Predictive Dynamic Digital Holography

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Mechanical and Aerospace Engineering Beam Control Laboratory

Sennan Sulaiman Steve Gibson

Mechanical and Aerospace Engineering UCLA [email protected]

1 1 Mechanical and Aerospace Engineering Beam Control Laboratory

Predictive Dynamic Digital Holography

Sennan Sulaiman Steve Gibson

Mechanical and Aerospace Engineering UCLA [email protected]

Thanks to Mark Spencer and Dan Marker AFRL, Kirtland AFB

2 2 Mechanical and Aerospace Engineering Beam Control Laboratory

Motivation

• Digital holography has received heightened attention in recent applications: – Optical Tweezing – Adaptive Optics – Wavefront Sensing

• Major advantages of digital holography over classical AO: – Fewer optical components required – No wavefront sensing device required (Shack-Hartmann or SRI) – Versatility of processing techniques on both amplitude and phase information

• Digital holography drawback: – Iterative numerical wavefront propagation and sharpness metric optimization • Computationally intensive

3 Mechanical and Aerospace Engineering Beam Control Laboratory

Classical Adaptive Optics

Optical Beam with Corrected Wavefront Adaptive Image of Wavefront (Phase) Optics Distant Point Source Sensor Algorithm Sharpened (Beacon) Image

Adaptive Optics with Wavefront Prediction

Optical Beam with Wavefront Corrected Wavefront Adaptive Image of Prediction Wavefront (Phase) Optics Distant Point Source Sensor Adaptive Algorithm Sharpened (Beacon) & Optimal Image

Wavefront prediction compensates for loop delays due to sensor readout, processing, bandwidth of deformable mirrors.

4 Mechanical and Aerospace Engineering Beam Control Laboratory Digital Holography

Optical Beam with Digital Wavefront Image of Holography Correction Extended Object Image Sharpened (Target) Sharpening Image

Predictive Digital Holography

Optical Beam with Digital Wavefront Image of Holography Correction Extended Object Image Sharpened (Target) Sharpening Image

Wavefront Prediction

Wavefront prediction accelerates image sharpening to enable real-time digital holography for adaptive optics and target tracking.

5 Mechanical and Aerospace Engineering Beam Control Laboratory Digital Holography Interferometry Overview

Illuminating Beam

Object Plane Pupil Plane Image Plane

6 Mechanical and Aerospace Engineering Beam Control Laboratory Digital Holography Interferometry Overview

Object Beam •Object actively illuminated at Object Plane. Hologram Object Beam and LO •Passes through phase mask at Pupil Plane. interference pattern is recorded •Phase masks: Aero-optical wavefronts on Image Plane.

•Fourier propagation to Image Plane. Generates Hologram

Local Oscillator (LO) •Illuminating beam is split off direct to Image Plane •LO incidence is tilted

•Represents “Off-Axis” Digital Holography method

7 Mechanical and Aerospace Engineering Beam Control Laboratory Digital Holography Imaging a Point Source

Inverse True Wavefront Phase Fourier Hologram Reconstructed Image: Phase Fourier Propagation Propagation

Object Plane Image Plane Pupil Plane

Pupil Plane Estimate: Phase

Spatial separation of the Object Image occurs due to tilted LO.

Enables reconstructing the image from the Hologram.

Simple point source imaging of the pupil plane.

8 Mechanical and Aerospace Engineering Beam Control Laboratory Digital Holography Imaging a USAF Bar Chart through Phase Aberration

Inverse True Object Fourier Hologram Reconstructed Image: Phase Fourier Propagation Propagation

Phase Screen

Object Plane Image Plane Pupil Plane

Initial Estimate Image: Amplitude  The phase of the Object Image gets corrupted by the wavefront.

 The Sharpness algorithm will try to correct for the wavefront.

9 Image Plane Mechanical and Aerospace Engineering Beam Control Laboratory Sharpness Algorithm Sharpness Optimization • Sharpness optimization: – Phase correction applied to wavefront estimate on Pupil Plane – Correction is projected onto nth order Zernike Polynomial over a circular pupil

– Determined by optimizing over a sharpness metric criterion: • Criteria (I = Irradiance):  – S1 : S  I (Bright pixels brighter for β > 1) (Dark pixels darker for β < 1)

– S2 : S  I log( I) (Dark pixels darker)

• Local Optimizer: – Quasi-Newton method • Analytic gradient calculated. – Significantly improves computational performance.

10 Mechanical and Aerospace Engineering Beam Control Laboratory Sharpness Algorithm Sharpness Optimization Model

 Coefficients initialized to zero.

The iterative propagation is the costliest operation.

