The first generation of monocentric multiscale gigapixel cameras used Keplerian designs to enable full field coverage. This paper considers alternative designs that remove the requirement that adjacent subimages overlap. Removing this constraint enables Galilean designs that reduce system volume and improve relative illumination and image quality. The entrance aperture can also be moved to more closely approximate telecentricity and gaps in the field of view can be filled using multiple co-boresighted MMS cameras. Even with multiple cameras, Galilean systems can still reduce the total volume by 10 times relative to previous Keplerian designs.
© 2017 Optical Society of America
Lens cost and volume scale with pixel count. Conventionally, as aperture size increases field of view decreases to ameliorate the increasing challenge of geometric aberration . Field curvature is the most challenging aberration. While field curvature can be accommodated using spherical focal planes , such systems cannot be focused at arbitrary conjugates. Monocentric multiscale (MMS) optical design deals with this problem by amortizing the imaging task into two stages: initial collection on a spherical manifold followed by parallel local processing for focus, chromatic and spherical aberration. Under this strategy, field curvature is no longer a problem and focus can be adjusted independently for each field segment. Through local field processing, MMS design overcomes the scaling law for geometric aberration . As demonstrated in the AWARE project, MMS cameras can achieve high angular resolution and wide field of view (FOV) in gigapixel scale systems. In contrast with gigapixel astronomical telescopes   and lithographic lenses, MMS systems can be manufactured and assembled using commercial off-shelf components.
The architecture of a MMS system resembles that of a telescope. One layered monocentric spherical objective lens is shared by several microcameras. Each microcamera covers a small portion of overall FOV denoted as MFOV. Refractive telescopes may be classified into Keplerian systems with an internal image surface and Galilean systems with secondary optics placed before the objective focal surface. MMS cameras also can be designed in these two styles. Although Galilean systems achieve smaller physical size, previously constructed MMS cameras all adopt Keplerian design because such architectures more readily accommodate overlap between adjacent microcamera fields and because they are somewhat easier to construct.
2. MMS optics of Galilean style
Galilean telescopes include a positive objective lens and a negative secondary, separated by difference between their focal lengths. Compared with the Keplerian style, Galilean style features a shorter tube length and therefore smaller overall size. The disadvantage is a smaller FOV . However, this disadvantage is unimportant under our multiscale imaging approach, since this approach intrinsically captures a wide FOV by dividing it into large array of small FOVs.
2.1. First-order Galilean model
To create a simple first-order Galilean model, the configuration of a telescope is divided into three optical groups, the objective lens in front, eyepiece in middle and sensor optic in the rear. The objective here is a layered monocentric lens with each layer made of homogeneous material. Secondary optics are arrayed on a spherical surface concentric with the objective. The Galilean eyepiece is a negative power group displaced from the center of the objective by the sum of the objective and eyepiece focal lengths. Conventionally, the stop aperture of the Galilean style is located at the eyepiece to facilitate easy observation. This stop position causes reduced spherical symmetry and thus creates local aberrations . The location of the stop can be used to strike a balance between the aperture sizes of the objective lens and the microcameras array and to manipulate aberration.
Considering a MMS lens with angular resolution of IFOV, total system FOV and aperture size F/#. We choose an image sensor with pixel pitch p and active size of V × H with. From these parameters, the effective focal length of the overall system is f = p/IFOV. The vertical field of view of each micro camera is MFOV = V/F , we denote the MFOV half angle as α.
From these basic specifications, we can derive the critical parameters of the optics. Fig. 1 is a sketch of the first order Galilean design with one sub-imager channel. If the separation between the center of the objective and the stop surface is dos, the entrance pupil is located at lε= fodos/(fo − dos) behind the center of the objective. The diameter of the entrance pupil is Dε= f/(F/#), where fo is the focal length of the objective. The diameter of the stop is given by
The distance between the stop and the eyepiece is dse = fo − dos − fe, where − fe is the focal length of the eyepiece. The image of the stop from the eyepiece can be treated as an intermediate pupil with location at lip = fedse/(dse + fe) in front of the eyepiece and diameter of Dip = Ds fe/(fe + dse). The clear aperture of the eyepiece is
By separating the stop and the eyepiece, the required aperture diameters of the objective and the eyepiece for zero vegnetting, as shown in Eqn. (2) and (3), are made of two terms. The first term originates from off-axis field angle and is affected by the position of the stop, i.e. the value of dos. The second term is from F-number of the overall system and is independent of the stop position. As we increase dos by moving it away from the objective, the diameter of the objective expressed in the first term will become larger and larger while the diameter of the eyepiece will decrease on the contrary. The position of the stop provides a way for adjusting the sizes of the two optical groups. On one hand, if dos is too large, off-axis rays may not be able to reach the surface of the objective and those do reach the surface may be plagued with hopeless aberration. On the other, if we make dos too small, the required diameter of the microcamera is likely to cause physical interference between adjacent microcameras. At the extreme case, when the stop is placed at the center of the objective, the required diameter of microcameras is infinite for zero vegnetting, in contrast, putting the stop at the eyepiece will lead to an infinitely large objective.
