Geometrical interpretation of dioptric blurring and magnification in ophthalmic lenses

Sergio Barbero; Javier Portilla

doi:10.1364/OE.23.013185

1. Introduction

Images viewed through eyeglasses, for different gaze directions, are basically affected by three major optical phenomena: image displacement, blur and magnification. Image displacement is due to the direction change of the line of sight when viewing an object point through the eyeglasses. In optometry literature image displacement is usually referred to as prismatic error [1]. Image blurring is produced when, for a gaze direction through eyeglasses, the image of a viewing point is not punctual. In this case it is customary to analyze the blurring by first-order optics (i.e., aberrations due to pupil size are ignored [2]) and, consequently, to locally describe it as a linear transformation, performed by the dioptric matrix [1, 3, 4].

The term magnification is somewhat ambiguous because it may refer to different concepts. In rotationally symmetric optical systems, magnification is typically defined as the ratio between the object and the image height (transverse magnification). However, when analyzing vision through an optical system, the interest is not in the comparison between the object size to its conjugate image size, but rather in the ratio of the image angles at two different viewing conditions: first with the naked eye and second when using visual aids. If the optical system in front of the eye is rotationally symmetric (i.e. the optical properties are invariant under rotations around a fixed optical axis), and the paraxial approximation is used, then the ratio of retinal image sizes between both situations is equal to the ratio of the nodal angles. This is called angular magnification or, particularly when using eyeglasses, spectacle magnification [1, 5].

Spectacle magnification provides information on the retinal image size for a static line of sight but says nothing about what happens when the eyes look at different angles. When looking at a natural extended object (scene) the eye quickly scans around the object at different points, hence defining different gaze directions. In order to characterize how a scene is viewed Arnulf Remole introduced a new type of magnification which he coined dynamic spectacle magnification [6, 7]. Remole’s idea was to reinterpret prismatic effects as a kind of image magnification. Consequently the dynamic spectacle magnification is defined for each line of sight of a scene as the ratio between the ocular rotation when using spectacles with respect to not using them. However, when rotational symmetry is not assumed the eye rotation can not be completely described by a scalar, and it is necessary to use a two-dimensional vector.

In a general case, the distortion of an image scene can be mathematically described in terms of a mapping, a one-to-one correspondence of the eye rotation angles without and with lenses. Such a transformation can be illustrated as follows (see Fig. 1): Let us set a 2-D net of regular points, each one representing an eye rotation vector. Without eyeglasses this net is uniformly spaced. When eyeglasses are used this net is mapped onto a new one which, in general, will be no longer uniform. Hence this mapping provides a visualization of the global scene distortion produced by eyeglasses. On the other hand, the local magnification describes how a net in the close neighbor of a gaze direction is magnified and/or distorted. The fact that the local magnification changes from one to another line of sight implies that the mapping is necessarily non-linear (and non-affine). As such, it can not be globally described using a single matrix. However, it is possible to obtain local affine approximations with physical meaning, which mathematically can be described with a rotational magnification matrix. Note that if the mapping were known (which is usually not the case), the rotational magnification matrix could be simply obtained as the local Jacobian matrix of the mapping. Therefore, in a general case, the rotational magnification matrix only has a local physical sense. It characterizes the change of the differentials of the rotational vectors with and without eyeglasses [10, 11]. The use of the local magnification matrix turns out to be specially suitable when describing scene viewing through eyeglasses with variable power, such as progressive lenses (PALs) [12] or Alvarez-type lenses [13].

Fig. 1 The (global) mapping between 2D ocular rotation vectors (gaze directions) without (left) and with glasses (right) is a non-linear one-to-one correspondence. The rotational magnification matrix locally describes the approximately linear distortion of a small object (in this case, the letter A) in a neighborhood of an ocular rotation.

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The dioptric matrix can be obtained by tracing localized wavefronts and applying differential geometry [14, 15, 4, 16]. The same techniques of propagating localized wavefronts have been re-formulated to compute the local magnification matrix. If spectacles are composed of sphero-cylindrical surfaces and the incident rays describe parallel lines to the optical axis some closed expressions have been found to provide the magnification matrix as algebraic operations of some other matrices, including the dioptric one [17, 11]. In the general case, that is, having arbitrary optical surfaces and non-normal light incidence, still some matricial solutions are available [18, 19].

