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Single-shot compressive spectral imaging with a dual-disperser architecture

M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz

Open Access Open Access

Abstract

This paper describes a single-shot spectral imaging approach based on the concept of compressive sensing. The primary features of the system design are two dispersive elements, arranged in opposition and surrounding a binary-valued aperture code. In contrast to thin-film approaches to spectral filtering, this structure results in easily-controllable, spatially-varying, spectral filter functions with narrow features. Measurement of the input scene through these filters is equivalent to projective measurement in the spectral domain, and hence can be treated with the compressive sensing frameworks recently developed by a number of groups. We present a reconstruction framework and demonstrate its application to experimental data.

©2007 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. Schematic of the spectral imager.

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Fig. 2.
Fig. 2. Distribution of filter functions that arises from simple tiling of the fundamental codeword. No compact region contains all 15 filters.

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Fig. 3.
Fig. 3. Distribution of filter functions that arises from the more complicated unit cell. Any 3×5 region contains all 15 filters.

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Fig. 4.
Fig. 4. Sample partition of a spatio-spectral data cube. The spatial partition is the same at each spectral band, making it impossible for the estimation method to perform spatial smoothing at some spectral bands but not others.

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Fig. 5.
Fig. 5. The experimental prototype.

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Fig. 6.
Fig. 6. Experimental results from simple targets with monochromatic illumination. (a) Detector image recorded for 532 nm illumination. (b) Intensity image generated by summing the spectral information in the reconstruction for 532 nm illumination. (c) Spectral reconstruction at a particular spatial location for 532 nm illumination. (d) Spectral reconstruction at a particular spatial location for 543 nm illumination.

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Fig. 7.
Fig. 7. Experimental results from simple targets with narrow-band illumination. (a) Detector image recorded for illumination with a 10 nm FWHM bandpass centered at 560 nm. (b) Intensity image generated by summing the spectral information in the reconstruction for the 560 nm bandpass. (c) Spectral reconstruction at a particular spatial location for the 560 nm bandpass. (d) Spectral reconstruction at a particular spatial location for the 580 nm bandpass. The origin of the small peak near 520 nm is explained in the text.

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Fig. 8.
Fig. 8. Experimental results from real-world objects under broadband (white) illumination. (a) Detector image recorded by the system. (b) Reconstructed intensity image of the scene. (c) Spectral reconstructions for spatial locations in the three regions.

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Fig. 9.
Fig. 9. Slices through the reconstructed datacube at 8 particular spectral channels.

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Fig. 10.
Fig. 10. Animation showing all 36 reconstructed spectral channels. (Multimedia file, 1.1 MB.) [Media 1]

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Equations (31)

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S 1 ( x , y ; λ ) = ∬ dx ′ dy ′ δ ( x ′ − [ x + α ( λ − λ c ) ] ) δ ( y ′ − y ) S 0 ( x ′ , y ′ ; λ )
= S 0 ( x + α ( λ − λ c ) , y ; λ ) .
S 2 ( x , y ; λ ) = T ( x , y ) S 1 ( x , y ; λ ) = T ( x , y ) S 0 ( x + α ( λ − λ c ) , y ; λ ) ,
S 3 ( x , y ; λ ) = ∬ dx ′ dy ′ δ ( x ′ − [ x − α ( λ − λ c ) ] ) δ ( y ′ − y ) S 2 ( x ′ , y ′ ; λ )
= T ( x − α ( λ − λ c ) , y ) S 0 ( x , y ; λ )
= H ( x , y ; λ ) S 0 ( x , y ; λ ) .
I ( x , y ) = ∫ dλH ( x , y ; λ ) S 0 ( x , y ; λ ) .
I nm = ∭ dxdydλ rect x Δ − m y Δ − n H ( x , y ; λ ) S 0 ( x , y ; λ ) .
T x y = ∑ m ′ , n ′ T n′m ′ rect x Δ − m ′ y Δ − n ′ .
I nm = ∑ m′n′ ∭ dxdydλ rect x Δ − m y Δ − n rect x − α ( λ − λ c ) Δ − m ′ y Δ − n ′
× T n′m′ S 0 ( x , y ; λ ) .
I nm ∣ λ = λ c = ∑ m′n′ ∭ dxdydλ rect x Δ − m y Δ − n rect x Δ − m′ y Δ − n ′
× T n′m′ I 0 x y δ ( λ − λ c )
= ∑ m′n′ δ mm′ δ nn ′ T n′m′ I 0 , nm
= T nm I nm ,
I nm ∣ λ = λ c + Δλ = ∑ m′n′ ∭ dxdydλ rect x Δ − m y Δ − m rect x Δ − ( m′ + 1 ) y Δ − n ′
× T n′m′ I 0 x y δ ( λ − ( λ c + Δλ ) )
= ∑ m′n′ δ mm′ δ nn′ T n′ ( m′ + 1 ) I 0 , nm
= T n ( m − 1 ) I 0 , nm .
w nmp = ∑ m′n′ ∭ dxdydλ rect x Δ − m y Δ − n λ − λ c Δλ − p
× rect x − α ( λ − λ c ) Δ − m′ y Δ − n′ T n′m′ ,
S nmp = 1 Δ 2 Δλ ∭ dxdydλ rect x Δ − m y Δ − n λ − λ c Δλ − p S 0 ( x , y ; λ ) .
I nm = ∑ p w nmp s nmp .
I = Ws ,
d ~ Poisson ( I ) = Poisson ( Ws ) ,
p ( d ∣ Ws ) = ∏ n = 1 N ∏ m = 1 M e − ∑ p w nmp s nmp ( ∑ p w nmp s nmp ) d nm d nm ! .
s ̂ = arg  min s ˜ ∈ S { − log p ( d | W s ˜ ) + pen ( s ˜ ) } ,
y ( t ) = s ̂ ( t ) . × W T ( d . / W s ̂ ( t ) ) ,
s ̂ ( t + 1 ) = arg  min s ˜ { − log p ( y ( t ) | s ˜ ) + pen ( s ˜ ) } .
P ̂ ≡ arg  min P { − log p ( y ( t ) ∣ s ˜ ( P ) ) + pen ( P ) }
s ̂ ≡ s ˜ ( P ^ ) ,
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