Phase Retrieval

1. Phase Retrieval Nickolaus Mueller University of Illinois at Urbana-Champaign “The mathematical sciences particularly exhibit order, symmetry, and limitations; and these are the greatest forms of the beautiful.” -- Aristotle

2. Goal • Recover Fourier phase angle given only a measurement of the Fourier Intensity to allow you to exactly determine an object. • Motivation: Crystallography, SAR, electron microscopy, and MRI all involve taking measurements of the intensity of a Fourier Transform of an object.

3. Mathematical Background and Definitions • Consider an object f(x) x = (x1,…,xn) xiє {0,…, N-1} • Most often n = 2 • N – point Discrete Fourier Transform (DFT) • Computed using FFT in O(N log N)n=1 O(N2 log N)n=2

4. Non-Uniqueness of Solution • F(u) = |F(u)|eiθ • There exist an infinite number of ‘trivial’ ways to modify the Fourier transform without changing the magnitude • Multiplying by a constant: eiφ • Taking the transform’s complex conjugate • Multiplying by a linear phase: ei(a1x1 + … + anxn) • Non-trivial solutions causing lack of uniqueness? • Uniqueness up to ‘trivial’ modifications when the function’s polynomial is irreducible in Cn • In n ≥ 2 dimensions, polynomials “almost always” irreducible

5. Gerchberg – Saxton Algorithm J.R. Fienup – “Phase Retrieval Algorithms: A Comparison” Initial estimate g0 g Transform G = |G|eiθ Satisfy Object Constraints Replace Fourier Modulus with Measured Modulus g’ Inv. Transform G’ = |F|eiθ

6. Gerchberg-Saxton an Error-Reduction Algorithm • Theorem: The error in the approximation of the function can only decrease or remain the same with each iteration of the Gerchberg-Saxton algorithm Parseval’ Parseval’s Theorem Parseval’s Theorem

7. Results for 1-dimensional Tests Actual Phase Angle 1 Iteration of Algorithm 1000 Iterations of Algorithm • f(x) = 1 + 2x2 + x4 + 2x6 • Estimate = 1 + x + 2x2 + x3 + x4 + x5 + 2x6 • Reasonable estimate converges quickly

8. Results for 1-dimensional Tests Actual Phase Angle 1 Iteration of Algorithm 1000 Iterations of Algorithm • f(x) = 1 + 2x2 + x4 + 2x6 • Estimate = 0 • Poor estimate approximates shape of graph, converges slowly

9. Non-Uniqueness in 1-Dimensional Tests By a process called “zero flipping,” the graphs of these phase angles come from two functions whose Fourier Transforms have the same magnitude. Results of Different Initial Estimates Single Iteration 1000 Iterations Single Iteration 1000 Iterations

10. Two – Dimensional Results • Using approximate support of object as object domain constraint

11. Two - Dimensional Results - Algorithm quickly and efficiently converges to correct object - Algorithm unable to converge to correct object

12. Conclusions • Functions of a single variable hard to reconstruct correctly without a very good estimate • Possible to achieve a good reconstruction in 2 – dimensions without a good estimate; a decent estimate almost guarantees perfect reconstruction • Gerchberg – Saxton algorithm quickly reduces error when close to actual object; other algorithms can be used to drive estimate towards correct object, and the G – S algorithm can then be used for speedy convergence • Questions?