11 Mechanical and Aerospace Engineering Beam Control Laboratory Sharpness Algorithm Sharpening a Single Wavefront (Frame) Uncorrected Image: Amplitude Corrected Image: Amplitude

Sharpness Optimization

Wavefront Aberration at Pupil Plane Wavefront Correction Estimate: Sharpness 

12 Mechanical and Aerospace Engineering Beam Control Laboratory Sharpness Algorithm Sharpening a Series of Wavefront (Frames)

Image Amplitude Error 60 Corrected Average: 8.57 • Sharpness optimization Uncorrected Average: 26.699 performed on a sequence of 50 aero-optical wavefront frames 40

30 • Average Image RMSE: 20 – Uncorrected frames: 27%

– Sharpened Frames: 8.6% ImageAmplitude Error) (% 10

0 0 50 100 150 200 250 300 350 400 450 500 Sequence of Frames

13 Mechanical and Aerospace Engineering Beam Control Laboratory Sharpness Algorithm Sharpening a Region of Interest (ROI) Corrected Image: Amplitude • Local Region of Interest: – Subset of the image plane pixels to concentrate sharpness optimization

• Wavefront correction focuses on optimizing one particular region of the image. Local Corrected Image: Amplitude – Generally at expense of the overall image.

14 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography System Identification and Minimum-Variance Prediction of Estimated Wavefronts

UCLA State-space Prediction Filter Estimated Adaptive Pupil Plane Wavefronts x(t 1)  Ax(t)  K[y(t)  yˆ(t)] Subspace (Sample Sequence) System ID yˆ(t)  Cx(t)

y(t)  estimated wavefront in Zernike modes ˆ y(t)  predicted wavefront

• Estimated pupil plane wavefronts constructed by digital holography

• Pupil plane imaged in the presence of aero-optical wavefronts (Notre Dame fight data)

• DH-constructed wavefronts projected onto Zernikes (or other modes)

Uses for Prediction Filter

• Minimum-variance (optimal) linear time-invariant (LTI) wavefront control in adaptive optics

• Integrate prediction with image sharpening in dynamics digital holography

15 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography PSDs for Prediction of Selected Zernike Sequences

Time Series PSD Mode 2 Mode 2 1 0

0.5 -20 0 -40 -0.5

-1 -60 0 500 1000 1500 2000 2500 3000 0 2000 4000 6000 8000 10000 Zernike Coefficient Mode 4 Mode 4 0.4 Prediction-10 Error

0.2 -20

0 -30

-0.2 -40

-0.4 -50 0 500 1000 1500 2000 2500 3000 0 2000 4000 6000 8000 10000

Mode 10 Mode 10 0.2 -20

0.1 -30 0 -40 -0.1

-0.2 -50 0 500 1000 1500 2000 2500 3000 0 2000 4000 6000 8000 10000 Frame Number 16 Frequency (Hz) Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Predictive Dynamic Model: Ultimate Goal

17 Mechanical and Aerospace Engineering Beam Control Laboratory

Predictive Dynamic Digital Holography Motivation

• Use optimal or adaptive prediction filter to improve sharpening for a fixed number of optimization iterations

• Equivalently, reduce number of iterations required to achieve near-optimal sharpening

18 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Simulations • Image USAF chart through sequence of aero-optical wavefront aberrations

• Employ digital holography and image sharpening to estimate wavefront – Correct pupil plane phase estimate projected onto first 15 Zernikes – Time series produced using various numbers of sharpening iterations

• A region of interest (ROI) is analyzed: – Global Sharpness: Using every pixel in the image plane – Local Sharpness: Using the localized pixels in the image plane for the ROI.

• Benchmark wavefronts: – Wavefronts obtained by projecting the true wavefronts onto first 15 Zernikes.

Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Wavefront Correction by Image Sharpening vs. Benchmark Wavefronts True Wavefront at Pupil Plane Wavefront Bias 40 Average:11.9952 35

WavefrontError) (% 10 Zernike Fitted Wavefront Correction 5

0 0 50 100 150 200 250 300 350 400 450 500 Sequence of Frames

 Average Error from True Benchmark Wavefront: 12%

 Sharpening Iterations: 10

20 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Sharpened Images vs. True (Unaberrated) Images

True Object Image Amplitude Bias 25 Average:9.1248

ImageAmplitude Error) (% Zernike Fitted Image Correction

0 0 50 100 150 200 250 300 350 400 450 500 Sequence of Frames

 Average Error from True Image: 9%

 Sharpening Iterations: 10

21 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Simulations • Two sharpness metrics are investigated: S  I  – S1 : , β =1 S  I log( I) – S2 :

• Three correction methods are investigated:

– M0 : For each frame, initial Zernike coefficient vector for sharpening = 0

– M1 : For frame t+1, initial Zernike coefficient vector for sharpening = final vector for frame t

– P∞ : For frame t+1, initial Zernike coefficient vector for sharpening = identified LTI prediction filter output • LTI filter is identified using the benchmark wavefronts for a number of frames prior to pediction starting frame at t0

Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Global vs. Local: Image Amplitude

Image Error: Global Image Error: Local 16 16 M , S : Global M , S : Local 15 0 2 15 0 2 M , S : Global M , S : Local 1 2 1 2 14 14 P , S : Global P , S : Local  2  2 13 13

12 12

11 11

10 10

9 9 IrradianceError) (% IrradianceError) (% 8 8

7 7

6 6

5 5 2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpening Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction requires far fewer iterations to converge.

