Image Enhancement of
Polycrystalline Aluminum
Electron Diffraction Patterns
Paul Larsen
William Stratton
ECE 533
Final Project
December 12, 2003
Introduction
Transmission electron microscopy (TEM) is an essential part of materials
characterization. Following L. de Broglie’s dual character theory, stating that particles
can be considered as waves, it was discovered the electron has a wavelength one hundred
thousand times smaller than visible light for an accelerating potential of 60 kV [1,2].
This smaller wavelength of the electron meant that electrons could image a specimen at a
much smaller resolution than any form of visible light. In 1931, Knoll and Ruska,
working at the Electrotechnical Institute of the Technological University-Berlin,
developed the first two-lens electron microscope. Siemens and Halske Company in
Germany first commercially manufactured this new invention in 1938.
Today’s TEMs work very similarly to the earlier versions. Using an electron
source (usually either a tungsten wire or a LaB6 crystal filament), electrons are boiled off
and accelerated to ~70% the speed of light along an optic axis of column under high
vacuum. These accelerated electrons are focused by a series of positive acting magnetic
lenses (figure 1), and coherently sent
through an electron transparent
Optic Axis
sample. Electron transparency
Ray 1
Ray 2
usually refers to specimens being
hundreds of Å thick. The electrons
interact with the sample and are
Lens 1
projected onto a viewing screen.
Imaging electrons can either be
viewed in imaging or diffraction
Back Focal Plane
mode.
Imaging mode in a TEM is
probably the first type of sample
viewing thought of by most people.
Lens 2
The electrons pass through the
specimen producing an image
Ray 2
Ray 1
similar to an image of a broken bone
taken by x-rays at a doctor’s office.
Crystals, impurities, and dislocations
Figure 1: Simplified ray diagram of a typical
in the sample all interact with the
TEM. The specimen would be placed above
electrons differently, therefore
lens 1. Electrons travel through the positive
generating an image with both high
lens 1, crossover at the back focal plane, and
and low contrast areas. Using
then continue through lens 2 to a viewing
images made by basic scattering,
screen. Diffraction patterns of specimens are
one can determine crystal size,
viewed at the back focal plane.
defect type, and defect size.
Diffraction mode can be thought of as the frequency domain representation of the
aforementioned projected image. Formed at the back focal plane of the TEM, the
diffraction pattern can give information on crystal structure, basic composition, and
lattice parameters. Various diffraction patterns can be seen in figure 2. Single crystal
samples generate a symmetric array of spots, polycrystalline samples generate an array of
concentric circles, and amorphous samples generate concentric fuzzy rings. In a single
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crystal diffraction pattern, the angle between the spots and the central beam (central beam
being at the vertex of the angle) is the angle that is between those reflecting planes.
While the distance from the center spot of the pattern to a feature, either a spot or ring is
proportional to the inverse of the distance between atomic planes of that feature. For
example, if a spot or ring corresponds to the {111} set of planes in a crystal lattice and
the distance from the spot to the feature is 2 cm, the spacing between the {111} planes
will be ξ * (1/2) cm-1, where ξ is the camera constant for the microscope [2,3].
Figure 2: Diffraction patterns of an A1 single crystal, polycrystalline
gold, and amorphous carbon respectively [3].
Diffraction patterns are generated in accordance with Bragg’s Law (equation 1
and figure 3).
n  2d  sin 
(1)
Where n is an integer, λ is the wavelength of the electron used for imaging, d is the
distance between two rows of atoms, and θ is the angle of reflection for the electron
waves. When a group of atoms is oriented at a Bragg condition (meaning the electrons
are reflected coherently), a bright spot will appear in the diffraction pattern, compared to
no generated spot when aligned out of the Bragg condition. This is how the various rings
and spots are generated in the diffraction patterns in figure 2.
Incoming coherent
electron waves (λ)
d
Outgoing coherent
electron waves
θ
Figure 3: Visual representation of Bragg’s Law, the dotted lines represents electron
wave paths while solid lines represent rows of atoms. Incoming electron waves
scatter coherently off columns of atoms, this coherence generates a bright spot in the
diffraction pattern.
