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

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

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): 209-224. ©LEIDOS. ALL RIGHTS RESERVED.   Target Volumetric Turbulence Detector

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. ©LEIDOS. ALL RIGHTS RESERVED.  

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, 701-703 (2009) ©LEIDOS. ALL RIGHTS RESERVED.   Target Detector …

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 ©LEIDOS. ALL RIGHTS RESERVED.  

Motivation for algorithmic differentiation animate [1]Tippie, Abbie E. Aberration correction in digital holography. University of Rochester, 2012. ©LEIDOS. ALL RIGHTS RESERVED.  

©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 ©LEIDOS. ALL RIGHTS RESERVED.  

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 ©LEIDOS. ALL RIGHTS RESERVED.  

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): 1348-1359. ©LEIDOS. ALL RIGHTS RESERVED.  

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): 1348-1359. ©LEIDOS. ALL RIGHTS RESERVED.  

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 ©LEIDOS. ALL RIGHTS RESERVED.   Turb Screen 2 𝐷 𝑟 0 =5 @ z = 75m Turb Screen 1 𝐷 𝑟 0 =5 @ z = 0m 0.1m 0.2m Phase planes 1 (Smooth phase) and 2 (PBP) @ z=0 m Phase planes 3 (Smooth phase) and 4 (PBP) @ z=75m Target @ z=150m

©LEIDOS. ALL RIGHTS RESERVED. Example sharpening problem (a) (b) (c) Aberrated Image Corrected Image Ideal image ©LEIDOS. ALL RIGHTS RESERVED.  

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)

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