Instead of propagating wavefronts a different approach is to confine ourselves within a ray-tracing methodology. Then to describe first-order optics (blurring and magnification) it is sufficient to trace a base ray for each gaze direction and closely spaced rays about this ray. This general scheme is called differential ray tracing [20, 21]. Our method, which is an extension of the one we used in [22] to compute the dioptric matrices, follows this approach. It only traces two additional rays in the close vicinity of each base ray.

The dioptric and rotational magnification matrices mathematically describe the optical performance of ophthalmic lenses for different gaze directions. A relevant question arises: Are both matrices related? If so, how do they depend on each other? The relationship between image blurring and magnification is of major interest in current ophthalmic lens design, specially in progressive lenses. In this paper we intend to connect both matrices by using a common geometrical framework. We empirically explore the relationship between the geometrical entities describing blur and magnification by studying some examples, first in two rotationally symmetric cases, and then using a highly asymmetric composite-lens.

2. Magnification

In ophthalmic lens characterization and design the eye is usually reduced to a single point: the eye rotation center (point C in Fig. 2). The optical axis z joins C with the optical center of the lens (point V in Fig. 2). A is the object viewing point. The object plane is the one containing A which is perpendicular to the optical axis. The intersection of this plane with the optical axis gives point T. The position vector of A with respect to T is denoted by r⃗ⁱ. The gaze direction using eyeglasses is determined by the line (green solid) joining the eye rotation center (C) and the point at the lens through which the eye looks. The direction associated to this gaze direction is denoted by the normalized vector k⃗₀ = (k_x, k_y, k_z). After refraction through the lens the direction vector changes to k⃗′₀ = (k′_x, k′_y, k′_z). When looking at the the same point (A) but with the naked eye the gaze direction is p⃗₀ = (p_x, p_y, p_z), along the line (red dashed) which joins C with A. We denote s the distance between C and A.

Fig. 2 Ray trajectories with and without eyeglasses

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Light propagates, in spectacles, in a specific positive z-axis and therefore the 3D normalized ray vectors can be uniquely characterized by their x and y components. Here we term k⃗ = (k_x, k_y) and p⃗ = (p_x, p_y) and from now on will be called ray-vectors.

In some previous works [8, 9, 22] a linear transformation has been postulated for the mapping of vector p⃗ onto k⃗. However a more careful inspection shows that, in absence of additional constraints, there is no unique linear transformation between these two 2D vectors. As a consequence, such linear transformation actually has no physical meaning. Next we propose an alternative formulation having a simple physical and geometrical interpretation.

The effect of eyeglasses over the line of sight is naturally described by the shortest-arc possible 3D rotation (G) of (p_x, p_y, p_z) onto (k_x, k_y, k_z) [24]:

{\vec{k}}_{0} = G {\vec{p}}_{0} .

Note that this equation does not imply a linear relationship between k⃗ and p⃗. For each line of sight (denoted here by index i) we have two equations directly derived from Eq. (1):

\begin{array}{l} k_{x}^{i} = G_{11}^{i} p_{x}^{i} + G_{12}^{i} p_{y}^{i} + G_{13}^{i} p_{z}^{i} \\ k_{y}^{i} = G_{21}^{i} p_{x}^{i} + G_{22}^{i} p_{y}^{i} + G_{23}^{i} p_{z}^{i} \end{array}

For a given line of sight p⃗ⁱ,

G_{13}^{i} p_{z}^{i}

and

G_{23}^{i} p_{z}^{i}

can be substituted by their corresponding numerical values, say

d_{x}^{i} = G_{13}^{i} p_{z}^{i}

and

d_{y}^{i} = G_{23}^{i} p_{z}^{i}

, d⃗ⁱ = (d_x, d_y), yielding

{\vec{k}}^{i} = G_{r}^{i} {\vec{p}}^{i} + {\vec{d}}_{i}

, where

G_{r}^{i}

is

G_{r}^{i} = (\begin{matrix} G_{11}^{i} & G_{12}^{i} \\ G_{21}^{i} & G_{22}^{i} \end{matrix})

In the vicinity of gaze direction i,

p_{z} \approx p_{z}^{i}

, so d⃗ = (G₁₃, G₂₃)p_z will change little (d⃗ ≈ d⃗ⁱ), and thus we can locally approximate the mapping as an affine transformation:

\vec{k} (\vec{p}) \approx G_{r}^{i} \vec{p} + {\vec{d}}^{i} .

