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Comparing signal processing methods for spectral bio-imaging

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Comparing signal processing methods for spectral bio-imaging

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Comparing Signal Processing Methods
for Spectral Bio-Imaging
By
M ark E. Arnoldussen
A Thesis Presented to the
FACULTY OF THE SCHOOL OF ENGINEERING
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER O F SCIENCE
BIOMEDICAL ENGINEERING
August 1998
© 1998 M ark E. Arnoldussen
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UMI Number: 1393165
UMI Microform 1393165
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This thesis, written by
Mark E. Arnoldussen _____________
under the guidance of his/her Faculty Committee
and approved by all its members, has been
presented to and accepted by the School of
Engineering in partial fulfillment of the re
quirements for the degree of
Master of Science
Biomedical Engineering
Date M y 22 ■ 1998
Faculty Committee
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Table of Contents
I. INTRODUCTION__________________________________________________________ I
II. METHODS________________________________________________________________ 5
A. Spectral Im a g in g ...............................................................................................................................................6
B. The Im aging Sy s t e m ........................................................................................................................................ 11
C. Wavelet An a l y sis............................................................................................................................................ 15
Continuous Wavelet T ra n sfo rm ........................................................................................................................ 17
Discrete Wavelet T r a n s f o r m .............................................................................................................................20
Wavelet Pa c k e t s.......................................................................................................................................................23
D. A Unique W av elet Sc h e m e..........................................................................................................................26
III. IMPLEMENTATION______________________________________________________ 28
A. Simulated Da t a ................................................................................................................................................30
B. Interferom etric Da t a ...................................................................................................................................35
IV. DISCUSSION OF RESULTS_________________________________________________38
V. CONCLUSION____________________________________________________________45
VI. REFERENCES____________________________________________________________47
ii
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List of Figures
Figure 1: Interferogram................................................................................................................................................8
Figure 2: Fou rier T ransform ation of Interfero g ra m ................................................................................... 10
Figure 3: Sagnac Interferom etric Sy stem .......................................................................................................... 12
Figure 4: Tilings of the tim e-frequency pla n esfo r m ......................................................................................16
Figure 5: Scaling function and wavelet (Daubechies length 4 )................................................................18
Figure 6: Continuous Wav elet T ra n sfo rm ........................................................................................................ 2 0
Figure 7: Discrete Wavelet Decomposition Tr e e.............................................................................................2 1
Figure 8: Discrete Wavelet Decom position....................................................................................................... 2 2
Figure 9: Wav elet Packet D ecomposition T ree................................................................................................ 2 3
Figure 10: Libra ry o f Wavelet Packets (size 8 ) ............................................................................................... 2 4
Figure 11: Wav elet Packet Decom position........................................................................................................ 2 5
Figure 12: 3-dim ensional representation of Figure 11...................................................................................2 7
Figure 13: Signal D ecomposition o f Simulated Sine Input for In pu t Lengths o f ...............................3 2
Figure 14: Signal Decomposition of Interferometric Da t a ........................................................................ 3 7
F ig u re 15: D e te rm in in g R e s o lu tio n , (a) F it t o F o u rie r d e c o m p o sitio n ...................................................39
Figure 16: r esolution vs. Da ta Length for (a) Sine, (b) Square, ..............................................................4 0
Figure 17: Resolution v s. D ata length for Interferom etric In p u t ........................................................41
Figure 18: Central Frequency Deviation for (a) Sine, (b) Sq ua re........................................................... 4 3
Figure 19: Cen tral Frequency D eviation for Interferom etric D a t a .................................................... 4 4
List of Tables
Table l : Elem ents o f a Spectral Imager (closed- image).................................................................................7
Table 2: Acquisition Param eters.............................................................................................................................14
Table 3: Frequency Reordering for Size 8 Wavelet Packet Lib r a r y .................................................... 25
iii
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I. Introduction
Many biological signals are fast, transitory events that pose certain difficulties
when trying to analyze them. Spectral imaging is a means of classifying the components
of a sample, such as living tissue, by looking at the colors of the light reflected or emitted
from its surface. In order to apply spectral image analysis on a sample, the duration of
the recorded light must be long enough for the signal processing algorithm to adequately
characterize the results. The limiting factor for any time-based method of analysis is
whether or not the resolution, or finest resolvable feature, provided by such spectral
analysis software is sufficient enough to describe the brief event. As the length of the
input signal decreases, the resolution becomes poorer. The trade-off between the desire
to decrease the time window of evaluation and the need to preserve enough resolution is a
question that the researcher must answer for himself based upon the specific type of
analysis. For samples that contain many spectrally overlapping components, this is a real
problem.
The intent of this project is to investigate the limits of two processing algorithms,
a unique wavelet-based signal decomposition method and a standard Fourier analysis
technique, in terms of their ability to resolve spectral features in a fast, short-windowed
