1. Or How I learned to stop worrying and love the analytic gradient
Word on the street is that at some point
Application of Algorithmic Differentiation to Turbulence Mitigation in Digital Holography
Or How I learned to stop worrying and love the analytic gradient
Wesley E. Farriss and James R. Fienup
Clrc 2018
Okinawa, Japan

2. Agenda Sharpness-based turbulence mitigation in digital holography
Introduction to sharpness maximization correction
Sharpness metric model
Motivation for algorithmic differentiation
Algorithmic differentiation
Concept
Application to sharpness-based turbulence mitigation
Advantages of algorithmic differentiation
Example sharpening problem
Future work
So one of the

3. Introduction to sharpness maximization correction
It has been shown that volumetric Kolmogorov turbulence can be well represented by 2D phase planes
Maximizing sharpness metric with respect phase planes shown to improve images
Also generates an estimate of phase screens
Cite Dainty Phase Screen paper. With respect 'to' phase in multiple planes
[1] Lane, R. G., A. Glindemann, and J. C. Dainty. "Simulation of a Kolmogorov phase screen." Waves in random media 2.3 (1992):
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Target
Volumetric Turbulence
Detector

4. Introduction to sharpness maximization correction
Atmospheric turbulence causes serious deleterious effect on remote imaging systems
Volumetric turbulence also causes anisoplanatic effects
i.e. aberration that is non-uniform across the image
Cite turbulence in imaging book Roggeman
[1] Roggemann, Michael C., Byron M. Welsh, and Bobby R. Hunt. Imaging through turbulence. CRC press, 2018.
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5. Sharpness metric model
𝛹 q ( 𝑥 𝑞 , 𝑦 𝑞 )= 𝑒𝑥𝑝 2𝜋 𝜆 𝑚 𝑐 q,𝑚 ∘ 𝑍 𝑚 ( 𝑥 𝑞 , 𝑦 𝑞 ) + 2𝜋 𝜆 𝛩 i,𝑝𝑏𝑝 ( 𝑥 𝑞 , 𝑦 𝑞 )
𝑭 𝒌, 𝒅𝒆𝒕
⋄ 𝚿 𝟏
⋄ 𝚿 𝟐
⋄ 𝚿 𝑸−𝟏
𝑭 𝒌, 𝒕𝒈𝒕
⋄ 𝚿 𝑸
𝒫 0→1
𝒫 2→3 … 𝒫 Q−2→𝑄−1
𝒫 1→2
𝒫 Q−1→𝑄
𝒫 Q→𝑡𝑔𝑡
𝑰 𝒙 𝒕𝒈𝒕 , 𝒚 𝒕𝒈𝒕 = 𝟏 𝑲 𝒌 𝑭 𝒌,𝒕𝒈𝒕 ( 𝒙 𝒕𝒈𝒕 , 𝒚 𝒕𝒈𝒕 ) 𝟐
Fix alpha and I breve
𝜳 𝒒 𝒙 𝒒 , 𝒚 𝒒 = 𝒆𝒙𝒑 𝒊 𝟐𝝅 𝝀 𝒎 𝒄 𝒒,𝒎 𝒁 𝒎 + 𝟐𝝅 𝝀 𝜣 𝒒
𝒔= 𝒙 𝒕𝒈𝒕 , 𝒚 𝒕𝒈𝒕 𝑰 𝜶 𝒙 𝒕𝒈𝒕 , 𝒚 𝒕𝒈𝒕
[1] Abbie E. Tippie and James R. Fienup, "Phase-error correction for multiple planes using a sharpness metric," Opt. Lett.34, (2009)
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Target
Detector …

