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We present an ultrawide angle catadioptric lens with a field of view (FOV) of 360° × 270° and F/# of 2.5. The lens consists of two optical configurations: the center configuration is all-refractive and has a FOV of ± 50° and the catadioptric configuration covers the remaining FOV. The MTF at 119lp/mm of the rear FOV (90° to 135°) can be improved by 0.15 via applying an annularly stitched aspherical surface (ASAS) to the rear surface of the catadioptric element. The developed lens presents smaller marginal distortions and higher relative illuminations compared with traditional panoramic lenses. A proof-of-concept prototype producing acceptable image quality is developed.
© 2016 Optical Society of America
1. Introduction
Sports cameras, driverless vehicles, and virtual reality (VR) and surveillance systems often include ultrawide angle systems. The advantages of these systems include full horizon coverage with fewer cameras, simultaneous tracking of multiple targets, and high immersion ability for VR application [1, 2 ]. Google, for example, recently announced an autonomous vehicle project [3]. Driverless cars are equipped with an excellent 360° panoramic system that makes the vehicles safer than ordinary vehicles; the same concept applies to autopilots [4]. Using a small and lightweight ultrawide angle system, unmanned air vehicles can obtain all-in-ones photograph through a single shot. The conciseness and accuracy of the vision system of the robots can be improved [5]. Panoramic aerial photography can provide timely meteorological services. Views of the changing panorama of streets are now available worldwide [6].
The fisheye lens and panoramic annular lens (PAL) are two typical types of ultrawide angle lenses [7]. An important function of the front group in an ultrawide angle lens is realization of the large inclination changes in incident rays so that they can pass the stop. In a fisheye lens, this inclination change is realized mostly by the refraction on the first meniscus element, which results in a bulky front element. The bulkiness of the lens can be measured by the diameter ratio, which is defined by the diameter of the largest element over that of the smallest. For example, the commercial Nikkor 6 mm F/# 2.8 fisheye lens has a field of view (FOV) and diameter ratio of 220° and 10.6, respectively [8]. The F/# 5.6 310° hyperfield fisheye lens designed by Chadwick B. Martin can achieve a diameter ratio of more than 30 [9]. However, in a PAL, reflections help change the direction of incident rays; thus, the size of the front element can be reduced. But conventional PAL structure creates a blind zone in the central region of the imaging plane; this blind zone can be suppressed but never eliminated [10].
Although very challenging to construct, small and lightweight PALs with a low F number and no blind area are of great demand; many attempts have been performed to achieve an optical system that satisfies the requirements above. One such attempt involves making the reflective front surface of the PAL semireflective so that another central optical channel can be established [11]. Pernechele presented a hyper-hemispheric lens using semireflective surfaces offering an FOV of 260°; however, in this lens, at least half of the light energy is lost because of semireflective surfaces [11].
Two-channel PALs without semireflective surfaces have also been constructed. Stürzl et al. described a catadioptric pinhole imaging system consisting of a reflective surface machined into a solid Perspex block of a specific profile and carrying an embedded lens; this system could produce an FOV of 260° with only a small separation of viewpoints [12]. However, this system could only offer good image quality in pinhole circumstances because the lens is actually divergent for the central channel. Kweon derived a new equation describing a panoramic mirror profile with an equidistance projection scheme based on the pinhole camera model. An FOV of 270° was obtained by adding an all-reflective attachment before a commercial camera. Unfortunately, the imaging quality of this system depended on the panoramic mirror profile obtained by the new equation, and the performance is not ideal [13].
In this paper, we present an ultrawide angle catadioptric lens with an FOV and F/# of 360° (azimuth) × 270° (elevation) and 2.5, respectively. The focal length of the system is 1 mm, its total track is less than 50 mm, and the diameter ratio of the system designed is controlled within 10. No semireflective element is used so that higher illumination can be achieved. Table 1 summarizes the specifications of some existing ultrawide angle designs described above.
Table 1. Specifications of some existing ultrawide angle designs
The remaining parts of the paper are organized as follows. Section 2 describes the design concept and system specifications. In Section 3, we discuss details of the design process as well as the system optimization methods. In Sections 4 and 5, the optical performance and tolerance analysis of the system are described, respectively. Finally, prototype and testing results are demonstrated in Section 6.
