Comparison of cubic B-spline and Zernike-fitting techniques in complex wavefront reconstruction

Novel Multivariable Evolutionary Algorithm-Based Method for Modal Reconstruction of the Corneal Surface from Sparse and Incomplete Point Clouds

Three-dimensional reconstruction of the corneal surface provides a powerful tool for managing corneal diseases. This study proposes a novel method for reconstructing the corneal surface from elevation point clouds, using modal schemes capable of reproducing corneal shapes using surface polynomial functions. The multivariable polynomial fitting was performed using a non-dominated sorting multivariable genetic algorithm (NS-MVGA). Standard reconstruction methods using least-squares discrete fitting (LSQ) and sequential quadratic programming (SQP) were compared with the evolutionary algorithm-based approach. The study included 270 corneal surfaces of 135 eyes of 102 patients (ages 11-63) sorted in two groups: control (66 eyes of 33 patients) and keratoconus (KC) (69 eyes of 69 patients). Tomographic information (Sirius, Costruzione Strumenti Oftalmici, Italy) was processed using Matlab. The goodness of fit for each method was evaluated using mean squared error (MSE), measured at the same nodes where the elevation data were collected. Polynomial fitting based on NS-MVGA improves MSE values by 86% compared to LSQ-based methods in healthy patients. Moreover, this new method improves aberrated surface reconstruction by an average value of 56% if compared with LSQ-based methods in keratoconus patients. Finally, significant improvements were also found in morpho-geometric parameters, such as asphericity and corneal curvature radii.

Freeform optics characterization with surface registration and fitting algorithms for optical point-based spatial path 3D topography metrology

Accurate characterization of the form error for freeform optics is critical for controlled manufacturing and evaluation of optical properties. To solve the difficulty of current surface registration and fitting algorithms, and improve characterization accuracy of the form error of freeform optics for optical point-based spatial path 3D topography metrology, in this paper, improved surface registration and fitting algorithms are proposed, including a B-spline surface description of freeform optics, point orthogonal projection, registration parameter optimization, and B-spline fitting. Limited by the angular characteristics of an optical point-based sensor, the slope and reference frame of freeform optics must be flexibly adjusted, and polynomial surfaces are described and fitted as B-spline surfaces based on B-spline geometric invariance. To efficiently determine corresponding points on B-spline surfaces, the point orthogonal projection algorithm without Bezier subdivision is proposed using the first-order Newton method. Then, the iteration procedure of coordinate adjustment in sphere space using Lie algebra registration parameters is proposed to solve the difficulty of the current registration parameter optimization procedure. To fit a spatial path form error, the least squares B-spline fitting method is proposed to improve the Zernike method. Through simulation experiments, the proposed registration algorithm with good convergence can improve accuracy by one to two orders of magnitude compared with current registration algorithms. Through repeated experiments of a freeform prism, the proposed method can significantly improve the peak-to-valley and RMS accuracies compared with the stylus method, and characterize the mid-high frequency form error (about 200 nm) of a freeform prism.

B-spline surface based 3D reconstruction method for deflectometry

In the field of optical three-dimension (3-D) measurement, reconstruction usually is completed by the integration of a two-dimensional (2-D) gradient data set. Position and posture of camera and shape of the surface under test determine the location of gradient data which usually is on quadrilateral grids. This paper proposes a B-spline surface-based 3D reconstruction method for deflectometry, which reconstructs the surface under test with its 2-D gradient data set. The 2-D gradient data set consists of gradient data and the 2-D location of the gradient data in the camera coordinate system. The 2-D gradient data set is first transferred to the cameras' virtual image plane, so it locates on rectangular grids. Then, based on the properties of the B-spline basis function and characteristics of the camera, linear equations are derived to solve control points along the virtual image plane. The solved control points reconstruct the surface under test in the camera coordinate system. The property of the B-spline basis function determines the relationship between the depth of the surface and its derivative. The characteristic of the camera determines the relationship between the depth of the surface and the 2-D location of the gradient data. Meanwhile, the accuracy of the 2-D location can also be improved by the linear equations. Finally, simulated and actual experiments show that the proposed method is accurate and efficient at reconstructing surfaces in deflectometry.

