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Corneal topography using a smartphone-based corneal topographer considering a biconical model for the corneal surface

Manuel Campos-García, Oliver Huerta-Carranza, Víctor Iván Moreno-Oliva, Daniel Aguirre-Aguirre, and Luis Ángel Pantoja-Arredondo

Open Access Open Access

Abstract

In this paper, we propose a generalization in both null-screen design and corneal surface topography retrieval algorithms to assess the nonsymmetric surface of the human cornea using a compact smartphone-based corneal topographer. The proposed new null-screen design considers the corneal surface to be a biconic surface, and the evaluation algorithms assume a general biconic shape with decentering, defocus, tilt, and rotation around the z-axis. We discussed how to integrate the measurement system to obtain geometric parameters of the corneal surface such as radius of curvature, conic constants, as well as elevation map, and sagittal and meridional curvature maps. To demonstrate that our proposed smartphone-based corneal topographer provides reliable measurements of corneal topography, we conducted measurements on a well-known biconic surface and two human corneas.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The cornea is the outer layer of tissue that protects the frontal part of the eye and contributes approximately two-thirds of the total refractive power of the human eye. Corneal shape assessment is a critical feature in the preoperative evaluation of patients for refractive surgery. It is known that due to various corneal pathologies such as keratoconus, pellucid marginal degeneration or other corneal dystrophies, the shape of the cornea is affected, thus reducing the visual capacity of the human eye. The shape of the cornea, for various applications, can be considered a spherical surface in its central region. It has also been considered as an oblate ellipsoid with an anterior radius of curvature between 7.7 to 7.9 mm and a conic constant between -0.18 to -0.26 [1,2].

It is worth mentioning that several corneal topography assessment tools start from the assumption that the cornea is shaped as a convex conical surface. However, some studies and analyses have been performed to describe the cornea as a more general surface without symmetry of revolution. Thus, starting from the raw data obtained from Placido disk-based corneal topographers, some least-squares fits have been performed to model the corneal surface as a biconic surface [3,4]. The importance of considering a biconical model of the corneal surface has recently been addressed and it has been shown that this model has advantages over traditional corneal models [5]. The Placido disk is a system composed of alternating black and white concentric rings with a central hole, along with a camera positioned behind the central hole of the disk system. The concentric ring system is reflected onto the corneal surface, creating a virtual image of the ring system, which is captured by the camera. Qualitative analysis of this virtual image allows for the estimation of corneal topography [1]. One of the main drawbacks of the Placido system is the oblique skew ray error, which has been demonstrated and discussed extensively [6,7]. Although the shape of the surface of the human cornea could be considered strictly a free-form surface, which varies from person to person, the fact that it adopts a quasi-biconic shape has allowed the development of various algorithms that have made it possible to better assess its shape.

On the other hand, the development of optical technologies based on mobile devices for ophthalmic applications has increased because they are very versatile, have high processing capacity, are portable, ubiquitous, and have advanced sensor technology [8,9]. Smartphones have sensors that allow high-quality images to be obtained; this feature has been exploited in the field of ophthalmology to develop various portable devices, including corneal topographers [10–12]. These devices require the integration of hardware and software technology with the cell phone to create algorithms that provide accurate corneal topography and can contribute to a correct diagnosis of each patient's corneal diseases.

Moreover, the method of evaluating the shape of the corneal surface, on which the design of the corneal topographer is based, is the so-called null-screen method. This technique consists of calculating the positions of the targets by exact ray tracing from a set of points on the image plane where the virtual image obtained by reflection of the targets is formed. If the design surface coincides with the evaluation surface, there will be no displacement of the reflected points in the image plane, but if there is a deviation of these points, it will be due to misalignments of the system and/or deformations of the surface under study. This method has been used to test surfaces of various geometries, such as convex surfaces [13], concave surfaces [14], off-axis surfaces [15], parabolic troughs [16], free-form surfaces [17,18], and the corneal surface [19,20]. The null-screen method, due to its simplicity, versatility, and low cost, has become a significant optical deflectometry test for testing spherical, aspheric, and fast free-form surfaces [21,22].

In a previous work, we proposed a design for the first smartphone-based compact corneal topographer to capture the images reflected from a conical null-screen by the corneal surface [20]. In that work, the null-screen was designed considering that the corneal surface is a convex aspheric surface. This proposal aims to address the challenges of providing quality eye care in resource-limited settings, such as rural and underserved areas. Moreover, it is a feasible option for obtaining corneal topographies due to its low cost, portability, and easy implementation. Figure 1 shows a schematic of the compact smartphone-based corneal topographer. Although important results have been obtained in corneal topography considering the shape of the cornea as an aspheric surface, we consider that it is more convenient to have an instrument that takes into account in its initial design and reconstruction algorithms that the corneal surface is a non-symmetric surface, specifically a biconic surface [23].

 figure: Fig. 1.

Fig. 1. Schematic of the conical null-screen attached to a mobile device.

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The aim of this work is to redesign the compact smartphone-based corneal topographer to capture the images of the null-screen produced by reflecting off the corneal surface, considering that the corneal surface can be represented in a more general way as a nonsymmetric surface, specifically a biconical surface. For this purpose, we describe in detail the design of the conical null-screen to test nonsymmetric human corneas, here we describe the procedure to obtain the targets that are reflected by the biconic surface, resulting in an array of circular dots on the detection plane. Next, we describe the procedure for evaluating surface topography, assuming that the cornea can be represented by a general biconic surface, it is important to mention that our algorithms were improved to obtain the topography of the corneal surface more accurately. In this part, we describe a procedure to model the shape of the cornea as a general biconic surface considering decentering, defocus, tilt in the x and y-axes, and rotation around the z-axis. Finally, to validate our proposal we conducted tests on a well-known biconic surface and obtained the topography of some human corneas with the new compact smartphone corneal topographer.

It is important to mention that this paper is an extension of the conference work given in Ref. [23]. In that work we proposed the design of a conical null-screen to test a biconical surface which is used to evaluate the corneal surface in this paper. Also, details on the improvements of the conical corneal topographer using the biconical null-screen are discussed in detail throughout the paper.

