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Structured illumination microscopy with unknown patterns and a statistical prior

Li-Hao Yeh, Lei Tian, and Laura Waller

Open Access Open Access

Abstract

Structured illumination microscopy (SIM) improves resolution by down-modulating high-frequency information of an object to fit within the passband of the optical system. Generally, the reconstruction process requires prior knowledge of the illumination patterns, which implies a well-calibrated and aberration-free system. Here, we propose a new algorithmic self-calibration strategy for SIM that does not need to know the exact patterns a priori, but only their covariance. The algorithm, termed PE-SIMS, includes a pattern-estimation (PE) step requiring the uniformity of the sum of the illumination patterns and a SIM reconstruction procedure using a statistical prior (SIMS). Additionally, we perform a pixel reassignment process (SIMS-PR) to enhance the reconstruction quality. We achieve 2Γ— better resolution than a conventional widefield microscope, while remaining insensitive to aberration-induced pattern distortion and robust against parameter tuning.

Β© 2017 Optical Society of America

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Figures (8)
Fig. 1
Fig. 1 Example experimental setup for structured illumination microscopy (SIM) using a deformable mirror device (DMD) to capture low-resolution images of the object modulated by different illumination patterns. Our IPE-SIMS algorithm reconstructs both the super-resoloved image and the unknown arbitrary illumination patterns.

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Fig. 2
Fig. 2 The first part of our algorithm, Pattern Estimation (PE), iteratively estimates the illumination patterns from an approximated object given by the deconvolved widefield image.

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Fig. 3
Fig. 3 The second part of our algorithm, termed structured illumination microscopy with a statistical prior (SIMS), estimates the high-resolution object from the measured images and the estimated illumination patterns obtained in Part 1.

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Fig. 4
Fig. 4 (a) Simulated reconstructions of a Siemens star target under a widefield microscope, deconvolved widefield, confocal microscope, deconvolved confocal, blind SIM [16], S-SOFI [20], our PE-SIMS and PE-SIMS-PR algorithms. (b) The effective modulation transfer function (MTF) of each method, given by the contrast of the reconstructed Siemens star image at different radii.

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Fig. 5
Fig. 5 Reconstructions of red fluorescent beads (Ex:580 nm/Em:605 nm) from the experiment using random pattern illumination with 20 Γ— 20 scanning step.

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Fig. 6
Fig. 6 Comparison of our algorithm on dataset from Multispot SIM (MSIM) which uses with Nimg = 224 scanned multi-spot patterns from [10]. We show the deconvolved widefield image and the reconstructions using MSIM with known patterns, as well as our blind PE-SIMS algorithm with and without pixel reassignment.

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Fig. 7
Fig. 7 Results with simulated and experimental (fluorescent beads) datasets comparing random speckle and multi-spot illumination patterns. (middle row) Shading maps overlaid on the object. Decreasing the number of random patterns results in shading artifacts in the reconstruction. The random patterns are scanned in 20 Γ— 20, 10 Γ— 10, and 6 Γ— 6 steps with the same step size of 0.6 FWHM of the PSF, while the multi-spot pattern is scanned with 6 Γ— 6 steps.

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Fig. 8
Fig. 8 (a) Comparison of the PSF and OTF for SIMS and SIMS with pixel reassignment (PR). (b) Comparisons of the deconvolved widefield image and the reconstructions of the 6 Γ— 6 multi-spot scanned fluorescent beads with and without pixel reassignment.

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Tables (2)

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Subroutine 1 Pattern Estimation

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Table 1 Achieved resolution for different algorithms

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Equations (19)