As illustrated in Fig. 2, the size of the sensor optic is limited by a physical cone angle, here denoted as 2θ, to avoid physical conflict between neighbors . As a result, the aperture diameter of the element of the microcamera is limited. Since the stop is placed in the front of each microcamera, the diameter of the stop is also constrained by Ds < 2θdos. Substituting into Eqn. (1), we have the lower bound of dos as dos > Dεfo/(2 foθ + Dε). Additionally, the stop cannot be positioned inside the objective lens, which put another lower boundary to dos. As an example, considering a two-layered monocentric lens consisting of a Schott F2 glass (flint) external shell and a BK7 (crown) inner sphere with the desired focal length of fo. The two radii achieving nearly both achromatic and third-order spherical free are found at R1 = 0.58 fo, where R1 is the radii of the outer surface and R2 is the radii of the inner surface. In this case the separation dos > 0.58 fo. Nonetheless, this bound can vary depending on the choices of the materials and the inner radii. This second restrict on lower bound of the stop position can be described by dos > R1.
To achieve a realistic design, a strict limit should also be imposed on the clear aperture size of the objective. Otherwise undesired ray path may occur, such as some rays may totally miss the aperture, or pass the outer shell while miss the inner sphere, or pass through both layers with hopeless large incident angles, as shown in Fig. 3(c). The first two cases lead to vignetting or even complete blackout. The third case introduces severe off-axis aberration which either demands highly complex secondary optics to correct or leads to failed imaging quality. According Eqn. (2), the clear aperture size of the objective with one imaging channel consists of two terms. One term is related to MFOV angle and the position of the entrance pupil, the other term is related to the size of the entrance pupil. Consequently, the clear aperture can be controlled by imposing constraint on the MFOV angle, F-number of the objective as well as the entrance pupil position. Empirically, the F-number of the spherical objective is between 5 to 8, the MFOV of the microcamera should be less than 10°, i.e., α < 5°. To prevent the described ray path failures from occurring, supposing that the clear aperture of the objective should not exceed one half of its actual physical diameter, this condition can be expressed as Do < R1. Applying the Eqn. (2) and rearranging this inequality, we have the upper bound of the stop position as
Based on these considerations, in practical designs the stop position must be placed within the range(6), the larger of the two terms within the parenthesis should be taken as the lower bound for dos. For example, we want an MMS design with overall effective focal length of f = 25mm and F/# = 3, then the entrance pupil diameter is Dε= 8.33mm, suppose the focal length of the objective lens is fo = 50mm, the half MFOV angle is α = 5.6° and the half cone angle is θ = 4°. By applying Inequality (6), we have max(27.16mm, R1) < dos < 27.24mm. Whatever the radius of the objective is, the location of the stop is almost fixed. When we take consideration of the supporting items, such as the opto-mehcanics and electronics, this design is likely to be impractical . There are several measures we can take to stretch the design space. Let’s First look at the left side of the Inequality (6). The focal length of a two-layered monocentric lens is related with the radii of the outer and the inner surfaces by  Eqn. (7), for given focal length, we can manage to reduce the radii of the outer surface R1 by decreasing the radii of the inner sphere R2 or trying different glasses. However, if R1 and R2 deviate too much from the spherical aberration free and achromatic condition, the imaging quality will degrade.
For a MMS imager, the final wide FOV panorama is obtained by stitching all the sub-images together, which requires that sub-images of adjacent microcameras must overlap on the boarder. This requirement is reflected by α > θ, in another word, the MFOV of the microcamera should be larger than the cone angle within which the microcamera can physically occupy. As this requirement limits the design space, we propose here a combination of image interleaving and MMS optical design. As shown in Fig. 4, we can use multiple MMS cameras co-boresightedly and arrange microcameras of each MMS camera in a way three array patterns complementing each other and composing a complete coverage of view. This interleaving strategy can be found in DARPA ARGUS-IS surveillance system . In this way, we can increase the cone angle θ as well as decrease the MFOV angle α. By doing this, the free space for positioning the stop is stretched.
