In many practical problems in applied sciences, the features of most interest cannot be observed directly, but have to be inferred from other, observable quantities. In particular, many important data acquisition devices provide an access to the measurement of the partial derivatives of a feature of interest rather than sensing its values in a direct way. In this case, the feature has to be recovered through integration which is known to be an ill-posed problem in the presence of noises. Moreover, the problem becomes even less trivial to solve when only a portion of a complete set of partial derivatives is available. In this case, the instability of numerical integration is further aggravated by the loss of information which is necessary to perform the reconstruction in a unique way. As formidable as it may seem, however, the above problem does have a solution in the case when the partial derivatives can be sparsely represented in the range of a linear transform. In this case, the derivatives can be recovered from their incomplete measurements using the theory of compressive sampling (aka compressed sensing), followed by reconstruction of the associated feature/object by means of a suitable integration method. It is known, however, that the overall performance of compressive sampling largely depends on the degree of sparsity of the signal representation, on the one hand, and on the degree of incompleteness of data, on the other hand. Moreover, the general rule is the sparser the signal representation is, the fewer measurements are needed to obtain a useful approximation of the true signal. Thus, one of the most important questions to be addressed in such a case would be of how much incomplete the data is allowed to be for the signal reconstruction to remain useful, and what additional constraints/information could be incorporated into the estimation process to improve the quality of reconstruction in the case of extremely under-sampled data. With these questions in mind, the present proposal introduces a way to augment the standard constraints of compressive sampling by additional information related to some natural properties of the signal to be recovered. In particular, in the case when the latter is defined to be the partial derivatives of a multidimensional signal (e.g. image), such additional information can be derived from some standard properties of the gradient operator. Consequently, the resulting scheme of derivative compressive sampling (DCS) is capable of reliably recovering the signals of interest from much fewer data samples as compared to the case of the standard CS. The signal recovery by means of DCS can be used to improve the performance of many important applications which include stereo imaging, interferometry, coherent optical tomography, and many others. In this proposal, we focus mainly on the application of DCS to the problem of phase unwrapping, whose solution is central to all the aforementioned applications. Specifically, it is shown both conceptually and experimentally that the DCS-based phase unwrapping outperforms a number of alternative approaches in terms of estimation accuracy. Finally, the proposal lists a number of research questions which need to be answered in order to attach strong theoretical guarantees to the practical success of DCS.

<p>Spectral&#8208;domain optical&#8208;coherence tomography (SDOCT) has recently emerged as a powerful new tool for noninvasive human retinal imaging. I have developed a low&#8208;cost, high resolution real&#8208;time Spectral Domain Optical Coherence Tomography (SDOCT) system optimized for rapid 3D imaging of the human retina in vivo. Then functional retinal OCT imaging such as polarization sensitive OCT (PSOCT) and Doppler OCT were also developed based on phase technique. Unique phase unwrapping method in retina is described to extract the total reflectivity, accumulative retardance and fast axis orientation of the retinal nerve fiber layer (RNFL). The polarization scrambling layer of the retinal pigment epithelium was segmented by employing single camera sequential scan bsed PSOCT. As an extension, synthetic wavelength method will be also introduced for phase unwrapping in cell imaging. Finally I present an algorithm for 3D refraction correction based on a vector representation which accounts for refraction of CT light in the cornea. Following 3D refraction correction of volumetric corneal datasets, we can estimate the corneal optical power, thickness and the individual wavefront aberrations of the epithelial and the refraction&#8208;corrected endothelial surfaces by using Zernike spectrum analysis.</p> / Dissertation

