Simulations and a Comparison of Noiseless Conventional and Compressive Imaging for Endo-microscopy

Bachelor thesis Physics & Astronomy

August 18, 2020

Amsterdam, the Netherlands

Simulations and a Comparison

of Noiseless Conventional and

Compressive Imaging for

Endo-microscopy

Author: Zartasheia Saeed

Student ID. (UvA): 12016047 Student ID. (VU): 2602010

Supervision:

Daily Supervisor: Dr. B. Lochocki Supervisor: Dr. L. Amitonova

Examiner: Dr. S. Witte

Project weight 15 ECTS

conducted between 11-04-2020 and 21-07-2020 research institute LaserLab Amsterdam department Physics and Astronomy

research group Biophotonics and Biomedical Imaging Group degree programme BSc Physics and Astronomy (Joint Degree) Faculties Faculty of Science, FNWI (UvA)

Samenvatting

In verschillende wetenschappelijke disciplines zoals Natuurkunde en Scheikunde wordt veelal gebruikt gemaakt van optische microscopie als het instrument voor beeldvorming. Deze traditionele manier van beeldvorming maakt ge-bruikt van laser licht uit het deel van het elektromagnetische spectrum dat zichtbaar is voor het menselijk oog. Door het laser licht te focussen op een object kan het object zichtbaar worden gemaakt. Vervolgens kan er met een scanner of een camera het object wordt afgebeeld. Deze manier van beeld-vorming wordt ook wel de traditionele manier van beeldbeeld-vorming genoemd. De spatiale en temporele resoluties van deze traditionele manier van beeld-vorming zijn fundamenteel gelimiteerd. Abbe-diffractie limiet beschrijft de afstand tussen twee naastgelegen punten waar de twee punten nog te onder-scheiden zijn met een optisch meetinstrument. Deze is afhankelijk van de golflengte van het licht en de numerieke apertuur van het meetinstrument. De fundamentele limiet van de temporele resolutie wordt gegeven door de Nyquist-limiet. Deze beschrijft de maximale haalbare meetsnelheid van het optische meetinstrument. Sinds een paar jaar is er een nieuwe methode van beeldvorming voorgesteld: compressieve beeldvorming. Deze techniek van beeldvorming berust niet op de fundamenten van de traditionele methode van de optische microscopie. Dit onderzoek simuleert de twee methodes van beeldvorming in MATLAB en vergelijkt de temporele en spatiele resoluties van beide methodes.

In dit onderzoek wordt aangetoond dat de spatiele resolutie van com-pressieve beeldvorming niet wordt bergend door de Abbe-diffractie limiet en de meetsnelheid niet wordt beperkt door de Nyquist-limiet. Dit betekent dat compressive beeldvorming zowel een hogere temporele resolutie als een hogere spatiele resolutie kan behalen. In korte tijd kan een hoger aantal beelden wor-den gemaakt die hogere spatiale resolutie hebben dan de beelwor-den gemaakt met de traditionele optische microscopie. Compressieve beeldvorming is dus een veelbelovend techniek waarmee super-resolutie en super-snelheid tegeli-jkertijd behaald kunnen worden.

Abstract

Far-field optical microscopy has been used in many different research fields. Since the last few decades it has been very much advanced, giving a higher imaging resolution. However, the higher resolution has been achieved with a compromise to the acquisition time.

Even the most modern microscopic technologies are fundamentally lim-ited in this paradigm. Abbe-diffraction limit still applies on the achievable spatial resolution and the Nyquist-Shannon theorema is still valid for the data-acquisition speed. An alternative imaging approach is needed to go beyond these spatial and temporal limits. Since a few decades, compressive imaging made it possible to go beyond the fundamental limits of conventional imaging. This has shown a paradigm-shift in the optical imaging technol-ogy. This thesis presents the simulations of a noiseless imaging setup for the two different approaches: conventional method of point-by-point-scanning and compressive imaging (CI), using speckle patterns for illumination. This research shows how the compressive imaging makes it possible to go be-yond the Abbe and Nyquist-limit. A comparison between the conventional method of point-by-point scanning and compressive imaging is demonstrated and shown how super-resolution and super-speed can be achieved with com-pressive imaging.

