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Reflective tomography using a TCSPC system –a study of current
limitations and possible
improvements
TOMAS OLOFSSON
Master’s thesis work at Umeå UniversitySupervisors: Christina
Grönwall and Markus Henriksson, FOI, LinöpingExaminer: Magnus
Andersson, Department of Physics, Umeå University
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AbstractTime-correlated single photon counting (TCSPC)
systemsare used for range profiling. The systems offer cm
precisionat kilometer ranges. This opens up for long range
imagingwith high resolution, for example by reflective
tomography.With range profiles from various aspect angles around a
tar-get reflective tomography can be used to create an image.The
tomographic image is a reconstruction of the boundaryof the
cross-section of the target. Images can be used forvarious
purposes, e.g. identification of satellites.
The quality of the tomographic reconstruction depends onthe
accuracy of the TCSPC system. Range profiles with acm precision
allows studies and reconstruction of complexobjects. With this work
we investigated the current limi-tations when reconstructing
complex targets with reflectivetomography and present possible
solutions to existing prob-lems.
The limitations were investigated by studying parameterssuch as
the intensity of the laser beam, SNR, center of ro-tation, angular
resolution, and the angular sector. We alsopresent methods that can
improve the tomographic image.A new pre-processing method that
adjusts range profilesafter estimating responses with RJMCMC was
introduced.We also studied different types of filters in the
reconstruc-tion process. Lastly we introduced two new
post-processingmethods. One that removes artifacts by considering
theconvex hull and one that sharpens edges in the
tomographicimage.
The performance study showed that reflective tomographyusing a
TCSPC system is robust in a controlled environ-ment. Details in the
low cm-range of an object can be re-constructed with high
precision. However, for some targettypes issues appear. Of the
tested performance parametersa high angle resolution was deemed to
be the most im-portant. When considering moving targets the
importanceof the center of rotation and integration time will also
in-crease. The study of improvement methods showed thatchoosing the
generalized ramp filter in the FBP more thendoubled the SNR.
Adjusting the range profiles, consideringthe convex hull, and
sharpening edges are methods thatwork well for specific signal
types. We showed that manyissues that arise when measuring on
complex objects canbe solved with signal processing. Therefore we
believe thatreflective tomography can be used in various
applicationsin the future.
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Sammanfattning
Tidskorrelerad räkning av fotoner (time-correlated singlephoton
counting, TCSPC) är en teknik som används föratt skapa
avståndsprofiler med cm-precision på upp emotflera kilometers håll.
Tekniken kan användas till att skapaavbildningar av föremål på
långa avstånd, till exempel medreflektiv tomografi. Reflektiv
tomografi kan användas närman har avståndsprofiler runt om ett
föremål. Den tomo-grafiska återskapningen beskriver de yttre
kanterna av ettföremåls tvärsnitt.
Bildernas kvalité är starkt beroende av noggrannheten
iTCSPC-systemet. En cm-precision möjliggör studier
ochåterskapningar av föremål med små detaljer. Detta arbetegår ut
på att underöka begränsningarna med återskapandetav detaljerade
föremål och framlägga metoder som förbätt-rar återskapningarna.
Begränsningarna undersöktes genom att studera olika pa-rametrar,
såsom intensiteten i lasern, SNR, rotationscent-rum,
vinkelupplösning och vinkelsektorer. Vi presenteradeockså nya
metoder som förbättrar återskapningarna. Vi tit-tade bland annat på
en metod som korrigerar avståndspro-filerna med hjälp av
anpassningar med RJMCMC. Sedanundersöktes olika filter i
återskapningen. Avslutningsvis in-troducerades två nya metoder i
efterbehandlingen. En somtar hänsyn till konvexa höljet och en
annan som gör kanteri bilderna skarpare.
Prestandaundersökningen visade att reflektiv tomografi ba-serad
på ett TCSPC-system är robust i en kontrollerad mil-jö.
Centimeterstora detaljer kan återskapas med hög upp-lösning. För
vissa föremål uppstår dock problem. Av de tes-tade
prestandaparametrarna var en hög vinkelupplösningden viktigaste.
Valet av rotationscentrum och integrations-tiden kommer spela en
större roll med föremål i rörelse.Studien av förbättringsmetoder
visade att bilders SNR merän dubblas om det generella rampfiltret
används i FBP. Attkorrigera avståndsprofilerna med RJMCMC, ta
hänsyn tillkomplexa höljet och att göra kanter skarpare är
metodersom fungerar bra med vissa signaltyper. Mer arbete finnsatt
göra men vi visade att många problem i reflektiv tomo-grafi går att
lösa med signalbehandling. Vi tror därför attreflektiv tomografi
går en ljus framtid till mötes.
iv
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Acknowledgments
First I would like to thank my supervisors, Christina Grönwall
and Markus Hen-riksson, for the support throughout the project. Not
only for providing informationand ideas about my work, but also for
giving me a lot of helpful tips about writingthe report. I would
also like to thank FOI (Swedish defence research agency)
forproviding me with a work space at FOI so I easily could perform
new measurementsand ask questions to my supervisors.
v
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Contents
Acknowledgments v
1 Introduction 11.1 Previous work . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 11.2 Goals and limitations . . . .
. . . . . . . . . . . . . . . . . . . . . . . 21.3 Report outline .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Method and theory
2 TCSPC 32.1 Principle . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 32.2 Resolution . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 42.3 Single-photon
avalanche diodes . . . . . . . . . . . . . . . . . . . . . 52.4
Experimental setup . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 52.5 Instrument response function . . . . . . . . . . . . .
. . . . . . . . . 6
3 Theory 93.1 Tomography . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 93.2 Reconstruction . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 113.3 Filters . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Signal
properties . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 153.5 RJMCMC . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 183.6 The convex hull . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 20
Results
4 Improvement methods 234.1 Adjustment of range profiles . . . .
. . . . . . . . . . . . . . . . . . . 234.2 Changing filter in FBP
. . . . . . . . . . . . . . . . . . . . . . . . . . 264.3 Removing
artifacts with the convex hull . . . . . . . . . . . . . . . .
274.4 Sharpening target edges . . . . . . . . . . . . . . . . . . .
. . . . . . 29
5 Performance study 315.1 Center of rotation . . . . . . . . . .
. . . . . . . . . . . . . . . . . . 31
vi
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CONTENTS
5.2 Angular sector . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 335.3 Angular resolution . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 355.4 The aim of the laser beam . . . .
. . . . . . . . . . . . . . . . . . . . 365.5 Integration time . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Combining methods of improvement 41
Discussion
7 Discussion 437.1 Summary . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 437.2 Future work . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 457.3 Conclusion . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Bibliography 47
vii
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Chapter 1
Introduction
Non-imaging laser radar (ladar) systems can be used for
high-resolution long dis-tance range profiling and are of interest
for both military and civil applications. Inthis project we will
study one of the applications: reflective tomography, a methodthat
can translate 1D range profiles of an object into a 2D image.
Range profiling can be performed with various ladar techniques,
we used time corre-lated single-photon counting (TCSPC). Ranges are
estimated by counting photonsreflected from an object. The time of
arrival of each photon is also registered. Ahistogram with arrival
times for many photons corresponds to a range profile. TC-SPC is a
very effective ladar method with a time resolution down to the ps
range.With range profiles collected from many aspect angles around
a target we can usereflective tomography to reconstruct its cross
section. High precision in the TCSPCsystem also makes the
resolution in the reconstructed images high. The recon-structed
images can be used for many purposes. Identification of objects far
awayis one example, where satellites, ships, airplanes, and
missiles are possible targets[1]-[3].
1.1 Previous work
Reflective tomography is not a new method, it has been used
since the late 80s[4]. However only recently reflective tomography
has become of interest in prac-tical applications. As a consequence
of the recent interest there are many effectsthat have not been
thoroughly studied yet. Feature tracking to find the center
ofrotation [5]-[6], alternative reconstruction techniques [7], and
phase retrieval [8] areexamples of methods that have been
documented. We will make use of the methodsfound in literature to
further study ways to improve the reconstruction, e.g.
artifactreduction.
A condition for obtaining high image resolution is high
precision in the TCSPCsystem. Therefore it is important to study
its limitations. The performance of
1
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CHAPTER 1. INTRODUCTION
lasers, detectors, and other components are continuously
improving. Because ofthat not only the current limitations, but
also characteristics and environmentaleffects are of interest. In
this regard many experiments and studies have beenperformed. System
characteristics [9]-[12] and how the range profile is affected
byturbulence [13]-[14] are examples of what has been studied. Also,
a model describingthe response as a function of the relevant
effects has been derived [15].
1.2 Goals and limitationsThe main goal of this project is to
study the current effectiveness of reflective to-mography using
TCSPC and to draw conclusions of the usefulness of the method.We
will also consider the effectiveness on complex objects that
typically are difficultto reconstruct. A secondary goal is to find
methods to improve the reconstruction.Interesting methods in
literature will be tested and, if possible, further developed.
The time frame for this project will be the spring semester
2012, which correspondsto 30 ECTS-credits. Because the main focus
of the project is to study reflectivetomography generally we will
not perform outdoor measurements. We will alsolimit ourselves to
rotating the object in one plane, so 3D reconstructions will notbe
possible.