manner. This comparison has been undertaken with particular consideration for the
SpectraCube™ SD200 system, a bio-imaging spectrometer developed by Applied
Spectral Imaging (Israel). The spectroscopic system developed by ASI collects the raw
data in the form of interferograms, or interference patterns, which reveal the underlying
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spectral components. The present signal processing technique uses Fourier analysis and
takes several seconds to acquire an image, making it difficult to get an accurate account
of quick time-based events. It is hoped that the compactness of the wavelet transform
will allow spectral data to be characterized over a shorter time span. While the intended
application of this study is for use in spectral bio-imaging, the results could be applied to
other time-dependent needs as well.
+ -----------------------
The most popular method of spectral data analysis used in signal processing is
based on the Fourier transform. While enabling good characterization for stationary
signals, the localization of frequencies in time for transitory signals cannot be measured
well because the Fourier decomposition is composed of infinite sinusoids. In other
words, the process of Fourier transforming a signal loses its time information because it
lumps the entire signal into one time window instead of breaking it into time-frequency
atoms. Although this project is not necessarily concerned with tracking the course of
changing frequencies over time, the power o f the Fourier transform will be tested on its
ability to recognize frequencies in as short a time window as possible. This process of
restricting the window within which the Fourier transform operates is akin to
characterizing the frequency spreading of a single time-frequency atom of the Short-Time
Fourier Transform (STFT). Shortening the length of the data window effects the
resolution in an inverse manner since the sinusoidal basis is forced to operate in a span
than is increasingly unsuitable.
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The wavelet transform, on the other hand, offers a more flexible approach. It is
based on finite-length “waves” that have zero value at each of its boundaries, which
provides the compact support necessary to operate on short time intervals. Shifting and
scaling (or stretching) of the mother wavelet can be used to obtain frequency information
during a given time interval. The function of scaling is analogous to the modulation by
complex exponentials in the Fourier transform and, therefore, scales are used in place of
frequency in a wavelet decomposition. The flexibility of the wavelet transform allows
special aspects of data to be analyzed, such as trends and discontinuities, which the other
techniques miss. It also provides a means for robust signal compression, which presently
is the most widespread application of wavelets. In this project, a unique wavelet method
is developed that is comparable to the Fourier transform in the sense that the processed
signals carry no time information. This is done in order to evaluate the effect of
shortening the data length on the spectral resolution.
This project will use several types of data to form a comparative evaluation of the
Fourier transform versus the wavelet decomposition method. By using different inputs
and varying their signal lengths, we will demonstrate each method’s
frequency/wavelength resolution capabilities as well as their accuracy based on the
deviation of the signals away from the center frequencies (the expected outputs). First,
simulated signals composed separately of sines, square waves, and sawtooth waves are
used for analysis to determine if either method is biased toward a certain kind of
waveform. After this preliminary investigation, experimental interferometric data
collected from the spectroscopic system is analyzed using wavelets and compared with
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the results from the present Fourier methods. In both simulated and experimental cases,
the source of the input is known, either by its frequency or emission wavelength, so that
the results can be compared to absolute values.
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n . Methods
This section reviews all the concepts that play a role in this project. The first part
is an introduction to spectral imaging, covering the ideas and mathematics underlying the
subject. It is followed by a description of how these thoughts have been incorporated into
the actual imaging system used in this project. Then an outline of wavelet analysis is
presented as an alternative to Fourier signal processing, the current method used by the
system. It starts with the conceptual premise of wavelets and then progresses from the
Continuous Wavelet Transform to the Discrete Wavelet Transform to Wavelet Packets,
the particular signal decomposition method used in this project. Finally, a unique
algorithm utilizing wavelet packets is described which allows a wavelet decomposition
method to be directly compared with a stationary Fourier transform.
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A. Spectral Imaging
Spectral imaging is the combination of point spectroscopy and imaging, allowing
the spectral response of a sample to be viewed as a two-dimensional spatial
representation. Spectroscopy is a technique that uses the intensity of light emitted,
transmitted, scattered, or reflected from a sample to identify its chemical constituents
based upon known characteristic absorption spectra. It can have high spectral resolution
over several wavelengths but is limited by the fact that it carries no spatial information.
Imaging, on the other hand, provides the high spatial resolution that spectroscopy
lacks. It allows a macroscopic view of a system and eliminates the need to base
information upon a singularity. The power of imaging also extends to the many
processing algorithms that exist. Images may be improved with enhancement techniques
such as edge detection and contrast enhancement or analyzed to extract regional
information with methods like segmentation and image feature extraction.
An example of spectral imaging is that of fluorescence imaging microscopy. It
uses the combination of spectroscopy and imaging to differentiate between the shape and
spectral ranges of a fluorescent dye and the background luminescence as well as separate
and map spectrally overlapping fluorescent probes. Another example is that of remote
sensing, which provides geologic information by scanning the Earth from satellites in
orbit. While both of these examples utilize spectral imaging, each uses a different
method to acquire data. The former example is that of a closed-image system because the
image size is fixed upon a certain region of a sample. In the case of remote sensing, the
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image is scanned continuously along the course of the tracking satellite so that the final
image can be as long as one wishes.
In this project, we are only concerned with the case of a closed-image system.
The hardware elements necessary to acquire a spectral image are listed in the table below.
Component Description
Dispersion element Separates the light into narrow bandwidths of color (or
wavelengths), like a tunable bandpass filter.
Camera Collects the image of the spectral data, such as a charge-
couple device (CCD).
Optical components
Collects the light from the sample and passes it through
to the dispersion element as well as the imaging optics in
order to produce a real image though the dispersion
element onto the CCD camera.
