Non-uniformity correction of wide field of view imaging system

Yiqun Ji; Yiqun Ji; Yiqun Ji; Chenxin Zeng; Chenxin Zeng; Chenxin Zeng; Fenli Tan; Fenli Tan; Fenli Tan; Anwei Feng; Anwei Feng; Anwei Feng; Jizhou Han; Jizhou Han; Jizhou Han

doi:10.1364/OE.458180

1. Introduction

Attaining a wide field of view (FOV) while retaining both uniform illumination and good image quality is a longstanding goal pursued by researchers in developing wide-FOV optical systems [1–7]. On one hand, the geometric aberrations dramatically increase with the increasing of FOV in optical systems, which contributes to non-uniformity of image quality. For example, fisheye lenses have long been used to achieve wide-FOV imaging, however, distortion and other aberrations severely limit the uniformity of image quality [8–10]. On the other hand, as is known to all that the irradiance of optical systems drop-off in accordance with the cosine-fourth of the field angle on the image plane [11,12]. As the FOV increases, the optical systems suffer a certain amount of illumination loss. Especially in wide-FOV endoscopes, the relative illumination at the edge FOV decreases to 0.6, compared to that at the central FOV [13,14].

To solve both the above issues, the MCIS is proposed. The MCIS divides the imaging task between a shared monocentric objective and a parallel array of relay imagers [15,16]. As the aberrations of the monocentric objective are field-angle-independent, a wide FOV spherical intermediate image with uniform residual aberrations is obtained. Then relay imagers convey sub-images with overlapping sections after sophisticated residual aberrations correction to their respective sensors. A wide-FOV image is obtained by stitching the sub-images. When the aperture stop lies in the center of the monocentric objective, the illumination value of each relay imager decreases dramatically with the increasing of FOV. To maintain a constant peak illumination for each relay imager, the aperture stop is designed at a certain optical surface of the relay imager [17]. The MCIS achieves uniform image quality and illumination during a full FOV range.

However, non-uniformity still happens in each relay imager. On one side, in order to obtain overlapping FOV for sub-image stitching, vignetting occurs at the entrance lens of each relay imager when adjacent relay imagers may not physically overlap. The AWARE-2 camera is a typical MCIS with an FOV exceeding 120° and an instantaneous FOV of $8 ^{\prime \prime}$ [18]. Because of vignetting, reduced illumination occurs in the overlapping regions and contributes to dissatisfactory performance in 20% of the external FOV. Similar issue exists in another MCIS, the AWARE-10 [19]. Due to vignetting, relative illumination of relay imager at the edge FOV decreases to 0.6, also the MTF curves drop almost to 0.15 at the Nyquist frequency. On the other side, because the aperture stop lies within each relay imager rather than the center of the monocentric objective, the ray path is not in an axial symmetrical form in the monocentric objective when it comes to the marginal FOV [20]. Asymmetric ray paths create the local aberrations on the spherical intermediate surface, which affects the imaging quality seriously. As is reported on a compact MCIS [21], because of the local aberrations, the spot diagram shows that the geometric spot radius of the marginal FOV attains 16 µm while the geometric spot radius of central FOV is only 1 µm. Similarly, in a report of a foveated MCIS, the MTF @30 lp/mm of the marginal FOV is less than half of that at the central FOV [22].

We verified the two issues of non-uniformity by optical imaging experiment with our developed prototype as shown in Fig. 1(a). The prototype is with 7 side-by-side channels and is designed to achieve a FOV of 33.11°. Figure 1(b) shows two sub-images of the clock tower formed by adjacent relay imagers. Obviously, under-filling and reduced resolution occurred near the overlapping region. Especially in the marginal FOV, the details of the window of the clock tower are hard to distinguish. As the result of non-uniformity, overlapping regions are highly noisy, which is also a challenge for the stitching process [23].

Fig. 1. Imaging experiment of the MCIS: (a)picture of the MCIS, (b)two sub-images formed by adjacent relay imagers.

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So far, there are few studies on non-uniformity correction of the MCIS in detail. The purpose of this study is to correct the non-uniformity of illumination and image quality for each relay imager. Firstly, we analyzed the cause of vignetting and improved vignetting by moving the aperture stop towards object space. However, the local aberrations further increased. Then, based on aberration theory, we discussed the relationship between the position of aperture stop and local aberrations on the spherical intermediate image. To balance the local aberrations, aspheric surfaces were introduced in the relay imager while the relationship between aberrations and the position of aspheric surface was studied. The performance evaluation shows that across the entire FOV, the proposed MCIS achieves a relative illumination exceeding 97% while the MTF is over 0.285 at the Nyquist frequency of 270 lp/mm.

