Abstract

A simple closed-form solution is derived for reconstructing a 3D spatial-chromatic image cube from a set of chromatically dispersed 2D image frames. The algorithm is tailored for a particular instrument in which the dispersion element is a matching set of mechanically rotated direct vision prisms positioned between a lens and a focal plane array. By using a linear operator formalism to derive the Tikhonov-regularized pseudoinverse operator, it is found that the unique minimum-norm solution is obtained by applying the adjoint operator, followed by 1D filtering with respect to the chromatic variable. Thus the filtering and backprojection (adjoint) steps are applied in reverse order relative to an existing method. Computational efficiency is provided by use of the fast Fourier transform in the filtering step.

© 2006 Optical Society of America

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References

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  1. J. M. Mooney, V. E. Vickers, M. An, and A. K. Brodzik, J. Opt. Soc. Am. A 14, 2951 (1997).
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    [CrossRef]
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  9. M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of Applied Mathematics Series (National Bureau of Standards, 1972).

1999

1997

1995

1988

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 4, 573 (1988).
[CrossRef]

1985

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 1, 301 (1985).
[CrossRef]

An, M.

Bernhardt, P. A.

Bertero, M.

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 4, 573 (1988).
[CrossRef]

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 1, 301 (1985).
[CrossRef]

Brodzik, A. K.

De Mol, C.

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 4, 573 (1988).
[CrossRef]

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 1, 301 (1985).
[CrossRef]

Goncharsky, A. V.

A. N. Tikhonov, A. V. Goncharsky, V. V. Steponov, and A. G. Yagola, Mathematical Methods for the Solution of Ill-Posed Problems (Kluwer, 1995).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 2.

Mooney, J. M.

Petrov, Y. P.

Y. P. Petrov and V. S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications (VSP, 2005).

Pike, E. R.

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 4, 573 (1988).
[CrossRef]

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 1, 301 (1985).
[CrossRef]

Sizikov, V. S.

Y. P. Petrov and V. S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications (VSP, 2005).

Steponov, V. V.

A. N. Tikhonov, A. V. Goncharsky, V. V. Steponov, and A. G. Yagola, Mathematical Methods for the Solution of Ill-Posed Problems (Kluwer, 1995).

Tikhonov, A. N.

A. N. Tikhonov, A. V. Goncharsky, V. V. Steponov, and A. G. Yagola, Mathematical Methods for the Solution of Ill-Posed Problems (Kluwer, 1995).

Vickers, V. E.

Yagola, A. G.

A. N. Tikhonov, A. V. Goncharsky, V. V. Steponov, and A. G. Yagola, Mathematical Methods for the Solution of Ill-Posed Problems (Kluwer, 1995).

Inverse Probl.

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 1, 301 (1985).
[CrossRef]

M. Bertero, C. De Mol, and E. R. Pike, Inverse Probl. 4, 573 (1988).
[CrossRef]

J. Opt. Soc. Am. A

Other

A. N. Tikhonov, A. V. Goncharsky, V. V. Steponov, and A. G. Yagola, Mathematical Methods for the Solution of Ill-Posed Problems (Kluwer, 1995).

Y. P. Petrov and V. S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications (VSP, 2005).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 2.

M.Abramowitz and I.A.Stegun, eds., Handbook of Mathematical Functions, Vol. 55 of Applied Mathematics Series (National Bureau of Standards, 1972).

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Figures (1)

Fig. 1
Fig. 1

Optical schematic of the sensor

Equations (17)

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g ( x ¯ , ϕ ) = f [ x ¯ a ( λ λ 0 ) p ¯ ϕ , λ ] d λ ,
G ( ξ ¯ , ϕ ) = d λ exp [ 2 π i a ( λ λ 0 ) ξ ¯ p ¯ ϕ ] F ( ξ ¯ , λ ) .
( F , F ) U = d 2 ξ ¯ d λ F * ( ξ ¯ , λ ) F ( ξ ¯ , λ ) ,
( G , G ) V = d 2 ξ ¯ 0 2 π d ϕ G * ( ξ ¯ , ϕ ) G ( ξ ¯ , ϕ ) .
G = A F ,
G ( ξ ¯ , ϕ ) = [ A F ] ( ξ ¯ , ϕ ) = d λ e 2 π i a ( λ λ 0 ) ξ ¯ p ¯ ϕ F ( ξ ¯ , λ ) .
( A F , G ) V = d 2 ξ ¯ 0 2 π d ϕ [ A F ] * ( ξ ¯ , ϕ ) G ( ξ ¯ , ϕ ) = d 2 ξ ¯ 0 2 π d ϕ d λ e 2 π i a ( λ λ 0 ) ξ ¯ p ¯ ϕ F * ( ξ ¯ , λ ) G ( ξ ¯ , ϕ ) = d 2 ξ ¯ d λ F * ( ξ ¯ , λ ) [ 0 2 π d ϕ e 2 π i a ( λ λ 0 ) ξ ¯ p ¯ ϕ G ( ξ ¯ , ϕ ) ] = ( F , A H G ) U .
F ( ξ ¯ , λ ) = [ A H G ] ( ξ ̃ , λ ) = 0 2 π d ϕ e 2 π i a ( λ λ 0 ) ξ ¯ p ¯ ϕ G ( ξ ¯ , ϕ ) .
F μ = A μ G = ( A H A + μ I ) 1 A H G ,
F μ = ( A H A + μ I ) 1 F ,
F = A H G ,
F ( ξ ¯ , λ ) = [ ( A H A + μ I ) F μ ] ( ξ ¯ , λ ) = 0 2 π d ϕ exp [ 2 π i a ( λ λ 0 ) ξ ¯ p ¯ ϕ ] d λ exp [ 2 π i a ( λ λ 0 ) ξ ¯ p ¯ ϕ ] F μ ( ξ ¯ , λ ) + μ F μ ( ξ ¯ , λ ) .
F ( ξ ¯ , λ ) = μ F μ ( ξ ¯ , λ ) + d λ M ( ξ ¯ , λ λ ) F μ ( ξ ¯ , λ ) ,
M ( ξ ¯ , λ ) = 0 2 π d ϕ exp [ 2 π i a λ ξ ¯ p ¯ ϕ ] = 0 2 π d ϕ exp [ 2 π i a λ ξ r cos ( ϕ ϕ ) ] = 2 π J 0 ( 2 π a ξ r λ ) .
F ̃ μ ( ξ ¯ , ω λ ) = F ̃ ( ξ ¯ , ω λ ) μ + M ̃ ( ξ ¯ , ω λ ) ,
M ̃ ( ξ ¯ , ω λ ) = d λ e 2 π i ω λ λ M ( ξ ¯ , λ ) = 2 ( a 2 ξ r 2 ω λ 2 ) 1 2 ,
0 e i k z J 0 ( k b ) d k = ( b 2 z 2 ) 1 2

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