Abstract

Macroscopic imagers are subject to constraints imposed by the wave nature of light and the geometry of image formation. The former limits the resolving power while the latter results in a loss of absolute size and shape information. The suite of methods outlined in this work enables macroscopic imagers the unique ability to capture unresolved spatial detail while recovering topographic information. The common thread connecting these methods is the notion of imaging under patterned illumination. The notion is advanced further to develop computational imagers with resolving power that is decoupled from the constraints imposed by the collection optics and the image sensor. These imagers additionally feature support for multiscale reconstruction.

© 2017 Optical Society of America

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2013 (2)

O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. USA 110, 21000–21005 (2013).
[Crossref]

S. Quirin and R. Piestun, “Depth estimation and image recovery using broadband, incoherent illumination with engineered point spread functions,” Appl. Opt. 52, A367–A376 (2013).
[Crossref]

2012 (1)

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref]

2011 (1)

2010 (1)

C. B. Müller and J. Enderlein, “Image scanning microscopy,” Phys. Rev. Lett. 104, 198101 (2010).
[Crossref]

2009 (2)

J.-Y. Lin, R.-P. Huang, P.-S. Tsai, and C.-H. Lee, “Wide-field super resolution optical sectioning microscopy using a single spatial light modulator,” J. Opt. A 11, 015301 (2009).
[Crossref]

V. Micó, Z. Zalevsky, and J. García, “Optical superresolution: imaging beyond Abbe’s diffraction limit,” J. Hologr. Speckle 5, 110–123 (2009).
[Crossref]

2008 (4)

A. Stemmer, M. Beck, and R. Fiolka, “Widefield fluorescence microscopy with extended resolution,” Histochem. Cell Biol. 130, 807–817 (2008).
[Crossref]

D. W. Tyler and E. B. Barrett, “Simulation of a passive grating-heterodyne super resolution concept,” Proc. SPIE 7094, 709403 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

A. Neumann, Y. Kuznetsova, and S. R. Brueck, “Structured illumination for the extension of imaging interferometric microscopy,” Opt. Express 16, 6785–6793 (2008).
[Crossref]

2007 (5)

2006 (5)

L. Zhang and S. K. Nayar, “Projection defocus analysis for scene capture and image display,” ACM Trans. Graph. 25, 907–915 (2006).
[Crossref]

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

R. Heintzmann and G. Ficz, “Breaking the resolution limit in light microscopy,” Brief Funct. Genomic Proteomic 5, 289–301 (2006).
[Crossref]

A. Greengard, Y. Y. Schechner, and R. Piestun, “Depth from diffracted rotation,” Opt. Lett. 31, 181–183 (2006).
[Crossref]

2005 (2)

M. G. L. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. USA 102, 13081–13086 (2005).
[Crossref]

A. Mudassar, A. R. Harvey, A. H. Greenaway, and J. Jones, “Band pass active aperture synthesis using spatial frequency heterodyning,” J. Phys. 15, 290–295 (2005).

2004 (2)

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recognit. 37, 827–849 (2004).
[Crossref]

P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
[Crossref]

2002 (3)

2001 (1)

2000 (3)

E. Sabo, Z. Zalevsky, D. Mendlovic, N. Konforti, and I. Kiryuschev, “Superresolution optical system with two fixed generalized Damman gratings,” Appl. Opt. 39, 5318–5325 (2000).
[Crossref]

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198, 82–87 (2000).
[Crossref]

J. T. Frohn, H. F. Knapp, and A. Stemmer, “True optical resolution beyond the Rayleigh limit achieved by standing wave illumination,” Proc. Natl. Acad. Sci. USA 97, 7232–7236 (2000).
[Crossref]

1999 (2)

M. G. L. Gustafsson, “Extended resolution fluorescence microscopy,” Curr. Opin. Struct. Biol. 9, 627–628 (1999).
[Crossref]