1.3 Report outlineThe first part of the report will cover theory
so that the area of reflective tomographyis well introduced and so
that it is easier to understand the methods we used toimprove the
tomographic reconstruction. Secondly, the performance of the
systemmade so the current limitations of reflective tomography
using TCSPC are wellunderstood. The center of rotation, different
angular sectors, the angular resolution,the importance of the aim
of the laser, and integration time will be investigated. Thecenter
of rotation decides at which distance from the center of the image
responsesare placed and is an important parameter in the
reconstruction. The angular sectoris the range of different aspect
angles that we measured from and the angularresolution decides how
many angles that separate each measurement. The aim of thelaser and
the integration time are factors that affect the number of counted
photonsin the range profiles. Lastly methods to improve the
tomographic reconstructionwill be presented. We will introduce pre-
and post-processing methods and studydifferent filters in the
reconstruction.
2
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Chapter 2
TCSPC
2.1 Principle
TCSPC is a method for high resolution range profiling. Ranges
are measured byemitting high repetitive laser pulses and by using
single-photon avalanche diodes tocount reflected photons and to
register their the time of flight (ToF). Each pulseis independent
and contributes with a new measurement. Summarizing over manypulses
will yield a histogram of the number of returning photons at
different timeinstants. The histogram corresponds to a range
profile. The general idea of themethod is shown in Figure 2.1.
Photons
Laser beam
Counts
Range
Detector
Laser Object
Histogram
Figure 2.1. A description of a TCSPC system setup.
With knowledge of the ToF and the fact that we are using a
laser, a target range rcan be calculated as
r = ct2 , (2.1)
3
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CHAPTER 2. TCSPC
where c is the speed of light and t is the time of flight. A
factor 1/2 is added becausethe light travels back and forth.
2.2 ResolutionWhen considering objects with cm-large details,
different targets will often very closedepth-wise. Using (2.1) we
see that targets at different distances will be separatedby
∆t = 2∆rc. (2.2)
Distances between targets correspond to a time interval (∆t) as
seen in (2.2). Forsmall distances the interval is of same magnitude
as the time jitter of the system(∆tsystem). Time jitter is random
variations in time. When ∆t ≈ ∆tsystem thepositions cannot be
resolved. Hence, ∆tsystem is a suitable representation of
thelimitation in accuracy. The total time jitter of the system can
be approximated to
∆tsystem =√
∆t2laser + ∆t2detector + ∆t2electronics, (2.3)
meaning that the laser (∆tlaser), the detector (∆tdetector), and
electronics (∆telectronics)are the relevant contributors to the
variations. The largest contribution usuallycomes from the detector
and is discussed in Section 2.3.
A simpler way of understanding the effects of time jitter is by
looking at the responsefrom a point source. A point source at a
fixed distance will theoretically result ina Dirac function when
summarizing photon counts from many pulses. However, inpractice a
distribution is obtained as there are uncertainties in the
measurements,i.e. time jitter. The distribution is commonly called
the instrument response func-tion (IRF) and is of interest for
deconvolution purposes. The width of the IRF iscommonly used as a
measure for ∆tsystem. A typical IRF is shown in Figure 2.2.In
Section 2.5 the specific IRF for our system is discussed.
0 100 200 300 400 500 600 700 8000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Typical IRF in TCSPC
Range [bins]
Norm
alize
d cou
nts
Figure 2.2. A typical IRF in a TCSPC system.
4
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2.3. SINGLE-PHOTON AVALANCHE DIODES
2.3 Single-photon avalanche diodesTo create histograms over
returning photons, an accurate and robust way of count-ing them is
necessary. This is accomplished with single-photon avalanche
diodes(SPADs). The principle is that you start a timer at a
specific time and measureover a time interval in which photons
might be detected. The interval correspondsto a certain range. In
TCSPC the photon is a portion of the reflected laser pulsefrom the
object. To avoid unwanted pile-up effects it is important that only
onephoton hits the detector in each time interval. If more than one
photon is de-tected the signal in the SPAD will be distorted [26].
The probability for detectinga reflected photon from an emitted
pulse is about 0.0001 − 0.001 for the systemused in this project.
This minimizes the risk of two detections happening in thesame
interval. A histogram is created by collecting results from many
time intervals.
When a photon hits the detector a self-sustaining current is
created, figuratively anavalanche is started. The SPAD measures the
current during the whole samplinginterval, resulting in a peak at
the detection of a photon. Ideally the response wouldbe a Dirac
function, however it takes some time to quench the avalanche
current.The process is explained in detail in [17]. In short, the
quenching is a discharge ofcapacitors that causes an exponential
decrease in current after the event.
In the TCSPC system the detection of a photon is recorded at the
time whenthe current suddenly increases, i.e. when the avalanche is
started. There is someuncertainty in the measurement which will
contribute to the IRF shown in Figure2.2. Still, the time jitter is
sufficiently small for making SPADs preferable to analogdetectors.
The accuracy is dependent on the setup but can venture down to the
psrange.
2.4 Experimental setupThe setup can be divided into two parts.
The first part is the components surround-ing the laser and the
detector and the other is the setup around the object. This isshown
schematically in Figure 2.1. In Figure 2.3 a photo of the first
part is shown.A Fianium SC450 was used. It is a supercontinuum
laser source with a 40 MHz rep-etition rate. The photon collecting
system consist of optical components to collectphotons effeciently.
The detector is a SPAD (Micro Photon Devices, PDM series).A 3 nm
bandpass filter with a center wavelength of 834 nm was used to
avoid noisesources like fluorescent lamps. A camera was used to see
where the laser was di-rected. A data acquisition card (PicoHarp
300, PicoQuant GmbH) translated theoutput from the SPAD to a
histogram with bin size 4 ps. The histogram was thenpassed on to a
PC where it is stored via a LabView routine.
5
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CHAPTER 2. TCSPC
In Figure 2.4 the second part of the system is shown. To isolate
reflections fromthe object black boards were placed in front and
behind the object. The boardswill give responses, however they are
well separated depth wise from the object sothey can easily be
excluded. A motion controller (Newport ESP301) together witha
rotating device makes it possible to turn the object 341 degrees
with a smalleststep of 1 degree. This means that the object will
not be seen from 19 degrees.
2.5 Instrument response function
As discussed in Section 2.2 the time jitter in the system will
yield an IRF. A goodmodel of the IRF is necessary for deconvolution
purposes. It is also a necessity inRJMCMC (Section 3.5) where it is
used to approximate the distribution. Typicallyin a TCSPC system
the IRF is modeled as a set of piece-wise exponential
functions[18]-[20],[12]. However, every experimental system is
unique. The behaviors in oursystem has been studied [25]. The model
used is
a
d
c
b
e
Figure 2.3. The components of the experimental setup. a) Laser
aperture, b)photon collecting setup, c) detector, d) camera, e)
filter.
6
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2.5. INSTRUMENT RESPONSE FUNCTION
fIRF (t) =
exp(−(t− t0)
2
2s2
)t < t1
exp(−(t1 − t0)
2
2s2
)
×(a× exp
(− t− t1
τ1
)+ b× exp
(− t− t1
τ2
))t ≥ t1
, (2.4)
where t1, s, a, and τ1 are optimizing parameters and t0 is the
time of the peak. band τ2 are defined as
b = 1− a, τ2 = b/((x1 − x0)/s2 − a/τ1). (2.5)
How the IRF looks like can be seen in Figure 2.2. Initial
choices for optimizingparameters are discussed in Section 4.1. This
model differs from [18]-[20],[12] inthe regard that the derivative
is continuous. By studying the FWHM of the IRFfrom a point source a
good representation of ∆tsystem is obtained. It is found to
beapproximately 60 ps, which corresponds to 9 mm.
a
b
c
d
e
Figure 2.4. Object rotational system. a) Background, b) object,
c) foreground, d)rotating device, e) motion controller.
7
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Chapter 3
Theory
In Figure 3.1 a description of the process can be seen.
Initially, range profiles areobtained with the TCSPC technique as
explained in Section 2. A tomography im-age is thereafter
reconstructed from the range profiles. To do this we used
filteredback projection (FBP), the most common reconstruction
method in reflective to-mography. Before and after the
reconstruction different methods can be used toimprove the image
quality. The theory behind a pre-processing method (based
onreversible jump Markov chain Monte Carlo) and a post-processing
method wherewe considered the convex hull will also be
discussed.
FBP
Range profilesTCSPC
LaserObject
SPADs
f(x,y) Pre-proce-ssing
Range
Post-proce-ssing
Counts
Angle
Image
f(x,y)
Figure 3.1. A description of the process.
3.1 TomographyOur goal is to create an image from the range
profiles obtained with TCSPC by ap-plying tomographic
reconstruction. Before we explain the reconstruction we beginwith
explaining the basics of tomography.
Tomography is a method that considers sections (greek tomos,
section) of an image.In signal processing, sections (or slices) are
usually profiles from different angles ofincidence. The profiles
can contain different types of information. Tomography isfurther
divided into different areas depending on the information. In our
case wehave profiles that contain range information of the object.
With range profiles re-
9
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CHAPTER 3. THEORY
flective tomography is used, an area which is still rather
unexplored. Most commontoday is transmission tomography, an area
where the profiles describes the transmis-sive properties of an
object. We first describe the more commonly used
transmissiontomography and then we describe reflective tomography.
The reconstruction willbe explained in Section 3.2.
3.1.1 Transmission tomography
In transmission tomography, an object is illuminated with an
electromagnetic wave.The energy transmitted through the object is
then measured. The profile obtainedis a measure of how easily
different parts of the object is penetrated. For example,a CT-scan
(computed tomography), commonly used in medicine, obtain profiles
byX-ray illumination. In Figure 3.2 a typical profile is shown.
