Yue Zhao Instructor ： Yangquan Chen Advisor: Changqing Li ME280 FISP Fall 2013

1. Application of Fractional Fourier Transform in structured light 3-D shape reconstruction for FMT imaging system Yue Zhao Instructor：Yangquan Chen Advisor: Changqing Li ME280 FISP Fall 2013

2. Contents • Basic Principles • Fourier Transform Profilometry (FTP) • Fractional Fourier Transform (FRFT) • Application of FRFT in FTP • Computer Simulations • Conclusion • Future Work • Acknowledgement • References • Q &Ａ

3. Basic Principles – Fourier Transform Profilometry Profilometry : method to retrieve 3D profile of an object. 3D surface reconstruction system

4. Basic Principles – Fourier Transform Profilometry Reference Fringe Pattern Beam Path Diagram : phase difference Distorted Fringe Pattern

5. Basic Principles – Fourier Transform Profilometry = FT Reference Fringe Pattern ) = FT Distorted Fringe Pattern

6. Basic Principles – Fourier Transform Profilometry Reference Fringe Pattern ) Distorted Fringe Pattern

7. Basic Principles – Fourier Transform Profilometry FT f x Reference Fringe Pattern ) FT f x Distorted Fringe Pattern

8. Basic Principles – Fourier Transform Profilometry FT f x IFT Reference Fringe Pattern ) FT f x IFT Distorted Fringe Pattern

9. Basic Principles – Fourier Transform Profilometry Reference Fringe Pattern = - Distorted Fringe Pattern

10. Basic Principles – Fourier Transform Profilometry Disadvantage of FTP: cannot deal with frequency overlapping

11. Basic Principles – Fractional Fourier Transform α

12. Basic Principles – Fractional Fourier Transform Noise Signal

13. Basic Principles – Fractional Fourier Transform Noise α Signal

14. Basic Principles – Fractional Fourier Transform Fourier transform: αth order fractional Fourier transform: F(ω) = Fα(u) = α = , Fα (u) = F(ω)

15. Computer Simulation + Chirp noise (SNR

16. Computer Simulation Spectrum distribution of 2D FT (3D view) Spectrum distribution of 2D FT (top view)

17. Computer Simulation Noise Signal Time-frequency analysis tool box from: http://www.mathworks.com/matlabcentral/fileexchange/11551-adaptive-time-frequency-analysis

18. Computer Simulation Time-frequency analysis tool box from: http://www.mathworks.com/matlabcentral/fileexchange/11551-adaptive-time-frequency-analysis

19. Computer Simulation Time Domain FT Spectrum Domain FRFT Domain ( = 0.76*) FRFT code from: http://nalag.cs.kuleuven.be/research/software/FRFT/

20. Computer Simulation Time Domain FT Spectrum Domain FRFT Domain ( = 0.76*) FRFT code from: http://nalag.cs.kuleuven.be/research/software/FRFT/

21. Computer Simulation Absolute Value of 2D FRFT (3D view) Absolute Value of 2D FRFT (top view) 2D-FRFT code from: http://nalag.cs.kuleuven.be/research/software/FRFT/

22. Computer Simulation Absolute Value of 2D FRFT (3D view) Absolute Value of 2D FRFT (top view) 2D-FRFT code from: http://nalag.cs.kuleuven.be/research/software/FRFT/

23. Computer Simulation Fractional Fourier Transform: Fα (u) Inverse Fractional Fourier Transform: F-α(u) Perform Fourier Transform again: Spectrum distribution of 2D FT (3D view, after denoising) Spectrum distribution of 2D FT (top view , after denoising)

24. Computer Simulation Fractional Fourier Transform: Fα (u) Inverse Fractional Fourier Transform: F-α(u) Perform Fourier Transform again: Spectrum distribution of 2D FT (3D view, after denoising) Spectrum distribution of 2D FT (top view , after denoising)

25. Computer Simulation Reconstruction Results: FTP with FRFT denoising FTP without FRFT denoising

26. Conclusion • Fractional Fourier Transform generates a domain between “time domain” and “frequency domain”, with a rotation angle α. • FRFT can help remove noise information that cannot be removed using Fourier transform only. • With the help of FRFT, 3D shape can be reconstructed correctly even though the fringe patterns are contaminated by noises.

27. Future Work • Develop FRFT 3D reconstruction algorithm. • Discuss FRFT’s ability to deal with more complicated noises. • Explorer the potential of fractional wavelet transform in 3D reconstruction. • Application of FRFT for FMT image processing.

28. Acknowledgement • Thanks for Dr. Chen’s instruction of “Fractional thinking”; • Thanks my advisor Dr. Li’s helpful suggestions; • Thanks all the useful discussions and encouragements from the classmates of this course.

29. References • B. H. Zhu, “Fractional profilometrycorrelator for three-dimensional object recognition,” Applied Optics, vol. 40, No. 35, pp. 6474–6478, Dec. 2001. • H. Ozaktas, “Introduction to the Fractional Fourier Transform and Its Applications,” Advances in Imaging and Electron Physics, vol. 106, pp. 239–291, 1999. • M. Takeda, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Applied Optics, vol. 22, No. 24, pp. 3977–3982, Dec. 1983. • X. Su, “Fourier transform profilometry: a review,”Optics and Lasers Engineering, vol. 35, pp 263-284, 2001. • L. Huang, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Optics and Lasers Engineering, vol. 48, issue 2, pp 141-148, 2010. • Y. Zhang, “Image-adaptive watermarking using 2D chirps”, Signal, Image and Video Processing, vol 4, issue 1, pp102-121, 2010. • A. Bultheel, “A shattered survey of the Fractional Fourier Transform,” 2002, manuscript [Online]. Available: http://www.cs.kuleuven.be/~nalag/papers/ade/frft/index.html. • Y. Chen, “An overview of fractional order signal processing(FOSP) techniques,” Proceedings of DETC’07ASME Design Engineering Technical Conferences, Las Vegas, Nevada, USA, September 4-7, 2007. • E Sejdic, “Fractional Fourier transform as a signal processing tool: An overview of recent developments,” Signal Processing, vol 91, pp1351-1369, 2011. • Y. F. Luchko, “Fractional Fourier transform and some of its applications,” Fractional Calculus and Applied Analysis, vol 11, number 4, pp457-470, 2008. • V.D.Sharma, “Two-dimensional generalized offset fractional Fourier transform,” International Journal of Advanced Scientific and Technical Research, vol 2, issue 1, pp337-344, 2011.

30. ThanksQ&A