2. Design of the null-screen for testing nonsymmetric corneas

In order to design a null-screen considering nonsymmetric corneal surfaces, we will use the mathematical expression representing a biconic surface,

$$z({x,y} )= \frac{{{c_x}{x^2} + {c_y}{y^2}}}{{1 + \sqrt {1 - {Q_x}c_x^2{x^2} - {Q_y}c_y^2{y^2}} }},$$
where ${c_x} = {1 / {{r_x}}}$ and ${c_y} = {1 / {{r_y}}}$ are the curvatures at the vertex to the surface (rx and ry are the radius of curvature); ${Q_x} = {k_x} + 1$ and ${Q_y} = {k_y} + 1$ (kx and ky are the conic constants) in x and y directions, respectively. This expression reduces to an equation that represents a conic surface (surface of revolution) if the curvatures and the conic constants are equal, i.e, cx = cy and kx = ky. For ease of computation, Eq. (1) can be conveniently written as
$$\phi ({x,y} )= ({{Q_x}c_x^2{x^2} + {Q_y}c_y^2{y^2}} ){z^2} - 2({{c_x}{x^2} + {c_y}{y^2}} )z\, + {({{c_x}{x^2} + {c_y}{y^2}} )^2} = 0,$$
or in the explicit form,
$$z({x,y} )= \frac{{({{c_x}{x^2} + {c_y}{y^2}} )\left\{ {1 - \sqrt {1 - ({{Q_x}c_x^2{x^2} + {Q_y}c_y^2{y^2}} )} } \right\}}}{{{Q_x}c_x^2{x^2} + {Q_y}c_y^2{y^2}}},$$
this expression is a biconic equation, which can represent corneal surfaces without symmetry of revolution and give information about astigmatism associated with the corneal surface. Equation (3) represent the same surface as Eq. (1) but with this explicit form the calculations on the null-screen design and corneal surface modeling are simpler. Equations (2) and (3) are detailed in Appendix 7.1.

In Fig. 2 we show a schematic diagram of the smartphone-based corneal topographer. Here, all the distances are not independent, they are related by

$$d = \frac{{aD}}{{b + e + \beta }},$$
where d is the diameter of circle that contains the reflected image on the detection plane, D is the diameter of the corneal surface, a is the distance from the pinhole to the detection plane, e is the distance from the vertex of the surface to the base of the cone, and β is the sagitta at the edge of the surface. In this case, the height h, the distance b and the radius s of the base of the cone are fixed.

 figure: Fig. 2.

Fig. 2. Schematic diagram on the design of the conical null-screen of the smartphone corneal topographer.

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On the other hand, to determine the coordinates of the target points of the null-screen that give us a semi-radial array of circular dots in the CMOS image plane, we perform an exact ray-tracing calculation starting from the detection plane points P1 = (x1, y1, -a-b-e). These rays pass through the pinhole at point P = (0, 0, -b-e) in a direction given by the straight line

$$\frac{{x - {x_1}}}{{ - {x_1}}} = \frac{{y - {y_1}}}{{ - {y_1}}} = \frac{{z + a + b + e}}{a}.$$

These rays strike the corneal surface given by Eq. (3) at the points P2 = (x2, y2, z2) whose coordinates are calculated by substituting Eq. (5) into Eq. (2). Thus, according to Appendix 7.2, we get

$$\begin{array}{l} {z_2} = \left[ {\frac{{a\{{({{c_x}x_1^2 + {c_y}y_1^2} )+ ({b + e} )({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )} \}}}{{{a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )+ {{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}}} - } \right.\\ \;\;\;\;\;\left. {\frac{{({{c_x}x_1^2 + {c_y}y_1^2} )\sqrt {{a^2} - ({b + e} )\{{2({{c_x}x_1^2 + {c_y}y_1^2} )+ ({b + e} )({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )} \}} }}{{{a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )+ {{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}}}} \right]a - b - e,\\ {x_2} ={-} \left( {\frac{{{z_2} + b + e}}{a}} \right){x_1},\\ {y_2} ={-} \left( {\frac{{{z_2} + b + e}}{a}} \right){y_1}. \end{array}$$

In addition, the direction of the reflected rays is obtained with the direction of the incident ray given by

$${\mathbf I} = \frac{1}{{\sqrt {x_1^2 + y_1^2 + {a^2}} }}({ - {x_1}, - {y_1},a} ),$$
and the normal N to the surface evaluated at points P2 of the corneal surface according to Eq. (2) is given by
$${\left. {{\mathbf N} = \frac{{\nabla \phi ({x,y,z} )}}{{|{\nabla \phi ({x,y,z} )} |}}} \right|_{{P_2}}} = \frac{{({\partial {\phi_x},\partial {\phi_y},\partial {\phi_z}} )}}{{{{\{{\partial \phi_x^2 + \partial \phi_y^2 + \partial \phi_z^2} \}}^{1/2}}}},$$
with
$$\begin{array}{l} \partial {\phi _x} = 2{Q_x}c_x^2{x_2}z_2^2 - 4{c_x}{x_2}{z_2} + 4{c_x}{x_2}({{c_x}x_2^2 + {c_y}y_2^2} ),\\ \partial {\phi _y} = 2{Q_y}c_y^2{y_2}z_2^2 - 4{c_y}{y_2}{z_2} + 4{c_y}{y_2}({{c_x}x_2^2 + {c_y}y_2^2} ),\\ \partial {\phi _z} = 2{z_2}({{Q_x}c_x^2x_2^2 + {Q_y}c_y^2y_2^2} )- 2{c_y}{y_2}({{c_x}x_2^2 + {c_y}y_2^2} ). \end{array}$$

So, the direction of the reflected rays is obtained by means of the reflection law [24] given by

$${\mathbf R} = {\mathbf I} - 2({{\mathbf I} \cdot {\mathbf N}} ){\mathbf N = }({{R_x},{R_y},{R_z}} ),$$
where
$$\begin{array}{l} {R_x} = \frac{{{x_1}({\partial \phi_x^2 - \partial \phi_y^2 - \partial \phi_z^2} )+ 2\partial {\phi _x}({{y_1}\partial {\phi_y} - a\partial {\phi_z}} )}}{{\partial \phi _x^2 + \partial \phi _y^2 + \partial \phi _z^2}},\\ {R_y} = \frac{{{y_1}({\partial \phi_y^2 - \partial \phi_x^2 - \partial \phi_z^2} )+ 2\partial {\phi _y}({{x_1}\partial {\phi_x} - a\partial {\phi_z}} )}}{{\partial \phi _x^2 + \partial \phi _y^2 + \partial \phi _z^2}},\\ {R_z} = \frac{{a({\partial \phi_x^2 + \partial \phi_y^2 - \partial \phi_z^2} )+ 2\partial {\phi _z}({{x_1}\partial {\phi_x} + {y_1}\partial {\phi_y}} )}}{{\partial \phi _x^2 + \partial \phi _y^2 + \partial \phi _z^2}}. \end{array}$$
are the components of the direction of the reflected rays. Thus, with Eq. (11) we obtain the prolongation of the reflect rays which are given by
$$\frac{{x - {x_2}}}{{{R_x}}} = \frac{{y - {y_2}}}{{{R_y}}} = \frac{{z - {z_2}}}{{{R_z}}}.$$

