Single snap-shot double field optical zoom
Optics Express, Vol. 13, Issue 24, pp. 9858-9868 (2005)
http://dx.doi.org/10.1364/OPEX.13.009858
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Abstract
In this paper we present a new approach providing super resolved imaging at the center of the field of view and yet allowing seeing the remaining of the original field of view with the original resolution. This operation resembles optical zooming while the zoomed and the non zoomed images are obtained simultaneously. This is obtained by taking a single snap shot and using a single imaging lens. The technique utilizes a special static/still coding element and a post processing algorithmic, without any mechanical movements.
© 2005 Optical Society of America
1. Introduction
1 . R. B. Johnson and C. Feng , “ Mechanically compensated zoom lenses with a single moving element ,” Appl. Opt. 31 , 2274 – 2280 ( 1992 ). [CrossRef] [PubMed]
2 . E. C. Tam , “ Smart electro optical zoom lens ,” Opt. Let. 17 , 369 – 371 ( 1992 ). [CrossRef]
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
2. Theory
2.1 Preliminary
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
2.2 Mathematical general description
- Left third S_{-1}(ν) with ν ∊ [-ν_{max}, -1/3 ν_{max}]
- Central third S_{0}(ν) with ν ∊ [-1/3 ν_{max}, 1/3 ν_{max}]
- Right third S_{1}(ν) with ν ∊ [1/3 ν_{max}, ν_{max}].
- Left third G_{-1}(ν) with ν ∊ [-ν_{max}, -1/3 ν_{max}]
- Central third G_{0}(ν) with ν ∊ [-1/3 ν_{max}, 1/3 ν_{max}]
- Right third G_{1}(ν) with ν ∊ [1/3 ν_{max}, ν_{max}].
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
14 . J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed]
- We shall first reconstruct the high frequency content S_{-1}(ν) and S_{1}(ν) by sampling I(ν): The spatial contents of S_{-1}(ν) and S_{1}(ν) occupy only a fraction of the field of view L_{T}. Therefore it is possible to keep only each 6-th (L_{T}/L_{C}) sample without losing information. Other samples are calculated using interpolation. Figure 5(a) illustrates the sampling grid. Note that at the sampling points of S_{-1}(ν) and S_{1}(ν) are orthogonal. On other hand, there is a certain noise added to the sampled high frequency content due to the S_{0}(ν). In order to minimize this noise effect, each sample value is taken to be as algebraic average in its neighborhood. Figure 5(b) shows the Fourier transform of the grating illustrated in Fig. 5(a). As one may note, it resembles 7 delta functions: the two pairs of delta functions appearing on both sides of the central delta resemble spatial derivative since each one of those two pairs contain one positive and one negative delta while small spatial shift is introduced between them. Those two pairs that make the derivative correspond to the two replications (the -1 and the 1 orders) related to the high frequencies [Fig. 4(a)]. The outer two deltas correspond to the two replications (again the -1 and 1 orders) of the low frequencies [Fig. 4(b)].Fig. 5.(a). Sampling high frequency content: S-1(ν) samples are marked with S1(ν) samples are marked with . (b). The Fourier transform of the grating.
- Next, we shall subtract the reconstructed S_{-1}(ν) and S_{1}(ν) from I(ν). This shall leave us ideally with only low frequency content. It is expressed in the spatial domain as:where s_{0} and g_{0} are the inverse Fourier transforms of S_{0}(ν) and G_{0}(ν), respectively and ‘*’ stands for convolution operation. rect (x/L_{T}) is defined as:The g_{0}(x) is in fact consists out of three Dirac impulse functions:
- Now we divide each i_{L}(x) and s_{0}(x) into sets of 6 equally-supported functions, denoted correspondingly as r_{j}(x) j=1,..,6 and f_{j}(x) j=1,..,6. These 2 sets of functions are related through 6 linear equations. Those equations can be well understood after observing Fig. 4(b):or alternately through a 6×6 matrix:By inverting the matrix we find f_{j}(x) and therefore s_{0}(x) - which is the low frequency content of the original image information. Note that f_{i}(x) are the original 6 spatial regions of s(x) while r_{i}(x) are the spatial distributions obtained in each of the 6 regions after generation of the replications on the CCD plane. Eqs. 11–12 correspond to the low frequency shift seen in Fig. 4(b). a_{i} are the coefficients with which each one of the 3 replication in Fig. 4(b) is multiplied.