Acknowledgements

First of all, I would like to thank my supervisor Lyuba and my daily super-visor Ben for making this project possible. During a time period when the research institutes were closed due to the Covid-19 pandemic, my supervi-sors provided me the right tools so I was able to work from distance and present this thesis. I also thank them for the weekly meetings which helped me staying on the right-track and reflecting back on my weekly progress. I enjoyed being part of the research group ‘Biophotonics and Biomedical Imaging’ during my Bachelor project. The weekly group meetings gave me more insight of the fundamental research done in the group.

I would also like to thank Kaoutar for sharing her results of the MM-fibre speckles illuminations, so I could implement these in my M AT LAB simula-tions and compare it with my findings.

Altogether, my Bachelor project increased my knowledge and understanding of optical imaging and gave me more insight of the cutting-edge research done in this field and increased my interest in this research field.

Contents

2.1 Conventional optical imaging . . . 3

2.1.1 Abbe-diffraction limit . . . 3 2.1.2 Nyquist-Shannon theorema . . . 4 2.2 Compressive imaging . . . 5 2.2.1 Sparsity . . . 5 2.2.2 Incoherence . . . 6 2.3 Number of measurements (Nm) . . . 6

2.4 Speckles Patterns in a MM-fibre . . . 7

2.4.1 Light Coupling in a MM-fibre . . . 7

2.4.2 Speckle pattern in MM-fibre . . . 8

3 Materials and Methods 9 3.0.1 Simulations and comparison in M AT LAB . . . 9

4 Results 12 4.1 Intensity profiles . . . 12

4.1.1 Euclidean Space . . . 12

4.1.2 Fourier Space . . . 13

4.2 Beyond the Nyquist-limit . . . 14

4.3 Different Pixel Sizes . . . 15

4.4 Beyond the Diffraction-limit . . . 16

4.5 MM-fibre Speckle patterns for CI . . . 17

4.7 1D intensity profiles in Fourier . . . 19 4.8 Symmetry effect in the MM-fibre . . . 21

5 Discussion 22

Chapter 1

Introduction

Far-field optical microscopy is the main tool used for fundamental research in different scientific disciplines such as physics, chemistry and bio-imaging. The optical imaging setup where laser beams are used to illuminate the sam-ple and a camera attached to the microscope is used to image the samsam-ple is called the conventional method of imaging. This method could also be used to image inside the human body, by using a fibre for illumination and de-tection, which is called endo-microscopy. This conventional optical imaging system has its fundamental limitations of diffraction and acquisition speed. This results in images which are limited in spatial and temporal resolution. During the past few decades, this conventional method of imaging has been advanced which has increased its resolution. However, the Abbe-diffraction limit and the Nyquist-limit for acquisition speed still apply. The need of high-resolution optical imaging leads to an alternative of the conventional imaging approach. With the emergence of compressive imaging, a new paradigm in the optical imaging, has been introduced. This opens the door for a new imaging approach in the fundamental research disciplines Compressive imag-ing operates in a paradigm where the reconstruction quality is fundamentally defined by sparsity and coherence/incoherence of speckle patterns [1].

Sparsity is defined by the number of non-zero elements present in the sample data and the correlation between two speckle patterns in the recon-struction domain defines the image reconrecon-struction. During the last few years a miniaturised endo-microscopy has been introduced which has given access to deep tissues so that in vivo bio-imaging has been possible [1]. The use of a multimode-fibre as a mini endo-microscope can also applied in scientific disciplines such as meteorology and geology [6]. For example, to collect the data of seismic and ground-motion, fibre imaging can be used to access un-derground areas which are not accessible with the conventional microscope with an attached camera setup. Furthermore, the speckle patterns generated

Figure 1.1: A MM-fibre-based experimental setup. The speckle patterns gen-erated in the MM-fibre can be used for image reconstruction with compressive imaging (CI)[1].

in a multimode-fibre can be used for image reconstruction with compres-sive imaging. Figure 1.1 presents the experimental setup where the speckle patterns in a MM-fibre are shown.

