# Amplitude demodulation of interferometric signals

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1 with a 2D Hilbert transform Warsaw University of Technology Faculty of Mechatronics University of Warsaw Faculty of Mathematics, Informatics and Mechanics Supervised by Prof. Krzysztof Patorski September 15, 2010

2 Interference - waves superposition Figure: Interferometry: double-slit experiment (picture by Lacatosias)

3 Time-average interference microscopy Figure: Scheme of the measurement system, source: L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A, Jacobelli, T. Dean, Active micor-elements testing by interferometry using time-average and quasi-stroboscopic techniques, Proc. SPIE 5145 (2003), 23-32;

4 Time-average interferogram Mathematical model of a time-average interferogram [ ( ) ] 4π I vib (x, y) = K 1 + C stat J 0 λ a 0 cos ϕ vib (x, y) (1) K = a 2 (x, y) + b 2 (x, y) (2) C stat = 2 a(x, y)b(x, y) (3) K Figure: Example time-average interferograms for frequencies 172kHz, 178kHz and 833kHz

5 Hilbert transform Hilbert transform - definition H{s(x)} = s H (x) = s(x) 1 πx = P.V. 1 + s(u) π x u du (4) S H (ω) = F{s H (x)} = i sign(ω)f{s(x)} = i sign(ω)s(ω) (5) Analytic signal s A (x) = s(x) + i s H (x) = [ δ(x) + 1 ] s(x) (6) πx S A (ω) = F{s A (x)} = [1 + sign(ω)] S(ω) (7)

6 Hilbert transform in 2D - spectral mask One obvious generalization H{s(x, y)} = 1 π 2 + s(u, v) (x u)(y v) dudv = 1 π 2 s(x, y) (8) xy S A (ζ, η) = [1 i sign(ζ)sign(η)] S(ζ, η) (9) Directional mask, Hahn mask and averaged mask S A (ζ, η) = [1 + sign(ζ)] S(ζ, η) (10) S A (ζ, η) = [1 + sign(ζ)] [1 + sign(η)] S(ζ, η) (11) S A (ζ, η) = [ ] (sign(ζ) + sign(η)) S(ζ, η) (12)

7 Quaternionic Fourier Transform approach The ring of quaternions QFT definition S q (ζ, η) = i 2 = j 2 = k 2 = ijk = 1 (13) + + QFT approach to the Hilbert transform e i2πxζ s(x, y)e j2πyη dxdy (14) S q A (ζ, η) = [1 + sign(ζ)][1 + sign(η)]s q (ζ, η) (15) S q A (ζ, η) = [ (sign(ζ) + sign(η)) ] S q (ζ, η) (16)

8 Spiral Phase Method Spectral spiral phase function This approach was developed by Larkin et al. Instead of anisotropic 2D signum function analog they define spectral spiral phase function P(ζ, η) = ζ + iη (17) ζ 2 + η 2 Spiral phase approach to the Hilbert transform s H (x, y) = i exp[ iβ(x, y)] F 1 {P(ζ, η)s(ζ, η)} (18) s A (x, y) = s(x, y) + i s H (x, y) (19) where β(x, y) is the local fringe orientation angle.

9 AM demodulation efficiency comparison Carrier data used s(x, y) = b(x, y) cos ( r + r 2) (20) Modulation data used ( r b 1 (x, y) = J 0 5 ( r b 3 (x, y) = 5x 2 y 2 2 exp 5 ) ; b 2 (x, y) = exp ) ; b 4 (x, y) = exp ( ) r 2 10 ( ) (x 6) 2 35 (21) (22) Error estimation Er(s A ) = ( s A (x, y) b(x, y) ) 2 (23) (x,y) Ω

10 Efficiency comparison results Table: Errors of presented methods applied to chirp signal with modulation b i. HS - spiral phase method, HQ - QTF method b(x, y) H1D H1 H2 H3 H4 HS HQ HQ2 b 1 (x, y) b 2 (x, y) b 3 (x, y) b 4 (x, y)

11 Demodulation examples Figure: Example results, a) chirp signal; b) modulation b 1 ; c) modulated chirp signal; d) demodulation result given by H3 algorithm; e) result given by HS algorithm; f) result given by HQ2 algorithm

12 Demodulation examples Figure: Example results, from left to right: Modulation b 2 ; result given by HS algorithm; result given by HQ2 algorithm

13 Demodulation examples Figure: Example results, from left to right: Modulation b 3 ; result given by HS algorithm; result given by H4 algorithm

14 Demodulation examples Figure: Example results, from left to right: Modulation b 4 ; result given by HS algorithm; result given by H2 algorithm

15 Towards the real signal demodulation Figure: Example results, demodulation of a real (preprocessed) signal with the HS algorithm, frequency 724kHz

16 Major references (1) K.G. Larkin, D.J. Bone, M.A. Oldfield, Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,,,journal of the Optical Society of America, 8(18), 2001, (2) R. Onodera, Y. Yamamoto, Y. Ishii, Signal processing of interferogram using a two-dimensional discrete Hilbert transform,,,fringe 2005, Springer Berlin, 2006, (3) T. Bulow, G. Sommer, Multi-dimensional signal processing using an algebraically extended signal representation, AFPAC LNCS, 1315, Springer, Heidelberg, (4) K. Patorski, A. Styk, Z, Sienicki,Time-average interference microscopy for vibration testing of silicon microelements, Proc. SPIE vol 6158 (2006),

17 Thank you for your attention!

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