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

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

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

Non-Uniqueness of Solution F(u) = |F(u)|e iθ F(u) = |F(u)|e iθ There exist an infinite number of ‘trivial’ ways to modify the Fourier transform without changing the magnitude There exist an infinite number of ‘trivial’ ways to modify the Fourier transform without changing the magnitude Multiplying by a constant: e iφ Multiplying by a constant: e iφ Taking the transform’s complex conjugate Taking the transform’s complex conjugate Multiplying by a linear phase: e i(a 1 x 1 + … + a n x n ) Multiplying by a linear phase: e i(a 1 x 1 + … + a n x n ) Non-trivial solutions causing lack of uniqueness? Non-trivial solutions causing lack of uniqueness? Uniqueness up to ‘trivial’ modifications when the function’s polynomial is irreducible in C n Uniqueness up to ‘trivial’ modifications when the function’s polynomial is irreducible in C n In n ≥ 2 dimensions, polynomials “almost always” irreducible In n ≥ 2 dimensions, polynomials “almost always” irreducible

Gerchberg – Saxton Algorithm J.R. Fienup – “Phase Retrieval Algorithms: A Comparison” Satisfy Object ConstraintsReplace Fourier Modulus with Measured Modulus Inv. Transform Transform g’ gG = |G|e iθ G’ = |F|e iθ Initial estimate g 0

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 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

Results for 1-dimensional Tests Actual Phase Angle1 Iteration of Algorithm1000 Iterations of Algorithm f(x) = 1 + 2x 2 + x 4 + 2x 6 Estimate = 1 + x + 2x 2 + x 3 + x 4 + x 5 + 2x 6 Reasonable estimate converges quickly

Results for 1-dimensional Tests Actual Phase Angle1 Iteration of Algorithm1000 Iterations of Algorithm f(x) = 1 + 2x 2 + x 4 + 2x 6 Estimate = 0 Poor estimate approximates shape of graph, converges slowly

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. Single Iteration 1000 Iterations Results of Different Initial Estimates

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

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

Conclusions Functions of a single variable hard to reconstruct correctly without a very good estimate 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 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 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? Questions?