This study is organized as follows. In Section 2, how the position of the aperture stop affects the vignetting and the local aberrations is studied. Moving laws of the aperture stop and its relationship with the local aberrations are presented. Further, to balance the local aberrations, the position of the aspheric surfaces is studied. In Section 3, an aspheric MCIS with non-uniformity correction is shown and the optical performance is evaluated. Then, the conclusion is presented in Section 4.

2. Analytical design

2.1. Non-uniformity correction of illumination

For sub-image stitching, adjacent relay imagers are required to be with overlaps of FOV. As it is shown in Fig. 2, the quantity β is half- FOV height of each relay imager. The half cone angle of single relay imager θ, is defined as the vertex angle of the minimum size cone needed to contain the relay imager optics with its vertex at the center of spherical intermediate image. Δ represents the overlapping area of FOV between adjacent relay imagers, which is imaged on the detectors of two relay imagers simultaneously. The focal length of monocentric objective is f_O, which is also the radius of the spherical intermediate image. When adjacent relay imagers are placed as close as possible, the overlapping area Δ and the ideal field overlap rate α are given by:

(1)$$\Delta \textrm{ = }2(\beta - \theta \cdot {f_O}),$$

(2)$$\alpha = \frac{{\beta - \theta \cdot {f_O}}}{\beta }.$$

Fig. 2. schematic of overlapping FOV between two adjacent relay imagers.

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Therefore, in order to make overlapping region of FOV between the adjacent relay imagers, β is greater than θ·f_O.

However, the packing on a sphere is not uniform, not all relay imagers can be packed as close as possible, when this irregularity is divided between two adjacent relay imagers, the minimum half FOV of the relay imager β is required to be greater than 1.261θ·f_O [24,25]. Thus, the ideal field overlap rate α should be greater than 20.7%. Unfortunately, in order to achieve enough field overlap, vignetting occurs in relay imager. Figure 3(a) shows the vignetting that occurs in relay imagers. The simplified relay imager consists of a fore lens and a rear lens and the entrance space focal point is on the spherical intermediate image. Obviously, imaging rays of the off-axis fields cannot be fully collected by the fore lens. As is shown in Fig. 3(b), as the aperture stop moves towards the object space, the vignetting is improved.

Fig. 3. Schematic of vignetting in relay imager: (a) the aperture stop is close to the rear lens, (b) the aperture stop is moved towards the object space

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Figure 4 shows the fraction of unvignetted rays and MTF curves of image on the sensor. As Fig. 4(a) and Fig. 4(b) show, by moving the aperture stop towards the object space, the vignetting is improved. However, as Fig. 4(c) and Fig. 4(d) show, after moving the aperture stop forward, the tangential image quality further decreases.

Fig. 4. Evaluation of the simplified relay imager: (a) the fraction of unvignetted rays before moving the aperture stop, (b) the fraction of unvignetted rays after moving the aperture stop, (c) MTF curves before moving the aperture stop, (d) MTF curves after moving the aperture stop forward

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2.2. Non-uniformity correction of image quality

Aiming at the problem of image quality deterioration, we study the relationship between the position of aperture stop and the local aberrations. Moreover, aspheric surfaces are introduced to balance the local aberrations.

The monocentric objective is a spherical lens with all surfaces sharing the same center of curvature, and forms a spherical intermediate image with aberrations independent of the field angle. Off-axis aberrations are minimal when aperture stop lies at the center of the monocentric objective. However, in the MCIS, the aperture stop lies at a certain surface of relay imagers rather than the monocentric objective, which contributes to the local aberrations on the spherical intermediate surface.

Figure 5 shows the change of chief rays when moving the aperture stop. The original aperture stop is located at P while the new aperture stop is located at P*. At the entrance surface, the incident height of original chief ray of object point B is h_z while the incident height of new chief ray is h_z*. h is the incident height of edge ray of on-axis object point A. There is a scale coefficient K given by Eq. (3)

(3)$$K = \frac{{{h_z}^\ast{-} {h_\textrm{z}}}}{h}.$$

Based on aberration theory [26], K is a constant factor for whole system. For the monocentric objective, when the position of aperture stop changes, the change of primary aberrations can be expressed by Eq. (4):