6. Motivation for algorithmic differentiation
Sharpness maximization accomplished by gradient-based optimization algorithm
Must use expression for analytic gradient
Finite differences is the enemy
Number of metric evaluations in the gradient scales as number of parameters
Constantly re-writing functions that involve similar mathematics in order to make small exploratory model changes
Openly available auto-differentiators are not reliable for precision needs of optics
Former graduate student who used only finite differences in optimization
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7. Motivation for algorithmic differentiation
animate
[1]Tippie, Abbie E. Aberration correction in digital holography. University of Rochester, 2012.
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8. ©LEIDOS. ALL RIGHTS RESERVED.
Algorithmic differentiation gains
Never have to calculate entire gradient at one time
Less error prone; even with respect to math engines derivations
Eliminates need for redundant calculations in gradient step
Modularity in function construction of function library
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9. Algorithmic differentiation concept
𝑥 𝑛 = 𝜕𝑠 𝜕 𝑥 𝑛 ,
𝑠≡some metric, 𝑥 𝑛 ≡independent parameter of 𝑠 we wish to optimize for
Rules of differentiation necessitate derivative of 𝑠 be taken with respect to all constituent sub-functions (chain rule)
𝑥 𝑛 = 𝜕𝑠 𝜕 𝑥 𝑛 = 𝜕𝑠 𝜕𝑎 𝜕𝑎 𝜕 𝑥 𝑛 = 𝑎 𝜕𝑎 𝜕 𝑥 𝑛 𝜕𝑠 𝜕 𝑥 𝑛 = 𝜕𝑠 𝜕𝑏 𝜕𝑏 𝜕𝑎 𝜕𝑎 𝜕 𝑥 𝑛 = 𝑏 𝜕𝑏 𝜕𝑎 𝑎 𝜕𝑎 𝜕 𝑥 𝑛 𝜕𝑠 𝜕 𝑥 𝑛 = 𝜕𝑠 𝜕𝑐 𝜕𝑐 𝜕𝑏 𝜕𝑏 𝜕𝑎 𝜕𝑎 𝜕 𝑥 𝑛 = 𝑐 𝜕𝑐 𝜕𝑏 𝑏 𝜕𝑏 𝜕𝑎 𝜕𝑎 𝜕 𝑥 𝑛
Cite Alden Paper
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10. Algorithmic differentiation concept
Practical Example: Forward Model: s= 𝑥,𝑦 𝐵(𝑥,𝑦) 𝐵 𝑥,𝑦 = 𝐼 𝛼 𝑥,𝑦 𝐼 𝑥,𝑦 = 𝑘 𝐼 𝑘 (𝑥,𝑦) 𝐼 𝑘 = 𝐹 𝑘 𝑥,𝑦 2
Gradient Model: B = 𝑠 ∀ 𝑥,𝑦 𝐼 = 𝛼 𝐼 𝛼−1 𝑥,𝑦 ∘ 𝐵 𝐼 𝑘 𝑥,𝑦 = 𝐼 (𝑥,𝑦) F 𝑥,𝑦 =2 𝐹(𝑥,𝑦)∘ 𝐼 (𝑥,𝑦)
Jurling, Alden S., and James R. Fienup. "Applications of algorithmic differentiation to phase retrieval algorithms." JOSA A 31.7 (2014):
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11. Application to sharpness-based turbulence mitigation
Forward Model
Gradient Model
Metric Block
Propagation Block
Phase Component
Block
Phase Object Block
𝑠= 𝑥,𝑦 𝐼 𝛼
F k,q =𝒫 F k,q−1 ′
Φ q = Φ 𝑍,𝑞 + Φ 𝑝𝑏𝑝,𝑞
𝐼 =α I α−1
F k,q−1 ′ = 𝒫 −1 F k,q
Φ q =Im Ψ q ⋄ Ψ q ∗
Intensity Block
F k,q−1 ′ = F k,q−1 ⋄ Ψ q−1
Φ 𝑝𝑏𝑝,𝑞 = 2π λ Θ 𝑝𝑏𝑝,𝑞
Ψ q−1 = k 𝐾 F k,q−1 ′ ⋄ F 𝑘,q−1 ∗
𝐼= 1 𝐾 𝑘 𝐾 𝐼 𝑘
Φ Z,q = 2π λ m c q,m Z m
𝐼 𝑘 = 1 K 𝐼
F k,q−1 = F k,q−1 ′ ⋄ Ψ q−1 ∗
Θ q,pbp = 2π λ Φ q
I k = F k,tgt 2
Ψ q = exp 𝑖 Φ q
F 𝑘,𝑡𝑔𝑡 =2 F k,tgt ⋄ I k
𝑐 𝑞,𝑚 = 2𝜋 𝜆 𝑥 𝑞 , 𝑦 𝑞 Φ 𝑞 ⋄ 𝑍 𝑚
Jurling, Alden S., and James R. Fienup. "Applications of algorithmic differentiation to phase retrieval algorithms." JOSA A 31.7 (2014):
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12. Example sharpening problem
Phase planes use Chebyshev type I polynomials
Each plane fit with 250 polynomials, 25 at a time
Point-by-point phase optimized after polynomial fit locked in
12 Speckle Realizations
Quadratic terms removed in center phase plane to avoid afocal telescoping/ oversharpnening errors
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Turb Screen 2 𝐷 𝑟 0 z = 75m
Turb Screen 1 𝐷 𝑟 0 z = 0m
0.1m
0.2m
Phase planes 1 (Smooth phase) and 2 z=0 m
Phase planes 3 (Smooth phase) and 4 z=75m
z=150m

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Example sharpening problem
(a) (b) (c)
Aberrated Image
Corrected Image
Ideal image
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14. Future Work
Use algorithmic differentiation metric library to simulate variety of metrics and system configurations using 2D digital holography
Determine ideal means to experimentally conduct turbulence mitigation using sharpness maximization
Optimize algorithmic differentiation library for fast computation
Use software library to clean up images take from lab data
Use modular components developed for 2D digital holography in effort to mitigate effects of turbulence in 3D holographic aperture ladar images (Shameless plug for my SPIE Optics and Photonics talk, August 2018, San Diego, CA, USA)

15. Questions