ELEN Wavelets and Sparsity XIV

“Refractive Index Map Reconstruction in Optical Deflectometry

Using Total-Variation Regularization”

L. Jacques1, A. González2, E. Foumouo2 and P. Antoine2

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

ELEN Wavelets and Sparsity XIV

1. Introduction

ELEN Wavelets and Sparsity XIV

‣ World Before Sparsity: Human Readable Sensing (up to an orthogonal transform, e.g., Fourier)

Nyquist sampling (wrt bandlimitedness)

Examples: Photography, First MRIs, Interferometry, ...

Sparsity in Wavefields

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World SensingDevice Human

Optimized

(1/2)

ELEN Wavelets and Sparsity XIV

‣ Paradigm shift allowed by Sparsity:Computer “Readable” Sensing + Prior information (sparsity)

Examples: Seismology, Radio-Interferometry, Compressed Sensing, MRI,Computed Photography, ... (Many in this Workshop!)

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World SensingDevice Human

Sparsity in Wavefields

Optimized

(2/2)

ELEN Wavelets and Sparsity XIV

2. Optical Deflectometry

ELEN Wavelets and Sparsity XIV

Optical Computed Tomography Classification:‣ Absorption Tomography (X-ray tomography)

‣ Phase Tomography (using interferometry)

‣ and Deflection Tomography ...6

(Check Session 12, Tue 23)

ELEN Wavelets and Sparsity XIV

‣ Problem:Reconstructing the refractive index map of transparent materials from light deflection measurements under multiple orientations.

‣ Advantages of deflectometry:‣ Insensitive to vibrations (vs. interferometry)‣ Less sensitive to dispersive medium‣ Precision: up to 10nm flatness deviation on 50mm FOV

Interests: ‣ (transparent) object surface topology ‣ default detection in glass, crystal growth study

Deflectometric Framework:

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multifocal lens

(e.g., on window glass)

ELEN Wavelets and Sparsity XIV

∆(τ, θ) ��

R2

�∇n(x) · pθ

�δ(τ − x · pθ) d

2x

Mathematical Model:

(first order approximation)

ELEN Wavelets and Sparsity XIV

Experimentally: Phase-Shifting Schlieren“Coding light deviation in intensity variations”

Incoherent Backlight Source

LCD based programmable spatial filter:

Telecentric Imaging Optics

Pinhole Spatial Filter

CCD camera

m(x) = sin( 2πΛ x)

Object Observation

Area

(1/2)

ELEN Wavelets and Sparsity XIV

∆x = f tanα

sin( 2πΛ ∆x)

∆x

Intensity change

Object

Phase-Shifting

Algorithm

m(x) = sin( 2πΛ x)

Experimentally: Phase-Shifting Schlieren(2/2)

ELEN Wavelets and Sparsity XIV

3. Reconstruction Methods

ELEN Wavelets and Sparsity XIV

Deflectometric Central Slice Theorem:

with �n the 2-D FFT of n.

∆(τ, θ) ��

R2

�∇n(x) · pθ

�δ(τ − x · pθ) d

2x

Sensing Model:

Continuous Facts:

z(ω, θ) :=

�

R∆(τ, θ) e−iτωdτ = iω �n

�ω pθ

�

⇒ z(ω, θ) = iω �n�ω pθ

�

ELEN Wavelets and Sparsity XIV

Refractive Index Prior‣ Heterogenous transparent materials

with slowly varying refractive indexseparated by sharp interfaces

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(e.g., optical fibers)

The perfect “cartoon shape” model (BV, TV, ...)

“Sparse” gradient, i.e.,small Total Variation norm

�n�TV =

�

R�∇n(x)� dxn0

n1

ELEN Wavelets and Sparsity XIV

‣ Weighting:

‣ Polar to Cartesian interpolation:

Discretization of Measurements

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y(ω, θ) :=z(ω, θ) + ε(ω, θ)

iω= �n

�ω pθ

�+ η(ω, θ).