Computer Regulates the dispersion element, stores the raw spectral
data, and performs signal processing on the data.
Table 1: Elements of a Spectral Imager (closed-image)
The power of a spectral imaging system is based on how the components are chosen.
Image acquisition speed, wavelength resolution, and stability are some of the factors that
are determined by the strengths and weaknesses of the individual components.
When the dispersion element used is an interferometer, the process of collecting
spectral images is known as Fourier spectroscopy because the Fourier transform is used
to calculate the spectrum from the raw data. This is particularly important because it is
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the method currently used by the spectral imager that is used in this project and,
therefore, will be the only dispersion method considered in this discussion.
Interferometry is a technique that utilizes the dual nature o f light as an electric
field to describe the effect of interfering wavefronts. When two coherent light beams
originating from a single source interfere, a pattern of light and dark streaks is produced
at a recording site, such as a CCD camera. This pattern is an oscillatory function of the
difference in path lengths traveled by the two beams, otherwise referred to as the optical
path difference (OPD). Figure 1 shows an example o f the output from an interferometer.
.o
b.
<
> »
500
450
400
350
300
250
200
150
100
50 100 150
Time
200 250 300
Figure 1: Interferogram
This pattern, also known as an interferogram, contains the spectral intensity distributions
of the sample originating from the source, but must first be transformed in order to reveal
this encoded information.
To understand how the interferogram is formed, we begin by considering light as
the combination of electromagnetic waves. A monochromatic light source can be written
as an electric field:
) = A(A.0)• cosf ^ * -e a t
(1 )
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For a general broadband source, the electric field can be described as the integral
summation of individual monochromatic light sources as
E = f A(jI ) • cos
' 2 * x >
- a t
\ ’
dX (2)
a ;
where a = 2k cf X. When a light source is put into the interferometer, the effect of
splitting the beam by a known path difference and then recombining it causes the electric
fields to interfere in a constructive (additive) and destructive (subtractive) manner,
creating the interferogram:
/(OPD) = | \ I{X)dX +J I(X) cos
(3)
where total intensity of the source is
/ Q =\A2(X)dX = \I(X)dX (4)
When the two beams in the interferometer travel the same distance so that OPD = 0, the
output intensity is maximum in Equation (3). When OPD = X/2, the two beams are
completely out of phase and the destructive interference makes the intensity minimum.
For broadband light, the interferogram is an oscillating function with an outer envelope
that decays to zero for large OPD’s.
The Fourier transform is used to obtain the spectral information hidden in an
interferogram. This is demonstrated in Figure 2, which is the result of transforming the
interferogram of Figure 1.
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3 5
2 4
< 3
d >
X J
2
500 1000 1500 2000 2500
wavelength (nm)
Figure 2: Fourier Transformation of Interferogram
In a Fourier spectroscopic imaging system, an interferogram is collected for every pixel
in an image. To form a spectral image, the Fourier transformation has to repeated at
every pixel, creating a cube of data that holds separate line spectra for each pixel. For a
512 x 512 pixel image, this amounts to 262,144 transformations!
There are several advantages of using the technique of Fourier spectroscopy with
respect to other spectral dispersion methods such as filters, gratings, or prisms. The
interferometer is able to collect the light intensities across all wavelengths
simultaneously, making it a more sensitive system. This is because the high optical
throughput, or efficient use o f light, results in measurements that have a larger signal-to-
noise ratio (SNR) than the others, which reject all photons that are outside a certain
narrow spectral band. At the same time, a Fourier spectroscopic system still has a wide
spectral range that can cover from the ultraviolet to the infrared regions.
Another characteristic of a Fourier spectroscopic imager is the ability to control
the spectral resolution by adjusting the image acquisition parameters. Because o f the
high optical throughput, faster measurements could be obtained by decreasing the
collection time to the point where the SNR was equal to the other methods.
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B. The Imaging System
The acquisition of spectral information in this project is done using a special
device, the SpectraCube™ SD-200. It uses a Sagnac interferometer as its dispersion
element, as shown in Figure 3, and thus utilizes the principles of Fourier spectroscopy
mentioned before. This type of interferometer is also know as a common path
interferometer since the two beams that are recombined to interfere with each other travel
along the same path, but in opposite directions. Not only does the Sagnac interferometric
system enjoy the advantages of other Fourier spectrometers described in the previous
section, but it also has a particular advantage over the others because of its intrinsic
stability. The resistance of the system to external disturbances, such as vibrations and
temperature fluctuations, is due to the fact that a shift in any of the optical elements
affects both beams equally. The SpectraCube™ SD-200 has already been successfully
implemented to identify human chromosomes by means of multicolor spectral
karyotyping, which is an efficient way to classify the entire genome by color()-
The operation of the spectral imager can be understood by following the path of
light through the interferometer and into the camera. In Figure 3, the incoming light is
shown to enter straight into the interferometer, as represented by two parallel rays, and
exit at 90° onto the imaging lens, after which it is focused on the central pixel of the CCD
detector. In this situation, the two split beams travel the same distance (zero OPD) and
the intensity is maximum due to purely constructive interference. This follows from
equation (3), where OPD = 0 making I(OPD) = I0 (maximum). When the incoming light
enters at an angle with respect to the interferometer, the resultant recombined output
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beam exits at an equal angle with respect to the camera. In this case, the two beams have
traveled different distances, and the effect of the OPD is a gradation of destructive
interference. The true operation o f the SpectraCube™ SD-200 is not to rotate the entire
interferometer so that an angle o f light entry is created, but instead the beamsplitter is
rotated slightly around a vertical axis (the plane out of the page in Figure 3) so that an
OPD is made. Once the interferometric data is collected by the camera and stored on
computer, it is then processed to evaluate the spectral information.
Collimated
light
from object
> ■
Beamsplit
Mirrors
Imaging Lens
CCD Camera
Wavelength >
Spectrum at
shaded pixel
(after FFT)
Figure 3: Sagnac Interferometric System
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There are several parameters concerning the imaging performance o f the
SpectraCube™ SD200 system that are determined before the signal processing occurs.
The finest resolvable feature in two dimensions, or spatial resolution, ultimately depends