In many practical problems in applied sciences, the features of most interest cannot be observed directly, but have to be inferred from other, observable quantities. In particular, many important data acquisition devices provide an access to the measurement of the partial derivatives of a feature of interest rather than sensing its values in a direct way. In this case, the feature has to be recovered through integration which is known to be an ill-posed problem in the presence of noises. Moreover, the problem becomes even less trivial to solve when only a portion of a complete set of partial derivatives is available. In this case, the instability of numerical integration is further aggravated by the loss of information which is necessary to perform the reconstruction in a unique way. As formidable as it may seem, however, the above problem does have a solution in the case when the partial derivatives can be sparsely represented in the range of a linear transform. In this case, the derivatives can be recovered from their incomplete measurements using the theory of compressive sampling (aka compressed sensing), followed by reconstruction of the associated feature/object by means of a suitable integration method. It is known, however, that the overall performance of compressive sampling largely depends on the degree of sparsity of the signal representation, on the one hand, and on the degree of incompleteness of data, on the other hand. Moreover, the general rule is the sparser the signal representation is, the fewer measurements are needed to obtain a useful approximation of the true signal. Thus, one of the most important questions to be addressed in such a case would be of how much incomplete the data is allowed to be for the signal reconstruction to remain useful, and what additional constraints/information could be incorporated into the estimation process to improve the quality of reconstruction in the case of extremely under-sampled data. With these questions in mind, the present proposal introduces a way to augment the standard constraints of compressive sampling by additional information related to some natural properties of the signal to be recovered. In particular, in the case when the latter is defined to be the partial derivatives of a multidimensional signal (e.g. image), such additional information can be derived from some standard properties of the gradient operator. Consequently, the resulting scheme of derivative compressive sampling (DCS) is capable of reliably recovering the signals of interest from much fewer data samples as compared to the case of the standard CS. The signal recovery by means of DCS can be used to improve the performance of many important applications which include stereo imaging, interferometry, coherent optical tomography, and many others. In this proposal, we focus mainly on the application of DCS to the problem of phase unwrapping, whose solution is central to all the aforementioned applications. Specifically, it is shown both conceptually and experimentally that the DCS-based phase unwrapping outperforms a number of alternative approaches in terms of estimation accuracy. Finally, the proposal lists a number of research questions which need to be answered in order to attach strong theoretical guarantees to the practical success of DCS.

Positioning an underwater object with respect to a reference point is required in diverse areas in ocean scientific and engineering undertakings, such as marine habitat monitoring, study of sedimentation processes, underwater searching and mapping, data collection, instrument placement and retrieval, and so on. Underwater acoustic positioning systems, including long baseline (LBL) systems, short baseline (SBL) systems, and ultra-short baseline (USBL) systems, are designed to operate from a reference point and employ external transducers or transducer arrays as aids for positioning. Traditional positioning methods rely on measuring of time-of-flight of an acoustic signal travelling from the target to the reference platform by means of the cross-correlation method. The positioning accuracy of LBL systems varies from a few centimeters to a few meters, depending on the operating range and working frequency. LBL systems provide a uniform positioning accuracy for a given transponder array setup, but they suffer the time-consuming instrument deployment on the seafloor, as well as the complicated operating procedures. SBL and USBL systems have relatively simple configurations. But their positioning accuracy is a function of water depth and operating range. To obtain absolute position accuracy, additional sensors such as the ship's gyro or a surface navigation system are needed. In this thesis, a novel positioning method is proposed which takes advantages of a tether cable between the reference platform and the target. This method conducts positioning via continuous phase measurement between a reference signal and the acoustic signal transmitted by the target to the reference platform. It is named the Positioning-based-on-PHase-Measurement method or PPHM method in short. Every 2π change in the phase difference between these two signals corresponds to a one-wavelength range increment along the radial direction from the target’s initial position to its new position. If a receiver array is used, with at least two hydrophones, the target’s bearing information can be also calculated by measuring the phases of the output signals from each of the array hydrophones. Under ideal conditions, the positioning error of the PPHM method is proportional to the phase measurement error. The PPHM method is very sensitive to changes in the underwater medium, such as sound speed variations, ocean currents and multipath interferences. Environmental fluctuations will degrade the positioning performance. These problems will be investigated and solutions will be proposed to minimize their effects. The PPHM method can be used to position an underwater moving object such as a remotely operated vehicle (ROV) or a bottom crawler. Also, it can be used to monitor the ocean currents speed variations over a path, or to monitor the movements of tectonic plates. The last two applications will be addressed in detail in this thesis, whereas the first one is very challenging and needs more work.