This thesis presents a comparison between the conventional imaging and compressive imaging approaches for a noiseless setup. First, the theoreti-cal background for both optitheoreti-cal imaging approaches has given, following by the simulations of both optical imaging approaches. The difference between the two imaging approaches shows how these two approaches fundamentally differ. The compressive imaging simulations has been shown by using an idealised randomly generated speckle patterns. A data-set of the speckle patterns in a MM-fibre, has also been used for compressive imaging and compared with the idealised compressive imaging.

Chapter 2

Theoretical Background

This chapter gives an overview of the physical fundamentals underlying the two imaging approaches: the conventional imaging and the compressive imaging. The conventional imaging method is based on the point-by-point scanning system, while compressive imaging used speckle patterns for data-recovery or image reconstruction.

2.1

Conventional optical imaging

The conventional method of imaging using an optical microscopy has been used for several centuries as the key instrument. The sample is then exposed to a laser beam with a wavelength in the visual light domain. The laser beam is then focused onto the sample and the reflective response is measured. The focused beam is raster scanned across the sample and the obtained response is constantly measured. The image of the sample is then reconstructed. However, the resolution and the acquisition speed of the conventional optical microscopy are fundamentally limited.

2.1.1

Abbe-diffraction limit

The resolution-limit of a conventional imaging setup can be described by the so called Abbe-diffraction limit. The wave nature of the propagating light and its diffraction results in the resolution-limited. Ernst Karl Abbe has reported that the smallest resolvable distance between two points is limited [2]. Equation 2.1 describes this smallest resolvable distance called d, where λ is the wavelength of the propagating light, n the refractive index of the medium and φ the angle of the cone of the light. The amount of light which coming from the objective focus is also called the numerical aperture (N A)

of the optical object.

Figure 2.1: Illustration of the diffraction-limit of the conventional method of imaging. Th smallest resolvable distance (d) between two points by using the conventional imaging is the diffraction limit.

d = λ 2n sin φ =

λ

2N A (2.1)

Figure 2.1 illustrates the maximum resolvable distance (d) between two points for a conventional imaging setup which is limited by the diffraction of the light. As a result of the limited N A, the image of the sample has a blurry appearance for any conventional imaging setup.

2.1.2

Nyquist-Shannon theorema

According to the Nyquist Theorem, the highest frequency which can be rep-resented accurately is half of the sampling rate [7]. This means that in order to represent the highest frequency, the data-acquisition speed must me twice the highest frequency. There is a minimum number of measurements required

in a time-interval for the data-recovery with the conventional method. For a noiseless imaging setup, this limitation in the acquisition-speed is also called the Nyquist-limit. This gives the theoretical fundamental limit in an ideal case, which can be used in the simulations of a conventional method. It is also known that in practice, the acquisition speed must be even higher then the Nyquist limit because of the quantisation errors in the data recovery pro-cesses [7]. A higher acquisition speed means a decrease in the value of the highest sampling frequency. The maximum sampling rate for the Nyquist-limit (Rmax,N) is given in equation 2.2 where B stands for the bandwidth of

the signal and M for the discrete level of a signal. (which can also be seen as the amplitude of the highest frequency present in the data-set) [4].

Rmax,N = 2B log2M (2.2)

Equation 2.3 gives the maximum sampling rate according the Shannon the-orem (Rmax,S), where S/N is the signal to noise ratio of the signal.

Rmax,S = 2B log2(1 + S/N ) (2.3)

For the purpose of this research, only the noiseless case is considered, which means that the Nyquist-limit applies for the simulations of the conven-tional imaging approach. In other words, the Nyquist-limits states that the data-acquisition speed must be at least twice the highest frequency present in the data-set.

2.2

Compressive imaging

The compressive imaging (called CI), has introduced a paradigm-shift in the optical imaging disciplines. This means that CI in not dependent on the fundamental limits of the conventional method. The method of Compres-sive imaging makes it possible to develop imaging systems which allow to image beyond the fundamental limits of the conventional imaging. It allows to break down the Abbe-diffraction limit, having a higher spatial resolution and a sampling rate beyond the limitations of the Nyquist-theorem. There-fore, by using compressive imaging, a higher spatial resolution and temporal resolution can achieved.