on the pixel size of the camera and the optical modulation transfer function of the system.
The total size of the sample that can be measured is limited by the field of view of the
system. The sensitivity is determined by the minimum detectable light level while the
dynamic range is determined by the largest difference in the intensities from the source.
The spectral performance of the spectroscopic system is also initially determined
by the hardware, after which the software takes over. The range of wavelengths across
the spectrum that the system can measure is defined as the spectral range and is a function
of the illumination source, the spectral sensitivity of the CCD, and the transmission
properties of the optics. The spectral resolution is specified by the full width at half
maximum of a narrow (monochromatic) source, which indicates the “spread” of the
spectrum. The polarization o f a system is a measure of the transmission as a function of
polarization of incident light.
The SD program is the software package that is used to acquire the images. This
acquisition software is able to control fringe density and alignment, fix the image
movement, and calibrate the interferometer and broad-band light source. Some of the
parameters that the program uses to acquire a spectral image are listed in Table 2. These
parameters can be changed according to the needs of the application in order to obtain
faster acquisition times or finer resolution.
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Acquisition Parameter Description
Step Size {x,} degree of rotation of the beamsplitter, which varies the OPD
Number of Frames {x2} total number of interference images acquired
Measurement Time {t(X j}} Collection Time + Calculation Time
Region of Interest size of pixel grid (N/A)
Fringe Density sampling parameter (N/A)
Table 2: Acquisition Parameters
Currently, SPCUB is the software that processes the raw data using the Fourier
transform. It is at this stage of signal processing where this particular project is
concerned with evaluating. The fundamentals of the Fourier transform can be reviewed
in many texts, such as Lathi’s Linear Systems and Signals, and is excluded from this
outline.
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C. Wavelet Analysis
At the heart of wavelet analysis is the concept of multiresolution, which is
basically the process of dividing a signal into different scales of resolution rather than
different frequencies. In a wavelet decomposition scheme, a signal at level m is filtered
into two parts at a lower level m-1: a smoothed average signal and a signal containing the
subtracted-out details. Each of the new byproduct signals is considered to be an
independent space which represents a separate scale of resolution, like a frequency band.
This decomposition can be repeated on each new signal and repeated iteratively to form a
decomposition tree. As each signal gets filtered further down the tree, the resolution at
each of the lowest level scales become finer and finer since each scale is the result of a
large function space being split into two smaller ones.
The power of wavelet analysis lies in the compact nature of its basis functions,
which allow a signal to be characterized in a short-windowed manner. The basis
functions that divide a signal into averages and details have finite resolution in scale as
well as time. Thus, instead of operating in the amplitude-frequency plane of the Fourier
Transform or the time-frequency plane of the STFT, the result of the wavelet
decomposition is shown in the time-scale plane (Figure 4).
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(a) (b) (c)
o
T 3
3
"5 .
E
<
u
c
u
3
( X
u
Frequency Time Time
Figure 4: Tilings of the time-frequency plane: (a) Fourier Transform, (b) STFT, and
(c) Wavelet Transform
Each rectangle in (b) and (c) has a constant area and is considered a time-
frequency tile or time-frequency atom. Even though wavelets operate in scale space, they
may still be referenced in terms of frequency. Each tile represents the region in the plane
which contains most o f that basis function’s energy. The difference between (b) and (c)
is that while the STFT tiles are squares representing equal resolution in both time and
frequency, the wavelet tilings are rectangles that can adapt to the specific needs of the
signal decomposition. The fact that the rectangles still occupy a constant area
demonstrates that Heisenberg’s uncertainty principle limits the product of the resolution
in time and frequency as
(5)
where A ,2 represents the time spread and A^2 represents the frequency spread. The
adaptive nature of wavelet analysis means that different bases can be formed, changing
the appearance of the tilings in Figure 4(c).
This way of characterizing a signal using tilings is analogous to the notation used
for music. Just as each note specifies a frequency at a given time, each tile has a specific
16
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scale at a given time. Although not exactly the same as the frequencies, the magnitudes
of the scales directly correlate with frequency. Small scales (fine resolution) correspond
to high frequencies and large scales (poor resolution) correspond to lower frequencies.
Signal transformations, whether Fourier or wavelet, are characterized by the bases
from which they are formed. These bases are sequences of vectors or functions that can
uniquely represent any signal, such as
in continuous notation. The V j represent the basis while the b; correspond to the
magnitude of the signal in each of the orthonormal spaces. A valid basis must be both
linear independent and complete. Linear independence is met when the zero vector can
only be represented with all b; = 0 and completeness is met when the basis “fills the
space.”
Continuous Wavelet Transform
To implement the Continuous Wavelet Transform (CWT), specific basis functions
must first be derived for low pass decomposition as well as high pass decomposition.
Corresponding to the lowpass filter is a continuous-time scaling function < f > ( t ) and
corresponding to the highpass filter is a wavelet w(t). The relations used to derive these
(6)
in vector form, or
v(t) = I 1 bi vi(t)
(7)
17
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functions are the dilation equation for < f i ( t ) and the wavelet equation for w(t), both of
which are derived from an initial scaling function by
m = 2 £ h 0( k M 2 t- k ) (8)
£=0
w(/) = 2 Y.hi(k)$(2t — k) (9)
k=0
Here, h0 and A , represent the lowpass and highpass coefficients, respectively. The
choice of the initial scaling function has certain constraints but can be tailored to the
needs of the designer. Below, the Daubechies scaling function (phi) and the wavelet (psi)
is shown as an example.
1 .4
1
0
0 .4
2
1
0
- 1
1 .5
Figure S: Scaling function and wavelet (Daubechies length 4)
The special property of these basis functions is that not only are these two
functions orthogonal (independent spaces), scaling (compressing by 2t) and shifting
(translating by k) each function produces an entirely new orthogonal basis. This is how
an input signal can be separated into banded frequencies: each level of the
highpass/lowpass decomposition is its own unique space.
Now that we have found the basis functions that will separate the signal into its
smooth and bumpy parts, we can implement the CWT. The decomposition of a function
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f(t) is formed from the matrix form of the orthogonal basis functions wjrft) in continuous
time, as demonstrates in the analysis function
bjK=\-o0m ^ j K^)dt (10)
where the set of coefficients bj l c are weights that relate the function to its wavelet
