2.1.

Optical Paths for Scheimpflug Imaging of the Corvis ST

In a common photographic system, the camera lens and the image plane are parallel, so the plane of focus is parallel to the camera lens. Image sharpness can be controlled by a suitable focused distance. Scheimpflug principle demonstrates another geometric layout that promises a clear record.³ The condition is when the plane of camera lens and the image plane extended meet at a line through which the plane of focus also passes, the subject can be all in focus, although it is not parallel to the plane of the camera lens. In Scheimpflug imaging, the subject plane should rotate rather than move along the lens axis, when adjusting the focus. The axis of rotation is the intersection of the camera lens’s front focal plane and the plane through the center of the camera lens perpendicular to the image plane. This concept was documented in an early British patent.²⁸

As per the principle, Corvis ST, like its precursor the Pentacam, substitutes a Scheimpflug camera for the slit-lamp camera of traditional anterior segment imaging devices in ophthalmology. Prior studies showed the three-dimensional optical path for Scheimpflug imaging.^26,27 In the system of Corvis ST, the camera lens is placed to form an angle of 45 deg with the object plane according to the specification from the manufacturer. Figure 1 illustrates the two-dimensional optical path with more details. It will be used to deduce the correction formulas later in this section. Here the proportion of the measurement distance to the corneal thickness is exaggerated, so the geometrical relationship can be seen more clearly. The blue arcs denote the anterior and posterior surface of the cornea, while the green arc is the virtual image of the posterior surface. It should be noted that the camera lens is replaced by a pinhole and does not induce any scaling. This simplification works because the effective image can always be found ignoring the real optical magnification. The intrinsic reason behind is that the image distortion from scaling stays stationary in different tests and subjects, so it can be eliminated when calibrating the real length for each pixel of the photographic record.

Fig. 1

The optical paths of Scheimpflug imaging in the Corvis ST: (a) a cornea at the preindentation status and (b) a cornea at its highest concavity.

According to the optical path, a one-to-one correspondence can be observed between the real CCT and its image length, denoted as $x$ and $y$, respectively. Ray tracing refers to a technique of generating an image by tracing the path of light.²⁴ Figure 1 shows the imaginary light beams emanating from the anterior and the posterior surface of the cornea in orange. Consider the example of the light beam from the posterior surface. It forms an angle ${\alpha }_2$ with the imaging plane. It refracts at the anterior surface of cornea, thus appearing as if it is coming from the light beam that forms angle ${\alpha }_1$ with the imaging plane. It continues to propagate passing through a pinhole and, finally, projects onto the CCD.

Figure 1(a) takes the cornea at its original status for an example; the relationship can be theoretically deduced between the object $x$ and image length $y$. According to the configuration of the optical system, angle ${\alpha }_1$ can be first calculated by

Accordingly, the true CCT $x$ can be finally calculated from its image length $y$

A similar one-to-one correspondence can be found when the cornea is reversed by air puff indentation. When air puff indentation is performed, the cornea is deformed from convex to concave and then rebounds to its normal convex status. Ambrosio et al. show a graphical presentation of corneal deformation under air puff indentation with detected applanation moments in their study.⁹ The whole process can be regarded as the radius of curvature changes. Figure 1(b) shows the optical layout for the cornea reversed. The only difference is that the spherical center shifts from inside to outside of the eyeball, so some signs in Eqs. (2), (4), and (6) should be modified as follows:

2.2.

Correcting the Distortion in Measuring the CCT

The essence of the CCT measurement is to estimate the real thickness from a raw photographic record. An intuitive-feasible approach is to conduct the calibration on standard samples and determine the effective length for each pixel of the photographic record. It can be analogical to reading a map in real life. If the distance between the two places is required, we can first measure the distance on the map and then calculate the real value using the scale of the map. Similarly, the estimated CCT can be calculated as the product of its pixel number and a total scaling factor. However, this method neglects the influence factors (the refractive index, the front curvature, and the measurement distance) differed from case to case. Here the method is proposed to better measure the CCT in two steps.

When the CCT is estimated from a measured image length, the units of the image length can be either in micrometers for the image captured by the CCD or in pixels for the image displayed on the screen. There is only a proportional scaling factor between them. It is determined by the electronic display system of the Corvis ST, and is therefore irrelevant to the optical path in different cases. In the first step of measurement, the scaling factor is experimentally determined by a calibration conducted on the samples with known thickness. However, the influence factors affect when the object length of CCT is transformed to the image length, since they change the optical paths and require to be considered. Accordingly, correction procedures should be performed in the second step. A real thickness is obtained by following Eqs. (1)–(7), if the cornea is convex. When air ejects onto the cornea, it will move inward and become concave, so substitute Eqs. (8), (9), and (10) in Eqs. (2), (4), and (6) to estimate the CCT at a concave status. All codes are implemented in a custom-developed program using MATLAB® (Version R2013a, MathWorks Inc., Massachusetts).

