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Distortion evaluation method for the progressive addition lens–eye system
Accepted Manuscript Distortion evaluation method for the progressive addition lens-eye system Yu Jing, Hua Fangfang, Jiang Weiwei
PII: DOI: Reference:
S0030-4018(19)30290-1 https://doi.org/10.1016/j.optcom.2019.03.079 OPTICS 23997
To appear in:
Optics Communications
Received date : 15 December 2018 Revised date : 26 March 2019 Accepted date : 31 March 2019 Please cite this article as: J. Yu, F. Hua and W. Jiang, Distortion evaluation method for the progressive addition lens-eye system, Optics Communications (2019), https://doi.org/10.1016/j.optcom.2019.03.079 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Distortion evaluation method for the progressive addition lens-eye system YU JING , HUA FANGFANG, JIANG WEIWEI *
School of Metrology and Measurement Engineering, China Jiliang University, ZheJiang HangZhou 310018, China *Corresponding author: [email protected] Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX
Distortion is one of the most significant defects in image optical systems. Herein, a method for comprehensively evaluating progressive addition spectacle lenses based on the modulation transfer function (MTF) and distortion of the lens-eye system is proposed. A lens-eye system is established, and the parameters of the system were simulated and measured simultaneously. The human eye model was compared with and without progressive addition lenses to obtain the image quality of the lenses. It was observed that eyes with different diopters and different lens designs show different trends; however, good improvement was observed in all cases after progressive addition lens correction. In terms of image clarity and deformation, the distant field was caused by a negative distortion value, the near area was caused by a positive distortion value. and the optical distortion was found to be proportional to the diopter. Moreover, the progressive corridor distortion value was minimal. With a larger field of view, the distortion of the lens became larger; thus, when an object was viewed through the lens, large distortion was observed at the edges .
Keywords: distortion; progressive addition lens; freeform lens; lens-eye system; MTF 1. Introduction The human eye is an important photosensitive organ. The image quality of objects in the retina directly affects people's perception of the world. The most common ophthalmological disorders include myopia, hyperopia, and astigmatism,
which results in low image quality. The traditional method of correcting ametropia is to wear corrective spectacle lenses. The quality of the lens influences the physical health, mental health, and sight quality of the wearer [1]. In recent years, with the development of freeform technology, freeform lenses have become increasingly popular, especially progressive
additional lenses (PAL). The wearer can obtain clearer and more continuous vision using PAL as they provide both near and far vision and can be customized. International standards for the measurement of the refractive parameters of
traditional lens have been established for many years. For freeform lenses, traditional methods, and equipment cannot provide comprehensive measurement, nor can the refractive parameters provide a comprehensive evaluation [2]. The
lack of standards and evaluation systems for freeform lenses has made it impossible for consumers to distinguish the
difference in imaging quality between designs, resulting in a confusing market. Thus, methods for evaluating the quality
of different modern freeform lens designs to achieve rational and efficient development of this growing market have become a key research topic.
For image optics, the modulation transfer function (MTF) is the main parameter for determining image quality, and it can be used to evaluate the quality of spectacle lenses. In recent years, major universities and optical companies and
universities such as Carl Zeiss and Tianjin university [3-4] have been working on a method for evaluating the imaging
quality of freeform spectacle lenses. They proposed evaluating the imaging quality of PAL using MTF [2], which expresses the transmit ability of optical imaging systems at different frequencies [6-7]. Distortion is common in optical image
systems and contributes significantly to MTF [8-9]. Distortion is introduced during the design and manufacture of
freeform lenses, especially at the edge of the lens. This distortion is an obvious way for the common wearer to distinguish between different spectacle lenses.
In this study, we establish a spectacle lens-eye system and propose using distortion as an image parameter for evaluating the quality of a freeform lens. In contrast to the traditional image quality evaluation method using MTF, using the root mean error of the distortion to evaluate the image quality together with other image quality parameters can be more intuitive and reasonable. Distortion of the spectacle lens-eye systems with different freeform spectacle lenses is used to
illustrate image quality. A distortion grid map and MTF curve qualitatively describe changes in the image quality. In order
to quantitatively evaluate image quality, it is necessary to calculate the root mean square error of the distortion value, compare the simulation results and experimental measurement results, and then visually and quantitatively characterize the image quality changes.
2. Spectacle Lens-eye system distortion model In the optical image system, as the intersection height of the chief ray and the Gaussian image plane is not equal to the ideal image height [10] and is often accompanied by distortion, it directly represents the maximum deformation of the entire
optical system. The spectacle lens -eye system is equivalent to an image system. The mathematical model is established
based on the rotation matrix model (RMM). As shown in Figure 1, the image distortion in the spectacle lens -eye system is established. Ow -XYZ denotes the world coordinate system (WCS) , OL -xyZ denotes the lens coordinate system (LCS), OM
-UVZ denotes the image coordinate system (MCS), where O is the lens center, and OZ is the optical axis. The point Om is at a distance f to the point O. P is a point from the WCS, which is at a distance d from point O.
Fig. 1 Image distortion in the spectacle lens- eye system.
The following transforming relationships exist in these coordinate systems [11,12]:
(1) Let (xw, yw, zw) and (x ,y, z)denotes the coordinates of the object point P expressed in WCS and LCS, respectively.
Transforming the coordinate of point P from LCS to WCS yields
xw x y R T y w = zw 0 1 z 1 1
（1）
where
R = Rot ( x, θ x ) Rot ( y, θ y ) Rot ( z , θ z )
0 1 = 0 cos θ x 0 sin θ x
0 cos θ y − sin θ x 0 cos θ x − sin θ y
0 sin θ y cos θ z 1 0 sin θ z 0 cos θ y 0
− sin θ z cos θ z 0
0 0 1
（2
） T
And T = t x , t y , t z . R and T denote the rotation matrix and translation vector from LCS to WCS, respectively, where θ x , θ y , θ z are the T
rotation angles of WCS with respect to LCS. t x , t y , t z is the coordinate of the vector
oow (d) expressed in LCS.