Antiderivative of gradient data by spline model integration

Numerous optical techniques describe the local slope of the functions at their discrete positions but do not report the actual functions. However, many applications require the description of the functions, which must be retrieved from the gradients by an integration process. This study shows a spline model function-based integration technique that can construct original functions from irregularly measured gradient data over general shape domains with high accuracy and speed.

Wavefront phase representation by Zernike and spline models: a comparison

A comparative analysis of spline and Zernike models is presented for wavefront phase construction. The techniques are analyzed on the basis of representation accuracy, computational costs, and the number of samples used for representation. The strengths and weaknesses of each model over a set of various wavefront phases with different domain shapes are analyzed. The findings show that both models efficiently represent a simple wavefront phase at irregular domain shapes. On the other hand, when complex wavefront phases at irregular domain shapes are represented, the spline model performs much better than the Zernike model. Further, results show that the spline model evaluation speed is significantly faster than the Zernike model.

Study and characterization of morphogeometric parameters to assist diagnosis of keratoconus

Background

In case of significant imperfections on the cornea, data acquisition is difficult and a significant level of missing data could require the interpolation of important areas of the cornea, resulting in a very ambiguous model. The development of methods to define in vivo customised geometric properties of the cornea based only on real raw data is extremely useful to diagnose and assess the progression of diseases directly related to the corneal architecture. The present work tries to improve the prognostic of corneal ectasia creating a 3D customised model of the cornea and analysing different geometric variables from this model to determine which variables or combination of them could be defined as an indicator of susceptibility to develop keratoconus.

Methods

A corneal geometric reconstruction was performed using zonal functions and retrospective Scheimpflug tomography data from 187 eyes of 187 patients. Morphology of healthy and keratoconic corneas was characterized by means of geometric variables. The performance of these variables as predictors of a new geometric marker was assessed and their correlations were analysed.

Results

The more representative variable to classify the corneal anomalies related to keratoconus was posterior apex deviation (area under receiver operating characteristic curve > 0.899; p < 0.0001). However, the strongest correlations in both healthy and pathological corneas were provided by the metrics directly related to the thickness, as deviations of the anterior/posterior minimum thickness points.

Conclusions

The presented morphogeometric approach based on the analysis and custom geometric modelling of the cornea demonstrates to be useful for the characterization and diagnosis of keratoconus disease, stating that geometrical deformation is an effective marker of the ectatic disease’s progression.

Zonal wavefront reconstruction in quadrilateral geometry for phase measuring deflectometry

There are wide applications for zonal reconstruction methods in slope-based metrology due to its good capability of reconstructing the local details on surface profile. It was noticed in the literature that large reconstruction errors occur when using zonal reconstruction methods designed for rectangular geometry to process slopes in a quadrilateral geometry, which is a more general geometry with phase measuring deflectometry. In this work, we present a new idea for the zonal methods for quadrilateral geometry. Instead of employing the intermediate slopes to set up height-slope equations, we consider the height increment as a more general connector to establish the height-slope relations for least-squares regression. The classical zonal methods and interpolation-assisted zonal methods are compared with our proposal. Results of both simulation and experiment demonstrate the effectiveness of the proposed idea. In implementation, the modification on the classical zonal methods is addressed. The new methods preserve many good aspects of the classical ones, such as the ability to handle a large incomplete slope dataset in an arbitrary aperture, and the low computational complexity comparable with the classical zonal method. Of course, the accuracy of the new methods is much higher when integrating the slopes in quadrilateral geometry.

Data processing for point-based in situ metrology of freeform optical surface

Freeform surfaces are widely used in advanced optical systems and the point-based metrology methods are often used for freeform surface measurement. In order to accurately process the measured data from point-based metrology system, the coordinate matching and surface fitting are essential. To ensure the matching efficiency and accuracy, we use a two-step coordinate matching method: pre-adjustment and accurate adjustment. We also develop an improved Newton iterative (INI) fitting algorithm to generate initial value and improve the fitting speed. The simulation and experiment verify the correctness and feasibility of the data processing.