transform. It is these coefficients which express the magnitude of the scale in time. This
decomposition serves much the same purpose as the Fourier series does describing a
signal with a sum of exponentials except that the wavelet basis function is a small, finite
wave (or pulse) that starts at time t = 0 and ends at time t = N.
The reconstruction, or synthesis, of a function is described by
/ ( 0 = X bjkxvjk(t) (11)
j
which completes the CWT as a method that has an inverse transformation as the Fourier
transform does.
The process of taking the CWT can be conceptualize in broad terms as
summarized in the following five steps:
1) Compare a wavelet to the first section in the original signal.
2) Calculate a number which expresses the correlation between the wavelet and
this section (the larger the number, the more the similarity).
3) Shift the wavelet in a continuous fashion along the signal, repeating the first
two steps until the whole signal is covered.
4) Scale (stretch or compress) the wavelet, and then repeat the first three steps.
5) Repeat the above four steps until the entire range of scales has been covered.
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The result is a three dimensional plot of time, scale, and magnitude of the scale at time t.
Figure 6 demonstrates a CWT analysis of the interferogram signal shown in Figure 1.
Absolute Values of Ca,b Coefficients for a - 64 63 62 61 60
0 20 40 60 80 100 120
Figure 6: Continuous Wavelet Transform
The darker areas denote where the frequencies are centered, as seen in the strip around the
scale equal to 4. The especially dark regions at each side of the plot are not true
frequencies but rather represent border distortions.
Discrete Wavelet Transform
Extending these concepts to the discrete case, wavelet analysis can also be
thought of as an iterative use of filter banks, or sets of filters. The decomposition begins
when the analysis bank separates an input signal into frequency bands. In the Discrete
Wavelet Transform (DWT), this consists of matrices that represent a lowpass filter and a
highpass filter. The lowpass filter (L) essentially performs a moving “average” o f the
signal while the highpass filter (B) performs a moving “difference.” In this way, the
analysis bank produces coefficients which form a smooth approximation signal and a
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bumpy details signal that can be combined together to add up to the original input. These
coefficients represent the weight assigned to each scale as a function of time, which can
then be used for the time-scale representation of the signal. Each filtered output may be
broken down many times into low and high frequency parts, extending into a tree
structure.
Lowpass
Signal
1
Highpass
1
A, D,
Lowpass | Higilipass
1
A,
Lowpass | Highpass
A3 d 3
Figure 7: Discrete Wavelet Decomposition Tree
In Figure 7, A represents the approximation of the higher level signal and D represents
the details left out of A. The level of decomposition, as noted by the subscript,
determines the bandwidth of frequencies or the resolution in that function space.
Figure 8 demonstrates a DWT analysis of the interferogram signal of Figure 1.
21
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Decomposition at level 5: s = a5 + d5 + d4 + d3 + d2 + d1.
Figure 8: Discrete Wavelet Decomposition
The magnitude of the details (dl-d5) at each level corresponds to the value o f the
scale (frequency) at that particular time. If one wished to reconstruct the signal from its
decomposition coefficients, the synthesis bank would be used, which is the inverse o f the
analysis bank. More specifically, since the decomposition is performed using an
orthogonal filter bank, the inverse is equal to the transpose of the lowpass and highpass
filter matrices according to
lF L + B TB = I
(12)
22
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In this project we are simply interested in the raw coefficients which reveal the frequency
information and, therefore, we will not be as concerned with the inverse transformation.
Wavelet Packets
So far we have discussed continuous and discrete wavelet analysis, where only the
approximations of the signal are iteratively expanded, as shown above. Wavelet packets,
on the other hand, decompose both the highpass and the lowpass branches to yield a
complete tree, which is analogous to the structure of the Short-Time Fourier Transform.
DD2 AD-
AADj ADD3
AA
AAAj
DA-
ADA3 DDD3 DDA3 DAA3 DAD
Signal
Figure 9: Wavelet Packet Decomposition Tree
Each split of the tree constitutes a separation of vector spaces, which combined
together forms a family of bases. These basis functions, otherwise known as a “Walsh
basis,” act like a discrete cosine basis. The number of zero-crossings of each Walsh
function corresponds directly with the frequency of an ordinary cosine. In other words,
the more times an individual basis function changes sign, the higher its relative
frequency.
23
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2
(1) 0
-2
2
(3) 0
-2
2
(5) 0
-2
2
(7) 0
-2
0 10 20 0 10 20
Figure 10: Library of Wavelet Packets (size 8)
As can be seen in the figure above, the library of wavelet packets (1) - (8) are not
ordered in terms of increasing zero-crossings (which denotes increasing frequency). A
numerical sequence, the inverse Gray code permutation, can be used to reorder this
library according to increasing. An example of this form is shown in Table 3.
Index Number
Natural
Order
0 1 2 3 4 5 6 7
Frequenc
y Order
0 1 3 2 6 7 5 4
24
-vJL -
T
(2) o
-2
(4) 0
2
0
2
0 20
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Table 3: Frequency Reordering for Size 8 Wavelet Packet Library
With this index adjustment, a wavelet packet decomposition can directly compete with
other frequency analysis techniques such as the STFT.
Figure 11 demonstrates a wavelet packet decomposition of the interferogram of
Figure 1. In this plot, the amplitude of a waveform is encoded by darkening the tile in
which the energy of the signal is located in time and scale.
©
S
in
50
100
200
250
20 30 25
time
range of
values ,
50 100 150 200
Figure 11: Wavelet Packet Decomposition
250
25
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D. A Unique Wavelet Scheme
The actual method of signal decomposition used in this project utilizes wavelet
packets in such a way that the results can be compared with that of the standard Fourier
transform. In this respect, we will consider all input signals to be stationary, although the
true nature may not be. For simulated data, this is not a concern because we can fix the
input frequencies to set values. However, for the interferometric data, the sample that is
being characterized may be from a dynamic system, yet the time course of the image
acquisition is fast enough to approximate the input as a stationary signal (usually in the
order of milliseconds).
Since the nature of a wavelet decomposition is to break the input signal up into
time-scale tilings, we can eliminate the time parameter by summating across equal scales
over the entire time duration. The effect is more or less like that of an averaging
function. A summation of wavelet packets can be used to characterize sinusoidal
frequencies as shown by a theorem demonstrated by Hess-NeilsenO- This theorem states
that a period wavelet packet can be used to approximate a function as
where the constants an, bn , and cqare the coefficients that represent the magnitude of
the frequency in a particular time-frequency atom. This is because the even-sequency
00 Q C
(13)
wavelet packets play the role of cosines and the odd-sequency periodic wavelet