2.2.1

Sparsity

Compressive imaging allows to sample the object with a fewer measurements by using the fact that a large number of the singles are in the sparse expan-sion. Not all natural signals are sparse in every basis, but most of them are

sparse in some basis. In this research, the Euclidean space has been used for the reconstruction. In this basis, having a low amount of non-zero elements, is called sparse in the Euclidean basis. All the signals with a sparse repre-sentation have many zero-values or close to the limit of zero with respect to the non-zero signals. This means that a lot of elements (the zero-elements) can be discarded so that CI can be used to reconstruct a sample with a limited amount of measurement, Which can still result in an accurate recon-struction. In order to control the accuracy of the reconstructed image, the correlation factor (corr.) between the reconstructed image and the sample can be calculated. A correlation factor between 0 and 1 can be calculated.

By achieving a satisfactory level of the correlation factor (for example 0.9), while using a limited number of measurements means that Nyquist-limit is not applied for compressive imaging.

2.2.2

Incoherence

For compressive imaging it is important to have the lowest correlation be-tween the elements of the imaging basis. Many mathematical functions such as the spikes (Dirac-delta functions), sinusoids can be used to generate speckle patterns, but the random function generates the idealised speckle patterns with a very low coherence. This means that the speckles used for compressive imaging are uncorrelated and thus highly random.

2.3

Number of measurements (N

)

For conventional imaging we need to scan through all the pixels to reconst the whole sample. For a sample having a size of 32x32 pixels,the total num-ber of the measurements is 1024. This total numnum-ber of measurements can also be called 100% of the measurements. For a responsible comparison be-tween the two imaging approaches, the total number of measurements for compressive imaging should also set to 1024. However, the total number of measurements for compressive imaging is not limited by this number. For compressive imaging, as a consequence of sparsity, less number of measure-ments can be used for reconstruction, which means that the percentage of the used/required measurements (Nm) can be less then 100%.

2.4

Speckles Patterns in a MM-fibre

The emergence of spatial wave-front shaping allows the MM-fibre to be used as an ultra-thin imaging probe [1]. In different scientific disciplines, the use of MM-fibres allows to image in areas which are not reachable with the conventional setup of a microscope. For the point by point scanning system, the Abbe-diffraction limit still applies and the sampling rate is still limited by the Nyquist theorema if the conventional method of imaging is still used. However, the speckle patterns generated in a MM-fibre could also be used for compressive imaging which makes the MM-fibre a good sensing tool as well as a good imaging tool.

2.4.1

Light Coupling in a MM-fibre

An optical fibre is a dielectric wave-guide which consists of a cylindrical central core, cladding and an outer coating. The fibre is made of a low-loss material such as silica glass [8] [5]. A schematic view of a cylindrical dialectic wave-guide is shown in figure 2.2.

Figure 2.2: A schematic view of an optical fibre showing its core, cladding and the coating.

The optical fibre is characterised by its index of refraction n. The refrac-tion at the interface of two media can be described with Snell’s law:

n1sin(φin) = n2sin(φr) (2.4)

where φin is the angle of incident and φr is the angle of reflection. To find

the critical angle for a total internal reflection, we set φr = 90◦ and find the

following equation with φc, the critical angle.

φc= arcsin(

n2

n1

The cladding of the fibre has just a slightly lower refractive index then the central core so that all the light rays, having an angle greater then the critical angle (φin> φc ) undergo total internal reflection.This way light rays

are guided through the core, without any refraction into the cladding. The guided light has a phase shift at every reflective boundary in the core of the fibre.There a is finite discrete number of paths which can be taken by the light rays. These paths are called the modes of the fibre. Depending on the diameter of the core of the fibre we can have single-mode or multi-mode fibres. In a single-mode fibre only the fundamental mode is transmitted, when light rays have different paths (different angle of incident), a multi-mode fibre is defined [8] [5]. Figure 2.3 shows the difference between a single-mode and a multi-mode fibre.

Figure 2.3: A schematic view of a single-mode with the fundamental mode and a multi-mode fibre.