E. Horn and N. Kiryati, “Toward optimal structured light patterns,” Image Vis. Comput. 17, 87–97 (1999).
[Crossref]

1996 (1)

1993 (1)

M. Chang and C. S. Ho, “Phase-measuring profilometry using sinusoidal grating,” Exp. Mech. 33, 117–122 (1993).
[Crossref]

1988 (1)

C. J. R. Sheppard, “Super-resolution in confocal imaging,” Optik 80, 53–54 (1988).

1985 (1)

1983 (2)

1982 (1)

J. Jahns and A. W. Lohmann, “OTF synthesis: low pass and band pass filtering realized by periodic replication of the pupil function,” Opt. Commun. 42, 231–236 (1982).
[Crossref]

1981 (1)

M. D. Altschuler, B. R. Altschuler, and J. Taboada, “Laser electro-optic system for rapid three-dimensional (3-D) topo-graphic mapping of surfaces,” Opt. Eng. 20, 953–961 (1981).
[Crossref]

1979 (1)

1972 (1)

Y. Shirai, “Recognition of polyhedron with a range finder,” Pattern Recognit. 4, 243–250 (1972).
[Crossref]

1965 (1)

1963 (1)

W. Lukosz and M. Marchand, “Optischen Abbildung Unter Uberschreitung der Beugungsbedingten Auflosungsgrenze,” J. Mod. Opt. 10, 241–255 (1963).

Aggarwal, M.

M. Aggarwal and N. Ahuja, “A pupil-centric model of image formation,” Int. J. Comput. Vis. 48, 195–214 (2002).
[Crossref]

Ahuja, N.

M. Aggarwal and N. Ahuja, “A pupil-centric model of image formation,” Int. J. Comput. Vis. 48, 195–214 (2002).
[Crossref]

Altschuler, B. R.

M. D. Altschuler, B. R. Altschuler, and J. Taboada, “Laser electro-optic system for rapid three-dimensional (3-D) topo-graphic mapping of surfaces,” Opt. Eng. 20, 953–961 (1981).
[Crossref]

Altschuler, M. D.

M. D. Altschuler, B. R. Altschuler, and J. Taboada, “Laser electro-optic system for rapid three-dimensional (3-D) topo-graphic mapping of surfaces,” Opt. Eng. 20, 953–961 (1981).
[Crossref]

Ambler, A. P.

R. I. Popplestone, C. M. Brown, A. P. Ambler, and G. F. Crawford, “Forming models of plane-and-cylinder faceted bodies from light stripes,” in Proceedings 4th International Joint Conference on Artificial Intelligence (1975), pp. 664–668.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Barrett, E.

E. Barrett, D. W. Tyler, P. M. Payton, K. Ip, and D. N. Christie, “New approaches to image super resolution beyond the diffraction limit,” Proc. SPIE 6712, 67120D (2007).
[Crossref]

Barrett, E. B.

D. W. Tyler and E. B. Barrett, “Simulation of a passive grating-heterodyne super resolution concept,” Proc. SPIE 7094, 709403 (2008).
[Crossref]

Batlle, J.

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recognit. 37, 827–849 (2004).
[Crossref]

Baumgartl, J.

Beck, M.

A. Stemmer, M. Beck, and R. Fiolka, “Widefield fluorescence microscopy with extended resolution,” Histochem. Cell Biol. 130, 807–817 (2008).
[Crossref]

Ben-Eliezer, E.

Bhakta, V.

P. Rangarajan, V. Bhakta, M. Christensen, and P. Papamichalis, “Perspective imaging under structured light,” in European Conference on Computer Vision, September2010, pp. 405–419.

P. Rangarajan, V. Bhakta, and M. Christensen, “Surpassing the diffraction limit of digital imaging systems using sinusoidal illumination patterns,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CTuC4.

Binford, T. O.

R. Nevita and T. O. Binford, “Structured descriptions of complex objects,” in Proceedings 3rd International Joint Conference on Artificial Intelligence (1973), pp. 641–647.