Robust fitting of Zernike polynomials to noisy point clouds defined over connected domains of arbitrary shape

A new method for fitting a series of Zernike polynomials to point clouds defined over connected domains of arbitrary shape defined within the unit circle is presented in this work. The method is based on the application of machine learning fitting techniques by constructing an extended training set in order to ensure the smooth variation of local curvature over the whole domain. Therefore this technique is best suited for fitting points corresponding to ophthalmic lenses surfaces, particularly progressive power ones, in non-regular domains. We have tested our method by fitting numerical and real surfaces reaching an accuracy of 1 micron in elevation and 0.1 D in local curvature in agreement with the customary tolerances in the ophthalmic manufacturing industry.

Resolving capacity of the circular Zernike polynomials

Circular Zernike polynomials are often used for approximation and analysis of optical surfaces. In this paper, we analyse their lateral resolving capacity, illustrating the effects of a lack of approximation by a finite set of polynomials and answering the following questions: What is the minimum number of polynomials that is necessary to describe a local deformation of a certain size? What is the relationship between the number of approximating polynomials and the spatial spectrum of the approximation? What is the connection between the mean-square error of approximation and the number of polynomials? The main results of this work are the formulas for calculating the error of fitting the relief and the connection between the width of the spatial spectrum and the order of approximation.

Geometrical custom modeling of human cornea in vivo and its use for the diagnosis of corneal ectasia

AIM

To establish a new procedure for 3D geometric reconstruction of the human cornea to obtain a solid model that represents a personalized and in vivo morphology of both the anterior and posterior corneal surfaces. This model is later analyzed to obtain geometric variables enabling the characterization of the corneal geometry and establishing a new clinical diagnostic criterion in order to distinguish between healthy corneas and corneas with keratoconus.

METHOD

The method for the geometric reconstruction of the cornea consists of the following steps: capture and preprocessing of the spatial point clouds provided by the Sirius topographer that represent both anterior and posterior corneal surfaces, reconstruction of the corneal geometric surfaces and generation of the solid model. Later, geometric variables are extracted from the model obtained and statistically analyzed to detect deformations of the cornea.

RESULTS

The variables that achieved the best results in the diagnosis of keratoconus were anterior corneal surface area (ROC area: 0.847, p<0.000, std. error: 0.038, 95% CI: 0.777 to 0.925), posterior corneal surface area (ROC area: 0.807, p<0.000, std. error: 0.042, 95% CI: 0,726 to 0,889), anterior apex deviation (ROC area: 0.735, p<0.000, std. error: 0.053, 95% CI: 0.630 to 0.840) and posterior apex deviation (ROC area: 0.891, p<0.000, std. error: 0.039, 95% CI: 0.8146 to 0.9672).

CONCLUSION

Geometric modeling enables accurate characterization of the human cornea. Also, from a clinical point of view, the procedure described has established a new approach for the study of eye-related diseases.

Optical surface reconstruction technique through combination of zonal and modal fitting

Videokeratometers and Scheimpflug cameras permit accurate estimation of corneal surfaces. From height data it is possible to adjust analytical surfaces that will be later used for aberration calculation. Zernike polynomials are often used as adjusting polynomials, but they have shown to be not precise when describing highly irregular surfaces. We propose a combined zonal and modal method that allows an accurate reconstruction of corneal surfaces from height data, diminishing the influence of smooth areas over irregular zones and vice versa. The surface fitting error is decreased in the considered cases, mainly in the central region, which is more important optically. Therefore, the method can be established as an accurate resampling technique.

Direct and inverse discrete Zernike transform

An invertible discrete Zernike transform, DZT is proposed and implemented. Three types of non-redundant samplings, random, hybrid (perturbed deterministic) and deterministic (spiral) are shown to provide completeness of the resulting sampled Zernike polynomial expansion. When completeness is guaranteed, then we can obtain an orthonormal basis, and hence the inversion only requires transposition of the matrix formed by the basis vectors (modes). The discrete Zernike modes are given for different sampling patterns and number of samples. The DZT has been implemented showing better performance, numerical stability and robustness than the standard Zernike expansion in numerical simulations. Non-redundant (critical) sampling along with an invertible transformation can be useful in a wide variety of applications.