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I β„“ ( r ) = [ o ( r ) β‹… p β„“ ( r ) ] βŠ— h det ( r ) = ∬ o ( r β€² ) p β„“ ( r β€² ) h det ( r βˆ’ r β€² ) d 2 r β€² .
I avg ( r ) = γ€ˆ I β„“ ( r ) 〉 β„“ = [ o ( r ) β‹… γ€ˆ p β„“ ( r ) 〉 β„“ ] βŠ— h det ( r ) β‰ˆ p 0 o ( r ) βŠ— h det ( r ) ,
o est ( r ) = β„± βˆ’ 1 { I ˜ avg ( u ) β‹… h ˜ det ( u ) | h ˜ det ( u ) | 2 + Ξ² } ,
minimize p β„“ f ( p β„“ ) = f diff ( p β„“ ) + I C ( p β„“ ) = βˆ‘ r | I β„“ ( r ) βˆ’ [ o est ( r ) β‹… p β„“ ( r ) ] βŠ— h det ( r ) | 2 + I C ( p β„“ ) , where I C ( p β„“ ) = { 0 , p β„“ ∈ C + ∞ , p β„“ βˆ‰ C , C = { p β„“ ( r ) | p ˜ β„“ ( u ) = 0 , βˆ€ u > 2 N A Ξ» illu } ,
g β„“ ( k ) ( r ) = βˆ‚ f diff ( p β„“ ( k ) ) βˆ‚ p β„“ = βˆ’ 2 o est ( r ) β‹… [ h det ( r ) βŠ— ( I β„“ ( r ) βˆ’ [ o est ( r ) β‹… p β„“ ( k ) ( r ) ] βŠ— h det ( r ) ) ] ,
∏ C ( y ) = β„± βˆ’ 1 { β„± { y } β‹… | h ˜ illu ( u ) | 2 | h ˜ illu ( u ) | 2 + Ξ΄ } ,
I cov ( r ) = γ€ˆ Ξ” p β„“ ( r ) Ξ” I β„“ ( r ) 〉 β„“ = ∬ o ( r β€² ) γ€ˆ Ξ” p β„“ ( r ) Ξ” p β„“ ( r β€² ) 〉 β„“ h det ( r βˆ’ r β€² ) d 2 r β€² ,
p β„“ ( r ) = ∬ t β„“ ( r β€² ) h illu ( r βˆ’ r β€² ) d 2 r β€² ,
γ€ˆ Ξ” p β„“ ( r ) Ξ” p β„“ ( r β€² ) 〉 β„“ = ∬ ∬ γ€ˆ Ξ” t β„“ ( r 1 ) Ξ” t β„“ ( r 2 ) 〉 β„“ h illu ( r βˆ’ r 1 ) h illu ( r β€² βˆ’ r 2 ) d 2 r 1 d 2 r 2 = ∬ ∬ Ξ³ t γ€ˆ Ξ” t β„“ 2 ( r 1 ) 〉 β„“ Ξ΄ ( r 1 βˆ’ r 2 ) h illu ( r βˆ’ r 1 ) h illu ( r β€² βˆ’ r 2 ) d 2 r 1 d 2 r 2 β‰ˆ Ξ± t ∬ h illu ( r βˆ’ r 1 ) h illu ( r β€² βˆ’ r 1 ) d 2 r 1 = Ξ± t ( h illu ⋆ h illu ) ( r βˆ’ r β€² ) ,
γ€ˆ Ξ” t β„“ ( r 1 ) Ξ” t β„“ ( r 2 ) 〉 β„“ = Ξ³ t γ€ˆ Ξ” t β„“ 2 ( r 1 ) 〉 β„“ Ξ΄ ( r 1 βˆ’ r 2 ) β‰ˆ Ξ± t Ξ΄ ( r 1 βˆ’ r 2 ) ,
I cov ( r ) = γ€ˆ Ξ” p β„“ ( r ) Ξ” I β„“ ( r ) 〉 β„“ = ∬ Ξ± t o ( r β€² ) [ ( h illu ⋆ h illu ) β‹… h det ] ( r βˆ’ r β€² ) d 2 r β€² .
I cov , dec ( r ) = β„± βˆ’ 1 { I ˜ cov ( u ) β‹… H ( u ) | H ( u ) | 2 + ΞΎ } ,
Ξ± t ( r ) = Ξ³ t γ€ˆ Ξ” t β„“ 2 ( r ) 〉 β„“ β‰ˆ Ξ³ t γ€ˆ Ξ” p β„“ 2 ( r ) 〉 β„“ .
I SIMS ( r ) = I cov , dec ( r ) β‹… Ξ± t ( r ) Ξ± t 2 ( r ) + Ο΅ ,
t β„“ ( r ) = Ξ› βˆ‘ m , n Ξ΄ ( r βˆ’ r m n βˆ’ r β„“ ) + t 0 Ξ” t β„“ ( r ) β‰ˆ Ξ› 2 βˆ‘ m , n Ξ΄ ( r βˆ’ r m n βˆ’ r β„“ ) ,
γ€ˆ Ξ” t β„“ ( r 1 ) Ξ” t β„“ ( r 2 ) 〉 β„“ = ∬ Ξ” t ( r 1 βˆ’ r β„“ ) Ξ” t ( r 2 βˆ’ r β„“ ) d 2 r β„“ = Ξ› 4 βˆ‘ m , n Ξ΄ ( r 1 βˆ’ r 2 βˆ’ r m n ) ⋆ βˆ‘ m , n Ξ΄ ( r 1 βˆ’ r 2 βˆ’ r m n ) β‰ˆ Ξ› 4 Ξ· βˆ‘ m , n Ξ΄ ( r 1 βˆ’ r 2 βˆ’ r m n ) ,
γ€ˆ Ξ” p β„“ ( r ) Ξ” p β„“ ( r β€² ) 〉 β„“ = ( h illu ⋆ h illu ) ( r βˆ’ r β€² ) βŠ— Ξ› 4 Ξ· βˆ‘ m , n Ξ΄ ( r βˆ’ r β€² βˆ’ r m n ) .
I cov s ( r , r s ) = γ€ˆ Ξ” p β„“ ( r βˆ’ r s ) Ξ” I β„“ ( r ) 〉 β„“ = ∬ o ( r β€² ) γ€ˆ Ξ” p β„“ ( r βˆ’ r s ) Ξ” p β„“ ( r β€² ) 〉 β„“ h ( r βˆ’ r β€² ) d 2 r β€² = ∬ Ξ± t o ( r β€² ) ( h illu ⋆ h illu ) ( r βˆ’ r s βˆ’ r β€² ) h det ( r βˆ’ r β€² ) d 2 r β€² .
I PR ( r ) = ∬ I cov s ( r + r s 2 , r s ) d 2 r s = ∬ Ξ± t o ( r β€² ) [ ∬ ( h illu ⋆ h illu ) ( r βˆ’ r s 2 βˆ’ r β€² ) h det ( r + r s 2 βˆ’ r β€² ) d 2 r s ] d 2 r β€² = ∬ Ξ± t o ( r β€² ) [ ( h illu ⋆ h illu ) βŠ— h det ] ( 2 ( r βˆ’ r β€² ) ) d 2 r β€²
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