2.4.2

Speckle pattern in MM-fibre

In a MM-fibre, light rays travel on different paths. The different angles of incident on a MM-fibre imaging setup gives different speckle patterns on the output. These speckles patterns are uncorrelated, having a large number of randomness. Compressive imaging is fundamentally dependent on the randomness of these speckle patterns. This means that speckle patterns, generated in a MM-fibre could be used for compressive imaging.

Chapter 3

Materials and Methods

This chapter describes the numerical method which is used to simulate the two different imaging methods: the conventional method of point-by-point scanning and compressive imaging CI by using the randomly generated speckles patterns.

3.0.1

Simulations and comparison in M AT LAB

The conventional method of optical imaging consists of a laser beam using the light in the visual range. Most of the laser beams propagate with a Gaus-sian profile. Thus the characteristics of the laser beams can described by an ideal Gaussian intensity profile which has a radially symmetrical distribu-tion. In this research, the conventional method of point-by point-imaging and compressive imaging is simulated by using M AT LAB. Furthermore, a set of MM-fibre speckle patterns is used to reconstruct the same sample called ‘original object’ and compared with the reconstruction of the idealised random speckle patterns. The data-set of MM-fibre speckle patterns has been provided by a fellow student. A sample with a size of 32x32 pixels and a sparsity of 4% has been used to reconstruct with the two different imaging approaches: diffraction-limited conventional imaging and compressive imag-ing.

The M AT LAB simulations represent an optical imaging setup where each pixel has a size of 300nm, the imaging object has a numerical aperture (NA) of 0.32. The wavelength of the light beams has been set to 532nm. The diffraction-limited imaging has been simulated by scanning a Gaussian profile, as it gives the most reliable numerical result for a point-by-point conventional imaging system. A set of randomly generated speckle patterns has been used to represent the compressive imaging approach. For image reconstruction with CI, l1 − minimisation algorithm is used [3]. In order to

physically compare the two imaging methods, the Gaussian peaks and the randomly generated speckle patterns has been transformed into the Fourier space. In the Fourier space the cut-off frequency of the speckle patterns is compared with the Gaussian function. A reasonable choice of the cut-off is necessary to make sure that both imaging setups are operating in the same physical domain. Furthermore, the set of MM-fibre has been used to test the working of the M AT LAB model and to reconstruct the sample with compressive imaging. A comparison between the conventional method of imaging and the idealised compressive imaging has given.A comparison between the idealised CI reconstruction and the reconstruction with MM-fibre speckle is also presented. Figure 3.4 shows a three-dimensional view of a central Gaussian peak and the distribution of both the randomly generated speckles and the MM-fibre generated speckles in the Euclidean space.

Figure 3.3 shows that the MM-fibre speckle patterns have no intensity at the edges of the space. Ideally the speckle patterns in a MM-fibre can be presented in the polar coordinates, because a MM-fibre has a circular geom-etry. Here the speckle patterns are projected in the Cartesian coordinates which shows the circular geometry of a MM-fibre. For the reconstruction with all the three methods, the size of the sample has been set to 32x32 pixels. Therefore, only a limited area of the MM-fibre speckle patterns has been used.

Figure 3.1: Central peak for a Gaussian function which is used to represent the diffraction-limited imaging.

Figure 3.2: The distribution of one of the randomly generated speckle pat-tern.

Figure 3.3: Distribution of a MM-fibre speckle patterns.

Figure 3.4: A three-dimensional view of a Gaussian peak, randomly gener-ated speckle patters and MM-fibre speckle patterns in the Euclidean space.

Chapter 4

Results

This chapter presents the simulations of the conventional imaging approach and the compressive imaging approach for an endo-microscopic setup. The simulations in M AT LAB are used to give a comparison between the two approaches. The compressive imaging uses speckle patterns for illumination of an image. Furthermore, this chapter includes the simulations while using two different types of speckle pattern: randomly generated speckle patters and speckle patterns in a MM-fibre.

4.1

Intensity profiles

The intensity profiles of the two imaging methods are compared both in the Euclidean space and the Fourier space.