packets 1act s^ nes- TMS ^ familiar form o f the usual Fourier series
26
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0 0 0 0
f = c0 + ^ an cos(2nnx) + ^ b n sin(2;znx)
rt= 1 n = l
(14)
Looking at Figure 12, we can consider the output of the wavelet packet
decomposition to be similar to 30 separate line spectra, since Figure 11 is really a three
dimensional plot where color represents the magnitude of the scale (frequency). To form
the composite stationary signal, the values of the coefficients at each of the 30 line
spectra are summated at every frequency bin, which is 256 in this case. The results of
this summation are shown in the following section.
Figure 12:3-dimensional representation of Figure 11
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III. Implementation
In this section, data of known frequency is processed using two signal processing
methods: the well-known Fourier Transform and the previously mentioned wavelet
decomposition method. To evaluate the strengths of each method, simulated signals of
known frequencies are presented as inputs before processing the actual interferometric
data. This will provide a larger sample set with which to compare the two methods.
In order to implement the wavelet technique described in the previous section, the
Wavelet Toolbox provided by MATLAB® was used because it has the versatility to
choose among several wavelet families. Two initial requirements in choosing a wavelet
family were that it had to have a discrete form of analysis (a DWT) and be compactly
supported orthogonal. This eliminated the Mexican hat and Morlet wavelets since they
are “crude” waveforms that only have continuous transforms and also eliminated all
biorthogonal wavelets, which are better suited for subband coding. The choice was
further narrowed down by selecting the wavelets that had the highest level of symmetry,
which were the Symlet and the Coiflet wavelets. Symmetry seems to be an advantageous
quality necessary for utilizing a wavelet transform as a frequency analysis tool.
The length of the wavelet filters was also an important criterion. As the filter is
lengthened, the frequency resolution is increased at the cost of poorer time support
because of Heisenberg’s uncertainty principle mentioned earlier. One would think this
would lead to creating a filter large enough to cover the entire input signal, but this would
cause erroneous measurements because the averaging effect of the intended algorithm
28
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would be lost. Since the Wavelet Toolbox already has predefined maximum lengths
which are sufficiently shorter than my intended inputs, I was satisfied using the longest
filter lengths supplied by the software package.
The Fourier transformations have also been performed using MATLAB®,
specifically the FFT function provided in the Signal Processing Toolbox. This is a one
dimensional discrete Fourier transformation that allows the convolution window to be set
to a predetermined size. This allows the user to control the number of frequency bins,
which also plays a role in determining the resolution limit of the transform.
To evaluate the FFT and the wavelet method on an equal basis, the absolute limit
of frequency spreading in each transformation has to be the same. This was done by
setting the level of decompositions for each transformation equal to 256 frequency bins.
In the Fourier case, performing a 512 point FFT achieves this since the transformation is
symmetric around the halfway point (512/2 = 256). For the wavelet case, this was done
by decomposing the signal to the eighth level, since the frequency bins created are
powers of two (28 = 256).
After trial and error, it appeared that the ‘coif5’ wavelet worked slightly better
than the ‘sym8’ wavelet. The ‘coif5’ wavelet had a filter length of 30 while the ‘sym8’
only had a filter length of 16. It is not readily apparent whether the difference in
frequency decompositions is due to the lengths o f the filters or the differences in the
shapes of each wavelet, which both approximate symmetric functions. Most likely it is a
combination of the two.
29
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A. Simulated Data
The simulation consists of testing each transform method based upon two criteria:
the type of input signal and the length of the input. The purpose of using different basic
inputs is to decorrelate the effectiveness of each method from the type of signal it sees. If
a basis is prejudiced toward a certain kind of signal, then a comparison to another basis
will not mean that much if the results are hedged in favor of one over the other. The
Fourier Transform is already derived from a basis of sines and cosines and, therefore,
might be expected to handle such input better than the wavelet method. The square
waves and sawtooths are independent functions that neither transform basis perfectly
matches and should provide an impartial comparison to see whether sine waves are a
biased input. Even though the FT is expected to do well given sine wave input, it is still
necessary to evaluate both methods with such input since it is these types of functions
that are most common.
The input signals presented to both methods were the same at each stage of
evaluation. Every signal was composed of four frequencies (50, 150, 300, and 450 Hz)
and sampled at 1000 Hz (enough to fulfill the Nyquist criteria). The purpose of using
multiple frequencies was to be able to average the processing “response” and cover the
full range of frequencies. This serves to obtain a rough measure of consistency of the
transformations in the form of deviation away from the expected center frequencies.
The only differences in the signals were in the lengths of the data segments.
These inputs were varied based on powers on two, from 28 = 512 points down to 2s = 32
points, in order to evaluate the “response” of each methods resolution capabilities.
30
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The results of the simulated signal analysis is presented in Figures 13-15. Only 3
of the data lengths for each type o f signal are shown for demonstration.
31
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S ines of frequency 50,150, 300,450 Hz (512 pts)
-5
150
100
50
150
100
50
0.1 0.2 0.3
time (sec)
0.4 0.5 0.6
Fourier Power Spectrum
I
1
1 1
Wavelet Power Spectrum
i .. >
100
1 i
200 300
frequency (Hz)
(a)
400 500
S in es of frequency 50,150, 300,450 Hz (128 pts)
-5
10
20
10
0.02 0.04 0.06 0.08
time (sec)
Fourier Power Spectrum
0.1 0.12 0.14
i_ _ _ A i 1
Wavelet Power Spectrum
i . . k
100
jL i
200 300
frequency (Hz)
(b)
400 500
Sines of frequency 50,150, 300,450 Hz (32 pts)
0.005 0.01 0.015 0.02 0.025 0.03 0.035
time (sec)
Fourier Power Spectrum
0.5 -
S
Wavelet Power Spectrum
L v J v w . . _ W V
) 100 200 300
frequency (Hz)
(C)
400 500
Figure 13: Signal Decomposition of Simulated Sine Input for Input Lengths of
(a) 512, (b) 128, and (c) 32 data points
32
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5
0
-5
400
200
0
400
200
S quare w aves of frequency 50,150,300,450 Hz (512 pts)
0.1 0.2 0.3 0.4
time (sec)
Fourier Power Spectrum
0.5 0.6
JL
elet Power Spectrum
1
ravi
JL . 1
100 200 300
frequency (Hz)
(a)
400
i
500
30
20
10
0
60
40
20
0
Square waves of frequency 50,150, 300,450 Hz (128 pts)
0.02 0.04 0.06 0.08
time (sec)
Fourier Power Spectrum
0.1 0.12 0.14
A lj A.
Wavelet Power Spectrum
.A,.. X
100
A i
200 300
frequency (Hz)
(b)
400 500
Square waves of frequency 50,150, 300,450 Hz (32 pts)
0.005 0.01 0.015 0.02 0.025 0.03 0.035
time (sec)
3
2
1
0
15
10
5
0,
Fourier Power Spectrum
\ / X x x