On the other hand, a similar affine local approximation of k⃗(p⃗) can be obtained without applying any particular geometrical interpretation of the problem at hand by using a differential approach [10, 11]. This is justified when the refractive surfaces forming the eyeglasses are continuous and smooth, so a differential perturbation δp⃗ = (δp_x, δp_y) will induce another differential perturbations δk⃗ = (δk_x, δk_y), which is a linear transformation of δp⃗.

We denote this linear transformation by

(\begin{matrix} δ k_{x} \\ δ k_{y} \end{matrix}) = (\begin{matrix} N_{11} & N_{12} \\ N_{21} & N_{22} \end{matrix}) (\begin{matrix} δ p_{x} \\ δ p_{y} \end{matrix})

and, for abbreviation sake, we will simply refer to N as the magnification matrix. Such linear transformation is directly determined by the local Jacobian matrix of the (unknown in most practical cases) global mapping.

3. Magnification and dioptric matrices

3.1. Computation method

In ophthalmic lenses the blurring of the image is usually described, in a local neighbor of a gaze direction, using the dioptric matrix. This matrix relates the ray position and the ray direction vectors at image space. A discussion about it and a method to compute the dioptric matrix for any optical system was previously proposed by Barbero [22]. The method is based on tracing two rays (with corresponding indices 1 and 2) close to the base ray along the gaze direction. It is valid for lenses comprising arbitrary surfaces, with the only condition of being continuous and smooth. Here we propose an extension of that work to considering both magnification and blurring, with some small differences in the way the two neighbor rays are constructed. The procedure comprises first computing the magnification matrix and then, based on the geometrical construction previously made, computing the dioptric matrix.

For computing the magnification matrix (see Fig. 3 as reference) we proceed as follows:

A base ray is traced through the lens. The intersection point of this ray with the lens surface closer to the eye is denoted by (x, y) and the ray vector by k⃗ = (ξ, η) (for notation convenience we set ξ = k_x, η = k_y).
Two points, lying at this surface, in the vicinity of (x, y) are selected with coordinates (x + δ_x, y) and (x, y + δ_y) respectively. δ_x and δ_y are carefully chosen small quantities [22].
Two new rays are constructed by joining the eye rotation center C with (x +δ_x, y) and (x, y + δ_y). The corresponding ray vectors are (ξ + δ_ξ, η) and (ξ, η + δ_η), respectively.
After propagating these three rays across the lens we obtain three ray intersections with the outer refractive surface: (x^s, y^s) for the base ray, plus ( $x^{s} + δ_{x 1}^{s}, y^{s} + δ_{y 1}^{s}$ ) and ( $x^{s} + δ_{x 2}^{s}, y^{s} + δ_{y 2}^{s}$ ) for its neighbor rays, and the corresponding refracted k⃗′ vectors.
The three rays are further traced, until reaching the object viewing plane, which yields: (xⁱ, yⁱ) for the base ray, plus ( $x^{i} + δ_{x 1}^{i}, y^{i} + δ_{y 1}^{i}$ ) and ( $x^{i} + δ_{x 2}^{i}, y^{i} + δ_{y 2}^{i}$ ) for its neighbor rays.
For computing p⃗ (the corresponding vectors for the base and the neighbor rays) we simply make use of the fact that p⃗ = r⃗/s [22], where s denote the distance without eyeglasses from the object viewing point to the eye center of rotation.
From the ray-coordinates computed in the previous steps we obtain the ray vector differentials appearing in Eq. (4): δk_x1 = δ_ξ, δk_y1 = 0, δk_x2 = 0, δk_y2 = δ_η, $δ p_{x 1} = (x^{i} + δ_{x 1}^{i}) / s_{1} - x^{i} / s_{0}$ , $δ p_{y 1} = (y^{i} + δ_{y 1}^{i}) / s_{1} - y^{i} / s_{0}$ , $δ p_{x 2} = (x^{i} + δ_{x 2}^{i}) / s_{2} - x^{i} / s_{0}$ and $δ p_{y 2} = (y^{i} + δ_{y 2}^{i}) / s_{2} - y^{i} / s_{0}$ . s₀, s₁ and s₂ denotes the s distance for the base ray and rays 1 and 2, respectively. Note that for far distance vision (s ↦ ∞) p⃗ ↦ k⃗′, as their corresponding lines intersecting the object point become parallel in that limit. Therefore, in that case p⃗ can be simply substituted by k⃗′.
Using matrix relation Eq. (4), a system of four equations is obtained containing the differential of p⃗ and k⃗ as computed in the previous step and the entries of the magnification matrix.