(2) Point Pc represents the image of point P. Let (xc yc, zc) denotes the coordinates of Point Pc expressed in LCS.
According to the pinhole imaging model, the relationship between the coordinates of point P and Pc can be expressed as
xc x s yc = y (3) zc z
where s is an arbitrary scalar related with the image system.
(3) Let (xm ym, zm) denote the coordinates of the point Pc expressed in MCS. Transforming the coordinates of point Pc
from LCS to MCS yields
xm y = RM L m 1
xc y (4) c zc
where α is the angle around the OX- axis, β is the angle around the OY- axis, and M L
R
1 0 = 0 1 0 0
0 0 1
cos β sin α sin β cos α sin β f
0 cos α − sin α
− sin β sin α cos β (5) cos α cos β
Considering the spectacle lens distortion, let point Pd (xd yd, zd), as opposed to point Pc , denote the real image of point P.
Then, the distortion model can be expressed as ;
Where,
xd + ∆xd x= c = y c yd + ∆yd (6) z = z + ∆z d d c
( (
) )
( (
∆xd = k1rd2 + k2 rd4 xd + 2 p1 xd yd + p2 rd2 + 2 xd2 2 4 2 2 ∆yd = k1rd + k2 rd yd + 2 p1 xd yd + p2 rd + 2 yd 0 ∆zd =
) )
(7)
2 and rd= xd2 + yd2 . k1 , k2 , p1 , p2 are the lens distortion coefficients.
For a PAL, the surface power can be obtained by the Gaussian curvature and mean curvature can be drawn as:
K = κ1κ 2 κ1 + κ 2 (8) H = 2
where, 𝜅𝜅1 and𝜅𝜅2 are called as the principal curvatures, and the surface power can be calculated.
In the spectacle lens -eye system. The following variables can be calculated
d O= f tan θ , = m Pc
d Pc P= f tan θ ' − f tan θ ∆= d
The radial distortion can be expressed as:
(9)
2 δ xr = k1 xd rd (10) 2 δ yr = k1 yd rd
The eccentric distortion can be expressed as:
2 2 δ x= d p1 ( 3 xd + yd ) + 2 p2 xd yd (11) 2 2 δ xd = p2 ( xd + 3 yd ) + 2 p1 xd yd
u = zc v 1
ax 0 0
0 ay 0
In summary,
u0 xw R T δ x v0 ⋅ ⋅ yw + (12) 0 1 δ y 1 zw
, ay f ax f= = x ∆ ∆y , is related to the eye, it can be calculated from the physiological parameters. where to the RMM model the distortion can be calculated and simulated.
3. Simulation of distortion correction in the spectacle lens-eye system
According
In this study, the optical design software ZEMAX is used to establish and optimize the spectacle lens-eye system. According to the physiological characteristics of the human eye, we choose the relatively common method of controlling the length
of the eye axis to simulate the refractive error of the eye. The selected individual human eye model displays axial myopia,
which is predominantly characterized by an axial refractive error. The refractive parameters are -1.0 D, -1.50 D, -2.0 D, and
-2.5 D. Liou [14] proposed a human eye model based on anatomical data, which is the most comprehensive human eye model to date. The basic parameters of the human eye model are shown in Table 1, where GradA and GradP indicate the gradient refractive indices of the first and second half of the lens, respectively [15]. Table 1 Parameters of the human eye model Surface
Radius (mm)
Thickness (mm)
Conic constant
Refractive index
aqueous
6.40
3.16
-0.60
1.336
cornea
pupil
anterior crystalline lens
7.77
Infinity 12.40
0.55 0.00 1.59
-0.18 0.00 0.00
1.376 1.336
GradA
Abbe number 50.23
posterior lens
crystalline
vitreous
Infinity
2.43
-8.10
retina
0.00
16.239
-12.0
GradP
0.96
-
1.336
-
-
-
The near-point distance visible to the human eye is calculated according to the refractive parameter of the eye, the object
distance is set to the near-point distance visible to the human eye in ZEMAX, and the distance of the length of the eye axis
is then optimized. Thus, the axial distance of the refractive parameter of the human-eye is obtained. The parameters for
simulating the refractivity of the human eye by controlling the length of the eye axis are listed in Table 2. Table 2 Corresponding distances of the eye axis Corresponding Diopter (D) (mm)
distances
Diopter (D)
Corresponding distances (mm)
-1.00
16.556
2.00
15.465
-2.00
16.924
1.00
15.816
-1.50
16.741
-2.50
1.50
17.109
15.644
0.50
16.001
The surface of PAL is non-rotationally symmetric, which is represented by the commonly used mathematical models
Zernike polynomial and XY polynomial. The XY polynomial is the typical mathematical model used to describe freeform
surfaces [12]. The described non-rotational symmetrical freeform surfaces are widely used in optical design. The XY
polynomial surface is used in the optical design software ZEMAX to describe the PAL. The freeform surface described by the XY polynomial is:
Z =
cr 2
N
1 + 1 − (1 + k ) c 2 r 2
+ ∑ Ci x m y n i=2
（13）
where c is the curvature of the basal plane (mm-1), k is the quadric surface coefficient, and r is the radial pupil coordinate.