Generally, the intensity of the diffraction patterns is proportional to the number of
electrons that can pass through the specimen. Hence, the intensity is dependent upon the
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thickness of the sample, assuming constant electron brightness. Thicker samples allow
only the central spot and a few other features to be visible, while thinner specimens allow
for more detail to be shown in the diffraction pattern. In both cases, however, the
outermost features are often lost due to the drastic difference in intensity between the
central spot and the outer features (rings or spots). The resulting loss of information
creates a problem for the analysis and presentation of the diffraction patterns.
Approach
Kirkland has suggested using digital image enhancement to make the lower
intensity portions of the diffraction patterns more visible and thereby extract as much
data as possible [4]. Researchers having little image processing background often apply
standard techniques in a trial-and-error fashion, which is often both ineffective and timeconsuming. For this project, we propose to develop a more concrete methodology for
digital enhancement of diffraction patterns. By applying image enhancement techniques
to a plethora of digitally captured diffraction patterns, we will identify those techniques
that are generally most effective for a given diffraction pattern, thereby assisting
researchers to apply image enhancement in a more systematic fashion.
Experimental
Digital diffraction images were obtained with the LEO 912 EFTEM at the
Materials Science Center at the University of Wisconsin – Madison. The TEM was
operated with an accelerating voltage of 120 kV, and has a spatial resolution of 16 Å.
Images were acquired in 8-bit TIFF format by AnalySIS ESIvision image acquisition
software.
A polycrystalline aluminum standard diffraction pattern manufactured by Ted
Pella Incorporated was used for the diffraction patterns. This type of polycrystalline
sample generates a diffraction pattern with multiple concentric rings about a bright
central spot. A beam blocker was used as to not damage the CCD camera from the high
intensity electrons traveling down the optic axis. Multiple images at varying exposure
times were taken, some with the bright central spot in the center, and some with the
image shifted to allow some of the dimmer rings of the diffraction pattern to be imaged.
Table 1 gives the combinations of exposure time and shift level for which images were
acquired.
Exposure Time
Table 1. Combinations of Exposure Time and Shift Level for which Images were Acquired.
50ms
100ms
500ms
1000ms
5000ms
10000ms
20000ms
30000ms
_________________Shift Level__________________
_0_
_1_
_2_
_3_
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
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The acquired images were each subjected to a power-law transformation, a log
transformation, and histogram equalization to enhance the lower intensity portions of the
diffraction pattern. The equations for the power-law and log transformations are,
respectively,
s  c  r
(2)
s  c  log( 1  r )
(3)
where c and γ are constants and r is the original pixel value. The value of c is computed
by first performing the transformation for c = 1, giving the output values s’, and then
letting c = max(s’)-1 such that subsequent application of the transform gives output values
that span the entire range of gray values. Histogram equalization is achieved by mapping
each original pixel with level rk into a corresponding pixel with level sk using
k n
j
(4)
sk  
j 0 n
in which nj is the number of pixels with gray-level rj and n is the total number of pixels in
the image. A median filter was also applied to each image to remove salt-and-pepper
noise. Finally, the images were visually inspected to determine which transform
provided the best contrast to the outer rings generated by the polycrystalline Al standard.
All image processing was done with the computer program Matlab.
Results
Table 2 gives the transformations that resulted in the best image enhancement for each
image. Although not mentioned explicitly in the table, there are two operations that
should be performed in addition to the power-law and log transformations in Table 2.
First, the constant c shown in Equations 2 and 3 should be calculated as explained in the
Experimental section. Second, the median filter should always be performed following
each transformation.
Table 2. Transform Giving the Best Enhancement for Each Image.