Notice that cavitiesand difference in absorption inside the object
shows in the intensity reading.
Figure 3.2. Example of an image slice in transmission
tomography.
Advantages of transmission tomography are the possibility to
reconstruct objectsconcealed to the naked eye (e.g. a kidney), the
high definition, and the fact that it isnoninvasive. It also has
its limitations. First, the measurements must be performedin a
controlled environment. Second, not all materials are easily
penetrated. Atlast you need detectors on the other side of the
object, which makes it difficult tomeasure on distant objects.
Profiles from many aspect angles are used to reconstructthe
cross-section of a targeted object.
3.1.2 Reflective tomography
As the name implies this method utilizes reflected signals. A
profile is obtained byilluminating an object with a laser and by
measuring the response. The profiles cancontain range information,
like in our case. Another example is Doppler measure-ments.
10
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3.2. RECONSTRUCTION
An example of how a profile looks in reflective tomography is
shown in Figure 3.3.It is noticeable that no information of the
interior or the back side of the object canbe obtained. In
addition, the information is perpendicular to the incident
wavefrontand not parallel like in transmission tomography. In the
range profile the intensityshows how much light that is reflected
at a specific distance. The intensity dependson various factors
including the angle of incidence, the type of material, the
exper-imental setup, and the size of the area illuminated. Also,
some materials reflectonly a small portion of the light. The
experimental setup is discussed in general inSection 2.3 and
reflection is explained further in Section 3.4.1.
rang
e
ProfileLaser beam
Object
Figure 3.3. An example of a range profile of an object from one
aspect angle.
As mentioned before reflective tomography and transmission
tomography are closelyrelated. This becomes clearer as the
reconstruction is explained. Although theslices differ in some
aspects the information they contain is enough to use a
similarreconstruction method.
3.2 ReconstructionAfter obtaining range profiles the question is
how to perform the reconstruction. Intomography the most common
choice is the filtered back projection (FBP), whichis a variant of
the inverse Radon transform. With measurements from one inci-dent
angle θ, the reconstruction process is illustrated in Figure 3.4.
Lr,θ is a lineat range r and angle θ, p(r, θ) is a function of the
responses in the range profileprojected along Lr,θ, and f(x, y) is
the boundary of the cross section that we wantto reconstruct. Note
that p(r, θ) is shifted 90 degrees from the incident ray. This
isthe only computational difference between transmission tomography
and reflectivetomography in the reconstruction phase. f(x, y) is
found in the reconstruction bysumming p(r, θ) from many angles
around the object.
3.2.1 Radon transformFBP is a variant of the inverse Radon
transform so we start with defining the Radontransform. It is
defined as a series of line integrals through a function u(x, y). A
line
11
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CHAPTER 3. THEORY
r
y
xθ
L r,θ
f(x,y)
ry
xθ
p(r,θ)
Counts
r
Range profile
TCSPC FBP
Lase
r bea
m
Figure 3.4. The different parameters and functions used in the
reconstruction.
is given by the set (r, θ), where r is the closest distance from
the origin to the lineand θ is the angle describing how much the
rθ-plane is rotated from the xy-plane.The Radon transform is then
defined as
Rf = p(r, θ) =∫ ∞−∞
∫ ∞−∞
u(x, y)δ(xcosθ + ysinθ − r)dxdy, (3.1)
where R is the Radon transform operator and δ is the Dirac delta
function. In otherterms, p(r, θ) is the projection of u(x, y) at an
angle θ. With the reconstructionprocess shown in Figure 3.4 in
mind, (3.1) can be more conveniently written as
p(r, θ) =∫Lr,θ
f(x, y)ds, (3.2)
where s represents the line along Lr,θ. We have now explained
the transform from(x, y) to (r, θ). Our goal is to obtain f(x, y)
from p(r, θ) so the reconstruction comesdown to making an inverse
transform. The different functions and parameters in(3.2) are shown
in Figure 3.4.
3.2.2 Fourier slice theoremWe have the projection data in polar
coordinates so we cannot use the inverse Fouriertransform directly
as it considers Cartesian coordinates. Instead we choose theinverse
Radon transform, which is closely related to the inverse Fourier
transform.The inverse is in itself a rather straightforward
procedure. However, beforehand oneshould be aware of a theorem that
supports our choice of reconstruction method.The Fourier slice
theorem (or the projection slice theorem) states that the
Fouriertransform of a projection is a slice of the Fourier
transform of the two-dimensionalimage. This fact will make it
possible to reconstruct an image with the 2D inverseFourier
transform. Because of the close relationship between the Fourier
and theRadon transform the theorem also holds for the Radon
transform. Mathematicallyit can be expressed as
F1pg = s1F2g, (3.3)
12
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3.2. RECONSTRUCTION
where F is the Fourier operator, g is the image, p is the
projection data, and s isthe operation that extracts a slice. The
indices denotes whether the operation is in1D or 2D.
3.2.3 Filtered back projection
An infinite number of projections from an infinite number of
angles will yield a per-fect reconstruction. In TCSPC we have a
finite number of angles and projections,hence we need a formula for
the discrete case. With that in mind and knowingthat the inverse
Radon transform is unstable in practice makes an adjusted
versiondesirable.
The most commonly chosen method to perform the reconstruction is
the filteredback projection. It is discrete and has a filter
function that can reduce artifacts andnoise. FBP is defined as
g(x, y) =m∑i=1F−1(c̃F(p(r, θi)))∆θ, (3.4)
where m is the number of range profiles and g(x, y) is the
desired image. F is theone-dimensional Fourier operator and F−1 is
its inverse. c̃ is a filter function thatconsists of a ramp filter
that appears in the derivation of the formula and a windowfunction.
The window function is added to reduce artifacts and noise.
Typicalchoices of filters behave as the ramp filter for low
frequencies with a decliningresponse for high frequencies.
Different filters are discussed in Section 3.3 and howthey perform
in the reconstruction is shown in Section 4.2. Also note that
thefiltering is performed in Fourier space. The filter function is
defined as
c̃ = |k|w(k), (3.5)
where w(k) is the window function and k is the spatial
frequency.
FBP is an intuitive method. To illustrate this Figure 3.5 shows
how each projec-tion p(r, θ) contributes to the image g(x, y).
According to (3.4) crossings of differentprojections will be
summarized and have increased intensity, emphasizing the
circle.Figure 3.5 also indicates a common problem with tomography,
namely artifacts. Aresponse will not only be visible in the "true"
position but is stretched all the wayto the image boundary (along
Lr,θ). In most cases artifacts will become relativelysmall as all
measurements are summarized. Some responses however are
relativelyvery large and will result in clearly visible artifacts.
Especially large surfaces per-pendicular to the incident wave will
create unwanted artifacts. It is important toknow that artifacts
are a part of the method. The question is how to reduce them.
13
-
CHAPTER 3. THEORY
Angle of incidence, θi
TCSPC & FBP
g(x,y)
f(x,y)
p(r,θ )i
Image boundary
y
x x
y
Figure 3.5. A description of FBP in reflective tomography. Data
readings are takenfrom various angles around a circular
cross-section (left). Each reading contributesto the reconstruction
(right).
3.3 FiltersThe ramp-filter (also known as the Ram-Lak filter)
was introduced by Ramachan-dran and Lakshminarayanan to reduce
artifacts. In that regard it works, howeverit does not account for
noise. Filters that reduce noise will also lower the accuracyin the
signal. Therefore there is a tradeoff between accuracy and noise
that has tobe considered. The optimal filter is dependent on the
signal type. To find a filterthat performs well in general we
studied four alternatives to the ramp filter: theShepp-Logan (S-L)
filter, a modified S-L filter, the Hann filter, and a
generalizedramp filter. The choices are mostly based on previous
studies [22]-[23].
The original ramp-filter, (3.5), was first adjusted by Shepp and
Logan to dampresponses while keeping the accuracy. The filter
response function is
HS−L(w) =∣∣∣∣2a sin wa2
∣∣∣∣ 0 ≤ w ≤ πa , (3.6)where w is the frequency and a is a
scaling factor. The function is mirrored inπ/a < w ≤ 2π/a. The
following filter response functions will also be mirrored inthe
same way. To reduce noise a modified version of the S-L filter was
introduced:
H̄S−L(w) = 0.4HS−L(w) + 0.6HS−L(w) coswa. (3.7)HS−L is the
response from the S-L filter and H̄S−L is the modified response.
Anothercommonly used filter is the Hann filter. It is defined
as
HHann(w) =12 cos
(w
a
)0 ≤ w ≤ π
a. (3.8)
What all the previous filters have in common is that they share
properties for lowfrequencies. For high frequencies the response
declines differently depending on
14
-
3.4. SIGNAL PROPERTIES
the filter. This is because high frequencies are typically
noise. With the sharedproperties of the previous filters in mind a
generalized ramp filter was proposed:
Hg.f.(w) =1a|w|e−ξ|w|p 0 ≤ w ≤ π
a, (3.9)
where ξ = |1/wc|p is a function of a parameter p and the cut-off
frequency wc. ξworks as a damping factor. With ξ = 0 the
generalized ramp filter will be identicalto the original ramp
filter. The benefit of the generalized ramp filter is that youcan
change the parameters to suit the needs for a specific signal type
instead ofconsidering another filter. In our case the parameters p
= 2.6, ξ = 1.874, and a = 1were chosen and used throughout the
project. The different response functions canbe seen in Figure 3.6.
