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Lens Designs for wide-angle Bokode
The purpose of this document is to describe what is ”Krill-eye” wide-angle Bokode.
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Principle of Krill-eye optics
Some kind of animals living in dim environments have a superposition compound eye. Most insects have apposition compound eyes, but the amount of incoming light which effects to a single photo receptor is very limited. Therefore some kind of moths, krills, scallops or lobsters have superposition compound eyes which gather entering light at a same point through several optics. As shown in Figure 1, the image formed by normal lenses (a) is upside-down (and left-toright as well), but we can flip the image upright by focusing the image at the next surface once more. To realize such imaging process, we can use following lens system (relay optics) proposed as Gabor Super Lens[3]. For such relay optics, gradient index (GRIN) rod lens (b) are now widely used. For example, GRIN lens array is widely used for imaging system of low-cost scanners. While the GRIN lens array is limited to a 1:1 (life-size) imaging system because all lenses are aligned parallel, we propose a novel optics design for omni-directional imaging system. As shown in Figure1(c), multiple GRIN rod lenses are aligned on a circle (or a sphere) with certain diameter, and all optical axes intersect at one point. If this case, images from several lenses are aligned on a sphere. Let us think about a object at infinity, B, as shown in figure 1(c). This object is just on the optical axis of upper lens, and it have a viewing angle s1 from lower lens. From the definition of focal length using thin lens approximation, the angle s3 is same to s1 and the height of the image is given by h = f · tan(s3 ). The image of the object B should be on a spherical surface with diameter r, and also it should be at a angle s2 = s3 = s1 , therefore r must be same to the focal length of GRIN lens, f . Such condition of superposition is independent from the angle between GRIN lenses s2 .
Figure 1: Superposition by GRIN lens. Contrary to the normal lens (a) which forms a image flipped upside-down, image by certain length of GRIN rod lens (b) is upright. Therefore the spherical array of GRIN lens (c) focuses into a image on a sphere.
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Lens Designs for wide-angle imaging
We briefly summarize pros and cons of wide-angle imaging system. Note that we can use monochromatic light source for our Bokode projector, so we do not have to care about chromatic aberration. Fish-eye The most common lens design for wide-angle camera / projector. Pros: • Registration is not necessary because it has a single mask. • Lenses are currently on the market. Cons: • Very expensive because the lens design is complex. • Angular resolution is not uniform. • It is difficult to cover the angle of view larger than 180deg. 220deg fish-eye lens had been on the market, but it is huge and very expensive. 360deg coverage is impossible. • It has very heavy light fall-off at the peripheral of the image when it is used for projector. Planetarium Currently used for omni-directional projectors (planetariums) and camera arrays. Pros: • High angular resolution with well compensated aberration. • It could cover whole angle-of-view. Cons: • The system will be large. • Registration of the all image is not easy, especially for the projectors. GRIN ball[1] Single spherical GRIN lens with a spherical mask. Pros: • It might be very small. • The mask is uniform to the angle of view. Cons: • GRIN ball lens is not available on the market. It is not easy to manufacture. • f/# is directly fixed by the diameter of the lens, so the spherical aberration must be compensated by the lens itself. • It does not cover whole direction. • Light source is not easy to align around the mask. Krill-eye Single mask is shared by several lens, as shown in Figure 3. It is inspired by the visual system of krills and moths[2]. Similar superposed imaging system have proposed[3, 4], but it is for a planar image plane. Pros: • It may cover whole direction. • The mask is uniform to the whole angle of view. • The cost of the GRIN rod lenses is now very cheap. Array of GRIN rod lens is widely used for entry-level document scanners under 100 USD. 2
Cons: • GRIN rod lenses should be aligned on the radial lines (the distribution of the lenses does not matter to the image quality).
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Some thoughts about wide-angle projectors
In general, complex lens system is required to compensate aberration caused by the shape and dispersion of the glass. Especially, very wide angle of view is one of the most common but difficult demands for the lens design. Therefore we used several special properties of Bokode projectors to simplify the lens design. The most useful aspect of the Bokode projector is the use of the monochromatic light source. Therefore we do not have to care about the chromatic aberration of the lens. It does not limit the use of multiple colors in the code, because the image of the BOKODE is captured and processed by machines. So we can compensate the displacement or magnification of the image between different colors while image is processed. For the Bokode projectors, we do not have to think about the focusing mechanism to the closer object. The mask should be always focused at the infinity. Fisheye lens (Fig.2) has an advantage of the ability of focusing (focusing is impossible for design (b), (c), (d) because of physical limitation or registration between the lenses is broken), but for us it is not necessary. Furthermore, pseudo large aperture (aperture array) has a good nature for Bokode, because it will enlarge the area of bokeh. If the gap between the lenses are smaller than the diameter of the camera’s aperture, bokehes are connected each other on the taken image as shown in Figure 3.
References [1] Yasuhiro Koike, Yuji Sumi, and Yasuji Ohtsuka, ”Spherical gradient-index sphere lens,” Appl. Opt. 25, 3356-3363 (1986) [2] M. F. Land and D-E Nilsson, Animal Eyes, Oxford unifersity press, 2002. [3] D. Gabor, UK Patent 541753, 1940. Imaging properties of the Gabor superlens, [4] C Hembd-Solner et al, Imaging properties of the Gabor superlens, J. Opt. A: Pure Appl. Opt. Vol. 1, pp. 94-102, 1999.
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Figure 2: Four alternatives for wide-angle camera / projector. (a) Fisheyelens : The most common lens design for wide-angle camera / projector, (b) Planetarium : To cover whole angle of view, multiple cameras or projectors are arranged and registered precisely. (c) GRIN ball : Spherical glass ball lens is currently used for laser coupling, but to compensate spherical aberration, gradient index lens is necessary. (d) Krill-eye : Similar to (b) but the image is inverted to use single spherical mask which is shared by the all lenses.
Figure 3: Prototype of Krill-eye projector and its superposed image with DataMatrix 2D code.
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