Braunecker, B.

Brown, C. M.

R. I. Popplestone, C. M. Brown, A. P. Ambler, and G. F. Crawford, “Forming models of plane-and-cylinder faceted bodies from light stripes,” in Proceedings 4th International Joint Conference on Artificial Intelligence (1975), pp. 664–668.

Brueck, S. R.

Bunt, G.

O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. USA 110, 21000–21005 (2013).
[Crossref]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Chang, M.

M. Chang and C. S. Ho, “Phase-measuring profilometry using sinusoidal grating,” Exp. Mech. 33, 117–122 (1993).
[Crossref]

Chiang, F. P.

P. S. Huang, S. Zhang, and F. P. Chiang, “Trapezoidal phase-shifting method for 3-D shape measurement,” Proc. SPIE 5606, 142 (2004).
[Crossref]

Chitnis, A. B.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref]

Christensen, M.

P. Rangarajan, M. Christensen, and P. Milojkovic, “Parsimony in PSF engineering using patterned illumination,” in Imaging and Applied Optics Postdeadline, OSA Postdeadline Paper Digest (Optical Society of America, 2013), paper CTh3C.1.

P. Rangarajan, V. Bhakta, and M. Christensen, “Surpassing the diffraction limit of digital imaging systems using sinusoidal illumination patterns,” in Computational Optical Sensing and Imaging, OSA Technical Digest (CD) (Optical Society of America, 2009), paper CTuC4.

P. Rangarajan, V. Bhakta, M. Christensen, and P. Papamichalis, “Perspective imaging under structured light,” in European Conference on Computer Vision, September2010, pp. 405–419.

P. Rangarajan, I. Sinharoy, P. Papamichalis, and M. Christensen, “Pushing the limits of digital imaging using structured illumination,” in International Conference on Computer Vision (2011), pp. 1315–1322

Christie, D. N.

E. Barrett, D. W. Tyler, P. M. Payton, K. Ip, and D. N. Christie, “New approaches to image super resolution beyond the diffraction limit,” Proc. SPIE 6712, 67120D (2007).
[Crossref]

Clever, M.

O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. USA 110, 21000–21005 (2013).
[Crossref]

Combs, C. A.

A. G. York, S. H. Parekh, D. D. Nogare, R. S. Fischer, K. Temprine, M. Mione, A. B. Chitnis, C. A. Combs, and H. Shroff, “Resolution doubling in live, multicellular organisms via multifocal structured illumination microscopy,” Nat. Methods 9, 749–754 (2012).
[Crossref]

Crawford, G. F.

R. I. Popplestone, C. M. Brown, A. P. Ambler, and G. F. Crawford, “Forming models of plane-and-cylinder faceted bodies from light stripes,” in Proceedings 4th International Joint Conference on Artificial Intelligence (1975), pp. 664–668.

Cremer, C.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Dholakia, K.

Dierking, M.

Dong, C.

Donoho, D.

D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Duncan, B.

Enderlein, J.

O. Schulz, C. Pieper, M. Clever, J. Pfaff, A. Ruhlandt, R. H. Kehlenbach, F. S. Wouters, J. Großhans, G. Bunt, and J. Enderlein, “Resolution doubling in fluorescence microscopy with confocal spinning-disk image scanning microscopy,” Proc. Natl. Acad. Sci. USA 110, 21000–21005 (2013).
[Crossref]

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IEEE Signal Process. Mag. (1)

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Link to supplementary material: http://lyle.smu.edu/~prangara/pubs/AO_Manuscript/ .

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Figures (28)

Fig. 1.
Fig. 1.

Illustrating the limits of macroscopic imaging.

Fig. 2.
Fig. 2.

Active computational imaging concept.

Fig. 3.
Fig. 3.

Mechanism underlying dual objectives surpassing the diffraction limit and recovering depth information.

Fig. 4.
Fig. 4.

