On Signal On Signal Reconstruction from Reconstruction from Fourier MagnitudeFourier Magnitude

Gil MichaelDepartment of Electrical Engineering

Technion - Israel Institute of Technology

Haifa 32000, ISRAELmg@aya.technion.ac.il

Advisor: Dr. Moshe Porat

02/01/20012

Lecture OutlineLecture Outline1. 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

02/01/20013

IntroductionIntroduction

• 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:

1

0

1

0

1

0

/21

0

],[],[]},[{

:2

][][][]}[{

:1

N

n

N

m

knN

lmN

nkN

N

n

NknjN

n

WWnmxlkXnmxDFT

D

WnxenxkXnxDFT

D

02/01/20014

Introduction - Cont’d (2)Introduction - Cont’d (2)

•Discrete Fourier Transform (DFT) notation

1-D: ])[exp(|][|]}[{][ kjkXnxDFTkX

2-D:]),[exp(|],[|]},[{],[ lkjlkXnmxDFTlkX

02/01/20015

Reconstruction from Reconstruction from Fourier Magnitude or Fourier Magnitude or PhasePhase

DFT

Magnitude

PhaseID

FT

IDFT

Original

02/01/20016

Introduction - Cont’d (3)Introduction - Cont’d (3)

The Fourier phase contains higher intelligibility than Fourier magnitude

BUT

Fourier phase is more difficult to obtain

02/01/20017

Applications of Applications of Reconstruction from Reconstruction from Fourier MagnitudeFourier Magnitude

Detector

FourierTransformer

S

Fixed Mirror

Movable MirrorE i(t)

E(t) E(t+ )

L2L

1

•Spectroscopy

After: A.Yariv,Optical Electronics

02/01/20018

Applications of Applications of Reconstruction from Reconstruction from Fourier MagnitudeFourier Magnitude•Diffraction Pattern

02/01/20019

Applications of Applications of Reconstruction from Reconstruction from Fourier Magnitude-Cont’dFourier Magnitude-Cont’d•Crystallography

2

S iS r

O

P

A

B

(AO-PB)= r r*r*= S r-S i

=ScatteringAtoms

After:Ramachandran & Srinivasan, Fourier Methods in Crystallography

r

rr*i dverr*F 2)(}{

02/01/200110

Reconstruction from Reconstruction from Fourier Magnitude - Fourier Magnitude - LimitationsLimitationsGlobal Fourier magnitude is, in general, insufficient information for reconstruction

Additional data is required:

Spatial data, finite support, FT Sign- information, localized Fourier magnitude

02/01/200111

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

02/01/200112

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

02/01/200113

Decimation-in-Time FFTDecimation-in-Time FFT

After: Cooley & Tukey, 1965

x[0]x[2]x[4]x[6]

x[1]x[3]x[5]x[7]

G[0]G[1]G[2]G[3]

N/2 PointFFT

N/2 PointFFT

H[0]H[1]H[2]H[3]

X[0]X[1]X[2]X[3]

X[4]X[5]X[6]X[7]

WN0

WN1

WN2

WN3

WN0

WN1

WN2

WN3

02/01/200114

Decimation-in-Time FFT Decimation-in-Time FFT (Cont’d)(Cont’d)

][][][ kHWkGkX kN

Hk

Gk jk

Nj ekHWekG ][][

•The problem of finding xodd [n] is equivalent to retrieving the N/2-point DFT values of xodd [n]:

120 Nk

(1)

02/01/200115

Decimation-in-Time FFT - Decimation-in-Time FFT - DetailDetail

][][][][...

][][][][][ 222

kHWkGkHWkG

kHkGkXkXkX

kN

kN

12,,2,1,0

][][][][...

][][]2/[ 222

Nk

kHWkGkHWkG

kHkGNkX

kN

kN

(a)

(b)

We may write the squared magnitude as:

02/01/200116

222

2 ][][2]2

[][ kHkGN

kXkX

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 - Decimation-in-Time FFT - DetailDetail

02/01/200117

120][][][ NkekHWekGkNXHk

Gk jk

Nj

Hk

Gk jk

Nj ekHWekGkX ][][][

)2

cos(][)cos(][)cos(][Nk

kHkGkX Hk

Gkk

)2

sin(][)sin(][)sin(][Nk

kHjkGjkXj Hk

Gkk

Summing and subtracting (d) and applying DFT properties of real signals,

(e)

(d)

Decimation-in-Time FFT - Decimation-in-Time FFT - DetailDetail

02/01/200118

)2

(cos][][2][][][ 222

Nk

kHkGkHkGkX Gk

Hk

Adding the square products of (e),

1202

][][2

][][][cos

222

NkNk

kHkG

kHkGkXa

Gk

Hk

And the resulting phase,

(f)

Decimation-in-Time FFT - Decimation-in-Time FFT - DetailDetail

02/01/200119

Decimation-in-Time FFT Decimation-in-Time FFT (Cont’d)(Cont’d)

120

2][][2

][][][cos

222

Nk

Nk

kHkG

kHkGkXa G

kHk

22

22 ][]2

[][21

][ kGN

kXkXkH

(2)

(3)

02/01/200120

Decimation-in-Time FFT Decimation-in-Time FFT (Cont’d)(Cont’d)

N

lk

kHjkHN

lH

kl

N

k kl

klevenevenodd

)(

,)cos()sin(

][][][

212

11 1

0

• Ambiguity in the phase of H[k] is resolved by using the interpolate by 2 formula:

(4)

02/01/200121

Decimation-in-Time FFT: Decimation-in-Time FFT: Extension to 2-D signals Extension to 2-D signals (Images)(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.