Fig. 3 Differential procedure to compute the magnification matrix, by using two auxiliary rays. For a better visualization only one differential ray (1) is plotted.

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Finally, the system of four equations is algebraically solved to obtain:

\begin{matrix} N_{11} & = & \frac{δ k_{x 1} δ p_{y 2} - δ k_{x 2} δ p_{y 1}}{δ p_{x 1} δ p_{y 2} - δ p_{x 2} δ p_{y 1}} \\ N_{12} & = & \frac{δ k_{x 2} δ p_{x 1} - δ k_{x 1} δ p_{x 2}}{δ p_{x 1} δ p_{y 2} - δ p_{x 2} δ p_{y 1}} \\ N_{21} & = & \frac{δ k_{y 1} δ p_{y 2} - δ k_{y 2} δ p_{y 1}}{δ p_{x 1} δ p_{y 2} - δ p_{x 2} δ p_{y 1}} \\ N_{22} & = & \frac{δ k_{y 2} δ p_{x 1} - δ k_{y 1} δ p_{x 2}}{δ p_{x 1} δ p_{y 2} - δ p_{x 2} δ p_{y 1}} \end{matrix}

The same base ray traced to compute the magnification matrix is now used to compute the dioptric matrix following these steps (see Fig. 4):

A pencil of rays is constructed from the object viewing point and reaching the eyeglasses. Two rays belonging to this bundle are constructed by joining the object point A with points located at the outer refractive surface computed in the procedure to evaluate the magnification matrix: ( $x^{s} + δ_{x 1}^{s}, y^{s} + δ_{y 1}^{s}$ ), ( $x^{s} + δ_{x 2}^{s}, y^{s} + δ_{y 2}^{s}$ ).
To compute the dioptric matrix, a new reference coordinate system (x′, y′, z′) is needed, where z′ is defined along the base ray traced backwards into the eye. To obtain x′ and y′ we apply the shortest-arc 3D rotation that goes from z to z′, which does not produce any torsion on the xy plane [23]. Finally we impose to x′, same as the absolute x axis, to be right-handed. Such choice for the new axes enables direct comparison of the magnification and dioptric matrices.
After tracing them through the lens, the corresponding intersection coordinates and ray vectors at the image plane (W) are computed with respect to the relative coordinate system (x′, y′, z′). These coordinates are denoted (x′₁, y′₁), (x′₂, y′₂), for the ray intersections, and (ξ′₁, η′₁), (ξ′₂, η′₂), for the ray vectors.

Fig. 4 Differential procedure to compute the dioptric matrix using two auxiliary rays. For a better visualization only one differential ray (1) is plotted.

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From these ray tracing data and following the same algebraic procedure as for the magnification matrix we obtain [22]:

\begin{matrix} R_{11} & = & \frac{ξ_{1}^{'} y_{2}^{'} - ξ_{2}^{'} y_{1}^{'}}{x_{1}^{'} y_{2}^{'} - x_{2}^{'} y_{1}^{'}} \\ R_{12} & = & \frac{ξ_{2}^{'} x_{1}^{'} - ξ_{1}^{'} x_{2}^{'}}{x_{1}^{'} y_{2}^{'} - x_{2}^{'} y_{1}^{'}} \\ R_{12} & = & \frac{η_{1}^{'} y_{2}^{'} - η_{2}^{'} y_{1}^{'}}{x_{1}^{'} y_{2}^{'} - x_{2}^{'} y_{1}^{'}} \\ R_{22} & = & \frac{η_{2}^{'} x_{1}^{'} - η_{1}^{'} x_{2}^{'}}{x_{1}^{'} y_{2}^{'} - x_{2}^{'} y_{1}^{'}} \end{matrix}

