PowerPoint-Pr

1
Sparsity-based sub-wavelength imagingand
super-resolution in time and frequency
Yoav Shechtman
Physics Department, Technion, Haifa 32000, Israel
Alex Szameit, Snir Gazit, Pavel Sidorenko, Elad
Bullkich, Eli Osherovic, Michael Zibulevsky, Irad
Yavneh, Yonina Eldar , Oren Cohen, Moti Segev
FRISNO 2011
Nonlinear Optics Laboratory
2
Sub-wavelength images in the microscope
3
Optical cut-off for high spatial frequencies
field propagation (z 0 ? z gt 0)
propagating waves
evanescent waves
4
Hardware solutions for sub-wavelength imaging

• Scanning near-field optical microscope
• Methods using florescent particles
• Structured Illumination
• Negative-index / metamaterials structures
  superlens, hyperlens
• Hot-spot methods nano-hole array,
  super-oscillations
• Require scanning, averaging over multiple
  experiments
• Is it possible to have real-time, single exposure
  sub-wavelength imaging using a regular
  microscope?

5
Analytic Continuation

• The 2D Fourier transform of a spatially bounded
  function is an analytic function.

 

• Problem Existing analytic continuation methods
  are not very robust
• sampling theorem based extrapolations yield a
  highly ill posed matrix.
• Iterative methods (Gerchberg - Papoulis) are
  sensitive to noise

6
Common wisdom

• All methods for extrapolating bandwidth beyond
  the diffraction limit are known to be extremely
  sensitive to both
• noise in the measured data and
• the accuracy of the assumed a priori knowledge.
• It is generally agreed that the Rayleigh
  diffraction limit represents a practical frontier
  that cannot be overcome with a conventional
  imaging system.

J. W. Goodman, Introduction to Fourier Optics,
2005
7
Bandwidth extrapolation problem
infinite number of possible solutions!
Measurements
How to choose the right one?
7
8
Problem non-invertible filter
signal
Filtered Fourier transform (non-invertible)
FT(signal)
Fourier transform (invertible)
Measurements
8
9
Under-determined system of equations

• Under-determined system of equations
• more variables than equations
• Infinite number of solutions (x)
• Choose the one that makes the most sense
• Based on what?
• Based on the knowledge that
• the object is sparse
  in a known basis.

We choose the solution with maximum sparsity
the one with the fewest nonzero elements.
9
10
Why sparsity?

• General Many objects are sparse in some
  (general) basis.
• Powerful
• Robust to noise.Without noise, in a sparse
  enough case the sparsest solution is unique
• Sparsity is used successfully for image
  denoising, deconvolution, compression,
  enhancement of MR images and more.
• However has never been used for sub-? imaging,
  or temporal bandwidth extrapolation.
• Attainable Efficient algorithms exist for
  estimating the sparsest solution.

11
Sparsity a general feature of information
Sparsity in real space image
Sparsity in another basis
Electronic chips Sparsity in gradient basis
few
biological species Real-space sparsity 2- 5
12
How to do it for example - Basis Pursuit

• Solve the (convex) optimization problem
• x unknown image
• y measured image
• ALow-pass filter sparsity basis
• e Noise parameter
• The requirement on the l1 norm is to promote
  sparsity.
• Find the sparsest x that is consistent with the
  measurements.

S.S. Chen et al., SIAM Journal on Scientific
Computing, 20, 33 (1998)
13
Proof of concept
diffuser (optional)
tunable filter
original image
filtered image
reconstructed image
Gazit et al., Opt. Exp. Dec. 2009 Shechtman et
al., Opt. Lett. Feb. 2010
14
Proof of concept
Original
Recovered
Gazit et al., Opt. Exp. Dec. 2009 Shechtman et
al., Opt. Lett. Feb. 2010
15
True sub-l experiments 1D _at_ l 532 nm
Chromium
SEM image
Glass
Width 150 nm Length 20 µm Spacing 150 nm
(left/right pair) 300 nm (center) diffraction
limit
Fabrication Kley group University of
Jena kley_at_iap.uni-jena.de
16
Best possible microscope image (NA 1)
17
Microscope image far-field
18
Experimental result (with hand-made microscope)
reconstruction
microscope image
150 nm
19
Comparison original - reconstruction
real space
spatial spectrum
20
True sub-l experiments 2D _at_ l 532 nm
100 nm
100 nm
21
Best possible microscope image (NA 1)
22
Microscope image far-field
23
Loss of power in the far-field
more than 90 of the intensity is lost
24
Experimental results
microscope image
reconstructed image
SEM image
25
Sub-l imaging from far-field intensity
measurements

• Can we do sub-wavelength reconstruction
• based on intensity
  measurements only?
• Without measuring phase at all?
• Yes, indeed. The knowledge of sparsity is
  powerful.
• First Fourier phase recovery using iterative
  algorithm given the blurred image intensity
  and Fourier intensity.
• Second sparsity-based reconstruction using
  recovered phase.
• or, better, combine the two!