Low frequencies are found in the beginning and in the endof the
responses according to the FFT output in Matlab.
0 pi/2 pi 3pi/2 2pi
0
pi/2
pi
w [rad]
H(w
)
Frequency response function for different filters
rampS−Lmod. S−Lgen. filterHann
Figure 3.6. The frequency response of the filters investigated
in the project witha = 1. Parameter choices for the generalized
ramp filter are p = 2.6 and ξ = 1.874.
3.4 Signal propertiesMany issues with reflective tomography can
be explained with the reflective prop-erties of the illuminated
object. Issues can be details that is not appearing in
thereconstruction and surfaces that dominate the image. Therefore
it is important toknow how the signal behaves in different
scenarios. For example how the responsewill change when looking at
a surface from different aspect angles. That is a com-monly
occurring scenario as we are rotating the object in our
measurements. We
15
-
CHAPTER 3. THEORY
start with explaining how surface characteristics affects the
response.
3.4.1 ReflectionThere is three possible scenarios when light
hits a surface; the light is absorbed,transmitted, or reflected.
When the light is absorbed the energy is transformedto another
form, like heat. How much light that is absorbed differs between
ma-terials and is commonly described by the attenuation
coefficient. The absorptioncan become important to consider in
reflective tomography if the target consists ofdifferent materials.
If some parts of the target absorbs relatively much more lightit
will typically be hard to see them in the reconstruction.
The second scenario is when light is transmitted through an
object. In that case wewill not be able to measure the signal.
Therefore we usually consider opaque objectsin reflective
tomography. Important to note is that a material do not interact
withlight in only one way. What you consider is the portion of
light that is absorbed,reflected, or transmitted. So even if light
is transmitted through some material itdoes not necessarily mean
that some light has not been reflected.
Lastly, the light can also be reflected on the surface. It is
the reflected light thatwe detect in reflective tomography so it is
the most important scenario.
Specular and diffuse reflections
Reflection of light is commonly divided into two separate types,
specular and diffuse.The physics behind the reflection is the same
in both cases. Energy is retained andthe law of reflection
holds:
θi = θr, (3.10)
where θi is the angle of incidence and θr is the angle of
reflection. The differencelies in the material. For specular
reflection the surface is smooth relative to thewavelength of the
light. Outgoing light will have the same angle relative the
normalof the surface as the incoming light according to (3.10). A
common example ofspecular reflection is when light is reflected in
a mirror.
If the surface of the object is not smooth the light will be
spread out. The reflectionis then diffuse. The two cases are shown
in Figure 3.7. In a TCSPC system thedetector is placed at the same
place at the laser, meaning that a specular surface onlywill give a
response when it is approximately perpendicular to the incident
light.It is then important to have a high angular resolution. The
effects of changing theresolution is discussed in Section 5.3. A
diffuse surface will give a response for allincident angles but the
response will be stretched out in time as it is tilted fromthe
perpendicular position. As it is stretched out the intensity will
also be lowered.
16
-
3.4. SIGNAL PROPERTIES
These properties of specular and diffuse surfaces leads to that
perpendicular surfaceswill dominate the responses, making tilted
and/or relatively small surfaces hard todistinguish.
θ
θ
r
i
Incident light
Reflected light
Specular reflection
Incident light
Reflected light
Diffuse reflection
SurfaceSurface
Figure 3.7. Examples of specular (left) and diffuse (right)
reflection.
3.4.2 Mathematical modelWhen analyzing a range profile there is
also other effects to take in considerationthan the type of
material. Here we will present a mathematical model of a
rangeprofile so that it becomes clearer how the response is
affected by, e.g., turbulence.A model has been derived in this
regard in a direct-detection ladar system [15]. Ina TCSPC system we
want to detect single photons so some effects, like dark countsin
the detector, cannot be neglected. Therefore we have to slightly
adjust the modelderived in [15].
The emitted laser pulse Ss(x, y, t) will be received as a signal
Sr(x, y, t), which isthe range profile. Sr(x, y, t) is affected by
a series of factors that can be modeledseparately. We start from
the beginning with the emitted laser pulse. Ideally thetemporal
shape of Ss(x, y, t) would be a Dirac function, however that is not
possiblein practice. Instead the temporal shape is exponentially
distributed. Spatially theshape is usually modeled as Gaussian.
The signal is affected the most by the interaction with the
object. The reflected sig-nal depends on the geometry and the
reflective properties of the object. Reflectiveproperties are
described by the bidirectional reflectance distribution [24]. The
totalinteraction with the object is described by an impulse
response function, h(x, y, t).That is, the reflected signal is the
convolution in time between Ss(·) and h(·).
17
-
CHAPTER 3. THEORY
Aside from the object the signal is also affected by atmospheric
disturbances, re-ceiver properties, and time jitter and noise in
the system. All these effects aremultiplicative factors and will be
represented by a function F (t). Atmospheric dis-turbances and the
receiver properties and more thoroughly described in [15]. Lastlya
term to account for external noise (n(t)) is added. The final model
is
Sr(t) =N∑i=1
(∫Ω
([Ss,i(x, y, t) ? h(x, y, t)]F (t) + n(t)) dxdy), (3.11)
where ? is the convolution operator, N is the number of emitted
laser pulses, andΩ denotes the area in which we detect light. Sr(t)
is reduced to one dimension aswe are only interested in when
photons are detected and not where. We summarizethe responses from
many laser pulses because we want to create a histogram withmany
counts. In TCSPC we typically detect none or only one photon from
eachpulse.
If we measure on a point source and let N → ∞ the response will
be the IRF. Amodel of the specific IRF our system is discussed in
Section 2.5.
3.5 RJMCMC
Specular surfaces will only give a response when the surface is
perpendicular tothe laser beam. If the whole surface is at the
approximately same distance theresponse will have the form of the
IRF. A diffuse surface will yield the approximatesame result for a
small range of angles, starting from the perpendicular position.As
explained in Section 3.4.1 these are the dominating responses. In
other words,dominating responses will have the form of the IRF. By
fitting the IRF to theTCSPC readings it should then be able to
separate the responses and treat themdifferently. It will then be
possible to make smaller, less distinguishable surfacesmore
emphasized. Different methods have been investigated in this regard
[18]-[11]. In this project a method based on reversible jump Markov
chain Monte Carlo(RJMCMC) is investigated. RJMCMC combines
different methods so for simplicitywe start from the beginning.
3.5.1 Monte Carlo methods
Monte Carlo methods use random samples from one distribution to
obtain samplesof another distribution. The main characteristic is
the random sampling. Forexample, complex models often have some
stochastic influences. In most casesit is very difficult or
impossible to obtain a deterministic solution. Monte Carlomethods
offer a way of avoiding that issue. First a random number is
generatedfrom a probability distribution specific for the problem.
The sample is used to solvethe problem deterministically. A new
sample is then generated and the problem is
18
-
3.5. RJMCMC
solved again. By solving for many random samples an estimation
of the solution isobtained.
3.5.2 Markov Chain
A Markov Chain is a way of describing a chain of events with
different transitionprobabilities. The different events are also
independent of each other. For example,suppose the weather is
cloudy. According to specific transition possibilities theweather
will be clear, rainy, or the same the day after. The Markov
property statesthat the events are independent. This means that the
possibility for rain after acloudy day is always the same, even if
it rained the whole week beforehand. Theonly deciding factor is the
current state. In other words, a Markov Chain has nomemory. By
exploring these chain of events the probabilities for the system to
bein each state will converge. The method has a burn-in period
before it stabilizes,usually resampling continues until the changes
in the estimated probabilities hasreached a certain threshold.
3.5.3 Markov Chain Monte Carlo
Combining the concept of Markov chains and Monte Carlo methods
offers a methodpossible of finding an estimation F̂est(φest) of the
desired distribution q based ona set of parameters φest. In our
case is q the range profile (a histogram) obtainedfrom the TCSPC
system. Parameters are typically the number of peaks and
theirposition and heights. First an initial choice F̂est with
parameters φest is made.By sampling new parameters φprop from
specific distributions based on φest MonteCarlo contributes with a
proposal distribution F̂prop.
Comparison of F̂est with q will give a probability of the
current state. In the sameway a probability of the proposal
distribution F̂prop is obtained. By comparingthe two probabilities
you can check if the proposed distribution is better. If
thecomparison measurement is above some acceptance threshold α,
F̂prop will be chosenas the new estimation of q, ergo F̂est =
F̂prop (only the current state is stored). Inother words, the
initializing transition in a Markov Chain has been performed.After
a number of transitions F̂est and the corresponding parameter
values φ̂estwill converge. Consequently the most probable
estimation of q is obtained. Theprobability will be limited by the
possible choices of parameters.
3.5.4 Reverse Jump Markov Chain Monte Carlo
In a TCSPC histogram the number of relevant peaks is often hard
to decide before-hand. Therefore RJMCMC is introduced, making it
possible to vary the number ofparameters. An event can be to
randomly remove or add a peak. A big advantagewith this is that the
initial estimation is not important. RJMCMC can find thenumber of
peaks automatically. In MCMC the number of parameters is decided
in
19
-
CHAPTER 3. THEORY
the initial estimation.
Each event is accepted by an acceptance probability (as stated
in Section 3.5.3).In [18]-[20] a Bayesian approach was considered.