The explicit expression for the points that give us the targets points P3 = (x3, y3, z3) of the null-screen depends on its geometry. For a conical arrangement of the null-screen given by

$${x^2} + {y^2} = \frac{{{s^2}}}{{{h^2}}}{({z + h + e} )^2},$$
where s is the radius of the base of the cone and h is its height, the target points P3 are obtained by substituting Eq. (12) into Eq. (13), resulting
$$\begin{array}{l} {z_3} = \frac{{({R_x^2 + R_y^2} ){z_2} - ({{R_x}{x_2} + {R_y}{y_2}} ){R_z} + \frac{{({h + e} )}}{{{h^2}}}{s^2}R_z^2}}{{({R_x^2 + R_y^2} )- \frac{{{s^2}}}{{{h^2}}}R_z^2}}\\ \,\,\,\,\,\,\,\, - \frac{{{R_z}{{\left\{ {\frac{{{s^2}}}{{{h^2}}}\{{{{[{({h + e + {z_2}} ){R_x} - {R_z}{x_2}} ]}^2} + {{[{({h + e + {z_2}} ){R_y} - {R_z}{y_2}} ]}^2}} \}+ {{({{R_x}{y_2} - {R_y}{x_2}} )}^2}} \right\}}^{1/2}}}}{{({R_x^2 + R_y^2} )- \frac{{{s^2}}}{{{h^2}}}R_z^2}},\\ {x_3} = \frac{{{R_x}}}{{{R_z}}}({{z_3} - {z_2}} )+ {x_2},\\ {y_3} = \frac{{{R_y}}}{{{R_z}}}({{z_3} - {z_2}} )+ {y_2}. \end{array}$$

The detailed development of these equations is shown in Appendix 7.3. Equation (14) gives us the coordinates of the targets on the conical null-screen topographer to evaluate the shape of non-symmetric corneal surfaces.

3. Corneal topography evaluation

3.1 Evaluation of the normal field

The null-screen method is an optical deflectometry technique [21,22] that requires knowledge of the deflection direction of a beam of light incident on the test surface for retrieving the shape of the surface under study. Thus, by knowing the directions of these beams (incident and deflected) the normal to the corneal surface can be easily calculated. Therefore, to evaluate the corneal topography of nonsymmetric corneas it is essential to assess the normal field to the corneal surface given by

$${\mathbf{N}^{\prime}} = \frac{{{\mathbf R^{\prime}} - {\mathbf I^{\prime}}}}{{|{{\mathbf R^{\prime}} - {\mathbf I^{\prime}}} |}},$$
where I’ and R’ are the directions of the incident and the reflected rays on the corneal surface, respectively. The evaluation of these rays is obtained by an approximated algorithm involving exact three-dimensional ray-racing. According to the test geometry (see Fig. 3), the reflected ray R’ is easily obtained because all rays starting from the coordinates Pc = (xc, yc, -a-b-e) of the centroids of the reflected target points in the detection plane pass through the center of the lens stop at P,
$${\mathbf R^{\prime}} = \frac{{({x_c},{y_c}, - a)}}{{\sqrt {x_c^2 + y_c^2 + {a^2}} }} = ({R{^{\prime}_x},R{^{\prime}_y},R{^{\prime}_z}} ).$$

 figure: Fig. 3.

Fig. 3. Evaluation of the normal vector field to the surface under test.

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Thus, the reflected rays are given by the straight line

$$\frac{{x - {x_2}}}{{R{^{\prime}_x}}} = \frac{{y - {y_2}}}{{R{^{\prime}_y}}} = \frac{{z - {z_2}}}{{R{^{\prime}_z}}}.$$

On the other hand, the direction of the incident ray I’ could be approximated by

$${\mathbf I^{\prime}} = \frac{{({x_s} - {x_3},{y_s} - {y_3},{z_s} - {z_3})}}{{\sqrt {{{({{x_s} - {x_3}} )}^2} + {{({{y_s} - {y_3}} )}^2} + {{({{z_s} - {z_3}} )}^2}} }}.$$

Here the points P3 are known because they are the null-screen targets given by Eq. (14). However, the points Ps = (xs, ys, zs) must be computed approximately by considering a biconic reference surface given by

$$\begin{array}{l} z = \frac{{\left\{ {[{{c_x}{{({x - {x_0}} )}^2} + {c_y}{{({y - {y_0}} )}^2}} ]\left\{ {1 - \sqrt {1 - [{{Q_x}c_x^2{{({x - {x_0}} )}^2} + {Q_y}c_y^2{{({y - {y_0}} )}^2}} ]} } \right\}} \right\}}}{{{Q_x}c_x^2{{({x - {x_0}} )}^2} + {Q_y}c_y^2{{({y - {y_0}} )}^2}}}\\ \,\,\,\,\,\,\,\,\,\, + A({x - {x_0}} )+ B({y - {y_0}} )+ C({x - {x_0}} )({y - {y_0}} )+ {z_0}, \end{array}$$
where (xo, yo, zo) are the surface vertex coordinates; (xo, yo) are the decentering terms, zo is the defocus, A and B are the tilt terms in x and y-axes, respectively, and C represents a rotation around the z-axis. Thus, the coordinates Ps where the rays intersect the reference biconic surface [Eq. (18)] are obtained by solving Eqs. (17) and (19), i.e., intersecting the reflected rays with the reference biconic surface. Finally, with the incident and the reflected rays we can approximately calculate the normal field vector of the corneal surface with Eq. (15).

3.2 Corneal topography

As is well known, corneal topography requires the evaluation of the shape of the surface and its curvatures (sagittal and meridional). The shape of the corneal surface can be evaluated by means of

$$z - {z_i} = \int\limits_{{P_i}}^{{P_f}} {\sqrt {{{\left( {\frac{{{n_x}}}{{{n_z}}}} \right)}^2} + {{\left( {\frac{{{n_y}}}{{{n_z}}}} \right)}^2}} d\rho } .$$

To evaluate Eq. (20) we need to define a set of integration trajectories through the discrete domain (x, y) of the surface under test. Here zi is the starting point of the integration trajectories, N’ = (nx, ny, nz) is the approximate normal to the corneal surface [Eq. (15)], as is schematized in Fig. 3, and ρ = (x2 + y2)1/2. The integral in Eq. (20) has been obtained without considering any approximation; however, the integrand is discrete and approximate, so the calculation of Eq. (20) must be performed by numerical integration [19,20]. As result, we obtain a sagittal points cloud that gives the shape of the corneal surface in the first instance.

Next, considering a general description for modeling the biconic corneal surface, the sagittal point cloud is fitted to the general biconic surface given by Eq. (20), where the main geometrical parameters describing the corneal shape can be retrieved. The geometrical parameters are the radius of the principal curvatures, the conic constants in two perpendicular meridians and a rotation parameter. Other parameters are the tilt in x- and y-axes, and the decentering and defocusing of the corneal surface; these parameters only affect the orientation and position of the surface under test. It is then proposed that this best-fit biconical surface serves as a reference surface for calculating a normal vector field closer to the actual one, using Eq. (15). Subsequently, the experimental normal vector and the gradient vector of the studied surface are analyzed. If the discrepancies between the two exceed a predefined threshold, the reconstruction process must be repeated until the desired level of accuracy is achieved, or until the model surface no longer changes between iterations. With this proposal we seek to obtain normal vectors close to the real normal vector, therefore the reconstructed surface will be close to the surface under study [19,20], in general, only about four iterations are enough to reach the desired value.