(4)$$\left. \begin{array}{l} \sum {S_I^\ast } = \sum {S_I^{}} \\ \sum {S_{II }^\ast } = {K_\textrm{m}}\sum {S_I^{}} \\ \sum {S_{III }^\ast } = {K_\textrm{m}}^2\sum {S_I^{}} \\ \sum {S_{IV }^\ast } = \sum {S_{IV }^{}} \\ \sum {S_V^\ast } = {K_\textrm{m}}^3\sum {S_I^{}} + {K_\textrm{m}}\sum {S_{IV }^{}} + \sum {S_V^{}} \end{array} \right\}.$$

where S_I, S_II, S_III, S_IV, S_V are the seidel aberration coefficients of spherical aberration, coma, astigmatism, field curvature, and distortion respectively when the aperture stop lies at the original position. S*_I, S*_II, S*_III, S*_IV, S*_V are the seidel aberration coefficients when the aperture stop lies at the new position. K_m is the scale coefficient of the monocentric objective.

Fig. 5. Schematic of the incident height of chief ray when the position of aperture stop changes.

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When the aperture stop lies within the center of the monocentric objective, as shown in Fig. 6, the incident height of h_zm is expressed as:

(5)$${h_{\textrm{zm}}} ={-} R \cdot \sin (w),$$

where R is the radius of the monocentric objective and w is the half-FOV angle of MCIS in one channel.

Fig. 6. Schematic of the chief ray when the aperture stop lies at the center of the monocentric objective.

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As shown in Fig. 7, when the aperture stop is designed within the relay imager, the chief ray is refracted at position I and position E at the spherical surface of the objective, respectively. According to the first-order principle, the distance between the entrance pupil and the monocentric objective s can be expressed by:

(6)$$s = \frac{{f_O^2 - l \cdot \frac{{f_O^2}}{{{f_f}}}}}{{{f_f}}} + {f_O} - R.$$

where f_O and f_f are the focal length of the monocentric objective and the fore lens respectively. From geometrical considerations, when the aperture stop lies in the relay imager, the incident height of chief ray h_zm* is given by:

(7)$${h_{\textrm{zm}}}^\ast{=} \frac{R}{{\tan w}} - \sqrt {\frac{{{R^2}}}{{{{\tan }^2}w}} - 2sR} .$$

Fig. 7. Schematic of the chief ray when the aperture stop lies within the relay imager

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According to Eq. (3), (5), (6) and (7), the aberration coefficient K_m of the monocentric objective is deduced as follow:

(8)$${K_\textrm{m}} = \frac{{{h_{\textrm{zm}}}^\ast{-} {h_{\textrm{zm}}}}}{{{h_\textrm{m}}}}\textrm{ = }\frac{{\frac{R}{{\tan w}} - \sqrt {\frac{{{R^2}}}{{{{\tan }^2}w}} - 2R(\frac{{f_O^2 - l \cdot \frac{{f_O^2}}{{{f_f}}}}}{{{f_f}}} + {f_O} - R)} \textrm{ + }R \cdot \sin (w)}}{{{h_\textrm{m}}}}.$$

where h_m is the incident height of axial ray. According to Eq. (8), the relationship between the aberration coefficient K_m and the position of aperture stop is clear. For a prescription, assume that f_O of 100 mm, f_f of 30 mm, R of 69 mm, w of 3° and h_m of 8.6 mm, the relationship between the aberration coefficient K_m and the distance l is shown as Fig. 8. When the aperture stop is moved towards the object space, l decreases while K_m increases. Furthermore, according to Eq. (4), the local aberrations including coma, astigmatism and distortion increase.

Fig. 8. The relationship between K_m and l in the prescription.

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To balance the local aberrations, aspherical surfaces are introduced in relay imager [27]. According to the primary aberration contributions of aspheric surface, the relationship between aberrations and the position of aspheric surface is discussed.

The aspheric form follows the Eq. (9) [28]:

(9)$$Z\textrm{ = }\frac{{cr}}{{1 + \sqrt {1 - (1 + k){c^2}{r^2}} }} + {a_4}{r^4} + {a_6}{r^6} + {a_8}{r^8} +{\cdot}{\cdot} \cdot{+} {a_n}{r^n}.$$

where Z is the sag of the surface, c is the vertex curvature of the surface, r represents the r coordinate at the incident point, a_n is the nth-order aspheric coefficient and k denotes the conic constant. The primary aberration contributions of aspheric surface are written as [26]:

(10)$$\left. \begin{array}{l} \Delta {S_I } = k{c^3}(n^{\prime} - n){h_a}^4\\ \Delta {S_{II }} = \Delta {S_\textrm{I}}( \frac{{{h_c}}}{{{h_a}}}) \\ \Delta {S_{III }} = \Delta {S_\textrm{I}}{(\frac{{{h_c}}}{{{h_a}}})^2}\\ \Delta {S_{IV }} = 0\\ \Delta {S_V } = \Delta {S_\textrm{I}}{(\frac{{{h_c}}}{{{h_a}}})^3} \end{array} \right\}.$$

In Eq. (10), h_a is the height that the axial ray arrives on the surface and h_c is the height that the chief ray of the edge FOV arrives. Also, n and n’ represent the refractive index of object space and image space respectively.