Surpassing the diffraction limit by heterodyning.

Fig. 5.
Fig. 5.

Recovering depth using phase modulation.

Fig. 6.
Fig. 6.

Select experimental results highlighting specific capabilities of proposed active computational imagers. Please note that in each example the super-resolved image contains spatial detail beyond the diffraction limit of the collection optics.

Fig. 7.
Fig. 7.

Notional active stereo arrangement used to develop model for imaging under patterned illumination.

Fig. 8.
Fig. 8.

Illustration of imaging model for a canonical collection optic.

Fig. 9.
Fig. 9.

Expression for detector irradiance in an active stereo setup.

Fig. 10.
Fig. 10.

Camera image of scene under periodic sinusoidal illumination in the active stereo arrangement of Fig. 7.

Fig. 11.
Fig. 11.

Canonical active stereo arrangement with epipoles at infinity.

Fig. 12.
Fig. 12.

Collocated stereo arrangement with camera epipole at infinity.

Fig. 13.
Fig. 13.

Super-resolution of a diffraction-limited imager affected by computational PSF engineering.

Fig. 14.
Fig. 14.

Super-resolution of a 19 mm biconvex lens affected by computational PSF engineering.

Fig. 15.
Fig. 15.

Super-resolution in a collocated active stereo arrangement.

Fig. 16.
Fig. 16.

Spatial frequency response (SFR) plots.

Fig. 17.
Fig. 17.

Super-resolving singlet using a coincident stereo arrangement.

Fig. 18.
Fig. 18.

Measured PSF at select image field locations.

Fig. 19.
Fig. 19.

Results from attempt to super-resolve a singlet using sinusoidal patterns in the coincident stereo arrangement of Fig. 16.

Fig. 20.
Fig. 20.

Estimating depth in canonical/collocated stereo arrangement.

Fig. 21.
Fig. 21.

Super-resolution by spread spectrum modulation.

Fig. 22.
Fig. 22.

Results from attempt to super-resolve a well-corrected optic using a stochastic illumination pattern. (Zoom-in for a closer view.)

Fig. 23.
Fig. 23.

Spatial frequency response (SFR) plots.

Fig. 24.
Fig. 24.

Super-resolution by spot(s) scanning/lattice illumination.

Fig. 25.
Fig. 25.

Super-resolving a singlet using lattice illumination.

Fig. 26.
Fig. 26.

Space variance in the camera PSF over illuminated field.

Fig. 27.
Fig. 27.

Spatial frequency response (SFR) plots.

Fig. 28.
Fig. 28.

Taxonomy of active computational imaging concepts explored in this work.

Equations (57)