4.1.1

Euclidean Space

Figure 4.1 shows the intensity profiles for both the conventional imaging and compressive imaging in the Euclidean space. The behaviour of the light beams for the conventional method of point-by-point scanning is represented by a Gaussian intensity profile, while for the compressive imaging approach one of the randomly illuminated speckle pattern is shown. For a responsible comparison between the two imaging approaches, it is important that the two setups operates in the same physical domain. Which means that the pixel cut-off for compressive imaging should be chosen in such a way that it is comparable with the chosen numerical aperture (NA) of the conventional imaging setup. The intensity profiles in the Euclidean space do not present sufficient information to choose the ’right’ cut-off value. For this reason, the intensity profiles are also projected in the Fourier space.

Figure 4.1: Left: A 2D intensity profile of a diffraction-limited conventional imaging, having a pixel size of 300nm and N A = 0.32. Right: The 2D intensity profile of a speckle pattern.

4.1.2

Fourier Space

Figure 4.2 shows the 2D intensity profiles in the Fourier space for both of the imaging methods. The colour bars present the intensity on the logarithmic scale. Figure 4.3 shows the 1D intensity profiles for both the Gaussian func-tion and the speckle patterns. The 1D profile is a line profile through the centre of the 2D plot. The 1D profiles presents the normalised intensities on the logarithmic scale, as a function of the its frequency (relative to the cut-off frequency of the setups). The pixel cut-off for the speckles is set between 1 and -1. Due to the the exponential behaviour of the Gaussian function, it is reasonable to set the ideal cut-off between 1 and -1. This makes the two sensing setups physically comparable.

Figure 4.2: left: A 2D intensity profile of a diffraction-limited conventional imaging, projected in the Fourier space. Right: 2D intensity profile for ran-domly generated speckle patterns in the Fourier space.

Figure 4.3: Normalised intensity profile for the Gaussian function and one of the speckle patterns, as a function of its frequency (relative to the cut-off frequency of the setup). The normalised intensities are given on the logarithmic scale.

4.2

Beyond the Nyquist-limit

The simulations of the compressive imaging approach show that the number of measurement (Nm) or its percentage is not defined by the Nyquist-limit.

Which states that (for the conventional method) the minimum number of measurement should be twice the maximum intensity present in the data. A less number of measurement means that the acquisition time can be shorten with the compressive imaging approach. Figure 4.4 shows in total 8 re-construction of a sample called ‘original object’ with a size of 32x32 pixels, having each pixel of 300nm. This reconstruction series is comparable to a conventional setup which has a N A of 0.32 and laser beams with a wave-length of 532nm. The reconstructions show that using a compression of 7.81%, a correlation of more then 0.9 can be achieved. Furthermore, for a compression of 11.7% the correlation value gets first above the 0.95. This shows that using 9 times less number of measurements then needed for the conventional method, a correlation of 0.951 can be achieved. However, this relation between the number of measurements and the correlation factor is only valid for the described parameters. This research also investigates the relation between the pixel size of a sample and the correlation for different compression rates.

Figure 4.4: Compressive imaging of a 32x32 pixels sample (1 pixel = 300nm) which has a sparsity of 4. This compressive imaging is comparable to a conventional imaging setup with λ = 532 nm and NA = 0.32. The title of each reconstruction gives the compression percentage (com.) and the corresponding correlation factor (corr.) between the reconstructed image and the sample.

4.3

Different Pixel Sizes

Figure 4.5 shows the relation between the pixel size of a sample the correla-tion factor between the reconstructed image and the sample. The black curve represents the conventional method of point-by-point imaging. It is obvious to set Nm percentage on 100 for the conventional method as we state that

for the point-by-point imaging each pixel should be scanned to capture the whole sample. On the other hand, for compressive imaging a less number of measurements can be sufficient for the same correlation factor or even a higher correlation. The curve with a compression percentage (called Nm)

of 1.9% shows that the speckle patterns are not enough for the reconstruc-tion. The curve with 6.8% compression is comparable with the conventional method for a pixel size between 100 and 200 nm. For a higher value of the pixel size, the curves declines. However, this result shows that all the curves with a compression percentage of 11.7% and higher, gives a higher correlation value, independent of the pixel size of a sample.