2.3.

Measuring the Scaling Factor for Unit Converting

Like any other digital photographic system, the image projected onto the CCD is visualized on the display screen of a Corvis ST system. The calibration experiment described here aimed to find the invariable scaling factor, which calculates the image length in micrometers from the pixels read on the display screen or the exported image. Figure 2 shows the experimental setup for the calibration experiment. A piece of scotch tape seals the air hole to prevent the potential shift induced by air puff indentation for the tested samples. The base of the custom-made platform is a magnetic block with an on/off switch. It can be firmly fixed onto an iron surface. The rail slider is mounted on the base, so the mobile plate above can move along under the control of a micrometer screw gauge. A tube mount is attached to the mobile plate in order to fix the lens holder. When using the eye contact lens, it should be dampened to adhere to the head of lens holder. The rod side is then plugged into the tube mount. The eye contact lens is kept 7 mm away from the camera lens in the calibration experiment. Taking the advantage of the custom-made platform, the eye contact lens can be precisely moved, but it needs to be kept fixed in this calibration experiment. The ability of the custom-made platform for controlling the position will be used later in the experiment discussed in Sec. 2.5.

Fig. 2

The experimental setup for measuring the scaling factor.

Three customized hard contact lens and four human eyes were used to estimate the scaling factor. The contact lenses were made of polymethylmethacrylate (PMMA) with the refractive index ($n$) of 1.432. The thickness was measured by the thickness gauge (G. Nissel & Company Limited, United Kingdom), and the front curvature was measured by the Ophthalmometer OM-4 (Topcon Corporation, Tokyo, Japan) in the Optometry Clinic at the Hong Kong Polytechnic University. Two healthy human subjects were tested in the General Hospital of People’s Liberation Army (the Chinese PLA General Hospital, Beijing, China). Both the left and right eyes were included. The thickness and front curvature were measured in vivo by an OCT anterior segment scanner SS-1000 CASIA (TOMEY Corporation, Nagoya, Japan). Table 1 details the geometrical parameters of all seven samples.

Table 1

Geometrical parameters of the hard contact lens and human corneas.

Not considering the optical magnification, the effective image length of CCT in micrometers was derived from the true thickness measured with the theoretical object–image curve. It can also be measured manually in pixels on the display screen by the custom-developed program. The scaling factors were calculated in all cases and averaged for further applications.

2.4.

Theoretical Simulations for the Influence Factors

Theoretical object–image curve of the CCT can predict measurement errors in the intuitive-feasible calibration aforementioned. This calibration actually assumes the scaling factor invariable from case to case for transforming image length in pixels to object length in micrometers. Accordingly, the factor can be determined experimentally and applied to each measurement. However, three factors may break this expectation: the refractive index, the front curvature, and the measurement distance. The simulation experiments here were designed to predict these influences. It should be noted that the simulation needs the inverse process, which calculates the image length from the real CCT. It cannot be directly obtained through Eqs. (1)–(10), but can be realized by interpolating²⁹ the image–object curve derived from the forward process. A real experiment was additionally conducted to confirm the prediction concerning the refractive index later in Sec. 2.5.

When investigating the influence of the refractive index, we simulated the materials of air ($n=1$), human cornea ($n=1.375$),³⁰ and contact lens ($n=1.432$). The front curvature was fixed to 7.5 mm, and the distance ($d$) between the sample and the camera lens was 7 mm. Another simulation imitated that a cornea moved toward the camera lens. The distances were set to be 5, 7, and 9 mm, while the front curvature was fixed to 7.5 mm. In the final simulation, the front curvature of a cornea varies from 7 mm at the convex status to the concave status of the same value, and the distance was set to 7 mm. It imitated the influence of the air puff indention, which changes the front curvature of cornea with time. All these simulations assumed that the intuitive-feasible calibration for determining the total scaling factor was conducted on a 550-um-thick cornea of the 7.5-mm front curvature measured 7 mm away from the lens.

2.5.

Confirming the Prediction of the Refractive Index

A real experiment was conducted on the Corvis ST to confirm the prediction concerning the refractive index. It is that the human cornea should be adopted in the intuitive-feasible calibration; otherwise the sample with a different refractive index will induce errors in the CCT measurement. In the following experiment, the total scaling factors were measured for the material of air, the human cornea, and the PMMA material. And then the measurement errors can be calculated for a human cornea, assuming that the calibration was conducted in air or the hard contact lens made of PMMA.