23 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Global vs. Local: Wavefront Correction

Phase Error: Global Phase Error: Local 40 40 M , S : Global M , S : Local 0 2 0 2 M , S : Global M , S : Local 35 1 2 35 1 2 P , S : Global P , S : Local  2  2

30 30

25 25

20 20

WavefrontError) (% WavefrontError) (% 15 15

10 10

2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpness Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction requires far fewer iterations to converge.

24 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Sharpness Metrics: Image Amplitude

Image Error: S Image Error: S 16 2 16 1 M , S : Global M , S : Global 0 2 0 1 15 M , S : Global 15 M , S : Global 1 2 1 1 P , S : Global P , S : Global  2  1 14 14

13 13

12 12

11 11 IrradianceError) (% 10 IrradianceError) (% 10

9 9

8 8 2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpness Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction requires far fewer iterations to converge.

25 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Sharpness Metrics: Wavefront Correction

Phase Error: S Phase Error: S 40 2 40 1 M , S : Global M , S : Global 0 2 0 1 M , S : Global M , S : Global 35 1 2 35 1 1 P , S : Global P , S : Global  2  1

30 30

25 25

20 20

WavefrontError) (% WavefrontError) (% 15 15

10 10

2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpness Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction requires far fewer iterations to converge.

26 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Simulations • In practice, the true incoming wavefronts will not be known a priori.

– The prediction LTI filter, P∞ , used the true wavefronts to identify the filter.

• Can a prediction filter be identified using a limited number of sharpness iterations as input?

– P4 : LTI filter is identified using the wavefronts produced by 4 sharpnening iterations for a number of frames prior to prediction starting frame at t0

– P6 : LTI filter is identified using the wavefronts produced by 6 sharpnening iterations for a number of frames prior to prediction starting frame at t0

Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Global vs. Local: Image Amplitude

Image Error: Global Image Error: Local 11 11 P , S : Global P , S : Local  2  2 P , S : Global P , S : Local 10 6 2 10 6 2 P , S : Global P , S : Local 4 2 4 2

9 9

8 8

7 7

IrradianceError) (% IrradianceError) (%

6 6

5 5 2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpness Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction filter does not require ID with true wavefronts to perform well.

28 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Global vs. Local: Wavefront Correction

Phase Error: Global Phase Error: Local 28 28 P , S : Global P , S : Local  2  2 26 26 P , S : Global P , S : Local 6 2 6 2 24 P , S : Global 24 P , S : Local 4 2 4 2

22 22

20 20

18 18

16 16

14 14

WavefrontError) (% WavefrontError) (% 12 12

10 10

8 8 2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpness Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction filter does not require ID with true wavefronts to perform well.

29 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Sharpness Metrics: Image Amplitude

Image Error: S Image Error: S 11 2 11 1 P , S : Global P , S : Global  2  1 P , S : Global P , S : Global 10.5 6 2 10.5 6 1 P , S : Global P , S : Global 4 2 4 1

10 10

9.5 9.5

9 9

IrradianceError) (% IrradianceError) (%

8.5 8.5

8 8 2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpness Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction filter does not require ID with true wavefronts to perform well.

30 Mechanical and Aerospace Engineering Beam Control Laboratory Predictive Dynamic Digital Holography Sharpness Metrics: Wavefront Correction

Phase Error: S Phase Error: S 28 2 28 1 P , S : Global P , S : Global  2  1 P , S : Global P , S : Global 26 6 2 26 6 1 P , S : Global P , S : Global 4 2 4 1 24 24

22 22

20 20

18 18

WavefrontError) (% WavefrontError) (%

16 16

14 14 2 4 6 8 10 2 4 6 8 10 Sharpness Iterations Sharpness Iterations

 Time-average errors vs. number of sharpening iterations

 Prediction filter does not require ID with true wavefronts to perform well.

31 Mechanical and Aerospace Engineering Beam Control Laboratory

Predictive Dynamic Digital Holography

Future Research

• Adaptive prediction filter with no a priori information about wavefront statistics

• Determine best image sharpening methods or integrated digital holography and prediction

• Subspace phase correction for parallel local sharpenig • Analyze nonlinear dyamics of predictive digital holography • Minimize effects of speckle for extended objects