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x0=mpΔZdZoX0+cx,y0=mpΔZdZ0Y0+cy.
[x0y01]=1Z0[mpZd/Δ0cx0mpZd/Δcy001]K[X0Y0Z0].
[x´0y´01]=γ´[m´pZ´d/Δ´0c´x0m´pZ´d/Δ´c´y001]K[r´11r´12r´13r´21r´22r´23r´31r´32r´33]R[X0bXY0bYZ0bZ],
[x´0y´01]=γ´Z0K´R´K1[x0y01]γ´K´R´K1K[bXbYbZ],
[x0y01]=1γ´1Z0KR´TK´1[x´0y´01]+1Z0K[bXbYbZ].
[x´0y´01]=γ´Z0H[x0y01]γ´H[tXtYtZ],
[x0y01]=1γ´1Z0H´[x´0y´01]+1Z0[tXtYtZ].
x´0=h11(Z0x0tX)+h12(Z0y0tY)+h13(Z0tZ)h31(Z0x0tX)+h32(Z0y0tY)+h33(Z0tZ),y´0=h21(Z0x0tX)+h22(Z0y0tY)+h23(Z0tZ)h31(Z0x0tX)+h32(Z0y0tY)+h33(Z0tZ).
x0=(Z0tZZ0)(h´11x´0+h´12y´0+h´13h´31x´0+h´32y´0+h´33)+1Z0tX,y0=(Z0tZZ0)(h´21x´0+h´22y´0+h´23h´31x´0+h´32y´0+h´33)+1Z0tY.
i(x,y)=p(u,v)hcam(xu,yv;u,v)dudv.
iθ(x,y)=(s(u,v)×r(u,v))hcam(xu,yv;u,v)dudv.
s(X,Y,Z)pθ(u´,v´)hill(x´u´,y´v´;u´,v´)du´dv´.
pθ(x´,y´)=A´+B´sin(2π(ξ0x´+η0y´)+θ).
pθ(x´,y´)=A´+B´sin(2π(ξ0h´11x´+h´12y´+h´13h´31x´+h´32y´+h´33+η0h´21x´+h´22y´+h´23h´31x´+h´32y´+h´33)+θ).
s(X,Y,Z)=A´(x´,y´)+|B´(x´,y´)|sin(2π(ξ0x´+η0y´)+B´(u´,v´)+θ).
iθ(x,y)={r(u,v)[A´(u´,v´)+|B´(u´,v´)|sin(2π(ξ0u´+η0v´)+B´(u´,v´)+θ)]hcam(xu,yv;u,v)}dudv.
iθ(x,y)={r(u,v)[A(u,v)+|B(u,v)|sin(φ(u,v)+θ+B(u,v))]hcam(xu,yv;u,v)}dudv.
φ(u,v)=def2πξ0h11(WutX)+h12(WvtY)+h13(WtZ)h31(WutX)+h32(WvtY)+h33(WtZ)+2πη0h21(WutX)+h22(WvtY)+h23(WtZ)h31(WutX)+h32(WvtY)+h33(WtZ),A(u,v)=defA´hill(u´u´,v´v´;u´,v´)du´dv´,B(u,v)=def{exp(j2π(ξ0(u´u´)+η0(v´v´)))×hill(u´u´,v´v´;u´,v´)}du´dv´.]
iθ(x,y)={r(u,v)[A(u,v)+|B(u,v)|sin(φ(u,v)+θ+B(u,v))]hcam(xu,yv;u,v)}dudv.
φ(u,v)=def2π(WtZW)(ξ0u+η0v)(2πWtZ)(ξ0tX+η0tY)A(u,v)=defA´hill(u´u´,v´v´;u´,v´)du´dv´B(u,v)=def{exp(jφ(u´u´,v´v´))×hill(u´u´,v´v´;u´,v´)}du´dv´].
φ(u,v)=2πκo(ξ0u+η0v)+φo+1W2πκd(ξ0bX+η0bY),
φ(u,v)=2πκo(ξ0u+η0v)+φo+1W2πκd(ξ0bX+η0bY),
φ(u,v)=2πκo(ξ0u+η0v)+φo,
κo=def(m´pmpZ´dZdΔΔ´),φo=def2π[ξ0(c´xκocx)η0(c´yκocy)+].
φ(u,v)=2πκo(ξ0u+η0v)+φo,
φo=def[2πξ0(c´x(m´pmpZ´dZdΔΔ´)cx)+2πη0(c´y(m´pmpZ´dZdΔΔ´)cy)2πκoW0(ξ0tX+η0tY)].