N represents the total number of polynomial coefficients in the series, Ci represents the coefficient (mm) of the polynomial of the i-th term, m represents a power series in the x direction, and n represents a power series in the y direction. Equation
(3) is composed of two main parts, the first part predominantly describes the conic surface and the second part describes the freeform surface type described by the XY polynomial in the Cartesian coordinate system [16]. In ZEMAX, the freeform surface is usually described by an XY polynomial using ten steps, where
i=
(( m + n ) 1+ N
y ∑C x = i=2
i
m
n
2
+ m + 3n
)
2
C2 x1 y 0 + C3 x 0 y1 + + C66 x 0 y10
（14）
（15）
The PAL are evaluated by comparing and analyzing the imaging quality of the human eye model and the spectacle lenseye system. Taking ametropia of -1.0 D as an example, the MTF curve of the lens-eye system is shown in Fig. 2, where the
MTF curve of the ametropia model corrected without the corresponding PAL is as follows. As shown in Fig. 2(a), according to the human eye characteristics, the cutoff frequency of the human eye can reach approximately 20–30 lp/mm and the
human eye resolution limit MTF value is 0.2. However, in Fig. 2(a), all the views are 30 lp/mm and the MTF value is
approximately 0.1, which does not reach the human eye resolution limit. Therefore, the image quality of the human eye model is poor and should be corrected by the PAL. The MTF curve of the spectacle lens-eye system after the PAL correction
is shown in Fig. 2(b), in which all the field-of-view MTF values at a spatial frequency of 30 lp/mm are greater than 0.2. The resolution limit of the human eye is reached, so the imaging quality of the eye whose ametropia is corrected by wearing the corresponding progressive multifocal lens is further improved.
（a）
（b）
Fig. 2 MTF curve of the lens-eye combined optical system: (a) eye model and (b) spectacle lens-eye system
This result shows that the image quality of the human eye model with ametropia exhibits improved correction with
the corresponding PAL. However, distortion can be introduced during the design and processing of the PAL, especially at the edge of the lens, which may cause the image quality of the lens to decrease. Comprehensive evaluation of the imaging quality of the PAL must be discussed along with lens distortion.
When analyzing the influence of distortion on the imaging quality of the PAL, it is necessary to separately discuss the distortion changes caused by the basic diopter and those caused by the additional diopter. First, a simulation analysis is
carried out to calculate the distortion value of the spectacle lens-eye system, and the near, distant, and progressive corridor
areas are set with a basic diopter and an additional diopter in the range from +2.5 D to -2.5 D. Because there is a large difference in the imaging quality between the optical center and edge of the lens, after changing the simulation of the
relationship between the diopter and distortion, the angle of the human eye model is adjusted and the lens is simulated in the near and progressive corridor areas. The distortion of the field of view is changed from -10° to 10°.
4. Distortion measurement and analysis As a precise optical imaging instrument, the human eye is very complicated in its refractive system. In order to complete the relevant experimental and imaging analysis, we choose an optical lens to replace the human eye, termed the human
eye lens [17]. The human eye lens is selected according to the necessary constraints (diopter, field of view); in the spectacle
lens-eye system, the total diopter of the normal human eye is approximately 60 D, the overall diopter range is 50–70 D,
and the angle of view is 30–100°. The measurements were conducted on different areas of the PAL with different designs.
The selected diopter of the external PAL is shown in Table 3. The results show the distance, the progressive corridor, and the near area. The measured human eye lens simulates human axial myopia conditions.The refracting parameter of the
PAL is represented by BASE+ADD, where BASE is the basic diopter of the distance and ADD represents the additional
diopter of the distant to near field [2]. The curvature of the front surface of the outer progressive lens continuously changes whereas the curvature of the back surface is fixed. The front surface of the lens is set to an XY polynomial surface and the back surface is set as a standard spherical surface. Table 3 Lens parameters Num
Diopter
Num
Diopter
1 2
BASE-2.5D+ADD2.5D BASE-2.0D+ADD2.0D
3 4
BASE-1.5D+ADD1.5D BASE-1.0D+ADD1.0D
We use ImageMaster HR to measure and analyze the distortion of the spectacle lens-eye optical system. ImageMaster HR is typically used for the measurement of the MTF, distortion, and other parameters of small-diameter lenses. It has good
accuracy and repeatability, and simple and easy operation of the software interface [16]. The on-axis measurement accuracy of ImageMaster HR is ±0.02, the off-axis measurement accuracy is ±0.03, the repeatability is ±0.01, the
measurable spatial frequency range is 0–300 lp/mm, and the maximum off-axis angle is ±90°. A schematic diagram of the
experiment is shown in Fig. 3. The distortion measurement protocol for the spectacle lens-eye system is designed
according to the simulation analysis. Firstly, the distortion of the spectacle lens-eye system is measured for the same basic
diopter and an additional diopter varying from +2.5 D to -2.5 D. Then, the angle between the light source and the human
eye lens is adjusted, and the visual state of the different field of view of the spectacle lens-eye system is simulated when measuring the relationship between the field of view and the distortion. That is, the variation in the field of view of the simulated lens between the near area, progressive corridor, and distance ranges of the angle from -10° to 10°, and the
distortion value is calculated. Finally, the experimental measurement results are compared with the simulation results to obtain the distortion relationship caused by the diopter and the field of view.
Target generator part
Collimator tube
Measured lens Connecting part Eye lens
Light source
Measured optical system
Measuring software and computer
Image analysis part CCD camera
Fig. 3 Distortion measurement experiment of the lens-eye combined optical system As shown in Fig. 3, the measurement device mainly includes the target generator component (light source, collimator), the
optical imaging system to be tested (the test PAL, connecting component, and human eye lens), the image analysis component (CCD camera), and the operation measurement software. The measurement first uses a high-performance
CCD camera in the analysis component to take a digital image of the spectacle lens-eye system in a star image, then inputs
it into a computer to compare it with a standard star image. The distortion of the spectacle lens-eye system is then obtained
by the measurement software.
The fitting curve of the experimental results of distortion after lens correction with the diopter change is shown in Fig. 4,
where the black curve represents the theoretical simulation result and the red curve represents the measurement experiment result. Figures 4(a), (b), and (c) show the result of the distant area, near area, and progressive corridor,
respectively. According to the trend of lens correction, the theoretical simulation and measurement results for the distance
area are essentially the same and the distortion value is negative because the optical distortion is caused by the distance. The size of the distortion is proportional to the basic diopter; as the basic diopter increases, the optical distortion also
increases, and the final imaging deformation is relatively large. In Fig. 4(b), the theoretical simulation results indicate that the optical distortion decreases with the increase of additional diopter, then increases after a certain point. As the
additional diopter increases, the optical distortion also increases. The measurement results are corrected as shown by the
red curve. The distortion caused by the near area is positive and is proportional to the additional diopter. With a reduction of the additional diopter, the optical distortion value also decreases. In Fig. 4(c), the theoretical simulation results show a
nonlinear relationship between the optical distortion and the diopter. After measurement, the corrected result is shown
by the red curve. In the progressive corridor, the distortion of the human eye lens after wearing the PAL is the smallest and essentially consistent.