___________________________Shift Level_____________________________
_______0______ _______1______ _______2______
_______3______
50ms
Exposure Time
100ms
Log or Power-law
with γ = 0.3
Log or Power-law
with γ = 0.3
500ms
1000ms
5000ms
10000ms
20000ms
30000ms
Log or Power-law
with γ = 0.3
Power-law with
γ = 0.4
Power-law with
γ = 0.4
Log or Power-law
with γ = 0.3
Log or Power-law
with γ = 0.3
Log or Power-law
with γ = 0.3
Power-law with
γ = 0.5
Power-law with
γ = 0.45
Power-law with
γ = 0.45
Power-law with
γ = 0.4
Power-law with
γ = 0.35
Power-law with
γ = 0.35
Power-law with
γ = 0.35
Power-law with
γ = 0.55
Power-law with
γ = 0.55
Power-law with
γ = 0.55
Power-law with
γ = 0.55
Power-law with
γ = 0.5
Power-law with
γ = 0.4
Power-law with
γ = 0.4
Power-law with
γ = 0.35
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The value of γ used for the power-law transformation was determined in a trial-and-error
fashion with the objective of minimizing noise without losing information. The result of
applying these “best” transformations are given for a few of the images in Figures 4-7.
original
power-law with median filter, gamma = 0.3
Figure 4. Result of applying power-law transform on image for 50 ms exposure time and
zero shift.
original
power-law with median filter, gamma = 0.4
Figure 5. Result of applying power-law transform on image for 1000 ms exposure time
and one shift.
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original
power-law with median filter, gamma = 0.4
Figure 6. Result of applying power-law transform on image for 5000 ms exposure time
and two shifts.
original
power-law with median filter, gamma = 0.4
Figure 7. Result of applying power-law transform on image for 10000 ms exposure time
and three shifts.
Discussion
Median Filter
The median filter should be used always because it greatly enhances lower exposure time
images and does no harm to higher exposure time images, as shown in Figures 8 and 9.
Histogram Equalization
Histogram Equalization should not be used because it usually gives results that are
considerably less desirable than the log and power-law transformations and never gives
results that are noticeably better. This fact is also seen in Figures 8 and 9.
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a) histogram equalization for 50ms s hift0
b) histogram equalization and median filter
c) log transformation for 50ms s hift0
d) log transformation with median filter
Figure 8. Effect of median filter for low exposure times (50 ms and zero shift).
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a) histogram equalization for 1000ms s hift0
b) histogram equalization and median filter
c) log transformation for 1000ms s hift0
d) log transformation with median filter
Figure 9. Effect of median filter for high exposure times.
Log Transformation
Log transformations give basically the same results as power-law transformations with γ
= 3. The only difference is that the log transformation displays the outer rings with
slightly better contrast while the power-law transformation gives better contrast for the
inner rings, as seen in Figure 10. The difference is nearly imperceptible, such that there
should not ever be a need to apply the log transformation.
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log transformation for 5000ms s hift1
power-law with gamma = 0.3
Figure 10. Comparison of log transformation and power-law transformation with γ =
0.3.
Power-law transformation
The foregoing discussion establishes that the power-law transformation achieves either
the best or very close to the best enhancement under all circumstances investigated. The
focus should therefore be not on the type of transformation to employ, but rather on
which value of γ gives the best enhancement. To this end, the values given in Table 2
serve as a guide to the best values of γ for a given exposure time and shift level.
References
1. Hall, C. E., Introduction to Electron Microscopy, Robert Krieger Publishing Co.,
Malabar, Florida, 2nd edition, reprinted, 1983
2. Class Notes for MSAE 748 – Structural Analysis of Materials, Spring 2002,
University of Wisconsin – Madison, Dr. Babcock, notes prepared by Dr. T.F.
Kelly
3. Williams, David, Carter, C Barry, Transmission electron microscopy, Plenum
Press, New York, 1st edition, 1996.
4. Kirkland, Earl, Advanced computing in electron microscopy, Plenum Press, New
York, 1st edition, 1998
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Tasks
Planning
Experimental &
Analysis
Writing/Presentation
Overall
Team Members Percent Contribution
William Stratton
Paul Larsen
50
50
50
50
50
50
50
50
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