Refractive Index Map Reconstruction in Optical Deflectometry Using Total-Variation Regularization

Transcription

1 Refractive Index Map Reconstruction in Optical Deflectometry Using Total-Variation Regularization L. Jacques 1, A. González 2, E. Foumouo 2 and P. Antoine 2 1 Institute of Information and Communication Technologies, Electronics and Applied Mathematics (ICTEAM) 2 Institute of Condensed Matter and Nanosciences (IMCN) University of Louvain, Louvain-la-Neuve, Belgium

2 1. Introduction

3 Sparsity in Wavefields (1/2) World Before Sparsity: Human Readable Sensing (up to an orthogonal transform, e.g., Fourier) Nyquist sampling (wrt bandlimitedness) World Sensing Device Human Optimized Examples: Photography, First MRIs, Interferometry,... 3

4 Sparsity in Wavefields (2/2) Paradigm shift allowed by Sparsity: Computer Readable Sensing + Prior information (sparsity) World Sensing Device Human Examples: Optimized Seismology, Radio-Interferometry, Compressed Sensing, MRI, Computed Photography,... (Many in this Workshop!) 4

5 2. Optical Deflectometry

6 Optical Computed Tomography Classification: Absorption Tomography (X-ray tomography) (Check Session 12, Tue 23) Phase Tomography (using interferometry) and Deflection Tomography... 6

7 Deflectometric Framework: Problem: Reconstructing the refractive index map of transparent materials from light deflection measurements under multiple orientations. Interests: (transparent) object surface topology multifocal lens default detection in glass, crystal growth study Advantages of deflectometry: Insensitive to vibrations (vs. interferometry) Less sensitive to dispersive medium Precision: up to 10nm flatness deviation on 50mm FOV (e.g., on window glass) 7

8 Mathematical Model: (τ,θ) R 2 n(x) pθ δ(τ x pθ )d 2 x (first order approximation)

9 (1/2) Experimentally: Phase-Shifting Schlieren Coding light deviation in intensity variations Object Observation Area Pinhole Spatial Filter CCD camera Incoherent Backlight Source LCD based programmable spatial filter: m(x) =sin( 2π Λ x) Telecentric Imaging Optics

10 (2/2) Experimentally: Phase-Shifting Schlieren Intensity change x sin( 2π Λ x) Phase- Shifting Algorithm x = f tan α Object m(x) =sin( 2π Λ x)

11 3. Reconstruction Methods

12 Continuous Facts: Sensing Model: (τ,θ) n(x) pθ δ(τ x pθ )d 2 x R 2 Deflectometric Central Slice Theorem: z(ω, θ) := R z(ω, θ) = iωn ω p θ (τ,θ) e iτω dτ = iωn ω p θ with n the 2-D FFT of n.

13 Refractive Index Prior Heterogenous transparent materials with slowly varying refractive index separated by sharp interfaces (e.g., optical fibers) The perfect cartoon shape model (BV, TV,...) n 0 n 1 Sparse gradient, i.e., small Total Variation norm n TV = R n(x) dx 13

14 Discretization of Measurements (1/2) Weighting: Given z = F τ ( (,θ)) (noise error) z(ω, θ)+ε(ω, θ) y(ω, θ) := = n ω p iω θ + η(ω, θ). with ε additive white Gaussian noise (in τ and ω) Polar to Cartesian interpolation: (interpolation error) Experimental Coordinate System (ω, θ) ωp θ (k x,k y ) y(ω, θ) y(k x,k y ) k y k x Controlling error: Cartesian grid resolution (could be NFFT) 14

15 Discretization of Measurements (2/2) Recording interpolated frequency pixel locations: K = {k 1,, k M/2 } R 2 Final Discrete Inverse Problem: Tomography (but colored ) ỹ = ỹ(k 1 ). ỹ(k M/2 ) C M/2 R M ỹ = SF n + η Selection Operator R N noise (colored) interp. + meas. 2-D FT Ill-posed: M N 15

16 Filtered Back Projection In our notation: FBP amounts to solve arg min u 2 s.t. ỹ = SFu u R N n =(SF) ỹ = F S ỹ since SS = FF =Id 16

17 Regularized Whitened Reconstruction Using TV prior and whitening noise: TV-L2 arg min u TV s.t. W (ỹ SFu) 2 u R N with W = diag σ(η 1 ),,σ(η M/2 ) 1 Estimating W : Montecarlo on Normal Gaussian Estimating std. dev. from Solving the reconstruction: Proximal methods (τ,θ) Iterative Chambolle-Pock algorithm 17

18 4. Results

19 (1/4) Synthetic Images Generating realistic maps: Using Image Reconstruction Toolbox (IRT 1 ) for computing (τ,θ) Varying #θ, Fixed#τ τ Adding noise and reconstructing FBP & TV-L2 θ

20 (2/4) Synthetic Images No meas. noise M/N = 10% 4.94 db db 20

21 (3/4) Synthetic Images Meas. noise: 11.51dB M/N = 10% 4.34 db 5.85 db 21

22 (4/4) Synthetic Images Measurement noise = 20 db (another image) TV-L2 SNR (db) FBP # angles 22

23 (1/3) Experimental Data bundle of 10 fibers immersed in an optical fluid working on one z-slice (2-D problem) #τ = 696 Origin of τ must be calibrated! Noise is estimated on (τ,θ) 23

24 (2/3) Experimental Data #θ = 60,M/N 30% FBP 24

25 (2/3) Experimental Data #θ = 60,M/N 30% FBP TV-L2 25

26 (3/3) Do not forget calibration! If misaligned τ origin FBP and TV-L2 will look something like this: 26

27 5. Conclusion

28 Conclusion and Perspectives Recent sparsity-driven reconstruction methods can be applied to Optical Deflectometry. 28

29 Conclusion and Perspectives Recent sparsity-driven reconstruction methods can be applied to Optical Deflectometry. Outside of easy synthetic Cartezian worlds, hard time with gridding/interpolation/noise/calibration/... 29

30 Conclusion and Perspectives Recent sparsity-driven reconstruction methods can be applied to Optical Deflectometry. Outside of easy synthetic Cartezian worlds, hard time with gridding/interpolation/noise/calibration/... Same framework valid for: Phase-Contrast X-Ray Tomography (& no first order approx!) 30

31 Conclusion and Perspectives Recent sparsity-driven reconstruction methods can be applied to Optical Deflectometry. Outside of easy synthetic Cartezian worlds, hard time with gridding/interpolation/noise/calibration/... Same framework valid for: Phase-Contrast X-Ray Tomography (& no first order approx!) Future Improvements: Integrating positivity constraint Using NFFT and reduce interpolation noise Testing other reconstruction (Dantzig Fidelity?) Φ (ỹ Φn) ρ forces spatially stationary noise 31

32 Thank you!