Note that there are two equations providing R₁₂. Theoretically both equations should give the same result, but numerical procedures could lead to slight discrepancies, so we just take the average [22]. Some possible cases of degeneracy could occur (see Ref. [22] for a discussion).

We also note that the location of the image plane (W) is usually taken after the posterior surface at a distance where that plane is tangent to the vertex sphere (the sphere center on C with radius CV) in order to obtain paraxial properties related to the back focal length.

3.2. Geometrical interpretation

If a square matrix A is not defective it can be always diagonalized (eigen decomposition) as A = QDQ⁻¹, where the elements of the diagonal matrix D are the eigenvalues and the columns of Q are the normalized eigenvectors. This matrix can be geometrically interpreted as anisotropic scaling. The eigenvectors represent the directions of pure stretching or scaling, whith corresponding scaling factors given by the eigenvalues. When A is symmetric, the eigenvectors are orthogonal (Q^T = Q⁻¹), that is, Q is a pure rotation (or, in a general case, a combination of rotation and reflection). This is the case of the dioptric matrix R, where the eigenvalues represent the two principal powers and Q represents the rotation of the principal axis where principal powers are defined. Also, the eigenvectors provide the directions of principal curvatures. However in the general case, as it happens with the N magnification matrix, matrices are not generally symmetric and, thus, their eigenvectors are not orthogonal. Still, the eigenvectors of the magnification matrix provide information on the directions of pure scaling. However, unlike for the symmetric matrix case, there may be a global rotation component. In analogy with the nomenclature used in the dioptric matrix Harris [25] called these the principal meridians of the magnification matrix and eigen-magnifications to the eigen-values.

A geometrical interpretation of the magnification matrix is provided by its polar decomposition [25, 10, 11]. N can be factored as the product of an orthogonal matrix and a symmetric positive definite matrix. The polar decomposition (N = OS), where S is a symmetric matrix and O is a rotation matrix) can be directly obtained from the singular valued decomposition (SVD).

We recall that the SVD decomposition is as follows: N=UD₁V^T, being U and V orthogonal matrices containing the left and right singular vectors. So V^T V = I. We can now make N = UV^T VD₁V^T. Then it must be O = UV^T. S, being positive symmetric, can be decomposed as S = VD₁V^T. The polar decomposition can also be done from the right (N = S’O’).

The orthogonal component of the polar decomposition implies that the image can suffer a pure rotation or reflection. The last one was analyzed as a theoretically possibility by Harris [25]. However, in spectacles it is an unrealistic scenario, so in practice a rotation can always be assumed.

Several geometrical entities can be defined [10, 11]:

Mean magnification: The average of the eigenvalues of matrix D₁
Anamorphic distortion: The difference between the eigenvalues of matrix D₁
Torsion angle: The angle of the rotation matrix O.

In this study we compare the mean magnification and anamorphic distortion with their counterparts in the dioptric matrix: the mean power and astigmatism, respectively. Considering the torsion component of the magnification matrix one may think that the amount of blur and magnification along different directions are not comparable. However, we observed in real eyeglasses designs (following sections) that the torsion angle is quite small, a fact which enables a meaningful comparison.

4. Study cases

We first study two rotationally symmetric lenses, and then a highly asymmetric one. To graphically illustrate the lenses’ behavior, we represent the blurring and magnification as oriented ellipses. The ellipses are oriented along the directions of maximum and minimum scaling, and scaled proportionally to these scaling factors.

