Damian Siedlecki

Optical distortion correction in Optical Coherence Tomography for quantitative ocular anterior segment by three-dimensional imaging

A method for three-dimensional 3-D optical distortion (refraction) correction on anterior segment Optical Coherence Tomography (OCT) images has been developed. The method consists of 3-D ray tracing through the different surfaces, following denoising, segmentation of the surfaces, Delaunay representation of the surfaces, and application of fan distortion correction. The correction has been applied theoretically to realistic computer eye models, and experimentally to OCT images of: an artificial eye with a Polymethyl Methacrylate (PMMA) cornea and an intraocular lens (IOL), an enucleated porcine eye, and a human eye in vivo obtained from two OCT laboratory set-ups (time domain and spectral). Data are analyzed in terms of surface radii of curvature and asphericity. Comparisons are established between the reference values for the surfaces (nominal values in the computer model; non-contact profilometric measurements for the artificial eye; Scheimpflug imaging for the real eyes in vivo and vitro). The results from the OCT data were analyzed following the conventional approach of dividing the optical path by the refractive index, after application of 2-D optical correction, and 3-D optical correction (in all cases after fan distortion correction). The application of 3-D optical distortion correction increased significantly both the accuracy of the radius of curvature estimates and particularly asphericity of the surfaces, with respect to conventional methods of OCT image analysis. We found that the discrepancies of the radii of curvature estimates from 3-D optical distortion corrected OCT images are less than 1% with respect to nominal values. Optical distortion correction in 3-D is critical for quantitative analysis of OCT anterior segment imaging, and allows accurate topography of the internal surfaces of the eye.

Optical coherence tomography for quantitative surface topography

We address the measurement and correction of fan distortion in optical coherence tomography (OCT). This effect arises from the scanning system configuration and prevents one, in general, from obtaining quantitative topographic data from OCT. Computer simulations allowed us to quantify the effect and evaluate its dependence on the scanning mirror separation and design of the collimating lens, as well as to estimate the optimal axial position of that lens to minimize the fan distortion. We also developed a numerical algorithm based on 3-D ray propagation for the correction of the residual fan distortion. The effect was studied experimentally using a custom developed time-domain OCT in a Michelson configuration provided with a confocal channel, and the accuracy of the fan distortion correction algorithm tested on samples of known dimensions (flat surfaces and spherical lenses). With a proper calibration of the system with use of an onfocal channel, this algorithm makes it possible for time-domain OCT devices to be used as topographers.

Three-dimensional reconstruction of the crystalline lens gradient index distribution from OCT imaging

We present an optimization method to retrieve the gradient index (GRIN) distribution of the in-vitro crystalline lens from optical path difference data extracted from OCT images. Three-dimensional OCT images of the crystalline lens are obtained in two orientations (with the anterior surface up and posterior surface up), allowing to obtain the lens geometry. The GRIN reconstruction method is based on a genetic algorithm that searches for the parameters of a 4-variable GRIN model that best fits the distorted posterior surface of the lens. Computer simulations showed that, for noise of 5 μm in the surface elevations, the GRIN is recovered with an accuracy of 0.003 and 0.010 in the refractive indices of the nucleus and surface of the lens, respectively. The method was applied to retrieve three-dimensionally the GRIN of a porcine crystalline lens in vitro. We found a refractive index ranging from 1.362 in the surface to 1.443 in the nucleus of the lens, an axial exponential decay of the GRIN profile of 2.62 and a meridional exponential decay ranging from 3.56 to 5.18. The effect of GRIN on the aberrations of the lens also studied. The estimated spherical aberration of the measured porcine lens was 2.87 μm assuming a homogenous equivalent refractive index, and the presence of GRIN shifted the spherical aberration toward negative values (-0.97 μm), for a 6-mm pupil.

Corneal topography from spectral optical coherence tomography (sOCT)

We present a method to obtain accurate corneal topography from a spectral optical coherence tomography (sOCT) system. The method includes calibration of the device, compensation of the fan (or field) distortion introduced by the scanning architecture, and image processing analysis for volumetric data extraction, segmentation and fitting. We present examples of three-dimensional (3-D) surface topography measurements on spherical and aspheric lenses, as well as on 10 human corneas in vivo. Results of sOCT surface topography (with and without fan-distortion correction) were compared with non-contact profilometry (taken as reference) on a spherical lens, and with non-contact profilometry and state-of-the art commercial corneal topography instruments on aspheric lenses and on subjects. Corneal elevation maps from all instruments were fitted by quadric surfaces (as well as by tenth-order Zernike polynomials) using custom routines. We found that the discrepancy in the estimated radius of curvature from nominal values in artificial corneas decreased from 4.6% (without fan distortion correction) to 1.6% (after fan distortion correction), and the difference in the asphericity decreased from 130% to 5%. In human corneas, the estimated corneal radius of curvature was not statistically significantly different across instruments. However, a Bland-Altman analysis showed consistent differences in the estimated asphericity and corneal shape between sOCT topographies without fan distortion correction and the rest of the measurements.