2. Design concept and specifications for an ultrawide angle catadioptric lens
2.1 Design concept
The PAL structure transforms a 3D omnidirectional cylindrical FOV between the minimum and maximum acceptance angles into a 2D annular image plane perpendicular to the optical axis [14]. Although the PAL has a FOV of 360° in the azimuthal direction, its FOV in the elevation direction commonly begins at a nonzero value, and the front view is lost as shown in Fig. 1(a) . We expect to overcome the disadvantage of losing part of the front view by developing a new type of catadioptric lens. Figure 1(b) shows the desired FOV of the ultrawide angle catadioptric lens. The new system has dual view configurations to realize a 270° FOV in the elevation direction.
For most wide-angle systems, specific efforts are made to improve the image quality of the central field, and the marginal field quality is allowed to degrade. However, for some applications, the marginal field is of more significance than the central field. In the case of aerial photographs obtained via UAVs, for example, the information on the perimeter of a scene is more pertinent than the information at the center because the former allows the taking of pictures of the ground below it. To maximize its optical performance, the novel lens should be designed to improve the picture image quality in the region of interest (ROI) (Fig. 1).
2.2 System specifications
The ultrawide angle catadioptric lens is composed of two optical paths as shown in Fig. 2(a) : the center and catadioptric view configurations, which are shown in Figs. 2(b) and 2(c), respectively. The center configuration consists of all-refractive lenses and half FOV of 50°. The catadioptric configuration contains a catadioptric lens and the semi-field angle, which can be divided into two portions: the side view is within the range of 50°−90° (green rays) and the rear view is within the range of 90°–135° (blue rays).
Fig. 2 The optical layout of (a) the ultrawide angle catadioptric lens (b) the center configuration and (c) the catadioptric configuration.
The image plane is divided into two zones according to the two optical paths: the image of the center configuration is a circle in the center, whereas the image of the catadioptric configuration is an annular zone. The focal length of both configurations is 1 mm, and the reflective surface causes the image of the catadioptric configuration to become inside-out. Between these two images is a gap area without any image, which can avoid disturbances between these two images. To reduce the complexity of stitching in the two images, the effective focal lengths of the two configurations are kept the same, which significantly adds difficulty to the optical design process. In the focal lengths of the two configurations, a slight amount of barrel distortion is needed in the center configuration to shrink images so that the gap can be formed. Table 2 provides the major specifications of the ultrawide angle catadioptric lens design. For an image capture system, the image sensor is an important element. The specifications for the complementary metal-oxide-semiconductor (CMOS) sensor used for this design are shown in Table 3 .
Table 2. Major specifications of the ultrawide angle catadioptric lens
Table 3. Specifications of the CMOS Sensor for the ultrawide angle catadioptric lens
3. Optical design strategies
3.1 Structure of the ultrawide angle catadioptric lens
Similar to conventional lenses, the ultrawide angle catadioptric lens is divided into two groups. The front group differs between the two configurations, whereas the rear group is shared by both configurations. For the center configuration, the structure is similar to the traditional wide angle lens as shown in Fig. 2(b). A reverse telephoto type with a negative former group and a positive rear group will be suitable for this application. For the catadioptric configuration, the lens structure is shown in Fig. 2(c). To avoid blocking the rays of the center configuration, a catadioptric element with only one reflective surface (S2) instead of the conventional PAL element is adopted. The optical region in S2 should be annular to create hole in the center for center configuration rays. The diameter of the hole is a trade-off between the two configurations. A larger hole yields a larger inclination angle in the catadioptric configuration, which adds to the design difficulty of the rear group, whereas a smaller hole contributes to the design difficulty of the front group of the center configuration.
3.2 Preliminary optimization
The catadioptric lens is the key component of the ultrawide angle system. Thus, we begin our design with the catadioptric configuration. We start the design with all spherical surfaces. Our goal here is to achieve a very compact, lightweight, easy installable, and wide FOV design; hence, the original configuration should be redesigned and further optimized.
During the catadioptric configuration optimization, the following parameters were set as variables: curvatures, thicknesses, and total length. The effective focal length was constrained to 1 mm. The diameter of the catadioptric lens is controlled within 55 mm, and the total length of the system is less than 50 mm. Optical plastics, such as polymethyl methacrylate (PMMA), can be selected as suitable material for catadioptric lens because it is cheap and convenient for processing. After several optimization iterations, we found that the stop set at the anterior surface of the third lens of the successive lens groups could lead to a more stable structure.
To simplify the mechanical structure, the former lens group of center configuration can be buried inside the catadioptric lens.