4.1. Rotationally symmetric lenses

In rotationally symmetric ophthalmic lenses, under the thin lens approximation, there exists a formula relating the scalar spectacle magnification with the spectacle lens power [1]: N_spec = 1/(1 − dP), d being the distance from the spectacles to the rotation center of the eye and P the optical power of the eyeglasses. This implies that negative lenses (for myopia correction) have magnification values less than 1 (retinal image size reduction) whereas positive lenses magnify retinal images [1, 5]. This equation can be generalized (keeping the thin lens approximation) to a matrix equation for centered sphero-cylindrical surfaces and incoming light parallel to the optical axis [11, 17, 9]:

N_{spec} = {(I - d R)}^{- 1},

where R and N_spec are the dioptric power matrix and the magnification matrix, respectively. For this case, due to the symmetry, N_spec is symmetric.

From Eq. (7) it is clear that principal meridians of the magnification and dioptric matrices are the same and that the magnification of each principal meridian is given by [9] $N_{spec}^{i} = {(1 - d P^{i})}^{- 1}$ , being Pⁱ the principal power for each principal meridian. So it is straightforward to relate the mean power and astigmatism with the mean magnification and the anamorphic distortion, respectively. However when the thin lens approximation is not valid exact formulae are not available.

Our first test was a positive +7D meniscus-type lens with spherical surfaces: 66.5 and 900 mm anterior and posterior radii of curvature respectively. A refractive index of 1.5 was assumed. Figure 5 shows mean power (Fig. 5(a)), astigmatism (Fig. 5(b)), scalar magnification (Fig. 5(c)) and anamorphic distortion (Fig. 5(d)) as a function of eye rotations.

Fig. 5 (a) Mean power (D) (b) Astigmatism (D) (c) Scalar magnification (d) Anamorphic distortion; as a function of eye rotations for the spherical +7D lens.

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For this lens the mean power increases from the center (+7D) up to +9.5D at the periphery. Additionally the astigmatism increases form center to the periphery. As expected by Eq. (7) (although only valid for thin lens approximation) the scalar magnification (always higher than 1) and the anamorphic distortion also increase radially from center to periphery. Figure 6 shows the orientation (elliptical representation of matrices) of blurring (a) and magnification (b) for this lens. In order to help visualizing the changes in the ellipses for each gaze direction a black circle is also plotted representing the mean power (a) and mean magnification (b) for central viewing directions.

Fig. 6 Elliptical representation of (a) dioptric and (b) magnification matrices; as a function of eye rotations for the spherical +7D lens.

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The fact that both mean power and astigmatism increase radially imposes that the direction of maximum blurring is also along the radial direction. Consequently the major axis of the dioptric ellipses are oriented in that direction. The same effect occurs with magnification. Therefore in lenses where the mean power and astigmatism increase radially, the axis of major blurring and magnification are the same.

We used as a second test a negative −6.5D spherical lens. The anterior and posterior radii of curvature were 800 and 70 mm respectively. As with the +7D lens the refractive index was 1.5. Figure 7 shows mean power (Fig. 7(a)), astigmatism (Fig. 7(b)), scalar magnification (Fig. 7(c)) and anamorphic distortion (Fig. 7(d)) as a function of eye rotations.

Fig. 7 (a) Mean power (D) (b) Astigmatism (D) (c) Scalar magnification (d) Anamorphic distortion; as a function of eye rotations for the spherical −6.5D lens.

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The mean power decreases to more negative values when moving towards to the periphery. As for the positive lens the astigmatism increases form center to the periphery. However, in contrast to the positive lens, the magnification is always below 1 and decreases from center to periphery due to the fact the lens radially increase towards more negative powers.

As in the case of the negative lens, for the spherical −6.5D lens the direction of maximum blurring is also along the radial direction. However, and somewhat anti-intuitively, in this case the axis of major magnification is orthogonal to that of maximum blurring. Or, in other words, the magnification is minimum in the radial direction. This occurs because magnification decreases along the radial direction, but, at the same time, the anamorphic distortion direction of maximum change is along that direction. Therefore, in lenses where the power becomes more negative when moving from the center to the periphery, the direction of maximum blurring is orthogonal to that of maximum magnification.

We note that both lens examples were chosen to illustrate the interrelation between blurring and magnification. In most of current spectacles power error and astigmatism towards the periphery have a much more modest effect, as, for example, in best-form spectacles.