In the form of images and videos, visual content has always attracted considerable interest and attention to itself since the early days of the computer era. Although, due to the high density of information in such contents, it has always been challenging to generate, process and broadcast videos and images. These challenges grew along with the demand for higher quality content and attained the research community's attention to themselves. Even though many works have been done by researchers and engineers in academic and industrial environments, the demand for high-quality content introduces new constraints on the quality, performance (speed) and energy consumption. This thesis focuses on a couple of image and video processing applications and introduces new approaches and tweaks to improve the performance and save resources while keeping the functionality intact. In the first part, we target Interferometric Synthetic Aperture Radar (InSAR), an imaging technique used by satellites to capture the earth's surface. Many algorithms have been developed to extract useful information, such as height and displacement, from such images. However, the sheer size of these images, along with the complexity of most of these algorithms, lead to very long processing time and resource utilization. In this work, we take one of the dominant algorithms used for almost every In-SAR application, Phase Unwrapping, and introduce an approach to gain up to 6.5 times speedups. We evaluated our method on InSAR images taken by the Radarsat-2 sensor and showed its impact on a real-world application. In the second part of this thesis, we look at a prevalent application, video streaming. These days video streaming dominates the internet traffic, so any slight improvement in terms of energy consumption or resource utilization will make a sizable difference. Although the streamers use various encoding techniques, the quality of experience of the clients prevents them from overplaying these techniques. On the other hand, there has been a growing interest in another venture of research which focuses on developing techniques that aim to restore the quality of the videos that have been subjected to compression. Although these techniques are used by many users on the receiver side, the streamers often ignore their capabilities. In our work, we introduce an approach that makes the streamer aware of the capabilities of the receiver and utilizes that awareness to reduce the cost of transmission without compromising the end user's quality of experience. We demonstrated the technique and proved our concept by applying it to the HEVC encoding standard and JCT-VC dataset. / Graduate

A three-by-three polarization ray tracing matrix method is developed to calculate the polarization transformations associated with ray paths through optical systems. The relationship between the three-by-three polarization ray tracing matrix P method and the Jones calculus is shown in Chapter 2. The diattenuation, polarization dependent transmittance, is calculated via a singular value decomposition of the P matrix and presented in Chapter 3. In Chapter 4 the concept of retardance is critically analyzed for ray paths through optical systems. Algorithms are presented to separate the effects of retardance from geometric transformations. The parallel transport of vectors is associated with non-polarizing propagation through an optical system. A parallel transport matrix Q establishes a proper relationship between sets of local coordinates along the ray path, a sequence of ray segments. The proper retardance is calculated by removing this geometric transformation from the three-by-three polarization ray trace matrix. Polarization aberration is wavelength and spatial dependent polarization change that occurs as wavefrontspropagate through an optical system. Diattenuation and retardance of interfaces and anisotropic elements are common sources of polarizationaberrations. Two representations of polarization aberrationusing the Jones pupil and a polarization ray tracing matrix pupil, are presentedin Chapter 5. In Chapter 6 a new class of aberration, skew aberration is defined, as a component of polarization aberration. Skew aberration is an intrinsic rotation of polarization states due to the geometric transformation of local coordinates; skew aberration occurs independent of coatings and interface polarization. Skew aberration in a radially symmetric system primarily has the form of a tilt plus circular retardance coma aberration. Skew aberration causes an undesired polarization distribution in the exit pupil. A principal retardance is often defined within (-π, + π] range. In Chapter 7 an algorithm which calculates the principal retardance, horizontal retardance component, 45° retardance component, and circular retardance component for given retarder Jones matrices is presented. A concept of retarder space is introduced to understand apparent discontinuities in phase unwrapped retardance. Dispersion properties of retarders for polychromatic light is used to phase unwrap the principal retardance. Homogeneous and inhomogeneous compound retarder systems are analyzed and examples of multi-order retardance are calculated for thick birefringent plates. Mathematical description of the polarization properties of light and incoherent addition of light is presented in Chapter 8, using a coherence matrix. A three-by-three-by-three-by-three polarization ray tracing tensor method is defined in order to ray trace incoherent light through optical systems with depolarizing surfaces. The polarization ray tracing tensor relates the incident light’s three-by-three coherence matrix to the exiting light’s three-by-three coherence matrix. This tensor method is applicable to illumination systems and polarized stray light calculations where rays at an imaging surface pixel have optical path lengths which vary over many wavelengths. In Chapter 9 3D Stokes parameters are defined by expanding the coherence matrix with Gell-Mann matrices as a basis. The definition of nine-by-nine 3D Mueller matrix is presented. The 3D Mueller matrix relates the incident 3D Stokes parameters to the exiting 3D Stokes parameters. Both the polarization ray tracing tensor and 3D Mueller matrix are defined in global coordinates. In Chapter 10 a summary of my work and future work are presented followed by a conclusion.