Wavelet Power Spectrum
jW k _ J L
100 200 300
frequency (Hz)
(c)
400 500
Figure 14: Signal Decomposition of Simulated Square Waves for Input Lengths of
(a) 512, (b) 128, and (c) 32 data points
33
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-5
100
50
0
150
100
50
0
5
0
-5
10
5
0
20
10
0,
Saw tooths of frequency 50,150, 300, 450 Hz (512 pts)
0.1 0.2 0.3 0.4
time (sec)
Fourier Power Spectrum
0.5 0.6
JL i i
Wavelet Power Spectrum
1
100
1
J II
200 300
frequency (Hz)
(a)
400 500
Sawtooths of frequency 50,150,300,450 Hz (128 pts)
0.02 0.04 0.06 0.08 0.1
time (sec)
______________ Fourier Power Spectrum
0.12 0.14
L a .A. JL i
Wavelet Power Spectrum
A A
A - . 1
100
L A jl I -A — M____Aw
200 300 400
frequency (Hz)
(b)
Sawtooths of frequency 50,150, 300,450 Hz (32 pts)
500
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
time (sec)
Fourier Power Spectrum
0.5
0
4
Wavelet Power Spectrum
100 200
frequency (Hz)
300 500 400
(C)
Figure 15: Signal Decomposition of Simulated Sawtooth Waves for Input Lengths of
(a) 512, (b) 128, and (c) 32 data points
34
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B. Interferometric Data
The SD200 spectral imager was used to collect the interferometric data from a
sample of known wavelength, a HeNe laser. Besides the fact that this data was collected
from a real source, the data also differs from the simulated data in the sense that instead
of packing multiple frequencies into the input signal, only the characteristic red
wavelength of the HeNe laser was used. This wavelength is recorded by pointing the
HeNe laser into the input aperture of the SD200. After the image was acquired, the
SPCUB software automatically processed the input using the FFT as well as recorded the
raw interferogram. From the processed spectral image, the maximum intensity pixel was
chosen and cross-referenced with the same pixel in the interferogram so that a nice
spectral response could be obtained and analyzed.
The interferometric data was processed in the same way as was done for the
simulated data. For the purpose of this demonstration, it is only necessary to show how
each method transforms from the time domain to the frequency domain. Yet for
evaluation and most practical concerns, the frequency axis can be changed to display
wavelength using the relation / = —. The frequency of the 632 nm HeNe laser is
— -— = 1.58MHz, which is the frequency around which all of the transformations are
632nm
centered. The frequency — ► wavelength transformation will be used to evaluate the two
methods as functions of wavelength resolution.
35
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Figure 11 displays the results of Fourier and wavelet transforms on different
lengths of an interferogram.
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Full Single-Pixel Interferogram (315 pts)
400
200
10 15 20
Time (Fs)
Fourier Power Spectrum
25 30
i
10
Wavelet Power Spectrum
2 3
frequency (MHz)
(a)
Partial Single-Pixel Interferogram (128 pts)
400
200
15
W W \A A A A a A A /W \A A a a A A /
20 40 60 80
time
Fourier Power Spectrum
100 120
10
5
Wavelet Power Spectrum
3x10’“
1
6x10™ 9x10™"
frequency (Hz)
(b)
12x1 O ’4 15x101 4
Partial Single-Pixel Interferogram (32 pts)
400
200
10 15
time
20 25 30
Fourier Power Spectrum
A
Wavelet Power Spectrum
A
3x10™" 6x10™"
frequency (Hz)
(C)
9x10™" 12x10’ 15x10’
Figure 14: Signal Decomposition of Interferometric Data
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IV. Discussion of Results
To quantify the effectiveness of each method, two measures of comparison are
made. The first is a measure of the resolution as a function of input data length. One
way to determine the resolution is by using the standard known as the Full Width at Half-
Maximum (FWHM). This is used when the input is known to be a spectrally narrow
source and the distance between the points at the half-max values can be measured.
However, this definition assumes a gaussian-like shape for which the maximum point is
clearly marked as the midpoint between two rolling-off curves. Since the output of a
wavelet decomposition disperses the energy of a narrow source in a pseudo-random
manner, the FWHM is not readily apparent.
To account for this dispersion in energy, the outputs from both the Fourier and the
wavelet decompositions have been least-squares fitted to gaussian functions so that the
comparison of resolution is standardized. This routine utilizes the MATLAB®
minimization function that uses the Nelder-Mead simplex search to optimize the solution.
The gaussian functions are made to fit in the region around which the output frequency is
expected. In the simulated case, this region is chosen as the midway point between two
frequencies in order to minimize the effects of overlapping spectra.
Figure 15 demonstrates the results of this fitting routine for the extreme case of
the interferometric processing, which is for 32 point data length. These results show how
the gaussian-fit approximates the resolution.
38
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0.8
0.6
0.4
0.2
400 600 800 1000 1200
1
0.8
0.6
0.4
0.2
800 600 1000 1200
wavelength (nm)
(a)
wavelength (nm)
(b)
Figure 15: Determining Resolution, (a) Fit to Fourier decomposition,
(b) Fit to wavelet decomposition.
Knowing that the normal gaussian function is o f the mathematical form:
, r -i2
1 x - x
1 T -
/(* ) = ■
'\ln c P '
(15)
the resolution can be found to be twice the standard deviation a, which is already defined
as the distance of the waveform from the central axis to a side lobe at the half-max level.
Figure 16 demonstrates the results of using the gaussian fit routine on the
transformed data from the simulated inputs.
39
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Resolution (H z) Resolution (H z) Resolution (Hz)
40
Fourier Decomposition
Wavelet Decomposition
25
15 -
100 200 300
Data Length
400 500 600
(a)
45
Fourier Decomposition
Wavelet Decomposition
200 400 300
Data Length
500 600 100
(b)
45
40
Fourier Decomposition
Wavelet Decomposition
200 100 300
Data Length
400 500 600
(C)
Figure 16: Resolution vs. Data Length for (a) Sine, (b) Square,
and (c) Sawtooth Input
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And for the interferometric data:
100
90
Fourier Decomposition
Wavelet Decomposition
E
S 60
I 50
3
§ 40
c a
20
1 00 150
Data Length
200 250 300 350
Figure 17: Resolution vs. Data Length for Interferometric Input
The trend in the amount of resolution attained as a function of data is apparently
the same for all input cases, whether simulated or experimental. When the length of the
data is sufficiently long, the Fourier Transform always resolves frequencies better than
the wavelet transform. The difference is very sleight, yet definitely considerable for
situations that require the finest resolution possible. When the data length shortens below
around 200 points, there is a shift in resolvability towards the favor of the wavelet
transform. While decreasing the data length gradually broadens the finest features in a
wavelet transform, the same condition drastically worsens the Fourier Transform
resolution as shown by the steep slope in Figures 16 and 17 as the data length approaches
zero.
The second quantifying measure is the deviation of the spectral outputs away from
the known center frequencies. Since the input frequencies are known, the output
41
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waveforms are expected to be centered at these same points, with the maximums
coinciding with the center frequencies. The average deviation is computed as
Avg. Dev. = - £ / „ - fn (0 ) (16)
n n
where f n is the peak magnitude centered around the known central frequency, f ni0). For