The acceptance probability isthen obtained with the use of the
likelihood function given different parameters. Itis a robust
approach that makes it possible to find peaks with low SNR. In
thisproject the SNR is typically high and only significant peaks
are targeted. Thereforea least squares approach is applied instead.
Mathematically we want to minimizethe squared residuals obtained
from the difference between F̂ and q. The event isaccepted if it
lowers the error. In other words, a proposal distribution F̂prop
withparameter set φprop is accepted if
N∑i=1
(F̂prop,i(φ1)− qi)2 ≤N∑i=1
(F̂est,i(φprop)− qi)2, (3.12)
where q is the TCSPC histogram and N is the number of bins. The
index i showswhat bin of the function that is considered. A typical
RJMCMC algorithm inreflective tomography is summarized in Algorithm
1.
Algorithm 1 An example RJMCMC algorithm in reflective
tomography.
1. Obtain an initial estimation F̂est with parameter set
φest.
2. Perform a sweep of events:
a) Update parameters in φest according to a specific event
toobtain φprop and F̂prop.
b) Find the likelihoods of F̂est and F̂prop.c) Compare the
likelihoods and obtain a measure of improvement.d) Set F̂est =
F̂prop if the improvement is above a threshold α.e) Perform steps
a-d until all events have been covered.
3. Continue with step 2 until F̂est has stabilized.
3.6 The convex hullAnother way to reduce artifacts is to
consider the convex hull. The convex hull isdefined as the smallest
convex set containing an object. The convex set is the set ofpoints
such that if straight lines are drawn between points in the set,
all the lineswill be inside of the set. An example is shown in
Figure 3.8. As there is no part ofthe object outside the convex
hull we can remove any artifacts in that region.
20
-
3.6. THE CONVEX HULL
Figure 3.8. The convex hull (gray area) of an object (solid
line). The dashed linesshows the boundary of the convex hull where
it is separated from the object.
21
-
Chapter 4
Improvement methods
In this part, methods to improve the tomographic reconstruction
is presented. First,a method that address the issue when some
targets dominates a reconstructed imageso that others are hard to
see, is introduced. It is a pre-processing method thatuses RJMCMC
to estimate responses. Second, different filters in the FBP
areinvestigated. Lastly, two post-processing methods, one that
removes artifacts byconsidering the convex hull and another one
that can sharpen edges, are introduced.
4.1 Adjustment of range profiles
Here we use RJMCMC to adjust high-intensity peaks in range
profiles to amplifylow-intensity target responses. The number of
parameters to optimize in RJMCMCcan be chosen arbitrarily. To
change the state of the distribution by altering oneof these
parameters is considered an event. It is relatively easy to add or
removeevents from the method so what you want to achieve with
RJMCMC and how youwant to do it can therefore vary a lot. We used
two different approaches. Onesimilar to that in refs. [18]-[20] and
another that was adjusted to suit our specificsituation. The
difference between the methods are mainly the possible events.
Theevents are listed in Table 4.1.
Table 4.1. Events in RJMCMC in the two approaches. Each event is
regarding aresponse.
Event Original method Adjusted methodPosition x xHeight xShape x
xBirth xDeath xMerging x
23
-
CHAPTER 4. IMPROVEMENT METHODS
One can see that heights, births, deaths and merging of peaks
are not regarded inthe adjusted version. The reason for this is
because the method is used only fordominating responses, so an
initial guess is sufficient. The initial guess was madewith the
Matlab function findpeaks. With it, positions and heights of peaks
can befound with appropriate settings. The position changes in the
adjusted method isonly small changes. With an initial guess the
solution stabilizes much quicker. Thatway we can use more shape
changing parameters at a low cost, obtaining a good fitin a short
time. The approach that does not have events that changes the
numberof peaks is strictly speaking not RJMCMC, but rather MCMC. We
will still referto that approach as RJMCMC to avoid confusion. A
more robust version will beable to change change dimensions (number
of responses). We did not include thatproperty in the adjusted
version to speed up the algorithm, making testing moreconvenient.
In this project only results from the adjusted version are
considered.
The shape parameters can all be connected to (2.4). To get a
better fit for non-idealresponses (2.4) was slightly adjusted so
that it is divided into three regions insteadof two. For x ≥ x1 the
equation looks the same. The region x < x1 was dividedinto x
< x0 and x0 ≤ x < x1, ergo to the right and to the left of
the peak. In bothregions the equation is still a Gaussian function,
however by dividing it into two wecan handle the standard
deviations separately. The derivative is not longer
strictlycontinuous in the location of the peak but we can obtain a
better fit for responsesthat not exactly follows the IRF of an
approximate point source. For example, wemight want to fit
responses from large surfaces that are slightly tilted. The
equationtherefore reads
f ′IRF (t) =
exp(−(t− t0)
2
2s2l
)t < t0
exp(−(t− t0)
2
2s2r
)t0 ≤ t < t1
exp(−(t1 − t0)
2
2s2r
)×(
a× exp(− t− t1
τ1
)+ b× exp
(− t− t1
τ2
))t ≥ t1
. (4.1)
The chosen shaping parameters are sr, sl and τ1, where τ1
determines the rate ofexponential decline for t ≥ t1. Initial
parameter choices are
t1 = FWHM/4 a = 0.5sl = FWHM/2 sr = FWHM/2τ1 = FWHM× 2
, (4.2)
where FWHM (full width, half max of beam width) were
approximated from apoint source. It was set to FWHM = 16 bins (9.6
mm).
24
-
4.1. ADJUSTMENT OF RANGE PROFILES
After fitting a response the next step was to adjust the signal
to suit our needs.Typically the signal is deconvoluted while
preserving the energy. However we tookanother approach. We are
looking at a special case where we want to reduce adominating
response to enhance others. Therefore we replace the response witha
model with reduced energy. Two models are considered, the first is
a Gaussianfunction and the other has the same shape as the original
signal, but with reducedheight. An example of the process of
adjusting a range profile is shown in Figure4.1. In the example a
Gaussian model is used.
1.5 4.5 7.5 10.5 13.5 16.50
0.2
0.4
0.6
0.8
1
Range [cm]
Nor
mal
ized
cou
nts
TCSPC histogram and threshold
1.5 4.5 7.5 10.5 13.5 16.50
0.2
0.4
0.6
0.8
1
Range [cm]
Nor
mal
ized
cou
nts
RJMCMC fit
1.5 4.5 7.5 10.5 13.5 16.50
0.2
0.4
0.6
0.8
1
Range [cm]
Nor
mal
ized
cou
nts
Model of fit
1.5 4.5 7.5 10.5 13.5 16.50
0.2
0.4
0.6
0.8
1
Range [cm]
Nor
mal
ized
cou
nts
Model added to residuals
ModelFit
FitTCSPC
Final modelTCSPCThreshold
Figure 4.1. The process in which RJMCMC is used. Peaks are
detected above thethreshold (upper left). A fit of the peak is
performed with RJMCMC (upper right).The response is replaced with a
model (lower left). The original response is removedfrom the TCSPC
histogram and is replaced with the model (lower right). The
heightis reduced to the threshold.
In Figure 4.1 a dataset with a highly reflective cylinder next
to a cylinder with lowreflection (circular cross sections) is
considered. When the process is performed onall histograms in the
dataset a reconstruction can be performed. The results areshown in
Figure 4.2. In this case both models are considered.
25
-
CHAPTER 4. IMPROVEMENT METHODS
Original
Distance [cm]
Dis
tanc
e [c
m]
3 9 15
0
6
12
18
Gaussian model
Distance [cm]D
ista
nce
[cm
]3 9 15
0
6
12
18
IRF model
Distance [cm]
Dis
tanc
e [c
m]
3 9 15
0
6
12
18
Figure 4.2. Reconstruction of two cylinders with different
reflective properties withand without RJMCMC. A Gaussian model
(left) and a IRF model (middle) is com-pared to the original image
(right).
4.2 Changing filter in FBP
The different filters discussed in Section 3.3 were tested in
the reconstruction of acomplex object. The object has large sides
that dominate the responses, makingdetails hard to distinguish.
There is details (relatively small targets) placed on thesides and
on top of the object. On the left side there is a wedge shaped
target, inthe front there is a half sphere, and on the right side
there is two small pieces witha quadratic cross section. On top
there is a "turret and a barrel". We performeda tomographic
reconstruction with all the filters discussed in Section 3.3.
Thereconstructed images are shown in Figure 4.3.To evaluate the
images the SNR was considered. It was chosen as
SNR = Aσ, (4.3)
where A is the maximum value in the image and σ is the standard
deviation in aregion where only noise should be present. The
results are shown in Figure 4.4.The results in Figure 4.4 shows
that the generalized ramp filter has much higherSNR than the other
filters. Many datasets were tested and the results are similar
forall. A higher SNR means that we have reduced the noise while
keeping most of theintensity in the responses in the range
profiles. This was our goal with the testing ofdifferent filters
and the general ramp filter was therefore considered as the best
forour applications. By looking directly at images reconstructed
with different filters(for example in Figure 4.3) this conclusion
was strengthened. The reconstructedimages are typically clearer
when the generalized ramp filter is used.
26
-
4.3. REMOVING ARTIFACTS WITH THE CONVEX HULL
Sketch of ideal reconstruction
Figure 4.3. A complex object reconstructed with different
filters in the FBP. Asketch of an ideal reconstruction is also
shown (lower right).
Ramp S−L mod. S−L Hann gen. filter0
20
40
60
80
100
Different filters
SN
R
SNR for different filters
rampS−Lmod. S−LHanngen. filter
Figure 4.4. The SNR in a reconstructed image for the different
filters.