Schematic eye with a gradient-index lens and aspheric surfaces

A new schematic eye with aspheric surfaces and a radially varying refractive-index distribution lens is proposed. Image quality and spherical aberration are determined by use of ray tracing, and the results are presented as spot diagrams and compared with five existing model eyes. The proposed model provides the best image quality and lowest spherical aberration.

Distortions of the posterior surface in optical coherence tomography images of the isolated crystalline lens: effect of the lens index gradient

We quantify the posterior surface distortions in optical coherence tomography (OCT) images of isolated crystalline lenses. The posterior radius of curvature and asphericity obtained from OCT images acquired with the beam incident first on the anterior, and then the posterior, surface were compared. The results were compared with predictions of a ray-tracing model which includes the index gradient. The results show that the error in the radius of curvature is within the measurement reproducibility and that it can be corrected by assuming a uniform refractive index. However, accurate asphericity values require a correction algorithm that takes into account the gradient.

Age-dependent variation of the Gradient Index profile in human crystalline lenses.

An investigation was carried out with the aim of reconstructing the gradient index (GRIN) profile of human crystalline lenses ex-vivo using optical coherence tomography (OCT) imaging with an optimization technique and to study the dependence of the GRIN profile with age. Cross-sectional images of nine isolated human crystalline lenses with ages ranging from 6 to 72 (post-mortem time 1 to 4 days) were obtained using a custom-made OCT system. Lenses were extracted from whole cadaver globes and placed in a measurement chamber filled with preservation medium (DMEM). Lenses were imaged with the anterior surface up and then flipped over and imaged again, to obtain posterior lens surface profiles both undistorted and distorted by the refraction through the anterior crystalline lens and GRIN. The GRIN distribution of the lens was described with three variables by means of power function, with variables being the nucleus and surface index, and a power coefficient that describes the decay of the refractive index from the nucleus to the surface. An optimization method was used to search for the parameters that produced the best match of the distorted posterior surface. The distorted surface was simulated with accuracy around the resolution of the OCT system (under 15 µm). The reconstructed refractive index values ranged from 1.356 to 1.388 for the surface, and from 1.396 to 1.434 for the nucleus. The power coefficient ranged between 3 and 18. The power coefficient increased significantly with age, at a rate of 0.24 per year. Optical coherence tomography allowed optical, non-invasive measurement of the 2D gradient index profile of the isolated human crystalline lens ex vivo. The age-dependent variation of the changes is consistent with previous data using magnetic resonance imaging, and the progressive formation of a refractive index plateau.

Distortion Correction of OCT Images of the Crystalline Lens: Gradient Index Approach

Purpose. To propose a method to correct optical coherence tomography (OCT) images of posterior surface of the crystalline lens incorporating its gradient index (GRIN) distribution and explore its possibilities for posterior surface shape reconstruction in comparison to existing methods of correction. Methods. Two-dimensional images of nine human lenses were obtained with a time-domain OCT system. The shape of the posterior lens surface was corrected using the proposed iterative correction method. The parameters defining the GRIN distribution used for the correction were taken from a previous publication. The results of correction were evaluated relative to the nominal surface shape (accessible in vitro) and compared with the performance of two other existing methods (simple division, refraction correction: assuming a homogeneous index). Comparisons were made in terms of posterior surface radius, conic constant, root mean square, peak to valley, and lens thickness shifts from the nominal data. Results. Differences in the retrieved radius and conic constant were not statistically significant across methods. However, GRIN distortion correction with optimal shape GRIN parameters provided more accurate estimates of the posterior lens surface in terms of root mean square and peak values, with errors <6 and 13 &mgr;m, respectively, on average. Thickness was also more accurately estimated with the new method, with a mean discrepancy of 8 &mgr;m. Conclusions. The posterior surface of the crystalline lens and lens thickness can be accurately reconstructed from OCT images, with the accuracy improving with an accurate model of the GRIN distribution. The algorithm can be used to improve quantitative knowledge of the crystalline lens from OCT imaging in vivo. Although the improvements over other methods are modest in two dimension, it is expected that three-dimensional imaging will fully exploit the potential of the technique. The method will also benefit from increasing experimental data of GRIN distribution in the lens of larger populations.

Radial gradient index intraocular lens: a theoretical model

A model for a gradient index intraocular lens (IOL) is presented. The distribution of the refractive index is Gaussian, varying radially in the equatorial plane and with distance from the optic axis in the sagittal plane. An algorithm for adjustment of the optimal gradient index distribution for individual parameters of a pseudophakic eye is also described. The performance, in terms of spherical aberration and resultant image quality that can be provided with the gradient index IOL, is compared with that provided by a commercially available IOL (Tecnis Z9000).

Three-dimensional ray tracing on Delaunay-based reconstructed surfaces

A method of ray tracing for free-form optical surfaces has been developed. The ray tracing through such surfaces is based on Delaunay triangulation of the discrete data of the surface and is related to finite-element modeling. Some numerical examples of applications to analytical, noisy, and experimental free-form surfaces (in particular, a corneal topography map) are presented. Ray-tracing results (i.e., spot diagram root-mean-square error) with the new method are in agreement with those obtained using a modal fitting of the surface, for sampling densities higher than 40 x 40 elements. The method competes in flexibility, simplicity, and computing times with standard methods for surface fitting and ray tracing.