The custom-made platform was applied to carry the lens holder to move horizontally along the line connecting the sample and the camera lens. The hard contact lenses were initially tested at the distance of 7 mm and then moved away from the camera lens at regular intervals of 0.2 mm until the distance of 7.8 mm. The whole procedure was repeated five times. Just like the calibration experiment described in Sec. 2.3, a piece of scotch tape sealed the air hole to prevent the potential shift of the testing sample due to air puff indentation. The image data of four human corneas collected in Sec. 2.3 were also included for further analysis.

Based on the data captured by the Corvis ST, we processed the collected images by the custom-developed program. On the images of hard eye contact lens, we manually recorded the apex of the anterior surface and the central thickness. Position records of the apex of the moving anterior surface were applied to calculate the total scaling factor in air, while the central thickness records were used to obtain that for PMMA. On the images of human corneas, the CCT were manually measured in pixels and averaged among five tests for calculating the total scaling factor in human cornea.

Using the calibration result of air, human cornea, and PMMA, we calculated the CCT of a human cornea multiplying the total scaling factor by the measured pixel numbers. Consequently, we obtained the measurement errors that the refractive index induced.

2.6.

Clinical Applications

Our previous study on the Corvis ST observed a thinning phenomenon with air puff indentation.²⁰ Now the proposed method was applied to correct the Scheimpflug distortion, which aimed to confirm this thinning. Sixty cases of healthy human corneas included in the study were from 35 patients visiting the Chinese PLA General Hospital for the eye examination from 2012 August to 2013 October. Twelve cases were excluded, since eyelashes degraded image quality, which may interfere with the estimation of front curvature at highest concavity. All the examinations were conducted in accordance with the Declaration of Helsinki, and the experimental protocol received the approval from the institutional review board of the Chinese PLA General Hospital.

The CCT of each case was measured three times on the first frame and the frame corresponding to highest concavity. It was manually recorded and averaged. These values of the pixel number gave the ratio of thinning without correction. Then the original CCT was calculated using the total scaling factor achieved in the intuitive-feasible calibration. The parameters of the calibration human cornea are as those in the simulation experiment ($n=1.375$, $R=7.5\mathrm{mm}$, $d=7\mathrm{mm}$, $\mathrm{CCT}=550\mathrm{um}$). The CCT at highest concavity was corrected with the curvature revealed by the Corvis ST at the measurement distance of 7 mm. The reasonableness of the approximation here will be presented in Sec. 4.

Statistical analyses on corrected CCTs were performed in IBM SPSS Statistics (Version 20, IBM Corporation, New York, USA). Statistical significance was set at the 5% probability level. The CCT differences before and after air puff indentation were first checked for their normality through the Kolmogorov-Smirnov test.³¹ Then the paired $t$ test was conducted to compare their mean values. Besides, Pearson correlation was applied to examine the contribution of the original thickness to the CCT variation. Many literature references provide useful instructions for the statistical analysis.^31,32

3.1.

Scaling Factor for Unit Conversion

In each measurement, the Corvis ST captures the whole process of cornea movement at a rate of 4330 image frames per second and records a 140-frame video clip in 30 ms. The image of each frame covers 8.5 mm horizontally of a single slit and has $200\times 576\text{pixels}$.⁹ All the collected clips were exported and the related frames were selected for further analysis.

The image length of any testing sample can be expressed both in pixels and in micrometers. We followed the protocol presented in Sec. 2.3 to experimentally derive the scaling factor for unit conversion. The average result of seven samples (three custom-made hard contact lens and four human corneas) manifested it to be $18.58\pm 0.43\mu \mathrm{m}/\text{pixel}$ in the Corvis ST, which means that each pixel in an exported image represents an image length of $18.58\mu \mathrm{m}$. This result was applied to estimate and correct the measurement error.

3.2.

Measurement Errors Attributed to Different Influence Factors

Figure 3 provides the theoretical predictions on the images captured by the Corvis ST. They are simulated based on the proposed equations deduced in Sec. 2.1. Object–image curves for different materials are given in Fig. 3(a). The simulation result is consistent to our common sense that the image and object length is equal in air. As for the material with a higher refractive index, it is observed that the image length is much smaller than the object length.