Figure 4.5: Correlation curves of the conventional method of point-by-point scanning and the curves for different compression percentage of compressive imaging.

4.4

Beyond the Diffraction-limit

Figure 4.6 shows a comparison between the optical imaging beyond the diffraction-limit for both the conventional method of imaging and compres-sive imaging. The diffraction-limit of conventional imaging can be calculated by using Equation 2.1. With λ = 532nm and N A = 0.32, the maximum resolvable distance (d) between two points is 0.83 microns. For each pixel having a size of 0.3 microns, a distance of 0.83 microns translates approxi-mately into 3 pixels. This means that for the conventional imaging approach, beyond a distance of 3 pixels the data-points are not distinguishable. This is exactly shown in Figure 4.6. However, compressive imaging does not have this limitation. While only using 9.7% of the measurements, imaging beyond the Abbe-diffraction limit is possible. This shows that, super-resolution can be achieved with compressive imaging.

Figure 4.6: Simulation of the reconstruction with conventional imaging and compressive imaging (Nm = 9.7%). This shows that the compressive imaging

is not diffraction-limited.

4.5

MM-fibre Speckle patterns for CI

A set of speckle patterns which are simulated to represent the speckles in a MM-fibre are also implemented in the M AT LAB model. Figure 4.7 shows the 2D intensity profiles of the MM-fibre speckles in both the Euclidean and the Fourier space. As stated earlier, the sample size in this research has set on 32x32 pixels. This means that a limited field of view (FOV) area of MM-fibre is needed for CI. The area highlighted with a blue box in Figure 4.7 (Euclidean space) is used for CI reconstruction.

Figure 4.7: Intensity profiles of the MM-fibre speckles in both the Euclidean and the Fourier space. A 32x32 pixel sized view (FOV) has been chosen for compressive imaging. (shown with a blue box in the Euclidean space).

Furthermore, Figure 4.8 shows the region of the speckle pattern which is used to reconstruct the sample called ‘original object’. Figure 4.8 shows the zoomed-in version of the blue highlighted area of Figure 4.7, both in the Euclidean and the Fourier space.

Figure 4.8: Field of view (FOV) of MM-fibre speckle patterns which is used to reconstruct the ’original object’.

4.6

Random speckles and MM-fibres speckles

This research also includes a comparison between the two different CI ap-proaches: with random speckle patterns and MM-fibre speckle patterns. Figure 4.9 shows three different reconstructed images for both the random speckles and the MM-fibre speckles. Image in the series called (A) are re-constructed by using the random speckle patterns while series (B) uses the speckle patterns of a MM-fibre. The compression percentage (com.) and the correlation factor (corr.) of each image has presented in the titles of the images. This figure shows that for the given compression percentage, the MM-fibre speckle patterns generates a higher correlation then the random speckle patterns. Furthermore, Figure 4.10 shows the relation between the percentage of Nm and the correlation factor for both the random speckles

and the MM-fibre speckles. This figure shows that the MM-fibre speckles give a higher correlation independent of the number of measurements. This means that the MM-fibre speckle patterns systematically create a ’better’ result then the random speckles. This can be explained by looking at the 1D profiles in Figure 4.11.

Figure 4.9: A comparison between the two different CI approaches: random speckle patterns and MM-fibre speckle patterns.

4.7

1D intensity profiles in Fourier

Figure 4.11 shows the ID intensity profiles for conventional imaging and for compressive imaging (for both the random speckle patterns and MM-fibre speckle patterns). This figure explain that the cut-off in the pixels which has set between -1 and 1 for random speckles, is less then what is needed for a reasonable comparison. The figure shows that the MM-fibre does have intensity outside the [-1,1] range. In fact, the MM-fibre speckles gives a better physical representation of an optical imaging setup. The randomly generated speckle patterns present a very idealised noiseless compressive imaging setup.

Figure 4.10: The correlation curves of randomly generated speckle patterns and the speckle patterns in a MM-fibre.

Figure 4.11: 1D intensity profile on the logarithmic scale. For conventional imaging the Gaussian function is used. Compressive imaging is done with random speckle patterns and with MM-fibre speckle patterns.