The calculation of the sagittal and meridional curvatures involves the curve φ(ρ) generated by the intersection between the surface ϕ(x, y) and a plane containing the normal vector to the surface ϕ(x, y) [20]. For this general case we propose that the curve φ(ρ) can be written as

$$\varphi (\rho ) = \frac{{r - {{\{{{r^2} - ({k + 1} ){\rho^2}} \}}^{{1 / 2}}}}}{{k + 1}} + \sum\limits_{i = 0}^{{M_p}} {{B_i}{\rho ^i}} ,$$
where r is the radius of curvature, and k is the conic constant of the base curve and Bi are the coefficients of a residual polynomial expansion. Considering a domain with a radial distribution of points, we calculate along each meridian the curvatures values. The coefficients r, k, and Bi of Eq. (21) can be calculated by a least-squares fit to the surface meridian data.

Thus, according to Ref. [1], the analytical expression for the sagittal curvature can be obtained by means of

$${k_s} = \frac{{\frac{{\partial \varphi }}{{\partial \rho }}}}{{\rho {{\left[ {1 + {{\left( {\frac{{\partial \varphi }}{{\partial \rho }}} \right)}^2}} \right]}^{\frac{1}{2}}}}},$$
and for the meridional curvature its expression is given by
$${k_m} = \frac{{\frac{{{\partial ^2}\varphi }}{{\partial {\rho ^2}}}}}{{{{\left[ {1 + {{\left( {\frac{{\partial \varphi }}{{\partial \rho }}} \right)}^2}} \right]}^{\frac{3}{2}}}}}$$
where the analytical expression for is given by Eq. (21).

4. Corneal topography

4.1 Null-screen for a biconic surface

For corneal topography measurements, a conical null-screen was designed considering that the cornea is a biconic convex surface with radius of curvature rx = 7.63 mm and ry = 7.40 mm, conic constants kx = -0.465 and ky = -0.481. These geometrical parameters are the mean cornea parameters reported by Navarro [3]. Table 1 shows the geometrical parameters for the design of the new conical null-screen.

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Table 1. Design parameters of the conical null-screen

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The null-screen for testing biconical corneas was designed as described in Section 2. Thus, starting from an ordered array of points P1 on the detection plane, by means of exact ray tracing the targets are obtained which are the objects that form the conical null-screen according to Eq. (15). To build the null-screen, a transformation to a two-dimensional plane is performed in order to print it on a sheet of paper as shown in Fig. 4(a). Figure 4(b) shows a frontal view of the null-screen inserted inside smartphone-based corneal topographer. Here we use LEDs to illuminate the null-screen in order to obtain well contrasted images when the null-screen is reflected on the corneal surface.

 figure: Fig. 4.

Fig. 4. (a) Flat conical null-screen, (b) Illuminated conical null-screen inside the compact smartphone topographer.

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4.2 Testing a biconic reference surface

In order to validate our proposal, we performed the shape evaluation of a biconic surface with the geometrical parameters listed in Table 1. The biconic surface has a diameter of D = 14.1 mm and was made of PDMS (Sylgard 184 from Dow Corning) [25]. The experimental image of the conical null-screen produced by reflecting light on the biconic surface is shown in Fig. 5(a), here we can see how the reflected targets have an almost circular shape, which agrees with the null-screen design that assumes that the reflected images must be composed of circular spots. Figure 5(b) shows the coordinates of the distortion-corrected centroids, which were obtained using previously developed image processing algorithms [20].

 figure: Fig. 5.

Fig. 5. (a) Experimental image generated by reflecting light on the biconic surface, (b) calculated centroids.

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Using the centroids and the conical null-screen’s target points, the direction of the incident and reflected rays can be calculated; in addition, the vector field normal is also obtained. Using these quantities, the shape of the biconic surface can be recovered, as is described in Section 3. Table 2 lists the geometrical parameters resulting from the least-squares fit to the data obtained from point cloud of the evaluated biconic surface. Calculating the differences between the evaluated points and the best-fit biconic surface, the peak to valley error in sagitta is Δzpv = 23.89 µm, and the rms error is Δzrms = 1.70 µm.

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Table 2. Parameters resulting from least squares fit of the point cloud

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Furthermore, Table 2 shows that both radius of curvature differ by 0.02 mm or approximately 0.26% and 0.15% in the x- and y-axes, respectively; and the conic constants differ by 0.005 and 0.046 or approximately 1.08% and 6.03% in the x- and y-axes, respectively, from the design values. These values agree with the value given by the manufacturer of the biconic surface (radius of curvature rx = 7.63 mm and ry = 7.40 mm, and conic constants kx = -0.465 and ky = -0.481) [25]. On the other hand, the obtained value of the conic constant on the y-axis differs significantly from the design value. We attribute this difference to imperfections in the surface manufacturing process. Figure 6(a) shows the height map of the evaluated biconic surface and Fig. 6(b) shows the differences in sagitta between the evaluated surface and the best-fit biconic surface, here the decentering and tilt have been removed.

 figure: Fig. 6.

Fig. 6. (a) contour map of the evaluated surface, (b) sagittal differences between the design surface and the best-fit biconic surface.

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4.3 Human corneal topography

With the new null-screen design, the next step was obtaining the topography of two human corneas of one subject by using our proposed smartphone-based corneal topographer. At the top of Fig. 7, we show the experimental images produces by reflecting light on the corneas, Fig. 7(a) right cornea, and Fig. 7(b) left cornea, respectively. In these Figures we can see that most of the reflected targets lack symmetry, and the asymmetric irregularities are more evident at the periphery of the corneal surface, so it is easy to see that the human cornea is not a biconic surface as we assumed in the null-screen design. Rather, we think that the anterior surface of the cornea is a more complex surface and, therefore, can be considered as a free-form surface. Moreover, the images show that some points of the array are lost due to the overlapping between some spots and the shadows produced by eyelashes and eyelids. Figure 7(c) and 7(d) show the centroid coordinates of the corneal reflected images, which were retrieved with our image processing algorithms.

 figure: Fig. 7.

Fig. 7. Resulting image of the null-screen targets after reflection: (a) right cornea, (b) left cornea. Calculated centroids: (c) right cornea, (d) left cornea.

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Once the centroids were calculated, according to Section 3, we proceeded to evaluate the corneal surface. The sagittal point cloud of each of the corneas were adjusted by least squares considering the general biconical surface of Eq. (20), Table 3 shows the results of the fitting. Additionally, for both corneas, the rms values of the sagittal differences Δzrms between the surface evaluated with our method and the best-fit biconical surface were calculated.