4.2. Highly asymmetric lens

In the previous examples all the geometrical entities describing blurring and magnification had z-axis rotationally symmetry. However a major goal of this paper is to analyze the behavior of lenses with arbitrary geometry.

We used as example (from now on the Asymmetric lens) a power adjustable composite lens; details of it can be found at pages 5–6 in [26]. Note that this is a non-optimized pre-design. A better design was also obtained [26], but here we are more interested in lenses producing high blur and distortion levels. Two lenses, with cubic-type geometries, when laterally shift one with respect to the other provide a power change. This configuration provides, for a specific lens movement, a central power of −5D but also a severe change in power for different gaze directions. As before, Fig. 8 shows mean power (Fig. 8(a)), astigmatism (Fig. 8(b)), scalar magnification (Fig. 8(c)) and anamorphic distortion (Fig. 8(d)) as a function of eye rotations. And Fig. 9 shows the orientation (elliptical representation of matrices) of blurring (a) and magnification (b).

Fig. 8 (a) Mean power (D) (b) Astigmatism (D) (c) Scalar magnification (d) Anamorphic distortion; as a function of eye rotations for the asymmetric lens.

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Fig. 9 Elliptical representation of (a) dioptric and (b) magnification matrices; as a function of eye rotations for the Asymmetric lens.

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The results show that even for lenses with extreme levels of power errors there is a strong correlation between mean power and mean magnification and between astigmatism and anamorphic distortion. The regions of maximum negative power match those of minimum magnification and vice versa. Relevantly, the torsion angle is negligible for all the gaze directions, so in practice the magnification matrix can be considered quasi-symmetric. The direction of maximum blur changes gradually across the different gaze directions and, as in the case of −6.5D spherical lens, the direction of maximum blur is in this case always close to orthogonal to that of maximum magnification.

We studied in more detail the correlation between the geometrical entities associated with the magnification matrix and those of the dioptric matrix (Fig. 10). Particularly we explored if it is possible to find a functional relationship between mean power and scalar magnification and between astigmatism and anamorphic distortion. Both pairs of data are function of each other for the +7D and −6.5D spherical lenses (see Fig. 10 (a)). This is a consequence of the lens axial symmetry. However this is not the case for the Asymmetric lens, which presents differences between measured and fitted astigmatism up to 0.04 D. The two pairs of data were approximated using a fourth order polynomial curve, which provided a good fit even for the Asymmetric lens case.

Fig. 10 Anamorphic distortion as a function of astigmatism (D) for +7D lens (a) and Asymmetric lens (c). Raw data are represented with blue circles and the fitted curved with a red line. The angle difference between the eigen-vectors of maximum scaling for the dioptric and magnification matrices as a function of astigmatism (D) is represented in the right column: +7D lens (b) and Asymmetric lens (d).

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At Figs. 10(b) and 10(d) we inspected more closely the relationship between the direction of maximum blurring and magnification. Whereas the difference is completely negligible (less than 0.03°) for symmetric lenses (+7D and −6.5D spherical lenses) and attributable to the accuracy errors of the two-rays numerical method, we found that the difference for a significant number of gaze directions is not negligible (values up to ±8°) for the Asymmetric lens though not large.

5. Discussion

It is clear that magnification and blurring, both degrading the image formation when looking through eyeglasses, are somehow related. However, contrary to some common misconceptions, the magnification matrix is not the effect of the dioptric matrix. For example in [12] it is stated that the “skew distortion”, which is essentially unequal magnification for different meridians, is due to the presence of astigmatism. Actually magnification and blurring are both effects of a specific ray mapping configuration. Our purpose has been to supply a geometrical interpretation of both transformations which permit a direct comparison.

The main contribution of this work is that we have shown, by doing numerical simulations, that effectively there is a strong correlation between geometrical entities describing blurring and magnification. Whenever mean power changes there is a change in mean magnification and the same can be said between astigmatism and anamorphic distortion. Particularly, the high correlation existing between astigmatism and anamorphic distortion, which is maintained even in highly asymmetric lens, has implications for ophthalmic lens design. The most important one being that if astigmatism is minimized (a typical target) the optical designer could be confident that the anamorphic distortion would be simultaneously minimized.

