1. On Signal Reconstruction from Fourier Magnitude Gil Michael Department of Electrical Engineering Technion - Israel Institute of Technology Haifa 32000, ISRAEL mg@aya.technion.ac.il Advisor: Dr. Moshe Porat

2. 02/01/2001 2 Lecture Outline 1.Introduction to Fourier Magnitude Reconstruction 2.Fourier Magnitude Applications 3.Reconstruction by Decimation-in-Time FFT 4.Reconstruction from Local Fourier Magnitudes 5.The Intermediate Fourier-Domain Algorithm (IFD) 6.Summary and Discussion

3. 02/01/2001 3 Introduction Reconstruction of signals from their Fourier transform properties is a useful task in science and engineering. Fourier transform properties consist of phase and magnitude information:

4. 02/01/2001 4 Introduction - Cont’d (2) Discrete Fourier Transform (DFT) notation 1-D: 2-D:

5. 02/01/2001 5 Reconstruction from Fourier Magnitude or Phase DFT MagnitudePhase IDFT Original

6. 02/01/2001 6 Introduction - Cont’d (3) The Fourier phase contains higher intelligibility than Fourier magnitude BUT Fourier phase is more difficult to obtain

7. 02/01/2001 7 Applications of Reconstruction from Fourier Magnitude Spectroscopy After: A.Yariv,Optical Electronics

8. 02/01/2001 8 Applications of Reconstruction from Fourier Magnitude Diffraction Pattern

9. 02/01/2001 9 Applications of Reconstruction from Fourier Magnitude-Cont’d Crystallography After:Ramachandran & Srinivasan, Fourier Methods in Crystallography

10. 02/01/2001 10 Reconstruction from Fourier Magnitude - Limitations Global Fourier magnitude is, in general, insufficient information for reconstruction  Additional data is required: Spatial data, finite support, FT Sign- information, localized Fourier magnitude

11. 02/01/2001 11 Reconstruction by Decimation- in-Time FFT From Nawab et al. (1983) we know, that a sequence x(n), which is zero outside 0  n  N-1, is uniquely specified by its spectral magnitude and at least N/2 samples of x(n). The proof relies on the relationship between the Fourier magnitude and the auto-correlation function. The proposed algorithm is a closed-form solution to a similar problem: Algorithm I

12. 02/01/2001 12 Reconstruction by Decimation- in-Time FFT (Cont’d) Given the Fourier magnitude of a real signal x(n), which is zero outside 0  n  N-1, and the even (odd) samples, find a closed-form solution to the odd (even) unknown samples. Use the Decimation-in-Time FFT algorithm * We will assume N is a power of 2, such that N=2 L for some integer L>0

13. 02/01/2001 13 Decimation-in-Time FFT After: Cooley & Tukey, 1965

14. 02/01/2001 14 Decimation-in-Time FFT (Cont’d) The problem of finding x odd [n] is equivalent to retrieving the N/2-point DFT values of x odd [n]: (1)

15. 02/01/2001 15 Decimation-in-Time FFT - Detail (a) (b) We may write the squared magnitude as:

16. 02/01/2001 16 And we obtain the absolute value of H[k] from: For the retrieval of the phase   K we rewrite X[k] in the polar form: (c) Decimation-in-Time FFT - Detail

17. 02/01/2001 17 Summing and subtracting (d) and applying DFT properties of real signals, (e) (d) Decimation-in-Time FFT - Detail

18. 02/01/2001 18 Adding the square products of (e), And the resulting phase, (f) Decimation-in-Time FFT - Detail

19. 02/01/2001 19 Decimation-in-Time FFT (Cont’d) (2) (3)

20. 02/01/2001 20 Decimation-in-Time FFT (Cont’d) Ambiguity in the phase of H[k] is resolved by using the interpolate by 2 formula: (4)

21. 02/01/2001 21 Decimation-in-Time FFT: Extension to 2-D signals (Images) An image may be fully reconstructed using the Decimation-in-Time FFT approach, with some constraints imposed on it. The even rows (columns) must be symmetric i.e., in each even row (column), x(m,n)=x(N-m,n). The 2-D Fourier magnitude of x(m,n) and the 1-D Fourier magnitude of the even rows (columns) are known.

22. 02/01/2001 22 Decimation-in-Time FFT: The Intermediate Fourier Domain The 2-D DFT may be implemented by an FFT on the image’s rows (columns) and then on the resultant columns (rows). The intermediate Fourier domain (IFD) is defined as the resultant matrix obtained after the first row (column) FFT.

23. 02/01/2001 23 Decimation-in-Time FFT: Extension to 2-D signals (Cont’d) Since the rows (columns) are symmetric, their (1-D) Fourier transforms are real. In addition: The columns (rows) of the image’s Fourier- transform are the (1-D) Fourier transforms of the IFD columns (rows).

