Transcript Document

Digital Holographic Microscopy for
Measurement of Cellular Refractive Indices
Robert Thomen
Creighton University
Master’s Thesis Defense
4-27-11
Motivation and Purpose


What we want: the refractive index of living cells, particularly bone
cells!
Why we want it: the refractive index is required to accurately
determine pressure measurements in the optical stretcher.
 S ( )  T (i )  nt cos t  ni cos i  R(i )  2  ni cos i  
I ( )
c
What is Holography?


Records 3-D images by taking advantage of the wave nature of light
Can also be used to record differences in optical path length through
transparent objects
d
Design of the Microscope
I  4 I 0 cos2 (abs   )
 
2

nc ( x, y )  h( x, y )
If we find Δφ,
we can find nc and h.
y
x
…so, how do we find Δφ?
Flowchart for Procedure
Cell Height
I 0 cos (( x, y))
2
1.
2.
3.
4.
( x, y)
Calculate phase using Hilbert Transform
Unwrap phase using Goldstein’s Algorithm
Calculate cell height and index for every pixel
Profit
Cell Index
1. Calculating the phase
Any analytic signal can be written in the form:
z( x)  f ( x)  iH{ f ( x)}  f ( x)  ifˆ ( x)
where the imaginary part is the Hilbert Transform of the real part.
The Hilbert Transform is defined

1 P f ( x' )
{ f ( x)}  fˆ ( x)  f   
dx'
x   x  x'
 


P  lim   f ( x)dx   f ( x)dx 
 
 o  

The phase of our signal at every point is then
 fˆ ( x) 
 Im[z ( x)] 

  arctan
 ( x)  arctan


f
(
x
)
 Re[ z ( x)] 


So what does the Hilbert transform actually do?...
What exactly does the HT do?

The Hilbert Transform has the effect of shifting each
frequency component in a signal by π/2

We can make this manipulation in frequency space!
Hilbert and Fourier are close relatives

The Hilbert Transform manipulates the phase of
Let φ=wt
the signal’s frequency space
F{ f ( x)}  F ( ) 


f ( x)ei 2x dx
The Fourier Transform

g ( )
nF
nF 1
f ( x)  F{ f ( x)}  F ( )  F{ fˆ ( x)}  fˆ ( x)
g ( )  i sgn( ) 
i for   0
0 for   0
 i for   0
i

ie
i

2
Two-Dimensional Phases
F (i ,  j ) 
 

f ( x, y)e
i 2 (i  x  j  y )
dxdy
The 2D Fourier Transform

DC centered
(  0)
Fringe Frequencies
FFT
n
g (k )  (i) n  sgn(k )  (1) sgn(1 ) sgn(1 )
k
nF
g ( k )
nF 1
f ( xk )  F{ f ( xk )}  F (k )  F{ fˆ ( xk )}  fˆ ( xk )
 Im[z (t )] 

  arctan
 Re[ z (t )] 
We have now found φ!.....or have we?....
2. Unwrapping the Phase

By nature of the arctan function, our phase map is
“wrapped”
cos2 (( x, y))
W {( x, y)}  ( x, y)
( x, y)
z(t )  f (t )  ifˆ (t )
 Im[z (t )] 

 Re[ z (t )] 
  arctan
We must then unwrap the phase
The Effects of Noisy Data
Digital noise can cause miscalculation of the phase in
The unwrapping algorithm.
6π
4π
4π
2π
2π
How can we deal with noisy data?
Filtering processes to Remove Noise
Several filtering techniques have been employed in our code
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
Median Filtering
 For each pixel, a surrounding
box of pixels is selected and
the median pixel intensity of
these is substituted
Fourier Filtering
 By taking the Fourier transform of an image, we can selectively
remove certain frequency information.
But filtering changes our data!
Residues and Branch Cuts
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Problematic pixels can be recognized as residues
Residues occur at singularities in otherwise entirely analytic functions
Res( f , c) 