3. Simulation investigation
15 . H. Dammann and E. Klotz , “ Coherent optical generation and inspection of two-dimensional periodic structures ,” Opt. Acta 24 , 505 – 515 ( 1977 ). [CrossRef]
4. Conclusions
References and links
1 . | R. B. Johnson and C. Feng , “ Mechanically compensated zoom lenses with a single moving element ,” Appl. Opt. 31 , 2274 – 2280 ( 1992 ). [CrossRef] [PubMed] |
2 . | E. C. Tam , “ Smart electro optical zoom lens ,” Opt. Let. 17 , 369 – 371 ( 1992 ). [CrossRef] |
3 . | H. Tsuchida , N. Aoki , K. Hyakumura , and K. Yamamoto , “ Design of zoom lens systems that use gradient-index materials ,” Appl. Opt. 31 , 2279 – 2286 ( 1992 ). [CrossRef] [PubMed] |
4 . | R. J. Pegis and W. G. Peck , “ First-order design theory for linearly compensated zoom systems ,” J. Opt. Soc. Am. 52 , 905 – 911 ( 1962 ). [CrossRef] |
5 . | G. Wooters and E. W. Silvertooth , “ Optically Compensated Zoom Lens ,” JOSA , 55 , 347 – 355 ( 1965 ). [CrossRef] |
6 . | T. ChunKan , “ Design of zoom system by the varifocal differential equation. I ,” Appl. Opt. 31 , 2265 – 2273 ( 1992 ). [CrossRef] [PubMed] |
7 . | Y. Ito , “ Complicated pin-and-slot mechanism for a zoom lens ,” Appl. Opt. 18 , 750 – 758 ( 1979 ). |
8 . | D. R. Shafer , “ Zoom null lens ,” Applied Optics , 18 , 3863 – 3870 ( 1979 ). [PubMed] |
9 . | K. Tanaka , “ Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type ,” Appl. Opt. 21 , 2174 – 2181 ( 1982 ). [CrossRef] [PubMed] |
10 . | D. Y. Zhang , N. Justis , and Y. H. Lo , “ Integrated fluidic adaptive zoom lens ,” Opt. Let. , 29 , 2855 – 2857 ( 2004 ). [CrossRef] |
11 . | A. Walter , “ Zoom lens and computer algebra ,” J. Opt. Soc. Am. A , 16 , 198 – 204 ( 1999 ). [CrossRef] |
12 . | M. N. Akram and M. H. Asghar , “ Step-zoom dual-field-of -view infrared telescope ,” Appl. Opt. 42 , 2312 – 2316 ( 2003 ). [CrossRef] [PubMed] |
13 . | A. Walther , “ Angle eikonals for a perfect zoom system ,” J. Opt. Soc. Am. A , 18 , 1968 – 1971 ( 2001 ). [CrossRef] |
14 . | J. Solomon , Z. Zalevsky , and D. Mendlovic , “ Geometrical super resolution by code division multiplexing ,” Appl. Opt. 44 , 32 – 40 ( 2005 ). [CrossRef] [PubMed] |
15 . | H. Dammann and E. Klotz , “ Coherent optical generation and inspection of two-dimensional periodic structures ,” Opt. Acta 24 , 505 – 515 ( 1977 ). [CrossRef] |
OCIS Codes
(100.6640) Image processing : Superresolution
(170.1630) Medical optics and biotechnology : Coded aperture imaging
ToC Category:
Research Papers
History
Original Manuscript: September 7, 2005
Revised Manuscript: September 3, 2005
Published: November 28, 2005
Citation
Zeev Zalevsky and Alexander Zlotnik, "Single snap-shot double field optical zoom," Opt. Express 13, 9858-9868 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-24-9858
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References
- R. B. Johnson and C. Feng, "Mechanically compensated zoom lenses with a single moving element," Appl. Opt. 31, 2274-2280 (1992). [CrossRef] [PubMed]
- E. C. Tam, "Smart electro optical zoom lens," Opt. Let. 17, 369-371 (1992). [CrossRef]
- H. Tsuchida, N. Aoki, K. Hyakumura and K. Yamamoto, "Design of zoom lens systems that use gradient-index materials," Appl. Opt. 31, 2279-2286 (1992). [CrossRef] [PubMed]
- R. J. Pegis and W. G. Peck, "First-order design theory for linearly compensated zoom systems," J. Opt. Soc. Am. 52, 905-911 (1962). [CrossRef]
- G. Wooters and E. W. Silvertooth, "Optically Compensated Zoom Lens," JOSA, 55, 347-355 (1965). [CrossRef]
- T. ChunKan, "Design of zoom system by the varifocal differential equation. I," Appl. Opt. 31, 2265-2273 (1992). [CrossRef] [PubMed]
- Y. Ito, "Complicated pin-and-slot mechanism for a zoom lens," Appl. Opt. 18, 750-758 (1979).
- D. R. Shafer, "Zoom null lens," Applied Optics, 18, 3863-3870 (1979). [PubMed]
- K. Tanaka, "Paraxial analysis of mechanically compensated zoom lenses. 1: Four-component type," Appl. Opt. 21, 2174-2181 (1982). [CrossRef] [PubMed]
- D. Y. Zhang, N. Justis and Y. H. Lo, "Integrated fluidic adaptive zoom lens," Opt. Let., 29, 2855-2857 (2004). [CrossRef]
- A. Walter, "Zoom lens and computer algebra," J. Opt. Soc. Am. A, 16, 198-204 (1999). [CrossRef]
- M. N. Akram and M. H. Asghar, "Step-zoom dual-field-of –view infrared telescope," Appl. Opt. 42, 2312-2316 (2003). [CrossRef] [PubMed]
- A. Walther, "Angle eikonals for a perfect zoom system," J. Opt. Soc. Am. A, 18, 1968-1971 (2001). [CrossRef]
- J. Solomon, Z. Zalevsky and D. Mendlovic, "Geometrical super resolution by code division multiplexing," Appl. Opt. 44, 32-40 (2005). [CrossRef] [PubMed]
- H. Dammann and E. Klotz, “Coherent optical generation and inspection of two-dimensional periodic structures,” Opt. Acta 24, 505–515 (1977). [CrossRef]
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