24. 02/01/2001 24 Decimation-in-Time FFT : Intermediate Fourier Domain Example DFT Performed On Columns DFT  Columns Rows

25. 02/01/2001 25 Decimation-in-Time FFT: Extension to 2-D signals (Cont’d) Using the Decimation-in-Time FFT algorithm, reconstruct the image from: 1) The calculated even (odd) samples of the IFD columns 2) Fourier transform magnitude of the image

26. 02/01/2001 26 Reconstruction from Local Fourier Magnitudes Recently, it was found that an image is represented by its Fourier magnitude and a quarter of its spatial samples (Shapiro & Porat, 1998) However, spatial samples may not always be available. In the following algorithm, image reconstruction from local Fourier magnitudes requires only a single spatial sample. Algorithm II

27. 02/01/2001 27 Reconstruction from Local Fourier Magnitudes - Cont’d Local Fourier magnitudes are the absolute values of Fourier transforms taken over specific sections of the image. Two methods of reconstructing an image from local Fourier magnitudes and a single spatial sample are presented here. In addition it is shown, that reconstruction quality is relatively unaffected by spatial data errors.

28. 02/01/2001 28 Reconstruction from Local Fourier Magnitudes - Cont’d The first algorithm is based on successively reconstructing larger and larger image blocks, with each newly reconstructed block being the spatial samples required for the next iteration. The second algorithm, iteratively reconstructs equal-sized blocks that overlap by ¼ of their samples.

29. 02/01/2001 29 Local Fourier magnitudes – Successive Block-Size Reconstruction Algorithm

30. 02/01/2001 30 Local Fourier magnitudes Reconstruction Algorithm– Successively larger Image Blocks method Known Spatial Block

31. 02/01/2001 31 Local Fourier magnitudes Reconstruction Algorithm– Overlapping Equal-Sized Blocks method (32x32 pixels) Known Spatial Block

32. 02/01/2001 32 Reconstruction from Local Fourier Magnitudes and an Arbitrary Spatial Sample Original Reconstructed Note: inconsistency

33. 02/01/2001 33 Example : 2x2 Block Reconstruction from Local Fourier Magnitudes Step 1: Calculate the Auto-correlation function, r(m,n), from the Local Fourier magnitude X(k,l):Step 1: Calculate the Auto-correlation function, r(m,n), from the Local Fourier magnitude X(k,l): where: N=2 is the block size, and |X(k,l)| is the 2N X 2N Fourier magnitude

34. 02/01/2001 34 Step 2: Calculate the diagonal value x(1,1) from:Step 2: Calculate the diagonal value x(1,1) from:with: 2x2 Block Reconstruction from Local Fourier Magnitudes - Cont’d

35. 02/01/2001 35 2x2 Block Reconstruction from Local Fourier Magnitudes - Cont’d Step 3: Retrieve the off-diagonal elements:Step 3: Retrieve the off-diagonal elements:with:

36. 02/01/2001 36 2x2 Block Reconstruction from Local Fourier Magnitudes - Cont’d Note: A is invertible iff: Step 4:Step 4: Use the reconstructed block as initial data for the next block, and repeat steps 1  3.

37. 02/01/2001 37 The Intermediate Fourier- Domain Reconstruction Algorithm Gerchberg-Saxton algorithm: iteratively imposes spatial constraints, known samples and Fourier magnitude in the spatial and Fourier domains As shown previously, the 2-D DFT may be implemented using the Intermediate Fourier Domain (IFD). Algorithm III

38. 02/01/2001 38 The Intermediate Fourier-Domain Reconstruction Algorithm - Cont’d A typical reconstruction scheme would require a 2N x 2N DFT for an N x N image, essentially zero-padding the additional spatial samples:

39. 02/01/2001 39 The Intermediate Fourier-Domain Reconstruction Algorithm - Cont’d The IFD, preserves some of the spatial constraints: 1.Finite support. 2.Nulling of the columns (rows) of the result.

40. 02/01/2001 40 The Intermediate Fourier-Domain Reconstruction Algorithm - Cont’d The proposed algorithm utilizes these properties to modify the Gerchberg-Saxton approach. It uses 1-D FFT’s only, and imposes twice as many magnitude constraints in each iteration, resulting in faster convergence to the desired image.

41. 02/01/2001 41 Intermediate Fourier-Domain Algorithm: Block Diagram

42. 02/01/2001 42 Intermediate Fourier-Domain Algorithm: Iterative Simulation Iteration: 12345 6 7 8910 25% known spatial samples

43. 02/01/2001 43 Intermediate Fourier-Domain Algorithm: Comparison After 10 Iterations

44. 02/01/2001 44 Intermediate Fourier-Domain Algorithm: Simulation Results After 100 Iterations 18dB

45. 02/01/2001 45Summary Three new approaches to signal reconstruction from Fourier magnitude were presented. I) Decimation-in-Time FFT Algorithm:  Closed-form solution: No iterations and convergence uncertainty.  Lower computational complexity compared to the auto-correlation approach.  Efficient implementation (FFT).  2-D Extension requires only Fourier information.

46. 02/01/2001 46 Summary - Cont’d II) Localized Fourier Transform Magnitude Algorithm:   Magnitude information and a single spatial sample fully reconstruct the image.   Two methods may be applied: constant block or successively larger block reconstruction.   This algorithm may prove useful in phase retrieval problems, where spatial information is unavailable (in contrast to Nawab et al.).

47. 02/01/2001 47 Summary - Cont’d III) Intermediate Fourier Domain Algorithm   A modified Gerchberg-Saxton algorithm utilizes the separability property of the 2-D DFT.   Imposes the Fourier magnitude constraints at twice the rate of the conventional approach, resulting in faster convergence rates.

48. 02/01/2001 48 Thank You For Your Attention !