4.3 Removing artifacts with the convex hull
In Section 3.6 we discussed how the convex hull can be used as a
post processingtool. To be able to remove the artifacts we consider
the TCSPC histograms. First
27
-
CHAPTER 4. IMPROVEMENT METHODS
we found the range to the first response (closest to the
detector) in all range profiles.Then we used that range as a limit
in the reconstructed image. By considering allrange profiles the
convex hull will be the area enclosed by these limits.
Anythingoutside that area in the image will be artifacts created in
the reconstruction andcan therefore be removed. The process can be
seen in Figure 4.5.
How well the convex hull is captured is mostly limited by the
angular resolution and∆tsystem. The range to a surface is found
with highest precision if the range profilefrom the angle of
incidence perpendicular to the surface is measured. Otherwise
theaccuracy is decided by the measurements closest to the specific
angle. Thereforea high angular resolution is favorable. The time
jitter makes it difficult to exactlydecide at what range a response
is placed because it is stretched out in time. Tomake a robust
method we considered a threshold value to decide where the
firstresponse is. In some cases, better precision can be found by
considering the timeof the peak of responses instead. This approach
works if there is clear peaks in therange profiles. However for a
complex object with small details, this is typicallynot the case.
The limitations from the angular resolution and ∆tsystem can be
seenin Figure 4.5 where measurements are performed every 5 degrees.
The corners arefound with high precision because they can be seen
from a wider range of angles.However there are still some artifacts
outside the surfaces.
Original image Lines representating the first response
The convex hull The convex hull applied to the image
Figure 4.5. An example of how an image can be processed with the
convex hull.Upper left is the original image, a reconstruction of a
number of square blocks. Thefirst response from each angle (upper
right) is used to find the convex hull (bottomleft). Everything
outside the convex hull is then removed (bottom right).
28
-
4.4. SHARPENING TARGET EDGES
4.4 Sharpening target edgesA flat surface perpendicular to the
laser beam will be reconstructed as an edge.Often are the edges
blurry after the reconstruction because of factors like time
jit-ter, angle resolution, and filtering. These factors are hard to
avoid so a method tosharpen edges is discussed in this section.
A diffuse surface will give a sharper response as the angle of
incidence gets closerto the perpendicular state. By looking at the
TCSPC histograms it will then bepossible to find at what angle the
surface was perpendicular to the incident wavefront. To illustrate
this an object with a quadratic cross section was considered.
InFigure 4.6 a surf plot of the TCSPC histograms are shown.
Figure 4.6. A surf plot of TCSPC measurments from an object with
quadratic crosssection.
Each peak corresponds to one side of the object. By finding the
maximum of eachpeak the angle and the range for each side can be
found. We can then performa reconstruction with that information
only and use that to emphasize the edges.Results are shown in
Figure 4.7. Image a and b was multiplied to reduce theartifacts.
These artifacts can also be removed by the method considered in
Section4.3.
29
-
CHAPTER 4. IMPROVEMENT METHODS
a
Distance [cm]
Dis
tanc
e [c
m]
12 18 24 30
12
18
24
30
b
Distance [cm]
Dis
tanc
e [c
m]
12 18 24 30
12
18
24
30
c
Distance [cm]
Dis
tanc
e [c
m]
12 18 24 30
12
18
24
30
d
Distance [cm]
Dis
tanc
e [c
m]
12 18 24 30
12
18
24
30
Figure 4.7. The process of finding and emphasizing diffuse
surfaces. a) Originalimage, b) Reconstruction of selected data, c)
Image a multiplied by image b, d) Imagec added to the original
image.
30
-
Chapter 5
Performance study
Practical applications of reflective tomography have
limitations. For example, if wewant to reconstruct an airplane that
flies by with measurements from the groundwe will not be able to
see the upper side of the plane. In other words, we will notbe able
to have an angular sector larger than 180 degrees. It is then
important toknow how the result is affected by this limitation.
Besides the angular sector wewill also investigate how a lower
angular resolution, the choice of center of rotation,and varying
SNR affects the results.
5.1 Center of rotationThe center of rotation (CoR) refers to the
distance in the range profiles that theobject rotates around. The
choice of CoR decides at which distance from the centerof the image
a response will be placed. Consequently the effects of varying
CoRare dependent of the geometry. With the geometry effects in mind
a cylinder withknown radius was studied. Measurements on a standing
cylinder will yield a circlein the reconstruction, ideally with the
same radius as the cylinder. With incorrectchoice of CoR a circle
will still be seen in the reconstruction, but with wrong radius.The
circle is intact but with varying radius in a wide range of choices
of CoR. Theradius is therefore a good measure of the dependency of
CoR. How the image isaffected is shown in Figure 5.1. Starting from
the correct choice of CoR the circleshrinks until it is a point and
then extracts back to a circle again, this time inverted.From the
images estimations of radii for a specific CoR can be obtained.
Resultsare presented in Figure 5.2.
31
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CHAPTER 5. PERFORMANCE STUDY
True CoR
0 3 6 9 12 15 18
0
3
6
9
12
15
18
CoR shifted 3 cm
0 3 6 9 12 15 18
0
3
6
9
12
15
18
CoR shifted 6 cm
0 3 6 9 12 15 18
0
3
6
9
12
15
18
CoR shifted 9 cm
0 3 6 9 12 15 18
0
3
6
9
12
15
18
Figure 5.1. A circular cross-section reconstructed with
different centers of rotation.The scale on the axes is in cm.
−0.15 −0.1 −0.05 0 0.050
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Shift from center of rotation [m]
Rel
ativ
e er
ror
Comparison of estimated radii and the true radius
Absolute error of estimated radiusTrue Center of rotation
Figure 5.2. A figure describing how the radius of a circle in a
reconstructed imagedepends on the choice of center of rotation.
As the circle is compressed and then expanded back to a circle
it will be recon-structed with roughly the correct radius twice
(where the error is close to zero inFigure 5.2). First at the
correct choice of CoR but also when the CoR has beenshifted
approximately 12.5 cm, which is the diameter of the cylinder. This
is dueto symmetry in the radial direction of the object and is
therefore a special case.
32
-
5.2. ANGULAR SECTOR
With changes occurring in all directions most objects will be
unrecognizable if thechoice of CoR is far off the true value.
However, an image can look equally goodor even better if the CoR is
slightly wrong, even when considering complex objects.It is
therefore good to know some distance in the object of interest
beforehand. Itis then possible to figure out the correct CoR by
measuring how far off differentdetails on the object are
placed.
5.2 Angular sectorThe angular sector refers to the range of
aspect angles from which we have measured.Because the object is
rotated the angular sector can be interpreted as a circular
arc.Effects of varying the size of the angular sector have been
studied theoretically [21].To study effects practically a steel rod
with a diameter of 6 mm was used. How thereconstructed image looks
for a specific angular sector can be seen in Figure 5.3.
a’a b’b
c’c d’d
e’e
Figure 5.3. Reconstructed images of a thin steel rod. The
angular sector used isshown in the respective primed image. The
sizes of the angular sectors are: a) 15◦,b) 30◦, c) 45◦, d) 80◦,
and e) 340◦. The primes shows the respective angular sector.
33
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CHAPTER 5. PERFORMANCE STUDY
To obtain more qualitative results cross sections through the
reconstructed steelrod were studied. The cross sections are shown
in Figure 5.4. Results are similarof [21], a rather small angular
sector is sufficient to make a good reconstruction ofan approximate
point source. There is no significant improvement in the quality
ofthe image after around 80 degrees.
Ideally the width of the peak in the cross section would be the
same as the diameterof the rod, 6 mm. This is not possible mostly
due to the time jitter of the systemand artifacts from FBP. So,
because of time jitter and artifacts the reconstructionwill not be
perfect even with a 360 degrees angular sector. This means that it
isnot so important to have a large angular sector when considering
an approximatepoint source. This is the main reason of why 80
degrees was deemed sufficient.
48 24 12 6 0 6 12 24 48−0.1
0
0.2
0.4
0.6
0.8
1
Distance from center [mm]
Nor
mal
ized
val
ues
Cross section of approx. point source image with different
section sizes
15 30 45 80340
Figure 5.4. Cross sections through the rod in the images in
Figure 5.3. The dashedlines represents the true diameter of the
rod.
It is important to remember that this study was performed on an
approximate pointsource. When considering a complex object you
might want to reconstruct detailson different sides of the object.
Then it is obvious that a large angular sector ismore important
because details can be hidden from many angles of incidence.
Theimportance depends on the particular application. For example,
if you want toidentify a ship or a boat by its shape, measurements
from just one side of the shipwould probably be sufficient.
34
-
5.3. ANGULAR RESOLUTION
5.3 Angular resolutionFor complex objects the angular resolution
is of great interest. Some details orsurfaces might be hidden and
can only be seen from some angles of incidence. Itis then important
that measurements are performed from those specific angles.
Toillustrate the problem a square, with a height, width, and depth
at approximately10 cm, was reconstructed for different resolutions.
The images can be seen in Figure5.5.
1 degree resolution Angles of incidence 2 degrees resolution
Angles of incidence
4 degrees resolution Angles of incidence 8 degrees resolution
Angles of incidence
16 degrees resolution Angles of incidence 32 degrees resolution
Angles of incidence
Figure 5.5. Reconstruction of a square with different angle
resolutions.