The phase and amplitude gradient estimator (PAGE) method has proven successful in improving the accuracy of measured energy quantities over the p-p method, which has traditionally been used, in several applications. One advantage of the PAGE method is the use of phase unwrapping, which allows for increased measurement bandwidth above the spatial Nyquist frequency. However, phase unwrapping works best for broadband sources in free-field environments with high coherence. Narrowband sources often do not have coherent phase information over a sufficient bandwidth for a phase unwrapping algorithm to unwrap properly. In fact, phase unwrapping processing can cause significant error when there is no coherent signal near and above the spatial Nyquist frequency. However, for signals at any frequencies up to the spatial Nyquist frequency, the PAGE method provides correct intensity measurements regardless of the bandwidth of the signal. This is an improved bandwidth over the traditional method. For narrowband sources above the spatial Nyquist frequency, additional information is necessary for the PAGE method to provide accurate acoustic intensity. With sufficient bandwidth and a coherence of at least 0.1 at the spatial Nyquist frequency, a relatively narrowband source above the spatial Nyquist frequency can be unwrapped accurately. One way of using extra information, called the extrapolated PAGE method, uses the phase of a tone below the spatial Nyquist frequency and an assumption of a propagating field, and therefore linear phase, to extrapolate the phase above the spatial Nyquist frequency. Also, within certain angular and amplitude constraints, low-level broadband noise can be added to the field near a source emitting a narrowband signal above the spatial Nyquist frequency. The low-level additive broadband noise can then provide enough phase information for the phase to be correct at the frequencies of the narrowband signal. All of these methods have been shown to work in a free-field environment.

The conventional light microscope is an indispensable tool for many physical and life science applications, but is of limited usefulness for three-dimensional imaging due to its increasingly narrow depth of field at high magnifications. Focused regions may be obscured by defocused neighbors or noise from extraneous light sources and subsurface scattering. By rejecting light originating from outside the depth of focus it is possible to minimize these problems. When a contiguous series of such focused slices, or optical sections, are obtained along an axis of an extended object they may be combined to form a complete, focused three-dimensional surface image. Here, a variety of methods to obtain optical sections in a reflective setup are presented. The first employs an optical feedback loop through a spatial light modulator (SLM) to selectively illuminate focused regions. The SLM is a flexible electro-optical device that also allows (non-feedback) experiments of an intensity modulated light source resulting in illumination with a linear structure. This structured illumination microscopy is an established sectioning technique, which requires three frame captures per axial position. By developing a color grid and exploiting the red, green, and blue channels of a CCD camera, the three frames have been reduced to one. The speed increase comes at a cost and the limiting effects of chromatic aberration are discussed. Digital holography offers an alternative to axial scanning by allowing the surface to be reconstructed from a single exposure. Use of multiple wavelength illumination with this extended focus imaging is proposed and preliminary results are shown.

In this dissertation, improved techniques in digital holography, that have produced high-resolution, high-fidelity images, are discussed. In particular, the angular spectrum method of calculating holographic optical field is noted to have several advantages over the more commonly used Fresnel transformation or Huygens convolution method. It is observed that spurious noise and interference components can be tightly controlled through the analysis and filtering of the angular spectrum. In the angular spectrum method, the reconstruction distance does not have a lower limit, and the off-axis angle between the object and reference waves can be lower than that of the Fresnel requirement, while still allowing the zero-order background to be cleanly separated. Holographic phase images are largely immune from the coherent noise commonly found in amplitude images. With the use of a miniature pulsed laser, the resulting images have 0.5um diffraction-limited lateral resolution and the phase profile is accurate to about several nanometers of optical path length. Samples such as ovarian cancer cells (SKOV-3) and mouse-embryo fibroblast cells have been imaged. These images display intra-cellular and intra-nuclear organelles with clarity and quantitative accuracy. This technique clearly exceeds currently available methods in phase-contrast opticalmicroscopy in both resolution and detail and provides a new modality for imaging morphology of cellular and intracellular structures that is not currently available. Furthermore, we also demonstrate that phase imaging digital holographic movies provide a novel method of non-invasive quantitative viewing of living cells and other objects. This technique is shown to have significant advantages over conventional microscopy.