the simulated data, n = 4 since there are four input frequencies at 50, 150,300, and 450
Hz.
Since n = 1 in the interferometric case (only the HeNe source is used), the result
would not be an average but simply a single point deviation. To remedy this situation,
four data segments at each of the partieil interferogram lengths (32, 64, and 12S points)
are used from different segments of the full interferogram and substituted into the
equation for average deviation. This averaging technique doesn’t apply to the full
interferogram, yet its deviation is not as questionable as the shorter data segments and,
therefore, is sufficient.
The results of applying the average deviation formula to the simulated data is
shown in Figure 18.
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3.5
N
I Fourier Decomposition
Wavelet Decomposition
2.5
E
o
c
o
5
’ >
u
a
0.5
400 500 100 200 300
Data Length
600
(a)
6
5
Fourier Decomposition
Wavelet Decomposition
4
3
2
1
0
100 400 500 600 200 300
Data Length
(b)
9
8
7
Fourier Decomposition
Wavelet Decomposition 6
5
4
3
2
1
0
100 200 300
Data Length
400 500 600
(C)
Figure 18: Central Frequency Deviation for (a) Sine, (b) Square,
and (c) Sawtooth Input
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And for the interferometric data, the average deviation is shown in Figure 20.
f 10 Fourier Decomposition
Wavelet Decomposition
150 200
Data Length
250 300 100 350
Figure 19: Central Frequency Deviation for Interferometric Data
These results suggest that the accuracy of the Fourier transformation is always
about twice as good as the wavelet transformation. One possible cause for the higher
level of variation is the method with which the natural order of the wavelet
decomposition has to be rearranged to fit into the frequency order. The frequency
adjustments may not be as easily distributed by a numerical sequence, but instead defined
by experimentally calibrating the wavelet decompositions into the correct frequency bins.
It could also be a case where a certain frequency is being dispersed into its harmonics, as
particularly demonstrated in the sawtooth input decompositions of Figure 15.
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V. Conclusion
A unique algorithm based on wavelets was developed for comparison with the
standard Fourier transform in an attempt to minimize the processing window with which
frequencies can be identified. The wavelet method seems to have better short-window
processing power than the Fourier transform for stationary signals in terms of resolution.
This is based on the results obtained from decomposing both simulated input signals and
experimental signals recorded from the SpectraCube™ SD200 spectral imaging system.
Not only does the wavelet method outperform the FFT’s resolving power for short data
lengths, but it does a fair job for long data lengths as well - almost as good as the FFT.
This performance verifies that a wavelet method of signal decomposition can, indeed,
represent a signal composed of sinusoidal frequencies and do it well.
However, the accuracy of the output for the wavelet method is consistendy twice
as far off as the Fourier based decomposition. This is the weakest part of wavelet
analysis and presently makes the FFT a more practical processing method. It is possible
that the inaccuracies could be reduced by refining the techniques used in this project, but
not for certain.
Even when considering the better characteristics of the wavelet analysis, the
resolution obtained at such short data lengths as used in this project may not be sufficient
for characterizing the spectra of samples in many applications. When the data length is
not a concern and the minimization of frequency spreading is required, there is no
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question that the Fourier transform is more robust than the WT, including accuracy of
results.
The future for this type of wavelet algorithm may not be to replace the FFT, but
rather to enhance its power. Because short-windowed signal evaluation is not very
accurate but is very resolvable, the WT could complement the FFT by giving a rough
estimate of the frequencies for short data lengths. Such an application in spectral imaging
would be a centering algorithm that uses the dominant frequencies to line up features in a
region of interest.
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VI. References
1. Cabib, D., Buckwald, R.A., Garim Y., Soenksen, D.G. (1995) Spatially resolved
Fourier Transform spectroscopy (Spectral Imaging): a powerful tool for quantitative
analytical microscopy. SPIE, Vol. 2678, pp. 278-291.
2. Garini Y., Katzir, N., Cabib, D., Buckwald, R.A., Soenksen, D.G., Malik Z. (1996)
Spectral Bio-imaging, from Fluorescence Imaging Spectroscopy & Microscopy,
edited by Xue Feng Wang and Brian Herman. John Wiley & Sons, Inc., pp. 87-124.
3. Delori, F.C. (1988) Noninvasive technique for oximetry of blood in retinal vessels.
Applied Optics, Vol. 27, No. 6, pp. 1113-1125.
4. Delori, F.C. (1994) Spectrophotometer for noninvasive measurement of intrinsic
fluorescence and reflectance of the ocular fundus. Applied Optics, Vol. 33, No. 31,
pp. 7439-7452.
5. Malik, Z., Cabib, D., Buckwald, R.A., Talmi, A., Garini Y., Lipson, S.G. (1996)
Fourier transform multipixel spectroscopy for quantitative cytology. Journal of
Microscopy, Vol. 182, Pt 2, pp. 133-140.
6. Novak, R.L. (1988) Microfiber reflection spectrophotometry of cytochrome oxidase
in the rat cerebral cortex: Relationships between brain oxidative metabolism and
function following direct cortical stimulation. Doctoral Dissertation, University of
Miami.
7. Soenksen, D.G., Sick, T.J., Garini, Y. (1995) Use of a novel spectral bio-imaging
system as an imaging oximeter in intact rat brain. SPIE, Vol. 2679, pp. 182-189.
8. Schrock, E., du Manoir, S., Veldman, T., Schoell, B., Wienberg, J., Ferguson-Smith,
M.A., Ning, Y., Ledbetter, D.H., Bar-Am, I., Soenksen, D., Garini, Y., Ried, T.
(1996) Multicolor Spectral Karyotyping of Human Chromosomes. Science, Vol.
273, pp. 494-497.
9. Misiti, M., Misiti, Y., Oppenheim, G., Poggi, J.M. (1996) Wavelet Toolbox For Use
with MATLAB. The MathWorks, Inc.
10. Strang, G., Nguyen, T. (1996) Wavelets and Filter Banks. Wellesley-Cambridge
Press.
11. Hess-Nielsen, N., Wickerhauser, M.V. (1996) Wavelets and Time-Frequency
Analysis. Proceedings of the IEEE, Vol. 84, No. 4, April pp. 523-540.
47
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12. Vetterli, M., Kovacevic, J. (1995) Wavelets and Subband Coding. Prentice Hall,
Inc.
13. Herley, C., Kovacevic, J., Vetterli, M. (1993) Tilings o f the Time-Frequency Plane:
Construction of Arbitrary Orthogonal Bases and Fast Tiling Algorithms. T F .F .F .
Transactions on Signal Processing, Vol. 41, No. 12, December pp. 3341-3359.
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Asset Metadata

Creator Arnoldussen, Mark Edward (author)

Core Title Comparing signal processing methods for spectral bio-imaging

School School of Engineering

Degree Master of Science

Degree Program Biomedical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, biomedical,OAI-PMH Harvest

Language English

Contributor Digitized by ProQuest (provenance)

Advisor [illegible] (committee chair), [illegible] (committee member), D'Argenio, David (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-24451

Unique identifier UC11341427

Identifier 1393165.pdf (filename),usctheses-c16-24451 (legacy record id)

Legacy Identifier 1393165.pdf

Dmrecord 24451

Document Type Thesis

Rights Arnoldussen, Mark Edward

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA