     

Mutual alignment errors due to wave-front aberrations in intersatellite laser communications

We analyze mutual alignment errors due to wave-front aberrations. To solve the central obscured problem, we introduce modified Zernike polynomials, which are a set of complete orthogonal polynomials. It is found that different aberrations have different effects on mutual alignment errors. Some aberrations influence only the line of sight, while some aberrations influence both the line of sight and the intensity distributions.     

基于V-矩的图像分类算法

基于一类不仅含有连续函数,还含有间断函数的正交完备函数系——V-系统,提出相应的V-矩函数,并将之应用到图像分类中.V-系统中基函数的间断特性,使得V-矩函数在描述含有多个闭合边界的形状时有特别的优势,这种优势表现为对这类复杂形状的特征提取更加准确.因此用V-矩可以得到一种图像分类的有效算法.在几个通用数据库中的图像分类实验表明,本文算法较Zernike矩、不变矩和几何中心矩有更高的准确率,对噪声不敏感,特别在含有多个闭合边界的复杂形状分类问题中,本文方法优势更为显著.