There are two factors that determine the primary aberration contributions of aspheric surface, including ΔS_I and the ratio of the chief ray height to the axial ray height at, h_c/h_a. ΔS_I mainly depends on the shape of aspherical surface, refractive index and axial ray height while h_c/h_a mainly depends on the the position of the aspheric surface. When:

(11)$$\left|{\frac{{{h_c}}}{{{h_a}}}} \right|\gg 1,$$

It’s obviously that

(12)$$\left. \begin{array}{l} {\left|{\frac{{{h_c}}}{{{h_a}}}} \right|^2} \gg \left|{\frac{{{h_c}}}{{{h_a}}}} \right|\\ {\left|{\frac{{{h_c}}}{{{h_a}}}} \right|^3} \gg \left|{\frac{{{h_c}}}{{{h_a}}}} \right|\end{array} \right\}.$$

On the contrary, when:

(13)$$\left|{\frac{{{h_c}}}{{{h_a}}}} \right|\ll 1,$$

There is

(14)$$\left. \begin{array}{l} {\left|{\frac{{{h_c}}}{{{h_a}}}} \right|^2} \ll \left|{\frac{{{h_c}}}{{{h_a}}}} \right|\\ {\left|{\frac{{{h_c}}}{{{h_a}}}} \right|^3} \ll \left|{\frac{{{h_c}}}{{{h_a}}}} \right|\end{array} \right\}.$$

According to the discussion above, when the absolute value of h_c/h_a is much bigger than 1, the main aberration contributions of aspheric surfaces are ΔS_III and ΔS_V. On the contrary, when the absolute value of h_c/h_a is much less than 1, the main aberration contributions are ΔS_I and ΔS_II. By introducing aspheric surfaces at the appropriate optical surfaces of the relay imager, local aberrations can be balanced.

3. Optimum design

To correct the field curvature of the spherical intermediate image, Petzval lens is selected as the initial structure of the relay imagers [29]. According to the discussion in section 2, the aperture stop is designed in the middle position of the relay imager for achieving a good balance between vignetting and local aberrations. Further, a MCIS with uniform illumination and excellent image quality is achieved [30]. In order to reduce the number of the optical/mechanical elements of the relay imager and the tolerance sensitivity, the aperture stop is designed on an optical surface rather than a separate position.

As the optimized relay imager shown in Fig. 9, the axial ray is blue while the chief ray is yellow. For balancing the spherical aberration and coma on the spherical intermediate image, one aspherical surface is introduced on the 15th surface where the absolute value of h_c/h_a is less than 1. For balancing the astigmatism and distortion, another aspherical surface is introduced on the 17th surface where the absolute value of h_c/h_a is greater than 1.

Fig. 9. The axial ray and the chief ray in the optimized relay imager.

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The optical layout of the MCIS is shown in Fig. 10. Figure 10(a) shows the coaxial structure with the tube length of 230.32 mm. Adjacent relay imagers and overlapping area Δ are shown in Fig. 10(b). In order to satisfy the mechanical packing requirements, the distance between the front lenes of adjacent relay imagers is designed to 2 mm. The overlap rate is 21.875%, which meets the requirement for sub-image stitching [24,25]. Figure 10(c) shows the MCIS with full channels. 23 channels are designed to achieve a wide-FOV of 116.4°.

Fig. 10. Optical layout of the MCIS: (a) coaxial optical layout; (b) adjacent relay imagers (c) full channels optical layout.

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The design specifications of the MCIS are shown in Table 1. Lens data of the MCIS is presented in Table 2 and the supplier is CDGM. The MCIS can also be achieved by the other glass catalogues, for example, Scott catalogue. The coefficients of aspheric surfaces are presented in Table 3.