（a）Distant
（b）Near
（c）Progressive corridor Fig. 4 Correlation results for distortion changes with the diopter of eye using PAL: (a) distant area, (b) near area, and
(c) progressive corridor
Figure 5 shows the fitting curves of the experimental results of distortion after correction with different lenses for different fields of view; (a) shows the theoretical simulation results and (b) shows the measurement results. After correction, the theoretical simulation results indicate that the optical distortions of lenses 1 and 2 are proportional to the field of view. As
the field of view increases, the optical distortion value also increases, and the deformation of the final edge portion is
relatively large. The optical distortion of lens 4 is inversely proportional to the field of view. As the field of view increases, the optical distortion value decreases. The distortion of lens 3 with respect to the field of view remains essentially constant.
Regarding the measurement results (Fig. 5b), the distortion increases as the field of view of the human eye model changes. The larger the field of view and the farther from the optical center of the lens, the more severe the change. When the lens is viewed through the object, the edge portion will cause a large distortion. The distortion of lens 4 is the smallest, whereas the distortion of lens 1 is the greatest.
（a）Theoretical simulation result
（b）Measurement result
Fig. 5 Correlation results for distortion changes with field of view using PAL: (a) theoretical simulation result and (b) measurement result
In order to visually and quantitatively evaluate the imaging quality of the ametropia model after wearing corrected PAL
1–4, we define the MTF root mean square (MRMS). First, we calculate the MRMS after wearing the PAL. The MRMS of the
human eye model after wearing the PAL correction describes the image quality intuitively and quantitatively. The larger
the MRMS value, larger is the MTF value. The clearer the final image of the human eye wearing the PAL, the better the
imaging quality of the PAL. The MRMS value is calculated by the following equation:
MRMS =
∑ (V )
2
mn
N
DRMSE = S
∑ (Y − X ) i =1
i
（16）
2
i
N
（17）
where Vnm is the MTF value in the m directions and n typical spatial frequencies; m represents the sagittal and meridian
directions, and n represents five typical spatial frequencies of 5, 10, 15, 20, and 30 lp/mm. In order to further numerically
describe the imaging quality of the PAL, we use the distortion root mean square error (DRMSE) to objectively and
numerically describe the deformation size of the image after wearing the PAL. In Eq. (16), S is a symbolic function operator, and N is the number of fields of view of each region, representing the distortion values after wearing the lens and of the
corresponding refractive parameter of the human eye lens. The smaller the DRMSE, the smaller the final imaging optical distortion of the human eye wearing the PAL, and the higher the imaging quality of the PAL. Fig.6 shows the experimental results of MRMS and DRMES after the human eye model is corrected by wearing PAL 1–4.
Value
MRMS
6 5 4 3 2 1 0 -1 -2 -3
1
2
3
distant area
4
1
DRMSE
2
3
near area
4
1
2
3
4
progressive corridor
Lens and area Fig. 6 Experimental results for progressive addition lenses 1–4
Fig.6 shows that, during the designing and processing of the PAL, distortion is introduced, especially in the near and
distance areas. Although the human eye has an automatic adaptation correction mechanism for distortion, when the distortion of the PAL is too large, the wearing comfort reduces, and the wearer's adaptation time increases. We combine the methods of MTF and distortion to comprehensively and objectively evaluate the image quality of the PAL. First, the
MRMS values are compared. The MRMS of lenses 1 and 4 are larger than that of lenses 2 and 3 in the distance; i.e., the
images are clearer for lenses 1 and 4. The MRMS values of all lenses in the near area are greater than 3.0, and lens 3 has
the clearest image. Lens 4 has the smallest MRMS value and the worst image resolution in the progressive corridor area. When the MRMS values are similar, the DRMS value is compared. In the distance, the DMRMS is larger but relatively small for lens 3. In the near area, the corresponding shape variable is larger for lens 1 but smaller for lens 2. In the progressive corridor area, the DMRMS of lens 3 is the largest. Based on the above analysis, the order of lens image quality is 1 > 4 > 3 > 2 in the distance, 3 > 2 > 4 > 1 in the near area, and 2 > 1 > 3 > 4 in the progressive corridor.
5. Conclusion
In the optical imaging system, distortion is typically introduced during the designing and processing of PAL; therefore, a
comprehensive evaluation of the PAL imaging quality and distortion is required. In this research, we built an individual human spectacle lens-eye model and evaluated the imaging quality of the PAL using the distortion value and the image quality evaluation parameter, MTF. Through theoretical simulation and experimental verification, we measured and
statistically analyzed the MTF and distortion values of the distance, near, and progressive corridor areas of the PAL. The
relationship between the distortion, refractive parameter, and field of view was evaluated for its effect on the PAL. According to a comparison between the spectacle lens-eye system and the human eye lens without PAL, we found that different diopter human eye models reveal different trends when wearing different lens designs, but the corrected human
eye model exhibits good image quality and low deformation. Distortion in the distance is negative and distortion in the near area is positive. The distortion is proportional to the diopter; as the diopter increases, the optical distortion also
increases, and the final imaging deformation is relatively large. In the progressive corridor, the human lens is corrected by
the PAL with the smallest distortion value and is essentially consistent. As the position of the human eye model changes, the distortion of the lens becomes increasingly large. The farther from the optical center of the lens, the larger the field of view, and the more severe the change; the edge portion in particular will cause a large distortion. This study describes a
method for analyzing a combination of distortion values, providing a more convenient basis for the evaluation and customization of the PAL.