irecon(x,y)=defibb(x,y)+(  cos(2πκo(ξ0x+η0y)+φ0)icos(x,y)+sin(2πκo(ξ0x+η0y)+φ0)isin(x,y)).
ibb(x,y)=def14[i0(x,y)+iπ/2(x,y)+iπ(x,y)+i3π/2(x,y)],
icos(x,y)=def12[iπ/2(x,y)i3π/2(x,y)],
isin(x,y)=def12[i0(x,y)iπ(x,y)].
irecon(x,y)=[r(u,v)o(xu,yv;u,v)hcam(xu,yv;u,v)]dudv,
o(x,y;u,v)=defA(u,v)+|B(u,v)|cos(2πκo(ξ0x+η0y)B(u,v)).
hengd(x,y;u,v)=defo(x,y;u,v)×hcam(x,y;u,v).
F{hengd(x,y;u,v)}=F{o(x,y;u,v)}F{hcam(x,y;u,v)}.
pθ(x´,y´)=12+12sin(2πη0(Π´21x´+Π´22y´+Π´23Π´31x´+Π´32y´+Π´33)+θ),
pθ(x´,y´)=12+12sin(2πξ0x´+θ),pθ(x´,y´)=12+12sin(2πη0y´+θ),
i^cos(x,y)=def|B(x,y)|cos(φ^(x,y))r(x,y),
i^sin(x,y)|B(x,y)|sin(φ^(x,y))r(x,y),
φ^(x,y)2πκo(ξ0x+η0y)+φ0+2πZ^1κd(ξ0bX+η0bY).
φ^wrapped(x,y)=defmod(2πκdZ^1(ξ0bX+η0bY),2π).
φ^unwrapped=φ^high+2πround(F  φ^lowφ^high2π).
pθlow(x´,y´)=12+12sin(2πξ0low(π´11x´+π´12y´+π´13π´31x´+π´32y´+π´33)+θ),pθhigh(x´,y´)=12+12sin(2πξ0high(π´11x´+π´12y´+π´13π´31x´+π´32y´+π´33)+θ),
{ξ0low=12592cycpixelξ0high=118cycpixel},θ[0,π/2,π,3π/2],{x[1,1400]y´[1,1050]}.
p(x´,y´)=k,=0M1N1ψ[k,]  g(x´kΔ´,y´Δ´),
i(x,y)=p(u,v)r(u,v)hcam(xu,yv;u,v)dudv.
i(x´,y´)=κo2p(u´,v´)r˜(u´,v´)h˜cam(x´u´,y´v´;u´,v´)du´dv´.
irecon(x´,y´)=κo2s,t=0M1N1{ps,t(x´,y´)[ps,t(u´,v´)r˜(u´,v´)h˜cam(x´u´,y´v´;u´,v´)du´dv´]},
ps,t(x´,y´)=k,=0M1N1ψ[mod(ks,M)mod(t,N),]g(x´kΔ´,y´Δ´).
s,t=0M1N1ps,t(x´,y´)ps,t(u´,v´)=k,=0M1N1m,n=0M1N1{(s,t=0M1N1ψ[mod(ks,M)mod(t,N),]ψ[mod(ms,M)mod(nt,N),])g(x´mΔ´,y´nΔ´)g(u´kΔ´,v´Δ´)},
s,t=0M1N1p(x´sΔ´,y´tΔ´)p(u´sΔ´,v´tΔ´)=k,=0M1N1g(x´kΔ´,y´Δ´)g(u´kΔ´,v´Δ´).
s,t=0M1N1p(x´sΔ´,y´tΔ´)  p(u´sΔ´,v´tΔ´)=g(x´u´2,y´v´2).
irecon(x´,y´)=r˜(u´,v´)g(x´u´2,y´v´2)h˜cam(x´u´,y´v´;u´,v´)du´dv´.
h˜engd(x´,y´;u´,v´)=g(x´2,y´2)h˜cam(x´,y´;u´,v´)g(x´2,y´2).
irecon[s,t]s,t=0M1N1{ps,t(u´,v´)r˜(u´,v´)(h˜cam(x´u´,y´v´;u´,v´)dx´dy´)}du´dv´,
ps,t(u´,v´)=def{g(sΔ´,tΔ´)u´(sΔ´±12Δ´),v´(tΔ´±12Δ´)0otherwise.
irecon[s,t]g(u´sΔ´,v´tΔ´)r˜(u´,v´)du´dv´.
ps,t(x´,y´)=k=1,=1848,480ψ[ks,t]g(x´kΔ´,y´Δ´),

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