This thesis is concerned with (a) the development of full-field, multi-axis and phase contrast wavelength scanning interferometer, using an electronically tuned CW Ti:Sa laser for the study of depth resolved measurements in composite materials such as GFRPs and (b) the development of temporal phase unwrapping algorithms for depth re-solved measurements. Item (a) was part of the ultimate goal of successfully extracting the 3-D, depth-resolved, constituent parameters (Young s modulus E, Poisson s ratio v etc.) that define the mechanical behaviour of composite materials like GFRPs. Considering the success of OCT as an imaging modality, a wavelength scanning interferometer (WSI) capable of imaging the intensity AND the phase of the interference signal was proposed as the preferred technique to provide the volumetric displacement/strain fields (Note that displacement/strain fields are analogous to phase fields and thus a phase-contrast interferometer is of particular interest in this case). These would then be passed to the VFM and yield the sought parameters provided the loading scheme is known. As a result, a number of key opto-mechanical hardware was developed. First, a multiple channel (x6) tomographic interferometer realised in a Mach-Zehnder arrangement was built. Each of the three channels would provide the necessary information to extract the three orthogonal displacement/strain components while the other three are complementary and were included in the design in order to maximize the penetration depth (sample illuminated from both sides). Second, a miniature uniaxial (tensile and/or compression) loading machine was designed and built for the introduction of controlled and low magnitude displacements. Last, a rotation stage for the experimental determination of the sensitivity vectors and the re-registration of the volumetric data from the six channels was also designed and built. Unfortunately, due to the critical failure of the Ti:Sa laser data collection using the last two items was not possible. However, preliminary results at a single wavelength suggested that the above items work as expected. Item (b) involved the development of an optical sensor for the dynamic monitoring of wavenumber changes during a full 100 nm scan. The sensor is comprised of a set of four wedges in a Fizeau interferometer setup that became part of the multi-axis interferometer (7th channel). Its development became relevant due to the large amount of mode-hops present during a full scan of the Ti:Sa source. These are associated to the physics of the laser and have the undesirable effect of randomising the signal and thus preventing successful depth reconstructions. The multi-wedge sensor was designed so that it provides simultaneously high wavenumber change resolution and immunity to the large wavenumber jumps from the Ti:Sa. The analysis algorithms for the extraction of the sought wavenumber changes were based on 2-D Fourier transform method followed by temporal phase unwrapping. At first, the performance of the sensor was tested against that of a high-end commercial wavemeter for a limited scan of 1nm. A root mean square (rms) difference in measured wavenumber shift between the two of ~4 m-1 has been achieved, equivalent to an rms wavelength shift error of ~0.4 pm. Second, by resampling the interference signal and the wavenumber-change axis onto a uniformly sampled k-space, depth resolutions that are close to the theoretical limits were achieved for scans of up to 37 nm. Access of the full 100 nm range that is characterised by wavelength steps down to picometers level was achieved by introducing a number of improvements to the original temporal phase unwrapping algorithm reported in ref [1] tailored to depth resolved measurements. These involved the estimation and suppression of intensity background artefacts, improvements on the 2-D Fourier transform phase detection based on a previously developed algorithm in ref [2] and finally the introduction of two modifications to the original TPU. Both approaches are adaptive and involve signal re-referencing at regular intervals throughout the scan. Their purpose is to compensate for systematic and non-systematic errors owing to a small error in the value of R (a scaling factor applied to the lower sensitivity wedge phase-change signal used to unwrap the higher sensitivity one), or small changes in R with wavelength due to the possibility of a mismatch in the refractive dispersion curves of the wedges and/or a mismatch in the wedge angles. A hybrid approach combining both methods was proposed and used to analyse the data from each of the four wedges. It was found to give the most robust results of all the techniques considered, with a clear Fourier peak at the expected frequency, with significantly reduced spectral artefacts and identical depth resolutions for all four wedges of 2.2 μm measured at FWHM. The ability of the phase unwrapping strategy in resolving the aforementioned issues was demonstrated by successfully measuring the absolute thickness of four fused silica glasses using real experimental data. The results were compared with independent micrometer measurements and showed excellent agreement. Finally, due to the lack of additional experimental data and in an attempt to justify the validity of the proposed temporal phase unwrapping strategy termed as the hybrid approach, a set of simulations that closely matched the parameters characterising the real experimental data set analysed were produced and were subsequently analysed. The results of this final test justify that the various fixes included in the hybrid approach have not evolved to solve the problems of a particular data set but are rather of general nature thereby, highlighting its importance for PC-WSI applications concerning the processing and analysis of large scans.