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Table 3. Parameters resulting from least squares fitting of corneal surfaces

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In Fig. 8, we show the differences in the height between the recovered surface and the best-fit biconic surface obtained by a least-squares fit, here the decentring and tilt have been removed. For the right cornea, the rms error sagitta value is Δzrms = 10.42 µm and the peak valley error is Δzpv = 63.16 µm, for the left cornea Δzrms = 8.91 µm, and Δzpv = 61.81 µm. From the values obtained in the asphericity of the corneas, we note that it corresponds to a prolate ellipsoid as expected for a normal corneal surface [3].

 figure: Fig. 8.

Fig. 8. Difference of height: (a) right cornea, (b) left cornea. Elevation maps: (c) right cornea, (d) left cornea.

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For completeness, Fig. 8 shows the elevation maps of both corneal surfaces. These elevation maps are obtained by numerical integration along the meridians passing through the geometric center of a reference spherical surface; then, the obtained point cloud is subtracted from the best-fit spherical surface [1]. Consequently, with these elevation maps it is not possible to make a comparison with the maps obtained with our method, since these are obtained from the height differences between the recovered surface and the best-fit biconical surface.

The plots of Fig. 9(a) and 9(b) show the axial or sagittal power maps calculated from the experimental data as proposed in Section 3 [Eq. (22)]. Axial power maps provide information about the corneal curvature measured with respect to the center of the circumference. The drawback of these maps is that the cornea is almost spherical in the central 3 mm or so, so moving away towards the marginal zones causes a change of geometry to aspheric and there will be a gradient of radius of curvature at that distance. These maps are accurate and very sensitive in that central area but lose validity in increasingly peripheral areas. A common feature of axial power maps is that, if corneal astigmatism is present, the maps adopt an hourglass shape, which is evident in the axial curvature maps obtained with our proposal. To compare our results, axial curvature maps were obtained with a Placido-based corneal topographer (Cornea 550 Essilor TM), the corresponding plots for the corneas of the same subject are shown in Figs. 9(c) and 9(d). As can be seen, the axial power maps obtained with both topographers are similar despite being two completely different topographers.

 figure: Fig. 9.

Fig. 9. Axial power maps. Smartphone-based corneal topographer: (a) left cornea, (b) right cornea. Placido-based corneal topographer: (c) left cornea, (d) right cornea.

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Lastly, the plots of Figs. 10(a) and 10(b) show the distribution of tangential power calculated with Eq. (23), as described in Section 3. In tangential power maps the radius of curvature and curvature are measured locally concerning a tangent at that point, independently of the center of the reference axis, i.e., tangential curvature maps provide information about instantaneous curvature changes. Of course, in this case, the sensitivity in the peripheral areas is extreme, however, in the central areas it loses efficiency and accuracy. Finally, Figs. 10(c) and 10(d) show the plots of the tangential power maps obtained with the Placido-based corneal topographer. In all the graphs, the tangential power maps obtained for both corneas of the same subject with both topographers are quite similar.

 figure: Fig. 10.

Fig. 10. Tangential curvature maps. Smartphone-based corneal topographer: (a) left cornea, (b) right cornea. Placido-based corneal topographer: (c) left cornea, (d) right cornea.

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5. Discussion

Throughout this work, we have proposed a new design null-screen which is used by the smartphone-based corneal topographer. The proposed device produces accurate results on the topography of human corneas considering them as a surface without revolution symmetry, these results are compared with those obtained with a corneal topographer (Cornea 550 Essilor TM). The results obtained in the measurement of the topography of human corneas agree with those obtained previously by other authors [3–5].

The manuscript describes a generalized technique for corneal topography measurements considering that the human cornea is an irregular asymmetric surface. This is particularly useful in the pathologies that are detected with corneal topography, such as keratoconus. To demonstrate the feasibility of corneal topography measurements with the smartphone-based topographer, we show that the proposed device can be used to reconstruct the shape of highly asymmetric and irregular surfaces; moreover, the method is used to test live human corneas.

In general, we should consider that the main model for measuring corneal topography is the spherical model because is the practical one and the commercial Placido-based corneal topographer uses this model. However, if the cornea is an irregular asymmetric surface the spherical model may be impractical and is better consider a conical or a biconical model to obtain accurate results in the measurements of the corneal topography.

It is important to mention that the prototype of the corneal topographer is not a clinical device at this moment, so the main objective of this work is to show that it can accurately measure the corneal surface even if the cornea has an asymmetric shape. On the other hand, once in the clinic we intend to perform topography measurements on several patients and compare the results with those obtained with commercial corneal topographers by means of correlation studies of the parameters obtained through a concordance study, for example, Bland-Altman plots for SimK values [26]. In addition, we intend to perform clinical measurements on patients who present some dystrophy that deforms the corneal surface, such as keratoconus.

6. Conclusions

The aim of the present work was focused on the design of a new-null screen to evaluate asymmetric corneal surfaces, specifically biconical surfaces. Also, we generalized both the algorithms for the evaluation of the corneal surface shape considering a general biconical surface as indicated in Eq. (19), and the algorithms for obtaining the curvature maps considering a conical surface plus deformation coefficients which is given by Eq. (21). All our square fitting algorithms were generalized to fit the sagittal point cloud obtained by the null-screen method with a general biconic model, these algorithms allow to recover the main geometrical parameters of the corneal surface including radius of curvature, conic constants in principal meridians, decentering, defocus, tilts, and rotation around the z-axis.

To validate the capability of the smartphone-based corneal topographer to obtain corneal topography, the topography of a biconical surface made of PDMS was measured. The results of this evaluation show that geometric parameters can be retrieved with accuracy. As shown the results of the biconic surface evaluation are very close to the design surface given by the manufacturer; here we found that the variations are approximately 1.70 µm rms measured with respect to the best-fit biconic surface, and the calculated radius of curvature differs about 0.26% and 0.15% in the x- and y-axes, respectively and the conic constants differs by 1.08% and 6.03% in the x- and y-axes, respectively, from the design values.

On the other hand, topography measurements of both corneas of one subject were performed. The results of these topographies agree with other similar studies on the evaluation of biconical surfaces but using a corneal topographer based on Placido disks [3–5]. With our method, corneal topographies of both corneas of one subject were obtained with an accuracy of 10.42 µm for the right cornea and 8.91 µm for the left cornea, respectively; these values were obtained considering the rms error of the differences in sagitta between the measured corneal surface and the best-fit biconic surface.

The quantitative results obtained on topographic maps and geometric values of the corneal surface (radius of curvature and conic constants) are like those obtained by a Placido corneal topographer used in an ophthalmologic clinic. In addition, we proposed a simple method to calculate the distribution of axial and sagittal powers; the results obtained are like those reported by the commercial topographer.

Note that the main feature of the smartphone-based corneal topographer is the use of a null-screen with a semiradial array of targets that avoids ambiguity produced by the oblique beam, which may be relevant if examining a cornea with dystrophy that alters the shape of the surface, such as keratoconus. On the other hand, the dimensions of the corneal topographer are such that it is possible to evaluate almost the entire corneal surface, which will make it possible to design contact lenses that provide greater comfort for the patient.