As the reconstruction is a square a projection of the image in
either the vertical orhorizontal direction will ideally yield a
Dirac function where the perpendicular edgesare found. The parallel
edges will yield a small constant value and everywhere else itwill
be zero. How the projection differs from the ideal scenario is a
more qualitativeresult and is shown in Figure 5.6.
35
-
CHAPTER 5. PERFORMANCE STUDY
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
cou
nts
Relative distance [cm]
Cross section with 1 degrees resolution
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
cou
nts
Relative distance [cm]
Cross section with 2 degree resolution
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
cou
nts
Relative distance [cm]
Cross section with 4 degree resolution
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
cou
nts
Relative distance [cm]
Cross section with 8 degree resolution
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
cou
nts
Relative distance [cm]
Cross section with 16 degree resolution
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
cou
nts
Relative distance [cm]
Cross section with 32 degree resolution
Figure 5.6. Projections of the image for different angle
resolutions.
Large surfaces are to a large extent characterized by
measurements from an angleof incidence perpendicular to the
surface. With lower angular resolution this anglemight not be used,
making the surface less distinguishable if the surface is diffuseor
not visible if the surface is specular. The effects on a diffuse
surface can be seenin Figure 5.6 where the surfaces at first become
less distinguishable and eventuallydisappears from the
reconstruction. It is important to note that a surface can beseen
even for low angular resolutions. There is however a lower
probability that itwill be measured.
In these examples the surfaces are relatively large and can be
seen from manyangles. Small or hidden targets might only be found
from a very narrow field ofview, making the resolution even more
important.
5.4 The aim of the laser beamMany lasers, including the one we
used, have a Gaussian intensity profile. As thenumber of photons
reflected depends on the intensity of the light it is important
toaim the laser beam so that the object is as centered as possible
in the laser beam.The whole object should also be within the beam
width, otherwise targets in theperiphery might not be detected.
Another important factor in our TCSPC systemis the optics that
collects the photons. Collecting photons further away from
thecenter of the laser beam becomes increasingly hard. This effect
might be even moredeciding than the laser beam intensity.
To study both effects the number of counts from a point source
measured at different
36
-
5.5. INTEGRATION TIME
distances from the beam center was studied. The results are
shown in Figure 5.7. Itis noticeable that the laser beam was not
focused at the center of rotation. In thatcase we expect to capture
the Gaussian profile of the laser beam intensity. Still,the effects
are clear. From the point where the object is closest to the beam
centerto where it is furthest away the number of counts decreases
by approximately 80%.Correctly aiming the laser beam is therefore
important. The results also indicatethat it is not simple to aim
the laser even in a controlled environment. In anapplication where
the system or the target is moving it will probably be even moreof
an issue.
0 2 4 6 8 10 12 14 160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Relative range from beam center [cm]
Nor
mal
ized
cou
nts
Number of counts vs. relative distance from laser beam
center
Figure 5.7. A plot showing how the intensity decreases as the
targets moves furtheraway from the laser beam center. The intensity
is plotted against a relative distanceand not against the distance
from the laser beam center.
5.5 Integration timeA longer integration time corresponds to a
larger amount of time intervals measuredby the SPAD. In turn it
will lead to a higher signal-to-noise ratio (SNR) as the noiseis
random and responses are not. In our setup the SNR is typically
high, howeverthere are cases where it is of interest. Small
surfaces, non-reflective materials, sur-faces tilted to the
incident light and other targets that gives low intensity
responsesare examples. A higher SNR also makes it possible to
distinguish peaks that are ata close range from each other.
To study the effects of different integration times a TCSPC
measurement was di-vided into subsets of data, each containing
sweeps from 1 second long intervals.
37
-
CHAPTER 5. PERFORMANCE STUDY
The whole measurement was 600 seconds long. Figure 5.8
illustrates that SNR ishigh even at low integration times for
highly reflective surfaces. In the region in theexample there were
approximately 10 000 counts per second.
3 6 9 120
50
100
150
200
250
300
350
400
1 second
Cou
nts
Range [cm]3 6 9 12
500
1000
1500
2000
2500
6 seconds
Cou
nts
Range [cm]3 6 9 12
0.5
1
1.5
2
2.5
x 105 600 seconds
Cou
nts
Range [cm]
Figure 5.8. A plot showing TCSPC histograms of a point source
after 1,6, and 600seconds.
The histogram containing the whole data set can be considered to
be the "true"signal. If the signals are normalized the root mean
square error (RMSE) can becalculated at different number of counts.
The results are shown in Figure 5.9.
105
106
0.01
0.012
0.014
0.016
0.018
0.02
RM
SE
Counts
RMSE vs Number of counts
Figure 5.9. A plot showing the RMSE of a point source as a
function of number ofcounts.
The error will eventually go to zero, however after
approximately 105 counts thechange in the error is relatively
small. Such a number of counts can be obtained
38
-
5.5. INTEGRATION TIME
after a short amount of time. In the example in Figure 5.9 it is
approximately 5seconds but with different settings, like a higher
laser intensity or pulse repetitionrate, it can be even shorter.
The maximum number of counts in our TCSPC systemis approximately
106 counts/s. That means that we can obtain 105 counts in 100ms.
The number of counts is highly dependent of the system and the
object so it isnot an informative number generally as it will
change a lot between measurements.However, one can deduce that
integration time is not an issue in a controlled en-vironment where
it is typically not an issue to have an integration time of a
fewseconds. It can start to become a problem in applications where
integration timesdown to 1− 10 ms are required.
As stated earlier the SNR in the range profiles can be an issue
in some cases. Anexample to illustrate this is shown in Figure
5.10. The measurements are done withsame settings and the range is
similar as in the example in Figure 5.8. Counts inthis region are
approximately 230 per second (compare with 10 000). These typesof
responses are typically difficult to see in a reconstruction,
especially when thereare more targets with relatively many more
counts in the same measurements.
6 12 180
1
2
3
41 second
Cou
nts
Range [cm]6 12 18
0
2
4
6
8
10
12
4 seconds
Cou
nts
Range [cm]
6 12 180
5
10
15
20
25
30
10 seconds
Cou
nts
Range [cm]6 12 18
200
400
600
800
1000
600 seconds
Cou
nts
Range [cm]
Figure 5.10. A plot showing TCSPC histograms of two targets with
a low responseand at close range after 1,4,10, and 600 seconds.
39
-
Chapter 6
Combining methods of improvement
This chapter will gather results to show the present usefulness
and flaws of theimprovement methods that we have studied. Complex
objects will be used to showthe usefulness generally and not only
for special cases. We will also show thatcombinations of different
methods are possible. In all examples we present in thischapter the
generalized ramp filter was used. Our studies have shown that it is
thebest filter for our applications.
In Section 4.3 we stated that the artifacts from the surface
sharpening method alsocan be reduced with the convex hull. This is
illustrated in Figure 6.1. Instead of anobject with a square cross
section a number of blocks were considered. The objectcan also be
seen in the example. Note that one block is "missing".
Figure 6.1. The image reconstructed with the generalized ramp
filter (upper left)was processed with the convex hull (upper
right). Then the surfaces were emphasized(lower left). A photo of
the object can also be seen (lower right).
41
-
CHAPTER 6. COMBINING METHODS OF IMPROVEMENT
We can see in Figure 6.1 that not all surfaces are sharpened.
With more surfaces itis typically more difficult to find them all
in the TCSPC histograms. However withsome development of the method
it will be able to find most surfaces. So far, onlyinitial work has
been performed. As we suggested, we can also see in the figure
thatartifacts are effectively removed.
In Figure 6.2 we show how a reconstruction of a complex object
can be improved byfirst enhancing details by adjusting the range
profiles and then removing artifactsoutside the convex hull. The
model used in the adjustment of the responses is theIRF. It is more
suited in this case because we have no specific need to
sharpenresponses.
Alteration of responses with RJMCMCReconstruction with gen. ramp
filter
Artifacts removed outside the convex hull Sketch of ideal
reconstruction
Figure 6.2. An example of how a reconstruction can be improved.
From the originalimage (upper left), detailes are enhanced by
adjusting large responses (upper right).Lastly artifacts are
removed outside the convex hull (lower left). A sketch of an
idealreconstruction is also presented (lower right).
We can see in Figure 6.2 that adjusting responses with RJMCMC
makes somedetails much clearer. Also, when applying the convex hull
it becomes more obviousthat there is a detail (a wedge) on the
upper side of the object in the reconstruction.Without the convex
hull the wedge could easily be disregarded as artifacts.
42
-
Chapter 7
Discussion
Here we will present a summary and discussion of the studies
that have been per-formed in the project. Possible future work will
also be discussed.
7.1 Summary
We can conclude from the results that reflective tomography is
an effective methodto reconstruct objects with details in the low
cm range. For example, see Figure 6.2where a lot of details are
present. It was an early conclusion that high
resolutionreconstructions can be obtained with reflective
tomography. Therefore the work hasbeen focused on improving the
reconstructions.
7.1.1 Improvement methods
First we introduced a new pre-processing method that adjusts
range profiles to en-hance details in a reconstruction. It performs
very well in that regard. To makeit even better more ways of
modeling the targeted responses can be studied. Alsomore types of
responses can be targeted and modeled separately. We
explicitlytargeted large responses that are similar to the IRF. To
make good estimations ofresponses we used RJMCMC. RJMCMC is very
robust and easily adaptable so weare confident that it can be used
in various methods that improve a tomographicreconstruction.