The former lens group of the center optical path is buried inside the catadioptric lens as shown in Fig. 2(a); consequently, the entire system becomes more compact in size, lower in weight, and easier to install. We can insert the former lens group into the center hole of the catadioptric lens directly, but doing so will significantly increase the design difficulty of the proposed system. Stray light is caused by ray overlaps and located at the contiguous region between the former and catadioptric lens groups. This overlap will also partially block the light. Controlling the semi-diameter of the lenses is insufficient to solve the issue; hence, we must introduce additional ray-based constraints during the optimization process.
3.3 Constraints in solving the overlap issue
During system design, additional constraints are needed to avoid the physical interference of the two configurations and ensure that all of the rays across the fields would be traced successfully to the image sensor without obstruction.
Figure 3(a) illustrates the overlap issue at the embedded area. In this case, extra vignetting or even loss of FOV occurs. To eliminate this problem, the specific structure control method is employed during optimization. Figure 3(b) demonstrates the solution to the ray overlap issue at the embedded area. During each optimization step, two feature rays are traced: the top marginal ray of the maximum field in the positive Y direction in the center configuration (Rt) and bottom marginal ray of the minimum field in the negative Y direction in the catadioptric configuration (Rb). As shown in Fig. 3, Pb1, and Pb2 denote the intersection points of the ray Rb with surfaces S2 and S1, respectively; Pt1, Pt2, Pt3, and Pt4 reflect the intersection points of the ray Rt with surfaces 1, 2, 3, and 4, respectively. Based on the physical structure requirements, the constraints are defined as follows:
where all of the Y, Z coordinates in the equations are referenced to the global coordinate system; here, the origin (point O) is located at the center of surface 1.In this paper, Eq. (1) ensures that the bottom marginal ray of the catadioptric lens can be traced through the system without obstruction of the center configuration by constraining the Y coordinates of points Pb1, Pb2 and Pt4, Pt1. By controlling the Z coordinates of points Pb1, Pb2 and Pt4, Pt1, Eq. (2) helps control the front part of the center configuration to be buried inside the catadioptric element. By limiting the Z coordinates of points Pb1, Pb2, Pt1, Pt2, Pt3, and Pt4, Eq. (3) ensures that the thicknesses of all of these lenses are suitable for fabrication, which helps control the size of the embedded structure further.
3.4 Design of the catadioptric lens with an annular stitched aspherical surface (ASAS)
To improve optical performance, the two spherical surfaces of the catadioptric lens (see Fig. 4 ) are converted to aspheres by adding a conic constant and 4th-order or higher aspheric coefficients. During optimization, the weighting factors of the sampled fields are set inversely proportional to their distance to the center of the field because the ROI is located at the rear region of the FOV in our case. The diameter of the catadioptric element should be constrained to obtain an appropriate diameter ratio. Furthermore, the derivative errors of an aspherical surface are typically accumulated during the optimization process; they force termination of the optimization without finding a valid solution [15]. In addition, the lenses are maintained with a desirable center and edge thicknesses.
The aspheric optical elements offer many degrees of freedom for the catadioptric lens; however, extra degrees of freedom may cause a dramatic increase in the complexity of the design and optimization process [16]. An inadequate method of representing and optimizing an aspherical surface may lead to discouraging and unpredictable results. Thus, specific constraint that ensures a reasonable shape of the aspherical surface should be adopted, which can be achieved by comparing the slopes of each adjacent point on one surface and restricting them in a fixed range. If the slopes of each adjacent are changed slightly, the entire surface would tend to be smooth. After a long period of global optimization, the polychromatic modulation transfer function (MTF) curves of the most suitable solution are shown in the first column of Fig. 5 .
Fig. 5 MTF graphs of the catadioptric lens (a) without ASAS (b) with ASAS and (c) after fitting ASAS to an aspheric surface. And from top to bottom, the first row represents center configuration with the semi FOV from 0° to 50°, the second row represents side view from 50° to 90°, and the third row represents rear view with the semi FOV from 90° to 135°.
The rays passing through the catadioptric element are first refracted by S1 and subsequently reflected by S2. Afterward, the rays transmit through S1 again at a lower height, and point C is located at the boundary curve (Fig. 6 ). Therefore, the region below point C is used twice in the ray path. The incident rays of the rear FOV also enter the system in this region. This region plays a very important role on improving the image quality of the marginal FOV. Hence, directly optimizing the shape of the region below point C is beneficial.
