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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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.

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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.

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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.

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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.

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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).

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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.

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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

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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.

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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

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