Funding Information.
National Science Foundation (NSF) (6160031108),( 51875543)
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PII: DOI: Reference:
S0030-4018(19)30290-1 https://doi.org/10.1016/j.optcom.2019.03.079 OPTICS 23997
To appear in:
Optics Communications
Received date : 15 December 2018 Revised date : 26 March 2019 Accepted date : 31 March 2019 Please cite this article as: J. Yu, F. Hua and W. Jiang, Distortion evaluation method for the progressive addition lens-eye system, Optics Communications (2019), https://doi.org/10.1016/j.optcom.2019.03.079 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Distortion evaluation method for the progressive addition lens-eye system YU JING , HUA FANGFANG, JIANG WEIWEI *
School of Metrology and Measurement Engineering, China Jiliang University, ZheJiang HangZhou 310018, China *Corresponding author: [email protected] Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX
Distortion is one of the most significant defects in image optical systems. Herein, a method for comprehensively evaluating progressive addition spectacle lenses based on the modulation transfer function (MTF) and distortion of the lens-eye system is proposed. A lens-eye system is established, and the parameters of the system were simulated and measured simultaneously. The human eye model was compared with and without progressive addition lenses to obtain the image quality of the lenses. It was observed that eyes with different diopters and different lens designs show different trends; however, good improvement was observed in all cases after progressive addition lens correction. In terms of image clarity and deformation, the distant field was caused by a negative distortion value, the near area was caused by a positive distortion value. and the optical distortion was found to be proportional to the diopter. Moreover, the progressive corridor distortion value was minimal. With a larger field of view, the distortion of the lens became larger; thus, when an object was viewed through the lens, large distortion was observed at the edges .
Keywords: distortion; progressive addition lens; freeform lens; lens-eye system; MTF 1. Introduction The human eye is an important photosensitive organ. The image quality of objects in the retina directly affects people's perception of the world. The most common ophthalmological disorders include myopia, hyperopia, and astigmatism,
which results in low image quality. The traditional method of correcting ametropia is to wear corrective spectacle lenses. The quality of the lens influences the physical health, mental health, and sight quality of the wearer [1]. In recent years, with the development of freeform technology, freeform lenses have become increasingly popular, especially progressive
additional lenses (PAL). The wearer can obtain clearer and more continuous vision using PAL as they provide both near and far vision and can be customized. International standards for the measurement of the refractive parameters of
traditional lens have been established for many years. For freeform lenses, traditional methods, and equipment cannot provide comprehensive measurement, nor can the refractive parameters provide a comprehensive evaluation [2]. The
lack of standards and evaluation systems for freeform lenses has made it impossible for consumers to distinguish the
difference in imaging quality between designs, resulting in a confusing market. Thus, methods for evaluating the quality
of different modern freeform lens designs to achieve rational and efficient development of this growing market have become a key research topic.
For image optics, the modulation transfer function (MTF) is the main parameter for determining image quality, and it can be used to evaluate the quality of spectacle lenses. In recent years, major universities and optical companies and
universities such as Carl Zeiss and Tianjin university [3-4] have been working on a method for evaluating the imaging
quality of freeform spectacle lenses. They proposed evaluating the imaging quality of PAL using MTF [2], which expresses the transmit ability of optical imaging systems at different frequencies [6-7]. Distortion is common in optical image
systems and contributes significantly to MTF [8-9]. Distortion is introduced during the design and manufacture of
freeform lenses, especially at the edge of the lens. This distortion is an obvious way for the common wearer to distinguish between different spectacle lenses.
In this study, we establish a spectacle lens-eye system and propose using distortion as an image parameter for evaluating the quality of a freeform lens. In contrast to the traditional image quality evaluation method using MTF, using the root mean error of the distortion to evaluate the image quality together with other image quality parameters can be more intuitive and reasonable. Distortion of the spectacle lens-eye systems with different freeform spectacle lenses is used to
illustrate image quality. A distortion grid map and MTF curve qualitatively describe changes in the image quality. In order
to quantitatively evaluate image quality, it is necessary to calculate the root mean square error of the distortion value, compare the simulation results and experimental measurement results, and then visually and quantitatively characterize the image quality changes.
2. Spectacle Lens-eye system distortion model In the optical image system, as the intersection height of the chief ray and the Gaussian image plane is not equal to the ideal image height [10] and is often accompanied by distortion, it directly represents the maximum deformation of the entire
optical system. The spectacle lens -eye system is equivalent to an image system. The mathematical model is established
based on the rotation matrix model (RMM). As shown in Figure 1, the image distortion in the spectacle lens -eye system is established. Ow -XYZ denotes the world coordinate system (WCS) , OL -xyZ denotes the lens coordinate system (LCS), OM
-UVZ denotes the image coordinate system (MCS), where O is the lens center, and OZ is the optical axis. The point Om is at a distance f to the point O. P is a point from the WCS, which is at a distance d from point O.
Fig. 1 Image distortion in the spectacle lens- eye system.
The following transforming relationships exist in these coordinate systems [11,12]:
(1) Let (xw, yw, zw) and (x ,y, z)denotes the coordinates of the object point P expressed in WCS and LCS, respectively.
Transforming the coordinate of point P from LCS to WCS yields
xw x y R T y w = zw 0 1 z 1 1
（1）
where
R = Rot ( x, θ x ) Rot ( y, θ y ) Rot ( z , θ z )
0 1 = 0 cos θ x 0 sin θ x
0 cos θ y − sin θ x 0 cos θ x − sin θ y
0 sin θ y cos θ z 1 0 sin θ z 0 cos θ y 0
− sin θ z cos θ z 0
0 0 1
（2
） T
And T = t x , t y , t z . R and T denote the rotation matrix and translation vector from LCS to WCS, respectively, where θ x , θ y , θ z are the T
rotation angles of WCS with respect to LCS. t x , t y , t z is the coordinate of the vector
oow (d) expressed in LCS.