1
2i
 f ( )d
C |  z 0 |
Residues can be calculated for a contour around each pixel
Residues can be connected by branch cuts which disallow
unwrapping progression through certain parts of the phase map
We have implemented Goldstein’s Algorithm to make these branch cuts.
This can be done in LabVIEW
Capture image
Take the Hilbert Transform
of that image
Inverse tangent to find phase
What do we get when we do this?...
Labview Program
We now have the phase image!
3. Taking the Data

We require 3 images
 1. Background  N 0
o Move cell into the frame
 2. Cell in media of index n
C 0
o Add media of higher index
 Must monitor phase shift dyn
 3. Cell in media of index n+δ C 

L
dyn
2

 C    C 0 
hL
2

 C 0   N 0   nm
nc 
2h
ΦC+
C0
N0
dyn
This procedure was developed by Rappaz et al. Opt Express 13, 9361 (2005)
Verifying the integrity of our results
Are the numbers accurate? Do we take good data? Does our program work?

We have constructed a program to synthesize perfect data to test
our algorithms
I ( x, y, t )  cos2 0 ( x0 , y0 )   ( x, y, t )
0 ( x, y )  S ( y0  j ) 2  ( x0  i) 2
 ( x, y, t ) 
2

n(t )L  hc ( x, y)  nc hc ( x, y)  hn ( x, y)  hc ( x, y)nn 
Examples of a Gaussian-shaped cell of index 1.38 and peak height 10,000nm
Now we can address the effects of filters…
Effects of Filtering on Ideal Data
So what are filters good for?...
Effects of Filters on Noisy Data
n=1.38874+/-.002827
Even if filtering is necessary, the average index is preserved!
radians
nm
nm
Single Cell Results (EMT6)
Frame #
Refractive Index Data for EMT6 cells
Average precision of
.003
n633 = 1.384 ± 0.041
n735 = 1.387 ± 0.023
n812 = 1.375 ± 0.013
What is the characteristic Index of EMT6 cells?
Conclusions
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DHM is operational and taking data
Image processing algorithm has been tested on
synthesized data to verify the integrity of results
There seems to be a great deal of index
variation from cell to cell (EMT6 cell data at
735nm)
Portable model has been developed
Soon to take data at 1064nm
Acknowledgments

Dr. Mike Nichols
Semere Woldemariam

NIH grant P20 RR016475 from the INBRE

Program of the National Center for
Research Resources.
Questions…
Extra slides follow
Holography and Interferometry

Holography can display 3D images
because it records not only the
light’s intensity but also its phase
We use this principle of holography to detect optical path length
variations in transparent media.
This difference in OPL is measured as a phase difference…
Cell images from Tomographic Phase Microscopy, Michael S. Feld et al. September 2007
Nature Methods Vol. 4, No. 9.
Design and Construction
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Mach-Zender Interferometer
Frequency Analysis
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Simply get rid of negative frequencies using a
Fourier transform! F{ fˆ ( x)}  i sgn(k )F (k )
This process will follow this procedure
nF
f ( x)  F{ f ( x)}  F (k )
 i sgn( k ) F ( k )

nF 1
F{ fˆ ( x)}  fˆ ( x)
Recall f(t) is two dimensional…
Let
nF
f ( x )  F (k )
and
sgn( k j )
nF 1

Fˆ (k )  fˆ ( x )
F ( k )( i ) n
 fˆ ( x) 
 Im[z (t )] 
 
  arctan
arctan


f
(
x
)
 Re[ z (t )] 


What good is phase?


We have phase data, but who cares about
phase?
In order to get n and h, we have developed
decoupling techniques
L

  2  1 
2
h  L

  4  3 
2
nc  nm 

 3  1 
2 h
Acknowledgments

Dr. Mike Nichols
Semere Woldemariam

NIH grant P20 RR016475 from the INBRE

Program of the National Center for
Research Resources.
Questions…