02/01/200122

Decimation-in-Time FFT: Decimation-in-Time FFT: The Intermediate Fourier The Intermediate Fourier DomainDomain•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.

)]}n,m(x[DFT{DFT)}n,m(x{DFT row.col

02/01/200123

Decimation-in-Time FFT: Decimation-in-Time FFT: Extension to 2-D signals Extension to 2-D signals (Cont’d)(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).

02/01/200124

Decimation-in-Time FFT : Decimation-in-Time FFT : Intermediate Fourier Intermediate Fourier Domain ExampleDomain Example

DFT Performed On Columns

DFT

Columns

Rows

02/01/200125

Decimation-in-Time FFT: Decimation-in-Time FFT: Extension to 2-D signals Extension to 2-D signals (Cont’d)(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

02/01/200126

Reconstruction from Local Reconstruction from Local Fourier MagnitudesFourier 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

02/01/200127

Reconstruction from Local Reconstruction from Local Fourier Magnitudes - Fourier Magnitudes - Cont’dCont’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.

02/01/200128

Reconstruction from Local Reconstruction from Local Fourier Magnitudes - Fourier Magnitudes - Cont’dCont’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.

02/01/200129

Local Fourier magnitudes Local Fourier magnitudes – Successive Block-Size – Successive Block-Size Reconstruction AlgorithmReconstruction Algorithm

x(0,0)

X3

X4

X1

X2

02/01/200130

Local Fourier magnitudes Reconstruction Local Fourier magnitudes Reconstruction Algorithm– Successively larger Image Blocks Algorithm– Successively larger Image Blocks methodmethod

Known Spatial Block

02/01/200131

Local Fourier magnitudes Reconstruction Local Fourier magnitudes Reconstruction Algorithm– Overlapping Equal-Sized Blocks Algorithm– Overlapping Equal-Sized Blocks method (32x32 pixels)method (32x32 pixels)

Known Spatial Block

02/01/200132

Reconstruction from Local Reconstruction from Local Fourier Magnitudes and Fourier Magnitudes and an Arbitrary Spatial an Arbitrary Spatial SampleSample

Original Reconstructed

Note: inconsistency

02/01/200137

The Intermediate Fourier- The Intermediate Fourier- Domain Reconstruction Domain Reconstruction AlgorithmAlgorithm• 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

02/01/200138

The Intermediate Fourier-The Intermediate Fourier-Domain Reconstruction Domain Reconstruction Algorithm - Cont’dAlgorithm - 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:

ImageN x N

0

00

0

2N

2N

02/01/200139

The Intermediate Fourier-The Intermediate Fourier-Domain Reconstruction Domain Reconstruction Algorithm - Cont’dAlgorithm - Cont’d

• The IFD, preserves some of the spatial constraints:

1. Finite support.

2. Nulling of the columns (rows) of the result.

02/01/200140

The Intermediate Fourier-The Intermediate Fourier-Domain Reconstruction Domain Reconstruction Algorithm - Cont’dAlgorithm - 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.

02/01/200141

Intermediate Intermediate Fourier-Fourier-Domain Domain Algorithm: Algorithm: Block Block DiagramDiagram

x_temp: K known rows,K known columnsX_row=DFT of x_temp rows

X_col=DFT of x_temp columnsX_temp=2-D DFT of x_temp

Impose Fourier magnitude onX_temp

Impose X_col on X_int_rNull columns N to 2N-1

X_temp = DFT{X_int_r} on rows

Impose Fourier magnitude onX_temp

X_int_r = IDFT{X_temp}on rows

X_int_c = IDFT{X_temp} on columns

Impose X_row on X_int_cNull rows N to 2N-1

X_temp = DFT{X_int_r} on columns

02/01/200142

Intermediate Fourier-Intermediate Fourier-Domain Algorithm: Domain Algorithm: Iterative SimulationIterative Simulation

Iteration: 1234567891

0

25% known spatial samples

02/01/200143

Intermediate Fourier-Intermediate Fourier-Domain Algorithm: Domain Algorithm: ComparisonComparison

After 10 Iterations

02/01/200144

Intermediate Fourier-Intermediate Fourier-Domain Algorithm: Domain Algorithm: Simulation Results Simulation Results After 100 IterationsAfter 100 Iterations

18dB

02/01/200145

SummarySummary• 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.

02/01/200146

Summary - Cont’dSummary - 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.).

02/01/200147

Summary - Cont’dSummary - 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.

02/01/200148

Thank You For Your Attention !Thank You For Your Attention !