2.2. Design example
For the convenience of comparison, in this example, we use the specifications of the AWARE-2 camera with glass optics . Our new design uses image sensor of Sony IMX274 (Type 1/2.5) 8.51 Megapixel, Color with 3840 × 2160 array and 1.62μm pixel size. The targeted angular resolution is IFOV = 62μrad and therefore the effective focal length f = 25mm. The aperture size is F/# = 3. To incorporate the new approach into the design, we increase the cone angle θ from 3.43° to 5.94°, while decrase the MFOV α from 5.6° to 4.8°. Under these changes, the gaps between adjacent microcameras is 2(θ − α) = 2.28°.
The outer shell of the monocentric objective lens is made of S-NBH8 glass from Ohara catalog and the center core element is made of fused silica. The effective focal length of this objective is fo = 48.08mm, compared with 67.30mm in the old design. This leads to a shrunken of the radii of the outer surface from 31.8mm to 20mm. Meanwhile the F-number of the intermediate image changes from 8.11 to 5.77. All these changes in the objective parameters indicates a more compact primary optics.
Substituting the relevant parameters into Inequality (6) we find the range of stop positions isFig. 5, in our design example, the stop aperture is positioned 24.5mm behind the center of the objective and 2.54mm before the microcamera, which results a 26.54mm separation between the objective lens and the microcamera, as opposed to be 125.52mm in the AWARE-2 design. This tremendous reduction of the space is mostly due to the implementation of the Galilean style and serves as the main contributor for a drastic reduction of the overall system volume.
As shown in Fig. 6, the microcamera of the new design only consists of 4 elements and each element has a slim shape featuring a proper diameter and thickness ratio. In comparison, the microcamera in the AWARE-2 consists of 5 elements with two thick doublets and four of them have awkward profiles. From the perspective of geometric aberration, the Galilean style resembles the configuration of the cook triplet with a negative power group in between. This leads to a better aberration characteristic and a simple optics.
Figure 7 and Fig. 8 show the MTF curves and the spot diagrams of the new design. The nominal MTF of all the fields is greater than 0.3@154cycles/mm, which is the half of the Nyquist resolution of the sensor. the spot diagrams show the imaging performance is reaching diffraction limit for the ray bundle falls inside the circle determined by the Airy disc. Another important aspect of the design is tolerancing. The tolerances in all dimensions of the AWARE-2 design falls into realistic level and ensures a relatively easy manufacturing and assembly. The tolerancing analysis of the new design shows the same property which guarantees a justifiable cost of fabrication and assembly. The tolerance for the objective elements deviating from concentric condition is 100μm, the placement error of the each microcamera relative to the objective lens is held to be 100μm axial, 50μm decenter and 50μm tilt.
At least 3 MMS imagers are required to cover a continuous panorama image. The microcameras array follows the hexagonal arrangement with arrays from different imagers complementing each other’s. The volume estimate of each imager is about 0.22L, accounting for three, the total volume should be around 0.66L, which is still more than ten times smaller than that of the AWARE-2 optics.
As shown on table 1, the radius of monocentric objective lens is reduced by one third. The track length is decreased by about two thirds. As a result, the volume is reduced by more than 10X.
In this paper, we have explored another class of monocentric multiscale cameras using Galilean style lens systems. The first-order calculation shows that the Galilean design space is more restricted than the Keplerian style. This tight constraint means that there is limited space for mechanical support and focus mechanisms and that there is less range for lens variation. However, the Galilean approach greatly reduces the overall size, which is highly desirable in miniaturizing and commercializing the gigapixel camera products. To stretch the design space of the Galilean style, we discard the idea of sub-image overlapping between neighboring microcameras. By allowing image gaps, we essentially increase the space budget as well as decrease the FOV angle for each sub-imager. The resulting gaps in the captured field of view can be resolved using multiple co-boresighted MMS imagers. These imagers must implement a complementary interleaving imaging pattern to cover the desired full FOV. In our design example, we have illustrated a Galilean style design with the same imaging specifications as that of the AWARE-2 camera, yet is more than ten times smaller in optical volume even after accounting for three parallel arrays. This drastic reduction in size does not compromise the imaging performance as illustrated by its tolerance data, MTF curves and spot diagrams.
Although this approach brings about a more compact optics volume, the co-boresighted idea suffers from parallax if the object is too close to the camera, which may cause serious problem in composing wide field panorama. The second limitation is the issue of independent focusing. The focusing capacity is related to the aperture size of the microcamera. As the aperture size of the microcamera gets reduced, we must make sure that a focusing solution is still available.
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