Our recently proposed ASAS [17] is applied to S1, so that the region below and above point C can have individual variables during optimization. The rotationally symmetric ASAS consists of a circular central zone and one or more annular zones; two neighbor zones are constrained to have the same derivatives on their joint curve, which means the ASAS is C 1 continuous. The ASAS formula is defined as follows [17]:
where the optic axis is presumed to lie in the Z direction and z(r) is the sag, which is the Z component of the surface displacement from the vertex. r is the radial coordinate that is measured perpendicularly from the optical axis (r 2 = (x 2 + y 2)); r 1 and r 2 means the r values of the joint curves. An aspheric surface behaves as a curve in the YOZ plane through this definition. For the central zone, c 1 is the curvature at the vertex of the surface, k 1 is the conic constant, and a 1 and b 1 are the 4th- and 6th-order deformation coefficients, respectively. The coefficients describe the surface deviation from the axially symmetric quadric surface specified by r and k. Δz 1 is the difference between the Z values of the joint curve for the central and outer zones. Coefficients with subscript 2 correspond to the parameters marked with subscript 1 for the outer zone.To use the ASAS on surface S1 fully, the first step is to determine the area of each aspheric zone and determine the intersection height of two zones. As shown in Fig. 4, S1 is divided into two zones. The first zone is the center zone, and the outer one is an annular zone. The two zones intersect with each other at the height where the highest ray is the passing point on the surface. Thus, the center zone is from the axis to 14.33 mm, and the second (annular) zone is within the range of 14.33–25.6 mm in the design (Fig. 4). The ASAS increases the number of optimization variables. While the 10th order ASAS with two zone areas adds seven variables: 1 radius, 1 conic and 4 aspheric coefficients (A4, A6, A8, and A10), and 1 z shift, but it needs two additional optimization constraints. The simple pseudocodes of the constraints are listed as below:
In these codes, the zeros1 constraint ensures that the two zones intersect with each other, and the zeros2 constraint controls the derivatives of the intersection point on the joint curve to the same value. The SAG evaluates the distance from the point to the vertex of the zone along Z axis. The Z indicates the global position of the vertex on zone. The last two terms of zeros1 eliminate the difference between the global positions of the two zones. DER is a function programmed by users to calculate the derivative of the point at the given height, and its features detailed in [16]. The codes mentioned above ensure two zones can be connected smoothly.The characteristic of each zone is defined by its aspheric parameters, and the two zones can be optimized individually by using the ASAS. The ASAS increases the number of optimization variables and helps the total system to achieve a more proper structure.
Figures 5(a) and 5(b) show a comparison of the MTF curves of the lens without and with the ASAS, respectively. By applying the ASAS during the design process, the novel lens provides a higher image quality in the full FOV, especially within the range of 180°–270°, which corresponds to the rear region in Fig. 1. Moreover, the two zones are finally fitted into one uniform aspheric expression for fabrication purposes. Figure 6 shows a comparison of the MTF curves of the three forms for the novel lens: without ASAS, with ASAS, and after fitting an aspheric surface to the ASAS. The average MTF curves for the three forms are also depicted in Fig. 6.
The average MTF values of the rear and side views increase by about 0.2 and 0.18, respectively, at the Nyquist frequency by using an ASAS during the design process (Fig. 6). The center view also increases by 0.02 because the center and catadioptric configurations share the same successive lenses; ASAS reduces the pressure of the catadioptric lens for transmitting rays so that the successive lenses could provide more contributions to the center configuration.
On the image plane of the novel system, the images have two parts on the same CMOS sensor. The image heights of the center and catadioptric configurations are within the range of 0–0.54 and 0.73–2.4 mm, respectively. The 0.19 mm gap is deliberately left out to avoid crosstalk between the two images of different configurations.
4. Optical performance and tolerance analysis
4.1 Modulation transform function
Image quality is the most important factor in judging system capability. The MTF represents the image quality in a lens design. MTF graphs for the catadioptric lens have been depicted in Section 3.4 and are shown in Figs. 5(c) and 6 . While the average MTF value at the Nyquist frequency exceeds 0.5 at the ROI, the MTF remains above 0.2 for the front and side views of most fields.
4.2 Distortion
Optical distortion is a major yet intractable issue for a typical wide-angle lens, specifically at the edge field. f-θ mapping is applied in the design, and the distortion of the system designed is shown in Fig. 7 . The horizontal ordinate represents the image height, whereas the vertical ordinate shows the distortion value. The f-θ distortion of this system is about −5%. Even at the marginal part of the image, the distortion is about −9%.
Fig. 7 f-θ distortion curves for the catadioptric lens of (a) center configuration and (b) catadioptric configuration .