4.8

Symmetry effect in the MM-fibre

This research also shows that a reflection-symmetry occurs when we use different field of view (FOV) then mentioned in Figure 4.7. The MM-fibre speckles has a reflection-symmetry which is also visible in the reconstruction if the FOV goes through the centre of the MM-fibre speckle patterns. Figure 4.12 illustrated this phenomena. To get rid of this symmetry effect in the simulations for a noiseless setup, an off-centred FOV seems to be a reasonable choice for the speckle patterns, as shown in Figure 4.7.

Figure 4.12: Symmetry effect in the MM-fibre speckle patterns. Going through the centre of the MM-fibre speckles gives the reflection-symmetry effect in the image reconstruction.

Chapter 5

Discussion

This research thesis has simulated the results of optical imaging for two dif-ferent approaches. The conventional method of point-by-point imaging and compressive imaging. The simulations has shown that compressive imaging is a very promising alternative to the conventional optical imaging approach. It has shown that for a noiseless case, compressive imaging makes it possible to image beyond the Abbe-diffraction limit and the Nyquist-limit. This means that compressive imaging is not limited by the diffraction of the light and a higher data-acquisition speed can be achieved then described by the Nyquist-limit. A comparison between the number of measurements shows that the compressive imaging approach needs at least 9 times less measurements then the conventional method of imaging. This shows how compressive imaging brings a paradigm-shift in the optical imaging technology.

However, the results presented in this thesis are only valid for a noiseless, almost an ideal compressive imaging simulation and the parameters used in this thesis such as the sample size, sparsity, the numerical aperture (N A) and the wavelength of the laser beam (λ) are chosen very strictly. Different pa-rameter values could affect the simulation results. Furthermore, simulations with a ‘signal to noise ratio’ (SNR) are needed to present a more realistic view and a more reliable comparison of the two optical imaging approaches. It is also recommended to find the relation between the sparsity of a sample and the correlation factor between the image and the sample. This com-parison could give an insight to the limitations of compressive imaging as it depends on the sparsity level of a sample. This thesis also presented the simultions of compressive imaging approach for a set of speckle patterns in a MM-fibre. The speckle patterns generated with a random function present an idealised compressive imaging case wheres the speckles in the MM-fibre are representing a more realistic setup. The simulations of the MM-fibre also show the presence of reflection-symmetry through the x-axis of the speckle

patterns. This symmetry is also seen in the image reconstruction with the MM-fibre speckle patterns. However, this symmetry-effect is only seen while going thorough the centre of the MM-fibre speckles. For a sample having a size of 32x32 pixels, a different range of the MM-fibre speckle patterns can be used. Speckle patterns starting from one of the edges of the MM-fibre, which does not go through the centre, can be used to get rid of this reflection-symmetry in the image reconstruction.

Altogether, this thesis shows that compressive imaging could be a good alternative of the conventional imaging in an idealised setup. Further re-search is needed to simulate a more realistic view of the two different optical imaging approaches and to get a deeper understanding of the limitations of compressive imaging.

Chapter 6

Conclusions

This research thesis presents the simulations of two different optical imaging methods. The conventional method of imaging operates in the diffraction-limited and speed-diffraction-limited paradigm, wheres the compressive imaging depends on the sparsity of a sample and the coherence of the speckle patterns which are used to illuminate the sample. Furthermore, this research gives a nu-merical comparison between the two optical imaging methods which shows that compressive imaging is a good alternative of conventional imaging. It makes possible to image beyond the diffraction-limit and to achieve a higher reconstruction speed then defined by the Nyquist-limit. For an idealised noiseless setup, the imaging speed can be increased by almost a factor of 9 and imaging beyond the Abbe-diffraction limit is possible. This research concludes that compressive imaging has brought the optical imaging technol-ogy into a paradigm where super-resolution and super-speed can be achieved simultaneously.

For further research, it is recommended to find the fundamental limi-tations of compressive imaging. Samples with different sparsity levels and speckle patterns with different coherence factors can be used to understand the fundamental limitations of compressive imaging.

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