Fig. 3

Theoretical predictions on the images captured by the Corvis ST: (a) object–image curve under different refractive indexes, (b) errors in the central corneal thickness (CCT) measurement related with the refractive index of calibrated samples, (c) errors in the CCT measurement at different measurement distances, and (d) errors in the CCT measurement with air puff indentation ($N$ for the convex cornea and $R$ for the revised cornea under air puff indentation).

Further images demonstrate the influence of the refractive index, the measurement distance, and the front curvature on the CCT measurement. According to Fig. 3(b), measurement errors are positively correlated with the absolute value of the CCT. The measured value of a $550\text{-}\mu \mathrm{m}$-thick human cornea is $29.3\mu \mathrm{m}$ larger than its real value when the calibration is conducted on the contact lens, while it is underestimated by $201.7\mu \mathrm{m}$ when the calibration is performed on a moving anterior surface of the contact lens in air. However, measurement errors can be confined to $7\mu \mathrm{m}$ within the whole thickness range between 450 and $650\mu \mathrm{m}$, if human corneas are adopted in the calibration. Figure 3(c) shows the measurement errors of a human cornea deviating from the original measuring position in the calibration. The measurement error due to the measurement position is $<1.82\mu \mathrm{m}$ for a $500\text{-}\mu \mathrm{m}$ thick cornea. The value are 2.17 and $2.56\mu \mathrm{m}$ for the CCT of 550 and $600\mu \mathrm{m}$, respectively. It is concluded that errors in a measurement are at the level of several micrometers, even if the testing cornea moves 2 mm away from the calibration position. According to our experience, the largest deviation would not exceed 0.5 mm; otherwise, the cornea would be out of the region of imaging and fail to be captured by the camera of Corvis ST. Figure 3(d) reveals the influence of air puff indention on the CCT measurement. The letters in parentheses denote the direction of corneal deformation. The letter N means the cornea is convex, and the letter R means the cornea is reversed under air puff indentation. The distance between the green line and the blue line shows only several micrometers error occurred due to the diversity of the front curvature of human corneas. However, when air puff indentation works and renders the cornea completely reversed, the measurement error could reach $\sim 50\mu \mathrm{m}$ referred to a $550\text{-}\mu \mathrm{m}$-thick human cornea.

3.3.

Calibration Errors Related with the Refractive Index

Former simulation results declared that a proper calibration could restrain the measurement errors of a stationary CCT to several micrometers. This conclusion will be further explained in Sec. 4. Here the experiment was performed to confirm the prediction concerning the refractive index shown in Fig. 3(b) that the CCT is underestimated $\sim 201.7\mu \mathrm{m}$ with an improper calibration conducted on the moving anterior surface of contact lens in air, and it is overestimated by $\sim 29.3\mu \mathrm{m}$ when the calibration is conducted on the contact lenses made of PMMA. We followed the experimental protocol presented in Sec. 2.5 to obtain the total scaling factor for the intuitive-feasible calibration. Among the three custom-made hard contact lenses, it was concluded that each pixel in the exported image represented $16.5\mu \mathrm{m}$ for air and $31.2\mu \mathrm{m}$ for contact lens, while the scaling factor was found to be $28.9\mu \mathrm{m}$ among four human corneas. Accordingly, we calculated the measurement error of a $550\text{-}\mu \mathrm{m}$-thick human cornea. It was found that the CCT was $44\mu \mathrm{m}$ thicker based on the calibration of a moving anterior surface of contact lens in air and $235\mu \mathrm{m}$ thinner based on the calibration of hard contact lens. Considering the pixel resolution of Corvis ST, which is $\sim 28.9\mu \mathrm{m}$ for the human cornea without air puff indentation, the result was in accordance with our simulation, which predicted $29\mu \mathrm{m}$ thicker and $202\mu \mathrm{m}$ thinner.

3.4.

Thinning Phenomena in Clinical Applications

The proposed correction method was applied to confirm the thinning phenomenon observed in our previous study.²⁰ It was observed that the CCT decreased by 7.6% of its original thickness when air puff indentation made the cornea totally reversed. After a correction, the CCT was found to increase by $66\pm 34\mu \mathrm{m}$ at highest concavity under air puff indentation, which was 2.5% of the original CCT. The corrected CCT was $576\pm 34\mu \mathrm{m}$ at the preindentation period and $590\pm 35\mu \mathrm{m}$ at highest concavity. Their differences in each case passed the Kolmogorov–Smirnov test for normality, and a significant difference was observed by the paired $t$ test ($p<0.001$). Pearson correlation further found a weak positive correlation between the uncorrected variation of CCT and the stationary CCT ($r=0.35$, $p=0.015$). However, no significant association was observed after the correction ($r=0.16$, $p=0.162$).