Second we considered different filters in the FBP. We came to
the conclusion thatthe generalized ramp filter works best. Its
biggest advantage is that it is very adap-tive as its response
function can be manipulated with shape parameters. It can beadapted
to specific scenarios by changing shape parameters. We used a
setting thatworks well generally. The SNR was much higher in a
reconstructed image when thegeneralized ramp filter was used,
compared to the other filters we tested.
43
-
CHAPTER 7. DISCUSSION
We also introduced two new post-processing methods that take
advantage of "hid-den" information in the range profiles. By
finding the range to the first responsein each range profile the
convex hull of the object could be estimated. Everythingoutside the
convex hull is artifacts so they can be disregarded. By applying
theconvex hull artifacts efficiently were removed. We could also
see that some de-tails were easier to notice after applying the
convex hull, see Figure 6.2. Lastly weconsidered a method that
takes advantage of the reflective properties of a diffusesurface.
When considering a plot of all range profiles the approximately
exact rangeand angle to a surface perpendicular to the incident
laser beam can be found. Weused this information to enhance
surfaces. The method is still raw but it can befurther developed to
perform specific tasks. Most important to note is that there
isinformation in the range profiles that the FBP do not
consider.
7.1.2 Performance
The performance study showed that reflective tomography using
TCSPC is robust ina controlled environment. Studies have also shown
that TCSPC works well outdoor,at longer distances [13].
Furthermore, as a long range imaging method reflectivetomography
has already proven to be effective [1]-[3]. However, even though
itworks in some outdoor applications, pushing the performance
limitations will openup for even more possibilities. For example,
obtaining results quickly could be ofinterest in many applications,
especially military. Reducing the integration timewhile keeping the
SNR will therefore be an attractive improvement. It will thenbe
easier to follow fast moving objects, like missiles. So even if we
have found themethod to be robust, improvements should not be
disregarded.
Our studies showed that the angular resolution is an important
factor. Without ahigh angular resolution the image quality is
drastically lowered, even in a controlledenvironment. Details
disappear from the reconstruction and edges are less distinct.These
effects will be even more apparent when the object or the system is
moving.Moving in a perfect circular arc around an object like in
the lab is not probable inan application. To effectively be able to
adjust for movement in the range profilesmany measurements will
probably be important. One way to adjust could be tolook at
correlation between subsequent range profiles. That would require a
highangle resolution.
To obtain a high angular resolution on a moving target we must
be able to performmeasurements quickly. This means that a short
integration time is of interest. Weconcluded that the integration
time is not a big issue in a controlled environment,however this
might not be the case outdoors.
A moving target and/or system will also make it harder to
determine the CoR be-cause separate range profiles will have
different CoR. Therefore it was unfortunatethat a robust method to
determine the CoR could not be found. In a controlled
44
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7.2. FUTURE WORK
environment it is possible to "guess" the CoR but it is
uncertain if this is possiblein an application. Finding CoR with
feature tracking [5]-[6] was tested but it isnot suited for complex
objects so it was disregarded. A further developed methodcan be of
interest. One possible way to adjust for CoR in the reconstruction
isto use phase retrieval [8]. Unfortunately we did not have enough
time to performinvestigations on our own.
7.2 Future work
A future objective to evaluate is the reflective tomography
performance on movingtargets outdooors. With moving targets the
method must be adjusted (as discussedin the previous section). So
how good can the performance be on a moving target?We have only
studied isolated objects so we have not considered disturbances
fromthe surroundings, like waves on the ocean if you are looking at
a ship. Will that bean issue? There is a lot of questions that have
to be answered but we are hopefulthat reflective tomography can be
used in many applications in the future. Onesimple experiment could
be to measure on a moving car. If even faster movingtargets, like
missiles, is of interest it would probably be necessary to simulate
anexperiment before considering a real missile.
It is also interesting to further develop the improvement
methods studied in thisproject. Using RJMCMC to find different
types of responses in range profiles is avery robust and flexible
approach. Because of the flexibility it would be interestingto see
if more types of responses can be modeled and adjusted to further
improvethe reconstruction.
Other future work can be to investigate if 3D reconstruction is
possible in an appli-cation. In a lab environment it has already
been tested [16]. However, to be ableto identify a target a 2D
reconstruction is sufficient. The natural next step in
theinvestigation is to perform 2D reconstructions outdoors on
moving targets.
7.3 Conclusion
We have investigated the performance of reflective tomography
using a TCSPCsystem. To make it easier to study limitations in a
possible application we consid-ered complex objects in our
measurements. A performance study showed that theangular resolution
is an important factor to consider when measuring on
complexobjects. Another important factor to consider is the
integration time. In a con-trolled environment it is not
significant but in an application it will very likely
beimportant.
45
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CHAPTER 7. DISCUSSION
We also studied different methods to improve the tomographic
reconstruction. Firstwe introduced a method that adjusts responses
in range profiles to enhance details.We used RJMCMC to obtain good
estimations of responses. With additional workmore types of
responses can be targeted so that this method becomes even
morepowerful. To further improve the reconstruction we studied
different filters in theFBP. With the generalized ramp filter the
SNR in the tomographic reconstructionsimproved significantly. To
remove artifacts we considered the convex hull of thereconstructed
object. Everything outside the convex hull in an image is
artifactsso we could disregard them from the image. Applying the
convex hull also made itintuitively easier to see some details.
Lastly we introduced a method that considersthe reflective
properties of diffuse reflecting surfaces. We showed that there
isinformation in the range profiles that is not used by the FBP. We
used it to sharpenedges in the reconstruction.
46
-
Bibliography
[1] Charles L. Matson, Eric P. Magee, Donald E. Holland,
Reflective tomographyusing a short-pulselength laser: system
analysis for artificial satellite imaging,Optical Engineering,
2811, (1995).
[2] Stephen D. Ford and Charles L. Matson, Projection
registration in reflectivetomography, Proc. of SPIE 3815,
(1999).
[3] James B. Lasche, Charles L. Mason, Staphen D. Ford, Weldon
L. Thweatt, Ken-neth B. Rowland, and Vincent N. Benham, Reflective
tomography for imagingsatellites: experimental results, Proc. of
SPIE 3815, (1999).
[4] F.K. Knight, S.R. Kulkarni, R.M. Marino, and J.K. Parker,
Tomography Tech-niques Applied to Laser Radar Reflective
Measurements, The Lincoln LaboratoryJournal 2, (1989).
[5] Xiaofeng Jin, Jianfeng Sun, Yi Yan, Yu Zhou, and Liren Liu,
Feature trackingfor projection registration in laboratory-scale
reflective tomography laser radarimaging, Optics Communications,
3475-3480, (2010).
[6] Jianfeng Sun, Xiaofeng Jin, Yu Zhou, and Liren Liu, Short
pulselength direct-detect laser reflective tomography, Proc. of
SPIE, 778017, (2010).
[7] Yi Yan, Jianfeng Sun, Xiaofeng Jin, and Liren Liu,
Two-dimension image con-struction for range-resolved reflective
tomography Laser radar, Proc. of SPIE,8162, (2011).
[8] Xiaofeng Jin, Jianfeng Sun, Yi Yan, Yu Zhou, and Liren Liu,
Application ofphase retrieval algorithm in reflective tomography
laser radar imaging, ChineseOptics Letters, 012801, (2011).
[9] Lars Sjöqvist, Markus Henriksson, Per Jonsson, and Ove
Steinvall, Time-of-flight range profiling using time-correlated
single-photon counting, Proc. of SPIE,67380N, (2007).
[10] Ove Steinvall, Lars Sjöqvist, Markus Henriksson, and Per
Jonsson, High reso-lution ladar using time-correlated single-photon
counting, Proc. of SPIE, 695002,(2008).
47
-
BIBLIOGRAPHY
[11] Thomas Neimert-Andersson, 3D imaging using time-correlated
single photoncounting, Thesis work, Uppsala University, (2010).
[12] Sara Pellegrini, Gerald S Buller, Jason M Smith, Andrew M
Wallace, and Ser-gio Cova, Laser-based distance measurement using
picosecond resolution time-correlated single-photon counting, Meas.
Sci. Technol., 712-716, (2000).
[13] Markus Henriksson and Lars Sjöqvist, Time-correlated
single-photon countinglaser radar in turbulence, Proc. of SPIE,
81870N, (2011).
[14] Lars Sjöqvist, Christina Grönwall, Markus Henriksson, Per
Jonsson, and OveSteinvall, Atmospheric turbulence effects in
single-photon counting time-of-flightrange profiling, Proc. of
SPIE, 71150G, (2008).
[15] Christina Grönwall, Ove Steinwall, Fredrik Gustafsson, and
Tomas Chevalier,Influence of laser radar sensor parameters on
range-measurement and shape-fitting uncertainties, Optical
Engineering, 106201, (2007).
[16] Yi Yan, Jianfeng Sun, Xiaofeng Jin, and Liren Liu,
Two-dimension imageconstruction for range-resolved reflective
tomography Laser radar, Proc. of SPIE,81620Y, (2011).
[17] S. Cova, M. Ghioni, A. Lacaita, and C. Samori, and F.
Zappa, Avalanchephotodiodes and quenching circuits for
single-photon detection, Applied Optics35, (1996).
[18] Gerald S. Buller and Andrew M. Wallace, Ranging and
Three-DimensionalImaging Using Time-Correlated Singe-Photon
Counting and Point-by-Point Ac-quisition, IEEE Journal of selected
topics in quantum