4.3 Relative illumination
The relative illumination (RI) curve shown in Fig. 8 is computed as the ratio of corner luminance to the center luminance. Two methods can be used to improve the illumination on the marginal area of an ultrawide angle lens, namely, introducing extra barrel distortion and increasing the entrance pupil of oblique fields. In our design, the entrance pupil of the catadioptric configuration is designed to be larger than that of the center configuration. Hence, illumination of the side and rear views is improved. Consequently, the RI decreases from 100% to 99% in the half-field range of the center configuration. The image of the catadioptric configuration is inside-out, thus, RI declines with decreasing field angle. In terms of marginal FOV, Fig. 8 illustrates that RI decreases from 98% to 78% at the semi-field angle within the range of 135°–50°; moreover, the RI valley occurs at 50°, which corresponds to the outermost position on the image plane.
4.4 Tolerance analysis
Tolerance analysis is the key preparation process for mass production. By utilizing the MTF tolerance function in the Code V program, manufacturing difficulty may be evaluated. Table 4 shows the tolerance data used in the present system. The precision level of the tolerance data is in accordance with economically attainable accuracy and assures high imaging quality. In the worst case scenario for the two configurations, the MTF value are higher than 0.18 and 0.34 at cumulative probabilities of 90% for most of the fields of the center configuration and the catadioptric configuration, respectively, as the dotted line shown in Fig. 9 .
Table 4. Tabulation of Precision Optical Fabrication Tolerances
Fig. 9 Cumulative possibility estimates MTF plots of the tolerance analysis at the sampled fields for the ultrawide angle lens optimized for best nominal performance, (a) center configuration and (b) catadioptric configuration.
5. Mechanical Structure and Prototype Results
To assemble the catadioptric element together with the successive lenses, a transparent plastic cover shell is designed. This plastic shell allows the wide angle rays to enter the system without any obstructions. Considering the fabrication accuracy and alignment difficulty, the distance between the shell and rear structure is adjustable. The external apertures of the successive lenses are also the same, which reduces the pressure for assembly. Figure 10 shows the mechanical structure.
Finally, we developed the prototype of the novel ultrawide angle lens. Figure 11(a) shows the PMMA catadioptric lens which is fabricated by a single-point diamond-turning machine; subsequently, the concave surface is coated with reflective film. The other lenses are made of glass, and all of the diameters of the successive lenses are 6 mm as shown in Fig. 11(b). Figure 11(c) is the experimental setup and the captured image is shown in Fig. 11(d).
Fig. 11 (a) the catadioptric lens; (b) the successive lenses; (c) the experimental setup; (d) the captured image.
The image in Fig. 11(d) can be decomposed into two concentric circular image regions as marked in the figure. Region A is captured through the center configuration, and it forms the front image. Region B is captured through the catadioptric configuration, and it forms the side and rear images. The top left corner of the figure has glare because of the strong intensity of the sun and this phenomenon will be analyzed in our future research. In order to reduce the stay light, it would be better to paint the edge of the lenses that buried inside the catadioptric lens with black ink. Experimental results demonstrate that the novel lens achieves satisfactory imaging performance for use in surveillance, robotics, automotives, pipeline inspection, remote meetings, augmented reality, panoramic videos, and 3D spherical projections.
6. Conclusion
In this study, an ultrawide angle catadioptric lens is designed and presented. ASAS is applied to the rear surface of the catadioptric element to improve the ROI performance. The FOV and F/# are 360° × 270° and 2.5, respectively. The average MTF value of all fields at 119 lp/mm is above 0.2; notably, the MTF values of semi FOV within the range of 90°–135° are over 0.5 for ASAS use. This feature may provide a high resolution in the ROI. The total track of the system is less than 50 mm, and the largest diameter is 51.2 mm. The proof-of-concept prototype is established, and experimental results show that the resultant image quality is acceptable. The system can be used on an UAV for its small size and light weight. Moreover, this lens presents advantages of smaller marginal distortion and higher relative illumination over the traditional panoramic lenses. The f-θ distortion is controlled within 9%, and the relative illumination is more than 0.78 within the whole image area. In future research, the analysis of stray light and image mosaicing method will be explored. Above all, this design may advance the development of high-definition ultrawide imaging techniques.
Acknowledgments
This work is partially supported by the National Basic Research Program of China (No. 2013CB328806), the National Natural Science Foundation of China (No. 61205024, 61178038), the National High Technology Research and Development Program of China (No. 2013AA013901), and the New Century Excellent Talents in University 2012 (NCET-12-0043). We thank Synopsys for education license of CODE V.
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