It has been realized, somewhat empirically, that in progressive lenses the unwanted cylindrical power axis (the direction of maximum blurring) is correlated with the axis of larger magnification (p. 248 [12]). However this is not always the case. We have proved that, generally, there are two possible situations: the first one (regions of increasing positive power), where the direction of maximum blurring is close to that of maximum magnification, and the second case (regions of increasing negative power), where the direction of maximum blurring is close to orthogonal to that of maximum magnification. These results have direct consequences in the selection of optical targets in ophthalmic lens design. For example one should try to avoid regions having simultaneously low magnification (magnification < 1) and high blurring, which are certainly possible, as shown in Fig. 8 and 9.

Rigorously speaking there are some non negligible differences between the direction of maximum blurring to either that of maximum or minimum magnification, being specially manifest with highly asymmetric lenses. ANSI standards [27] establish tolerances (to ophthalmic lenses manufactures) on cylinder axis of ±2° for astigmatism higher than 1.5 D. In consequence these differences are not negligible. They are specially interesting because they raise a new question: is it possible to talk about a direction of maximum degradation, i.e., blurring and magnification combined?

The quest of finding an explicit relationship between the magnification and dioptric matrix is still alive. We have clearly differentiated between the magnification defined as the ratio between object and image sizes in an optical system and that (with more meaning for ophthalmic lenses) relating the image angle through two different optical configurations. Rubinstein [10] has proved that there are mathematical equations relating geometrical quantities describing magnification and dioptric matrices connecting the object and image space. This could be a starting point for trying to find a theoretical connection between the dioptric matrix and the angular magnification as defined in this work.

Magnification in ophthalmic lenses has not been systematically studied. One reason is that it is well known that the visual system is more tolerant to magnification defects than to blur, because of neural adaptation [28]. However the existence of progressive lenses (PALs) and their never-ending demanding of better optical quality has meant an increasing awareness of its role. Some recent designs claim to take magnification into account. Some examples are Varilux S Series^TM from Essilor which claim to minimize anamorphic distortion [29], or the proposal [30] of using a target to reduce the difference in the mean magnification between the near and far portions of the PAL corridor. Also it has been argued that using a progressive rear surface can help to control magnification differences [19].

Moreover, magnification has more relevance when binocular vision is considered. For example, magnification differences between the two eyes (termed aniseikonia) may affect binocular fusion. However, overall the literature on visual effects of distortion on PALs is scarce [31]. Some clues could be extracted from recent clinical and visual studies. Changes in the scalar magnification are visually described as pumping effects, whereas anamorphic distortion as swimming effects [19]. Pure image rotations (a possible effect of the magnification matrix) has also been proved to have a visual effect. In particular, the visual system is sensitive to changes in the orientations of the image blurring [32, 33]. In this sense, the amount of tolerance to astigmatism depends on its orientation [34]. The method proposed in this paper provides an easy way to quantify (through the dioptric and magnification matrices) all these effects in specific lens designs.

We note that, although we have concentrated in this work in the distortion in the neighbor of a gaze direction, the global mapping between 2D ocular rotation vectors with and without glasses could be directly used in an optical design strategy, as done in rotationally symmetric lenses. In this case, the unequal scalar magnification (which causes the scene distortion) has been targeted in ophthalmic lens design, as proposed and showed by Katz [35].

Finally we note that the methods presented in this work can also be used to simulate vision through eyeglasses. In recent years a great deal of work has been done to characterize vision through ophthalmic lenses. Particularly, in the field of computer vision, the concept of vision-realistic rendering has been introduced [36, 37]. However all the attention has been paid to correctly simulated the blurring effect, whereas the effect of local magnification has been typically ignored.

Acknowledgments

This work was supported by grants FIS2012-30820 and TIN2012-38102-C03-01 from the Spanish Ministerio de Economía y Competitividad.

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Geometrical interpretation of dioptric blurring and magnification in ophthalmic lenses

Abstract

1. Introduction

2. Magnification

3. Magnification and dioptric matrices

3.1. Computation method

3.2. Geometrical interpretation

4. Study cases

4.1. Rotationally symmetric lenses

4.2. Highly asymmetric lens

5. Discussion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (8)

Optics Express