J. R. Fienup, Appl. Opt. 21 (1982)
26
Experimental sparsity-based recovery of
random distribution of circles

• Circles are 100 nm diameter
• Wavelength 532 nm

Diffraction-limited (low frequency) intensity
measurements
Assuming non-negativity
Model Fourier transform
27
Experimental incorrect reconstruction
with wrong number of
circles
22 circles left
30 circles left
11 circles left
12 circles left
28
Sparsity-based super-resolution in pulse-shape
measurements experimental
Slow Photodiode (t 1 ns)
Laser Pulse
Fast Photodiode (t 175 ps)
Impulse response functions
Transfer functions


(b)
100
1
Slow PD
Fast PD
0.8
10-2
0.6
Intensity a.u.
Intensity a.u.
0.4
0.2
10-4
0

10
1
0.1

-1.5
-0.5
0
0.5
1
1.5
-1
Frequency GHz
Time ns
29
Sparsity-based super-resolution in pulse-shape
measurements experimental
Measured signals
Spectra of measured signals

1
0.8
0.6
VOSC a.u.
0.4
0.2
Fast PD
0
Slow PD

-1.5
-1
-0.5
0
0.5
1
1.5
10
1
0.1

Frequency GHz
Time ns
Reconstruction


100
Deconv Fast PD
1
Reconstructed
0.8
Deconv Slow PD
10-2
0.6
Intensity a.u.
Intensity a.u.
0.4
Deconv Fast PD
0.2
Reconstructed
10-4
Deconv Slow PD
0

10
1
0.1

-1.5
-1
-0.5
0
0.5
1
1.5
Frequency GHz
Time ns
30
Sparsity-based super-resolution FTIR spectroscopy
Power (W)
X (cm)
Power (W)
? (nm)
Because the interferogram cannot be collected
from x -? to ?, it is always truncated,
hence some error arises in the resulting
spectrum the spectral line is broadened
side-lobes are added
Resolution of a F-T spectrometer ?? 1 / (path
difference 4x)
31
Sparsity-based super-resolution in FTIR spectrum
measurement experimental example
32
Conclusions

• method for recovering sub-l information
• from the optical far-field of images
• requires no additional hardware
• works in real time and with ultrashort pulses
• applicable to all microscopes (optical and
  non-optical)
• reconstruction also with incoherent / partially
  coherent light
• Ideas are universal can be used to recover
• shapes of
  ultrashort pulses in time
• spectral features
• quantum info!

33
Sub-? lensless imaging
Reference holes
34
Many thanks for your attention!
35
(No Transcript)
36
A little about uniqueness
An object comprising on n features is uniquely
determined by 2n(n1) measurements on a polar
grid in k-space, without noise.
Y. Nemirovsky, Y. Shechtman, A. Szameit, Y.C.
Eldar, M. Segev, in preparation
37
Comparison of approaches
Our CS-related approach
Original CS approach

• measurement in uncorrelated basis
• (commonly Fourier basis)
• sampling (randomly) over the
• entire measurement basis with
• low resolution
• reduction of required samples
• to retrieve the function

• measurement in far-field
• ( Fourier basis)
• OR blurred near field
• or in between
• sampling in a small part of the
• measurement basis (kx lt k) with
• high resolution
• obtain maximal info on the
• frequency region where we
• do NOT measure

We do NOT do CS. We do NOT use CS rules.
Why does it work for us?
38
Unique sparse solution
sparse
39
Reconstruction of the phase (Fienup-Algorithm)
real space
real space
far-field
Iteration
phase
Fienup, Opt. Lett. 3, 27 (1978).
40
Experimental holes on a grid
139 nm
41
Consider a function that can be
written as a superposition of spikes If it is
comprises of spikes, and N is a prime
number, then can be uniquely defined by
any of its Fourier measurements, defined
as Specifically, the low pass
Fourier coefficients will do.

• Candes, E.J. Romberg, J. Tao, T. , "Robust
  uncertainty principles exact signal
  reconstruction from highly incomplete frequency
  information," Information Theory, IEEE
  Transactions on , vol.52, no.2, pp. 489- 509,
  Feb. 2006

42
Sparsity-based super-resolution in pulse-shape
measurements theoretical example
Source laser pulse
Oscilloscope signal
1
1

Intensity
VOSC
0
0
0
50
100
150
time ps
time ps
100
10-2
10-4
Intensity
10-6
10-8
10-10
1
10
100
Freq. GHz
43
Sparsity-based super-resolution in pulse-shape
measurements theoretical example
Source laser pulse
Oscilloscope signal
1
1

Intensity
VOSC
0
0
Without noise De-convolution ? perfect
reconstruction
0
50
100
150
time ps
time ps
100

10-2
10-4
Intensity
10-6
10-8
10-10
1
10
100
Freq. GHz
44
Sparsity-based super-resolution in pulse-shape
measurements theoretical example
Source laser pulse
Oscilloscope signal
Wiener Deconvolution
1
1


1
Intensity
Intensity
VOSC
0
0
0
0
50
100
150
50
100
150

time ps
time ps
time ps
100
100

10-2
10-2
10-4
10-4
Intensity
Intensity
10-6
10-6
10-8
10-8
10-10
10-10
1
10
100
1
10
100
Freq. GHz
Freq. GHz
45
Sparsity-based super-resolution in pulse-shape
measurements theoretical example
Source laser pulse
Oscilloscope signal
Wiener Deconvolution
Sparsity-based reconstruction
1
1



1
1
Intensity
Intensity
Intensity
VOSC
0
0
0
0
0
50
100
150
50
100
150
50
100
150

time ps
time ps
time ps
time ps
100
100
100

10-2
10-2
10-5
10-4
10-4
Intensity
Intensity
Intensity
10-6
10-6
10-10
10-8
10-8
10-10
10-10
1
10
100
1
10
100
1
10
100
Freq. GHz
Freq. GHz
Freq. GHz

• 40 ps features are well reconstructed (t1 ns)
• Resolution is enhanced by gt10 times vs. Wiener
  de-convolution