(2) Point Pc represents the image of point P. Let (xc yc, zc) denotes the coordinates of Point Pc expressed in LCS.
According to the pinhole imaging model, the relationship between the coordinates of point P and Pc can be expressed as
xc x s yc = y (3) zc z
where s is an arbitrary scalar related with the image system.
(3) Let (xm ym, zm) denote the coordinates of the point Pc expressed in MCS. Transforming the coordinates of point Pc
from LCS to MCS yields
xm y = RM L m 1
xc y (4) c zc
where α is the angle around the OX- axis, β is the angle around the OY- axis, and M L
R
1 0 = 0 1 0 0
0 0 1
cos β sin α sin β cos α sin β f
0 cos α − sin α
− sin β sin α cos β (5) cos α cos β
Considering the spectacle lens distortion, let point Pd (xd yd, zd), as opposed to point Pc , denote the real image of point P.
Then, the distortion model can be expressed as ;
Where,
xd + ∆xd x= c = y c yd + ∆yd (6) z = z + ∆z d d c
( (
) )
( (
∆xd = k1rd2 + k2 rd4 xd + 2 p1 xd yd + p2 rd2 + 2 xd2 2 4 2 2 ∆yd = k1rd + k2 rd yd + 2 p1 xd yd + p2 rd + 2 yd 0 ∆zd =
) )
(7)
2 and rd= xd2 + yd2 . k1 , k2 , p1 , p2 are the lens distortion coefficients.
For a PAL, the surface power can be obtained by the Gaussian curvature and mean curvature can be drawn as:
K = κ1κ 2 κ1 + κ 2 (8) H = 2
where, 𝜅𝜅1 and𝜅𝜅2 are called as the principal curvatures, and the surface power can be calculated.
In the spectacle lens -eye system. The following variables can be calculated
d O= f tan θ , = m Pc
d Pc P= f tan θ ' − f tan θ ∆= d
The radial distortion can be expressed as:
(9)
2 δ xr = k1 xd rd (10) 2 δ yr = k1 yd rd
The eccentric distortion can be expressed as:
2 2 δ x= d p1 ( 3 xd + yd ) + 2 p2 xd yd (11) 2 2 δ xd = p2 ( xd + 3 yd ) + 2 p1 xd yd
u = zc v 1
ax 0 0
0 ay 0
In summary,
u0 xw R T δ x v0 ⋅ ⋅ yw + (12) 0 1 δ y 1 zw
, ay f ax f= = x ∆ ∆y , is related to the eye, it can be calculated from the physiological parameters. where to the RMM model the distortion can be calculated and simulated.
3. Simulation of distortion correction in the spectacle lens-eye system
According
In this study, the optical design software ZEMAX is used to establish and optimize the spectacle lens-eye system. According to the physiological characteristics of the human eye, we choose the relatively common method of controlling the length
of the eye axis to simulate the refractive error of the eye. The selected individual human eye model displays axial myopia,
which is predominantly characterized by an axial refractive error. The refractive parameters are -1.0 D, -1.50 D, -2.0 D, and
-2.5 D. Liou [14] proposed a human eye model based on anatomical data, which is the most comprehensive human eye model to date. The basic parameters of the human eye model are shown in Table 1, where GradA and GradP indicate the gradient refractive indices of the first and second half of the lens, respectively [15]. Table 1 Parameters of the human eye model Surface
Radius (mm)
Thickness (mm)
Conic constant
Refractive index
aqueous
6.40
3.16
-0.60
1.336
cornea
pupil
anterior crystalline lens
7.77
Infinity 12.40
0.55 0.00 1.59
-0.18 0.00 0.00
1.376 1.336
GradA
Abbe number 50.23
posterior lens
crystalline
vitreous
Infinity
2.43
-8.10
retina
0.00
16.239
-12.0
GradP
0.96
-
1.336
-
-
-
The near-point distance visible to the human eye is calculated according to the refractive parameter of the eye, the object
distance is set to the near-point distance visible to the human eye in ZEMAX, and the distance of the length of the eye axis
is then optimized. Thus, the axial distance of the refractive parameter of the human-eye is obtained. The parameters for
simulating the refractivity of the human eye by controlling the length of the eye axis are listed in Table 2. Table 2 Corresponding distances of the eye axis Corresponding Diopter (D) (mm)
distances
Diopter (D)
Corresponding distances (mm)
-1.00
16.556
2.00
15.465
-2.00
16.924
1.00
15.816
-1.50
16.741
-2.50
1.50
17.109
15.644
0.50
16.001
The surface of PAL is non-rotationally symmetric, which is represented by the commonly used mathematical models
Zernike polynomial and XY polynomial. The XY polynomial is the typical mathematical model used to describe freeform
surfaces [12]. The described non-rotational symmetrical freeform surfaces are widely used in optical design. The XY
polynomial surface is used in the optical design software ZEMAX to describe the PAL. The freeform surface described by the XY polynomial is:
Z =
cr 2
N
1 + 1 − (1 + k ) c 2 r 2
+ ∑ Ci x m y n i=2
（13）
where c is the curvature of the basal plane (mm-1), k is the quadric surface coefficient, and r is the radial pupil coordinate.