7. Appendix

7.1 Mathematical representation of a biconical surface

From Eq. (1) that represents a biconical surface, we can get

$$z\left( {1 + \sqrt {1 - {Q_x}c_x^2{x^2} - {Q_y}c_y^2{y^2}} } \right) = {c_x}{x^2} + {c_y}{y^2},$$
ordering the terms, it turns out that
$$z\sqrt {1 - {Q_x}c_x^2{x^2} - {Q_y}c_y^2{y^2}} = {c_x}{x^2} + {c_y}{y^2} - z,$$
squaring this expression, we have
$${z^2}({1 - {Q_x}c_x^2{x^2} - {Q_y}c_y^2{y^2}} )= {({{c_x}{x^2} + {c_y}{y^2}} )^2} + {z^2} - 2z({{c_x}{x^2} + {c_y}{y^2}} ),$$
regrouping the terms, we obtain Eq. (2).

On the other hand, Eq. (2) is a second order algebraic equation whose solution is given by

$$z = \frac{{2({{c_x}{x^2} + {c_y}{y^2}} )- \sqrt {4{{({{c_x}{x^2} + {c_y}{y^2}} )}^2} - 4({{Q_x}c_x^2{x^2} + {Q_y}c_y^2y} ){{({{c_x}{x^2} + {c_y}{y^2}} )}^2}} }}{{2({{Q_x}c_x^2{x^2} + {Q_y}c_y^2y} )}},$$
⇒
$$z = \frac{{2({{c_x}{x^2} + {c_y}{y^2}} )- 2({{c_x}{x^2} + {c_y}{y^2}} )\sqrt {1 - ({{Q_x}c_x^2{x^2} + {Q_y}c_y^2y} )} }}{{2({{Q_x}c_x^2{x^2} + {Q_y}c_y^2y} )}},$$
from which Eq. (3) is obtained. Thus, Eqs. (1) and (3) represent the same biconical surface.

7.2 Exact ray tracing from sensor to surface

From the line Eq. (5) we obtain the expressions

$$\begin{array}{l} x = \frac{{ - {x_1}}}{a}({z + b + e} ),\\ y = \frac{{ - {y_1}}}{a}({z + b + e} ). \end{array}$$

By direct substitution in Eq. (2), and after some algebraic manipulations, we obtain

$$({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )z_2^2 - 2({{c_x}x_1^2 + {c_y}y_1^2} ){z_2} + \frac{{{{({{z_2} + b + e} )}^2}{{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}}}{{{a^2}}} = 0,$$
⇒
$$\begin{array}{l} {a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )z_2^2 - 2{a^2}({{c_x}x_1^2 + {c_y}y_1^2} ){z_2} + {({{c_x}x_1^2 + {c_y}y_1^2} )^2}z_2^2 + \\ \,\,\,\,\,\,\,\,2({b + e} ){({{c_x}x_1^2 + {c_y}y_1^2} )^2}{z_2} + {({b + e} )^2}{({{c_x}x_1^2 + {c_y}y_1^2} )^2} = 0. \end{array}$$

Finally, regrouping the terms, we get

$$\begin{array}{l} \{{{a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )+ {{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}} \}z_2^2 - 2({{c_x}x_1^2 + {c_y}y_1^2} )\{{{a^2} - ({b + e} )({{c_x}x_1^2 + {c_y}y_1^2} )} \}{z_2}\\ \,\,\,\,\,\,\,\,\,\, + {({b + e} )^2}{({{c_x}x_1^2 + {c_y}y_1^2} )^2} = 0. \end{array}$$

Solving this quadratic equation, we get

$$\begin{array}{l} {z_2} = \frac{{({{c_x}x_1^2 + {c_y}y_1^2} )\{{{a^2} - ({b + e} )({{c_x}x_1^2 + {c_y}y_1^2} )} \}}}{{{a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )+ {{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}}} - \\ \;\;\;\;\;\frac{{a({{c_x}x_1^2 + {c_y}y_1^2} )\sqrt {{a^2} - 2({b + e} )({{c_x}x_1^2 + {c_y}y_1^2} )- {{({b + e} )}^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )} }}{{{a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )+ {{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}}}. \end{array}$$

Adding and subtracting the term ${a^2}({b + e} )({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )$ to the numerator of Eq. (33), after some algebraic calculations, the following result is obtained

$$\begin{array}{l} {z_2} = \left[ {\frac{{a\{{({{c_x}x_1^2 + {c_y}y_1^2} )+ ({b + e} )({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )} \}}}{{{a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )+ {{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}}} - } \right.\\ \;\;\;\;\;\left. {\frac{{({{c_x}x_1^2 + {c_y}y_1^2} )\sqrt {{a^2} - ({b + e} )\{{2({{c_x}x_1^2 + {c_y}y_1^2} )+ ({b + e} )({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )} \}} }}{{{a^2}({{Q_x}c_x^2x_1^2 + {Q_y}c_y^2y_1^2} )+ {{({{c_x}x_1^2 + {c_y}y_1^2} )}^2}}}} \right]a - b - e. \end{array}$$

This Equation represents the z2 coordinate of the point P2, where the rays strike the biconical surface.

7.3 Exact ray tracing from the surface to the cone

From Eq. (12), we get

$$\begin{array}{l} x = \frac{{{R_x}}}{{{R_z}}}z - \frac{{{R_x}}}{{{R_z}}}{z_2} + {x_2},\,\\ y = \frac{{{R_y}}}{{{R_z}}}z - \frac{{{R_y}}}{{{R_z}}}{z_2} + {y_2}, \end{array}$$

Substituting in Eq. (13), we have

$$\begin{array}{l} \frac{1}{{R_z^2}}\{{{{[{{R_x}{z_3} - ({{R_x}{z_2} - {R_z}{x_2}} )} ]}^2} + {{[{{R_y}{z_3} - ({{R_y}{z_2} - {R_z}{y_2}} )} ]}^2}} \}= \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{{{\left( {\frac{{h + e}}{h}} \right)}^2}{s^2}\{{z_3^2 + 2({h + e} ){z_3} + {{({h + e} )}^2}} \}}}{{{{({h + e} )}^2}}}, \end{array}$$
⇒
$$\begin{array}{l} \left\{ {({R_x^2 + R_y^2} )- {{\left( {\frac{s}{h}} \right)}^2}R_z^2} \right\}z_3^2 - \;2\left\{ {{R_x}({{R_x}{z_2} - {R_z}{x_2}} )+ {R_y}({{R_y}{z_2} - {R_z}{y_2}} )+ R_z^2\frac{{({h + e} )}}{{{h^2}}}{s^2}} \right\}{z_3}\\ \;\;\;\;\;\; + {({{R_x}{z_2} - {R_z}{x_2}} )^2} + {({{R_y}{z_2} - {R_z}{y_2}} )^2} - R_z^2{\left( {\frac{{h + e}}{h}} \right)^2} = 0. \end{array}$$