Table 1. Design Specifications

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Table 2. Lens Data of the aspherical MCIS

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Table 3. Aspheric Coefficients for the relay imager

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The performance evaluation results are provided in Fig. 11. As Fig. 11(a) shows, the maximum RMS radius of the diffuse spot is about 1.645 µm, which is smaller than the airy disk radius. The sensor selected is Sony IMX226 sensor with a pixel size of 1.85 µm, which achieves an instantaneous FOV of 0.0021°, the total number of non-redundant illuminated pixels of 6.34 million and a number of pixels in the overlap regions of 0.86 million between two neighboring relay imagers. The MTF shown in Fig. 11(b) is over 0.285 at the Nyquist frequency (270 lp/mm), which is close to diffraction limit. Figure 11(c) shows that the field curve is less than 30 µm while the distortion is less than 0.2%. Because of no vignetting, as is shown in Fig. 11(d), the relative illumination is higher than 97%. The evaluation shows that after the non-uniformity correction, the MCIS achieves both uniform illumination and excellent image quality across the entire FOV.

Fig. 11. Evaluation results of the MCIS: (a) spot diagram; (b) MTF curves; (c)field curve/distortion; (d)relative illumination.

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4. Conclusion

In response to the demand for both uniform illumination and excellent image quality in wide FOV imaging, we proposed a wide-FOV monocentric cascade imaging system. We studied how the position of aperture stop affects the vignetting and local aberrations in MCIS. Uniform illumination is achieved by moving the aperture stop towards object space while the local aberrations are further balanced by introducing aspherical surfaces on proper surfaces. The optimized MCIS achieves a relative illumination exceeding 97% and an instantaneous FOV of 0.0021°. The MTF value is over 0.285 at 270 lp/mm across the full 116.4° FOV. Good uniformity not only improves the resolution, but also is conductive to image stitching for the MCIS. This study will serve as a useful theorical reference to guide future research on MCIS design.

Funding

National Natural Science Foundation of China (61340007, 61405134); National Defense Basic Scientific Research Program of China (JCKY2018414C013); Natural Science Foundation of Jiangsu Province (BK20161512); Priority Academic Program Development of Jiangsu Higher Education Institutions.

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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Item	Specifications
FOV of single channel	6.4°
FOV of full channels	116.4°
Number of sensors	23
Field overlap rate for adjacent channels	21.875%
Working F-number	3.1
Focal length	50 mm
Wavelengths range	486 nm ∼ 656 nm
Distortion	<0.2%
Relative illumination	>0.95

Radius(mm)	Thickness(mm)	Glass	Comment
52.6066	23.5973	H-ZLAF52	Objective start
29.0093	29.0093	H-FK61
infinity	29.0093	H-FK61
-29.0093	23.5973	H-ZLAF52
-52.6066	57.4543		Objective end
-110.0609	31.8580		Spherical intermediate image
21.2112	1.7385	H-ZLAF2A	Relay imager start
-113.9010	3.6796
12.0455	3.1753	H-ZK10L
-20.4369	1.0797	H-TF8
20.1817	1.6807
-8.2905	3.8398	H-ZF6
17.9586	0.3569
14.7156	4.8886	H-ZK10L	Aperture stop
-7.6909	4.1761
7.1457	6.2124	H-LAK8B
4.9170	4.9682		Relay imager end
Infinity

Surface	Conic Constant	Fourth	Sixth	Eighth	Tenth	Twelfth
15	-1.2416	0	-8.6279E-6	1.2782E-6	-6.4938E-8	0
17	0.0713	0	-1.1594E-4	1.4790E-5	-1.7103E-6	0

Item	Specifications
FOV of single channel	6.4°
FOV of full channels	116.4°
Number of sensors	23
Field overlap rate for adjacent channels	21.875%
Working F-number	3.1
Focal length	50 mm
Wavelengths range	486 nm ∼ 656 nm
Distortion	<0.2%
Relative illumination	>0.95

Radius(mm)	Thickness(mm)	Glass	Comment
52.6066	23.5973	H-ZLAF52	Objective start
29.0093	29.0093	H-FK61
infinity	29.0093	H-FK61
-29.0093	23.5973	H-ZLAF52
-52.6066	57.4543		Objective end
-110.0609	31.8580		Spherical intermediate image
21.2112	1.7385	H-ZLAF2A	Relay imager start
-113.9010	3.6796
12.0455	3.1753	H-ZK10L
-20.4369	1.0797	H-TF8
20.1817	1.6807
-8.2905	3.8398	H-ZF6
17.9586	0.3569
14.7156	4.8886	H-ZK10L	Aperture stop
-7.6909	4.1761
7.1457	6.2124	H-LAK8B
4.9170	4.9682		Relay imager end
Infinity

Non-uniformity correction of wide field of view imaging system

Abstract

1. Introduction

2. Analytical design

2.1. Non-uniformity correction of illumination

2.2. Non-uniformity correction of image quality

3. Optimum design

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (11)

Tables (3)

Equations (14)

Optics Express