N represents the total number of polynomial coefficients in the series, Ci represents the coefficient (mm) of the polynomial of the i-th term, m represents a power series in the x direction, and n represents a power series in the y direction. Equation
(3) is composed of two main parts, the first part predominantly describes the conic surface and the second part describes the freeform surface type described by the XY polynomial in the Cartesian coordinate system [16]. In ZEMAX, the freeform surface is usually described by an XY polynomial using ten steps, where
i=
(( m + n ) 1+ N
y ∑C x = i=2
i
m
n
2
+ m + 3n
)
2
C2 x1 y 0 + C3 x 0 y1 + + C66 x 0 y10
（14）
（15）
The PAL are evaluated by comparing and analyzing the imaging quality of the human eye model and the spectacle lenseye system. Taking ametropia of -1.0 D as an example, the MTF curve of the lens-eye system is shown in Fig. 2, where the
MTF curve of the ametropia model corrected without the corresponding PAL is as follows. As shown in Fig. 2(a), according to the human eye characteristics, the cutoff frequency of the human eye can reach approximately 20–30 lp/mm and the
human eye resolution limit MTF value is 0.2. However, in Fig. 2(a), all the views are 30 lp/mm and the MTF value is
approximately 0.1, which does not reach the human eye resolution limit. Therefore, the image quality of the human eye model is poor and should be corrected by the PAL. The MTF curve of the spectacle lens-eye system after the PAL correction
is shown in Fig. 2(b), in which all the field-of-view MTF values at a spatial frequency of 30 lp/mm are greater than 0.2. The resolution limit of the human eye is reached, so the imaging quality of the eye whose ametropia is corrected by wearing the corresponding progressive multifocal lens is further improved.
（a）
（b）
Fig. 2 MTF curve of the lens-eye combined optical system: (a) eye model and (b) spectacle lens-eye system
This result shows that the image quality of the human eye model with ametropia exhibits improved correction with
the corresponding PAL. However, distortion can be introduced during the design and processing of the PAL, especially at the edge of the lens, which may cause the image quality of the lens to decrease. Comprehensive evaluation of the imaging quality of the PAL must be discussed along with lens distortion.
When analyzing the influence of distortion on the imaging quality of the PAL, it is necessary to separately discuss the distortion changes caused by the basic diopter and those caused by the additional diopter. First, a simulation analysis is
carried out to calculate the distortion value of the spectacle lens-eye system, and the near, distant, and progressive corridor
areas are set with a basic diopter and an additional diopter in the range from +2.5 D to -2.5 D. Because there is a large difference in the imaging quality between the optical center and edge of the lens, after changing the simulation of the
relationship between the diopter and distortion, the angle of the human eye model is adjusted and the lens is simulated in the near and progressive corridor areas. The distortion of the field of view is changed from -10° to 10°.
4. Distortion measurement and analysis As a precise optical imaging instrument, the human eye is very complicated in its refractive system. In order to complete the relevant experimental and imaging analysis, we choose an optical lens to replace the human eye, termed the human
eye lens [17]. The human eye lens is selected according to the necessary constraints (diopter, field of view); in the spectacle
lens-eye system, the total diopter of the normal human eye is approximately 60 D, the overall diopter range is 50–70 D,
and the angle of view is 30–100°. The measurements were conducted on different areas of the PAL with different designs.
The selected diopter of the external PAL is shown in Table 3. The results show the distance, the progressive corridor, and the near area. The measured human eye lens simulates human axial myopia conditions.The refracting parameter of the
PAL is represented by BASE+ADD, where BASE is the basic diopter of the distance and ADD represents the additional
diopter of the distant to near field [2]. The curvature of the front surface of the outer progressive lens continuously changes whereas the curvature of the back surface is fixed. The front surface of the lens is set to an XY polynomial surface and the back surface is set as a standard spherical surface. Table 3 Lens parameters Num
Diopter
Num
Diopter
1 2
BASE-2.5D+ADD2.5D BASE-2.0D+ADD2.0D
3 4
BASE-1.5D+ADD1.5D BASE-1.0D+ADD1.0D
We use ImageMaster HR to measure and analyze the distortion of the spectacle lens-eye optical system. ImageMaster HR is typically used for the measurement of the MTF, distortion, and other parameters of small-diameter lenses. It has good
accuracy and repeatability, and simple and easy operation of the software interface [16]. The on-axis measurement accuracy of ImageMaster HR is ±0.02, the off-axis measurement accuracy is ±0.03, the repeatability is ±0.01, the
measurable spatial frequency range is 0–300 lp/mm, and the maximum off-axis angle is ±90°. A schematic diagram of the
experiment is shown in Fig. 3. The distortion measurement protocol for the spectacle lens-eye system is designed
according to the simulation analysis. Firstly, the distortion of the spectacle lens-eye system is measured for the same basic
diopter and an additional diopter varying from +2.5 D to -2.5 D. Then, the angle between the light source and the human
eye lens is adjusted, and the visual state of the different field of view of the spectacle lens-eye system is simulated when measuring the relationship between the field of view and the distortion. That is, the variation in the field of view of the simulated lens between the near area, progressive corridor, and distance ranges of the angle from -10° to 10°, and the
distortion value is calculated. Finally, the experimental measurement results are compared with the simulation results to obtain the distortion relationship caused by the diopter and the field of view.
Target generator part
Collimator tube
Measured lens Connecting part Eye lens
Light source
Measured optical system
Measuring software and computer
Image analysis part CCD camera
Fig. 3 Distortion measurement experiment of the lens-eye combined optical system As shown in Fig. 3, the measurement device mainly includes the target generator component (light source, collimator), the
optical imaging system to be tested (the test PAL, connecting component, and human eye lens), the image analysis component (CCD camera), and the operation measurement software. The measurement first uses a high-performance
CCD camera in the analysis component to take a digital image of the spectacle lens-eye system in a star image, then inputs
it into a computer to compare it with a standard star image. The distortion of the spectacle lens-eye system is then obtained
by the measurement software.
The fitting curve of the experimental results of distortion after lens correction with the diopter change is shown in Fig. 4,
where the black curve represents the theoretical simulation result and the red curve represents the measurement experiment result. Figures 4(a), (b), and (c) show the result of the distant area, near area, and progressive corridor,
respectively. According to the trend of lens correction, the theoretical simulation and measurement results for the distance
area are essentially the same and the distortion value is negative because the optical distortion is caused by the distance. The size of the distortion is proportional to the basic diopter; as the basic diopter increases, the optical distortion also
increases, and the final imaging deformation is relatively large. In Fig. 4(b), the theoretical simulation results indicate that the optical distortion decreases with the increase of additional diopter, then increases after a certain point. As the
additional diopter increases, the optical distortion also increases. The measurement results are corrected as shown by the
red curve. The distortion caused by the near area is positive and is proportional to the additional diopter. With a reduction of the additional diopter, the optical distortion value also decreases. In Fig. 4(c), the theoretical simulation results show a
nonlinear relationship between the optical distortion and the diopter. After measurement, the corrected result is shown
by the red curve. In the progressive corridor, the distortion of the human eye lens after wearing the PAL is the smallest and essentially consistent.