By solving this quadratic equation, we get

$$\begin{array}{l} {z_3} = \frac{{({R_x^2 + R_y^2} ){z_2} - ({{R_x}{x_2} + {R_y}{y_2}} ){R_z} + R_z^2\frac{{({h + e} )}}{{{h^2}}}{s^2}}}{{({R_x^2 + R_y^2} )- {{\left( {\frac{s}{h}} \right)}^2}R_z^2}} - \\ - \frac{{{R_z}\sqrt {\frac{{{s^2}}}{{{h^2}}}\{{{{[{({h + e + {z_2}} ){R_x} - {R_z}{x_2}} ]}^2} + {{[{({h + e + {z_2}} ){R_y} - {R_z}{y_2}} ]}^2}} \}- {{({{R_x}{y_2} - {R_y}{x_2}} )}^2}} }}{{({R_x^2 + R_y^2} )- {{\left( {\frac{s}{h}} \right)}^2}R_z^2}}. \end{array}$$

This Equation represents the z3 coordinate of the point P3, where the reflected rays strike the cone surface.

Funding

Consejo Nacional de Ciencia y Tecnología (293411); Dirección General de Asuntos del Personal Académico, Universidad Nacional Autónoma de México (IT100321, IT103623, TA100521).

Acknowledgment

The authors would like to express their gratitude to Agustin Santiago-Alvarado and Angel S. Cruz-Félix for providing the PDMS surface used to obtain the shape of a reference biconical surface. Additionally, Oliver Huerta Carranza (CVU: 710606) thanks the support of the Consejo Nacional de Humanidades, Ciencias y Tecnologías (CONAHCYT) for providing a postdoctoral fellowship.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (10)
Fig. 1.
Fig. 1. Schematic of the conical null-screen attached to a mobile device.

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Fig. 2.
Fig. 2. Schematic diagram on the design of the conical null-screen of the smartphone corneal topographer.

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Fig. 3.
Fig. 3. Evaluation of the normal vector field to the surface under test.

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Fig. 4.
Fig. 4. (a) Flat conical null-screen, (b) Illuminated conical null-screen inside the compact smartphone topographer.

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Fig. 5.
Fig. 5. (a) Experimental image generated by reflecting light on the biconic surface, (b) calculated centroids.

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Fig. 6.
Fig. 6. (a) contour map of the evaluated surface, (b) sagittal differences between the design surface and the best-fit biconic surface.

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Fig. 7.
Fig. 7. Resulting image of the null-screen targets after reflection: (a) right cornea, (b) left cornea. Calculated centroids: (c) right cornea, (d) left cornea.

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Fig. 8.
Fig. 8. Difference of height: (a) right cornea, (b) left cornea. Elevation maps: (c) right cornea, (d) left cornea.

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Fig. 9.
Fig. 9. Axial power maps. Smartphone-based corneal topographer: (a) left cornea, (b) right cornea. Placido-based corneal topographer: (c) left cornea, (d) right cornea.

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Fig. 10.
Fig. 10. Tangential curvature maps. Smartphone-based corneal topographer: (a) left cornea, (b) right cornea. Placido-based corneal topographer: (c) left cornea, (d) right cornea.

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Tables (3)

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Table 1. Design parameters of the conical null-screen

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Table 2. Parameters resulting from least squares fit of the point cloud

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Table 3. Parameters resulting from least squares fitting of corneal surfaces

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Equations (38)