Given z = Fτ (∆(·, θ))

with ε additive white Gaussian noise (in τ and ω)

(ω, θ) ∼ ωpθ → (kx, ky)

Controlling error: Cartesian grid resolution

(could be NFFT)

y(ω, θ) → �y(kx, ky)

kx

ky

(noise error)

(interpolation error)

ExperimentalCoordinateSystem

(1/2)

ELEN Wavelets and Sparsity XIV

‣ Recording interpolated frequency pixel locations:

‣ Final Discrete Inverse Problem: Tomography (but “colored”)

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Discretization of Measurements

y = SF n+ η

K = {k1, · · · , kM/2} ⊂ R2

Selection Operator

2-D FT

noise (colored)interp. + meas.

y =

y(k1)

...y(kM/2)

∈ CM/2 � RM

∈ RN

Ill-posed: M � N

(2/2)

ELEN Wavelets and Sparsity XIV

Filtered Back Projection‣ In our notation: FBP amounts to solve

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argminu∈RN

�u�2 s.t. y = SFu

⇒ n∗ = (SF )†y = F ∗S∗ y

since SS∗ = FF ∗ = Id

ELEN Wavelets and Sparsity XIV

Regularized Whitened Reconstruction‣ Using TV prior and whitening noise: TV-L2

‣ Estimating W : ‣ Montecarlo on Normal Gaussian‣ Estimating std. dev. from

‣ Solving the reconstruction: ‣ Proximal methods‣ Iterative Chambolle-Pock algorithm

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argminu∈RN

�u�TV s.t. �W (y − SFu)�2 � �

with W = diag�σ(η1), · · · ,σ(ηM/2)

�−1

∆(τ, θ)

ELEN Wavelets and Sparsity XIV

4. Results

ELEN Wavelets and Sparsity XIV

Synthetic Images‣ Generating realistic maps:

‣ Using Image Reconstruction Toolbox (IRT1) for computing

‣ Adding noise and reconstructing FBP & TV-L2

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1 http://www.eecs.umich.edu/~fessler/code/

∆(τ, θ)

τ

θ

Varying #θ, Fixed #τ

(1/4)

ELEN Wavelets and Sparsity XIV

‣ No meas. noise‣ M/N = 10%

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Synthetic Images

4.94 dB 10.37 dB

(2/4)

ELEN Wavelets and Sparsity XIV

‣ Meas. noise: 11.51dB‣ M/N = 10%

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Synthetic Images

4.34 dB 5.85 dB

(3/4)

ELEN Wavelets and Sparsity XIV

‣ Measurement noise = 20 dB (another image)

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Synthetic Images

# angles

(4/4)

SNR

(dB

)

FBP

TV-L2

ELEN Wavelets and Sparsity XIV

Experimental Data‣ bundle of 10 fibers immersed

in an optical fluid‣ working on one z-slice (2-D problem)

‣ Origin of must be calibrated!‣ Noise is estimated on

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#τ = 696

τ

∆(τ, θ)

(1/3)

ELEN Wavelets and Sparsity XIV

‣

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Experimental Data

FBP

#θ = 60,M/N � 30%

(2/3)

ELEN Wavelets and Sparsity XIV

‣

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Experimental Data

FBP TV-L2

#θ = 60,M/N � 30%

(2/3)

ELEN Wavelets and Sparsity XIV

Do not forget calibration!‣ If misaligned origin

FBP and TV-L2 will look something like this:

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τ

(3/3)

ELEN Wavelets and Sparsity XIV

5. Conclusion

ELEN Wavelets and Sparsity XIV

Conclusion and Perspectives‣ Recent sparsity-driven reconstruction methods can be

applied to Optical Deflectometry.

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ELEN Wavelets and Sparsity XIV

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/...

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ELEN Wavelets and Sparsity XIV

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!)

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ELEN Wavelets and Sparsity XIV

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 ?)

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�Φ∗(y − Φn)�∞ � ρforces spatially stationary noise

ELEN Wavelets and Sparsity XIV

Thank you!