（a）Distant
（b）Near
（c）Progressive corridor Fig. 4 Correlation results for distortion changes with the diopter of eye using PAL: (a) distant area, (b) near area, and
(c) progressive corridor
Figure 5 shows the fitting curves of the experimental results of distortion after correction with different lenses for different fields of view; (a) shows the theoretical simulation results and (b) shows the measurement results. After correction, the theoretical simulation results indicate that the optical distortions of lenses 1 and 2 are proportional to the field of view. As
the field of view increases, the optical distortion value also increases, and the deformation of the final edge portion is
relatively large. The optical distortion of lens 4 is inversely proportional to the field of view. As the field of view increases, the optical distortion value decreases. The distortion of lens 3 with respect to the field of view remains essentially constant.
Regarding the measurement results (Fig. 5b), the distortion increases as the field of view of the human eye model changes. The larger the field of view and the farther from the optical center of the lens, the more severe the change. When the lens is viewed through the object, the edge portion will cause a large distortion. The distortion of lens 4 is the smallest, whereas the distortion of lens 1 is the greatest.
（a）Theoretical simulation result
（b）Measurement result
Fig. 5 Correlation results for distortion changes with field of view using PAL: (a) theoretical simulation result and (b) measurement result
In order to visually and quantitatively evaluate the imaging quality of the ametropia model after wearing corrected PAL
1–4, we define the MTF root mean square (MRMS). First, we calculate the MRMS after wearing the PAL. The MRMS of the
human eye model after wearing the PAL correction describes the image quality intuitively and quantitatively. The larger
the MRMS value, larger is the MTF value. The clearer the final image of the human eye wearing the PAL, the better the
imaging quality of the PAL. The MRMS value is calculated by the following equation:
MRMS =
∑ (V )
2
mn
N
DRMSE = S
∑ (Y − X ) i =1
i
（16）
2
i
N
（17）
where Vnm is the MTF value in the m directions and n typical spatial frequencies; m represents the sagittal and meridian
directions, and n represents five typical spatial frequencies of 5, 10, 15, 20, and 30 lp/mm. In order to further numerically
describe the imaging quality of the PAL, we use the distortion root mean square error (DRMSE) to objectively and
numerically describe the deformation size of the image after wearing the PAL. In Eq. (16), S is a symbolic function operator, and N is the number of fields of view of each region, representing the distortion values after wearing the lens and of the
corresponding refractive parameter of the human eye lens. The smaller the DRMSE, the smaller the final imaging optical distortion of the human eye wearing the PAL, and the higher the imaging quality of the PAL. Fig.6 shows the experimental results of MRMS and DRMES after the human eye model is corrected by wearing PAL 1–4.
Value
MRMS
6 5 4 3 2 1 0 -1 -2 -3
1
2
3
distant area
4
1
DRMSE
2
3
near area
4
1
2
3
4
progressive corridor
Lens and area Fig. 6 Experimental results for progressive addition lenses 1–4
Fig.6 shows that, during the designing and processing of the PAL, distortion is introduced, especially in the near and
distance areas. Although the human eye has an automatic adaptation correction mechanism for distortion, when the distortion of the PAL is too large, the wearing comfort reduces, and the wearer's adaptation time increases. We combine the methods of MTF and distortion to comprehensively and objectively evaluate the image quality of the PAL. First, the
MRMS values are compared. The MRMS of lenses 1 and 4 are larger than that of lenses 2 and 3 in the distance; i.e., the
images are clearer for lenses 1 and 4. The MRMS values of all lenses in the near area are greater than 3.0, and lens 3 has
the clearest image. Lens 4 has the smallest MRMS value and the worst image resolution in the progressive corridor area. When the MRMS values are similar, the DRMS value is compared. In the distance, the DMRMS is larger but relatively small for lens 3. In the near area, the corresponding shape variable is larger for lens 1 but smaller for lens 2. In the progressive corridor area, the DMRMS of lens 3 is the largest. Based on the above analysis, the order of lens image quality is 1 > 4 > 3 > 2 in the distance, 3 > 2 > 4 > 1 in the near area, and 2 > 1 > 3 > 4 in the progressive corridor.
5. Conclusion
In the optical imaging system, distortion is typically introduced during the designing and processing of PAL; therefore, a
comprehensive evaluation of the PAL imaging quality and distortion is required. In this research, we built an individual human spectacle lens-eye model and evaluated the imaging quality of the PAL using the distortion value and the image quality evaluation parameter, MTF. Through theoretical simulation and experimental verification, we measured and
statistically analyzed the MTF and distortion values of the distance, near, and progressive corridor areas of the PAL. The
relationship between the distortion, refractive parameter, and field of view was evaluated for its effect on the PAL. According to a comparison between the spectacle lens-eye system and the human eye lens without PAL, we found that different diopter human eye models reveal different trends when wearing different lens designs, but the corrected human
eye model exhibits good image quality and low deformation. Distortion in the distance is negative and distortion in the near area is positive. The distortion is proportional to the diopter; as the diopter increases, the optical distortion also
increases, and the final imaging deformation is relatively large. In the progressive corridor, the human lens is corrected by
the PAL with the smallest distortion value and is essentially consistent. As the position of the human eye model changes, the distortion of the lens becomes increasingly large. The farther from the optical center of the lens, the larger the field of view, and the more severe the change; the edge portion in particular will cause a large distortion. This study describes a
method for analyzing a combination of distortion values, providing a more convenient basis for the evaluation and customization of the PAL.
Funding Information.
National Science Foundation (NSF) (6160031108),( 51875543)
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