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z ( x , y ) = c x x 2 + c y y 2 1 + 1 − Q x c x 2 x 2 − Q y c y 2 y 2 ,
ϕ ( x , y ) = ( Q x c x 2 x 2 + Q y c y 2 y 2 ) z 2 − 2 ( c x x 2 + c y y 2 ) z + ( c x x 2 + c y y 2 ) 2 = 0 ,
z ( x , y ) = ( c x x 2 + c y y 2 ) { 1 − 1 − ( Q x c x 2 x 2 + Q y c y 2 y 2 ) } Q x c x 2 x 2 + Q y c y 2 y 2 ,
d = a D b + e + β ,
x − x 1 − x 1 = y − y 1 − y 1 = z + a + b + e a .
z 2 = [ a { ( c x x 1 2 + c y y 1 2 ) + ( b + e ) ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) } a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) + ( c x x 1 2 + c y y 1 2 ) 2 − ( c x x 1 2 + c y y 1 2 ) a 2 − ( b + e ) { 2 ( c x x 1 2 + c y y 1 2 ) + ( b + e ) ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) } a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) + ( c x x 1 2 + c y y 1 2 ) 2 ] a − b − e , x 2 = − ( z 2 + b + e a ) x 1 , y 2 = − ( z 2 + b + e a ) y 1 .
I = 1 x 1 2 + y 1 2 + a 2 ( − x 1 , − y 1 , a ) ,
N = ∇ ϕ ( x , y , z ) | ∇ ϕ ( x , y , z ) | | P 2 = ( ∂ ϕ x , ∂ ϕ y , ∂ ϕ z ) { ∂ ϕ x 2 + ∂ ϕ y 2 + ∂ ϕ z 2 } 1 / 2 ,
∂ ϕ x = 2 Q x c x 2 x 2 z 2 2 − 4 c x x 2 z 2 + 4 c x x 2 ( c x x 2 2 + c y y 2 2 ) , ∂ ϕ y = 2 Q y c y 2 y 2 z 2 2 − 4 c y y 2 z 2 + 4 c y y 2 ( c x x 2 2 + c y y 2 2 ) , ∂ ϕ z = 2 z 2 ( Q x c x 2 x 2 2 + Q y c y 2 y 2 2 ) − 2 c y y 2 ( c x x 2 2 + c y y 2 2 ) .
R = I − 2 ( I ⋅ N ) N = ( R x , R y , R z ) ,
R x = x 1 ( ∂ ϕ x 2 − ∂ ϕ y 2 − ∂ ϕ z 2 ) + 2 ∂ ϕ x ( y 1 ∂ ϕ y − a ∂ ϕ z ) ∂ ϕ x 2 + ∂ ϕ y 2 + ∂ ϕ z 2 , R y = y 1 ( ∂ ϕ y 2 − ∂ ϕ x 2 − ∂ ϕ z 2 ) + 2 ∂ ϕ y ( x 1 ∂ ϕ x − a ∂ ϕ z ) ∂ ϕ x 2 + ∂ ϕ y 2 + ∂ ϕ z 2 , R z = a ( ∂ ϕ x 2 + ∂ ϕ y 2 − ∂ ϕ z 2 ) + 2 ∂ ϕ z ( x 1 ∂ ϕ x + y 1 ∂ ϕ y ) ∂ ϕ x 2 + ∂ ϕ y 2 + ∂ ϕ z 2 .
x − x 2 R x = y − y 2 R y = z − z 2 R z .
x 2 + y 2 = s 2 h 2 ( z + h + e ) 2 ,
z 3 = ( R x 2 + R y 2 ) z 2 − ( R x x 2 + R y y 2 ) R z + ( h + e ) h 2 s 2 R z 2 ( R x 2 + R y 2 ) − s 2 h 2 R z 2 − R z { s 2 h 2 { [ ( h + e + z 2 ) R x − R z x 2 ] 2 + [ ( h + e + z 2 ) R y − R z y 2 ] 2 } + ( R x y 2 − R y x 2 ) 2 } 1 / 2 ( R x 2 + R y 2 ) − s 2 h 2 R z 2 , x 3 = R x R z ( z 3 − z 2 ) + x 2 , y 3 = R y R z ( z 3 − z 2 ) + y 2 .
N ′ = R ′ − I ′ | R ′ − I ′ | ,
R ′ = ( x c , y c , − a ) x c 2 + y c 2 + a 2 = ( R x ′ , R y ′ , R z ′ ) .
x − x 2 R x ′ = y − y 2 R y ′ = z − z 2 R z ′ .
I ′ = ( x s − x 3 , y s − y 3 , z s − z 3 ) ( x s − x 3 ) 2 + ( y s − y 3 ) 2 + ( z s − z 3 ) 2 .
z = { [ c x ( x − x 0 ) 2 + c y ( y − y 0 ) 2 ] { 1 − 1 − [ Q x c x 2 ( x − x 0 ) 2 + Q y c y 2 ( y − y 0 ) 2 ] } } Q x c x 2 ( x − x 0 ) 2 + Q y c y 2 ( y − y 0 ) 2 + A ( x − x 0 ) + B ( y − y 0 ) + C ( x − x 0 ) ( y − y 0 ) + z 0 ,
z − z i = ∫ P i P f ( n x n z ) 2 + ( n y n z ) 2 d ρ .
φ ( ρ ) = r − { r 2 − ( k + 1 ) ρ 2 } 1 / 2 k + 1 + ∑ i = 0 M p B i ρ i ,
k s = ∂ φ ∂ ρ ρ [ 1 + ( ∂ φ ∂ ρ ) 2 ] 1 2 ,
k m = ∂ 2 φ ∂ ρ 2 [ 1 + ( ∂ φ ∂ ρ ) 2 ] 3 2
z ( 1 + 1 − Q x c x 2 x 2 − Q y c y 2 y 2 ) = c x x 2 + c y y 2 ,
z 1 − Q x c x 2 x 2 − Q y c y 2 y 2 = c x x 2 + c y y 2 − z ,
z 2 ( 1 − Q x c x 2 x 2 − Q y c y 2 y 2 ) = ( c x x 2 + c y y 2 ) 2 + z 2 − 2 z ( c x x 2 + c y y 2 ) ,
z = 2 ( c x x 2 + c y y 2 ) − 4 ( c x x 2 + c y y 2 ) 2 − 4 ( Q x c x 2 x 2 + Q y c y 2 y ) ( c x x 2 + c y y 2 ) 2 2 ( Q x c x 2 x 2 + Q y c y 2 y ) ,
z = 2 ( c x x 2 + c y y 2 ) − 2 ( c x x 2 + c y y 2 ) 1 − ( Q x c x 2 x 2 + Q y c y 2 y ) 2 ( Q x c x 2 x 2 + Q y c y 2 y ) ,
x = − x 1 a ( z + b + e ) , y = − y 1 a ( z + b + e ) .
( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) z 2 2 − 2 ( c x x 1 2 + c y y 1 2 ) z 2 + ( z 2 + b + e ) 2 ( c x x 1 2 + c y y 1 2 ) 2 a 2 = 0 ,
a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) z 2 2 − 2 a 2 ( c x x 1 2 + c y y 1 2 ) z 2 + ( c x x 1 2 + c y y 1 2 ) 2 z 2 2 + 2 ( b + e ) ( c x x 1 2 + c y y 1 2 ) 2 z 2 + ( b + e ) 2 ( c x x 1 2 + c y y 1 2 ) 2 = 0.
{ a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) + ( c x x 1 2 + c y y 1 2 ) 2 } z 2 2 − 2 ( c x x 1 2 + c y y 1 2 ) { a 2 − ( b + e ) ( c x x 1 2 + c y y 1 2 ) } z 2 + ( b + e ) 2 ( c x x 1 2 + c y y 1 2 ) 2 = 0.
z 2 = ( c x x 1 2 + c y y 1 2 ) { a 2 − ( b + e ) ( c x x 1 2 + c y y 1 2 ) } a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) + ( c x x 1 2 + c y y 1 2 ) 2 − a ( c x x 1 2 + c y y 1 2 ) a 2 − 2 ( b + e ) ( c x x 1 2 + c y y 1 2 ) − ( b + e ) 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) + ( c x x 1 2 + c y y 1 2 ) 2 .
z 2 = [ a { ( c x x 1 2 + c y y 1 2 ) + ( b + e ) ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) } a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) + ( c x x 1 2 + c y y 1 2 ) 2 − ( c x x 1 2 + c y y 1 2 ) a 2 − ( b + e ) { 2 ( c x x 1 2 + c y y 1 2 ) + ( b + e ) ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) } a 2 ( Q x c x 2 x 1 2 + Q y c y 2 y 1 2 ) + ( c x x 1 2 + c y y 1 2 ) 2 ] a − b − e .
x = R x R z z − R x R z z 2 + x 2 , y = R y R z z − R y R z z 2 + y 2 ,
1 R z 2 { [ R x z 3 − ( R x z 2 − R z x 2 ) ] 2 + [ R y z 3 − ( R y z 2 − R z y 2 ) ] 2 } = ( h + e h ) 2 s 2 { z 3 2 + 2 ( h + e ) z 3 + ( h + e ) 2 } ( h + e ) 2 ,
{ ( R x 2 + R y 2 ) − ( s h ) 2 R z 2 } z 3 2 − 2 { R x ( R x z 2 − R z x 2 ) + R y ( R y z 2 − R z y 2 ) + R z 2 ( h + e ) h 2 s 2 } z 3 + ( R x z 2 − R z x 2 ) 2 + ( R y z 2 − R z y 2 ) 2 − R z 2 ( h + e h ) 2 = 0.
z 3 = ( R x 2 + R y 2 ) z 2 − ( R x x 2 + R y y 2 ) R z + R z 2 ( h + e ) h 2 s 2 ( R x 2 + R y 2 ) − ( s h ) 2 R z 2 − − R z s 2 h 2 { [ ( h + e + z 2 ) R x − R z x 2 ] 2 + [ ( h + e + z 2 ) R y − R z y 2 ] 2 } − ( R x y 2 − R y x 2 ) 2 ( R x 2 + R y 2 ) − ( s h ) 2 R z 2 .
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