Phase Unwrapping

This work presents a robust algorithm for phase unwrapping. The proposed algorithm is based on the expansion of the estimated phase through a linear combination of a set of Basis Functions. We present a novel weighted robust functional which is minimised using a two step strategy. This model allows us to reduce the influence of noise and to remove inconsistent pixels in the estimation of the unwrapped phase. The proposed model assumes that the phase is smooth. Under this assumption, experiments demonstrate that if the phase is corrupted by high levels of noise, our model presents a better performance than state of the art algorithms. For low levels of noise, the results are comparable.

We present a generic regularized formulation, based on robust half-quadratic regularization, for unwrapping noisy and discontinuous wrapped phase maps. Two cases are presented: the convex and the non-convex one. The unwrapped phase with the convex formulation is unique and robust to noise; however, the convex function solution is deteriorated by real discontinuities in phase maps; therefore, we also present a non-convex formulation which, with a parameter continuation strategy, shows a superior performance.

A discrete Fourier transform ͑DFT͒ based algorithm for solving a quadratic cost functional is proposed; this regularized functional allows one to obtain a consistent gradient field from an inconsistent one. The calculated consistent gradient may then be integrated by use of simple methods. The technique is presented in the context of the phase-unwrapping problem; however, it may be applied to other problems, such as shapes from shading ͑a robot-vision technique͒ when inconsistent gradient fields with irregular domains are obtained. The regularized functional introduced here has advantages over existing techniques; in particular, it is able to manage complex irregular domains and to interpolate over regions with invalid data without any smoothness assumptions over the rest of the lattice, so that the estimation error is reduced. Furthermore, there are no free parameters to adjust. The DFT is used to compute a preconditioner because there is highly efficient hardware to perform the calculations and also because it may be computed by optical means.

A powerful technique for processing fringe-pattern images is based on Bayesian estimation theory with prior Markov random-field models. In this approach the solution of a processing problem is characterized as the minimizer of a cost function with terms that specify that the solution should be compatible with the available observations and terms that impose certain ͑prior͒ constraints on the solution. We show that, by the appropriate choice of these terms, one can use this approach in almost every processing step for accurate and robust interferogram demodulation and phase unwrapping.

A robust procedure for analyzing fringe patterns obtained from speckle interferometric techniques is proposed. The fringes generally are observed only in a region S of a rectangular lattice L. We give a method for computing ͑from S͒ the region R, where the unwrapped phase is to be computed; this computation is done by use of a morphological filter ͑in particular, a closing filter͒. We then use a fast-unwrapping algorithm to compute the phase: a preconditioned conjugate-gradient algorithm that uses the discrete Fourier transform.

A new, to our knowledge, algorithm for the phase unwrapping ͑PU͒ problem that is based on stochastic relaxation is proposed and analyzed. Unlike regularization schemes previously proposed to handle this problem, our approach dispells the following two assumptions about the solution: a Gaussian model for noise and the magnitude of the true phase-field gradient's being less than everywhere. We formulate PU as a constrained optimization problem for the field of integer multiples of 2, which must be added to the wrapped phase gradient to recover the true phase gradient. By solving the optimization problem using simulated annealing with constraints, one can obtain a consistent solution under difficult conditions resulting from noise and undersampling. Results from synthetic test images are reported.

Damien Closson 1, Yvers Cornet 1, André Ozer 1, Herbert Hansen 2, Dominique Derauw 2, Christian Barbier 2 ... 1 Laboratoire de Géomorphologie et Télédétection, Université de Liège, Belgium 2 Centre Spatial de Liège, Université de Liège, Avenue du Pré-Aily, B-4031 Angleur, ...

We describe a new algorithm for phase determination from a single interferogram with closed fringes based on an unwrapping procedure. Here we use bandpass filtering in the Fourier domain, obtaining two wrapped phases with sign changes corresponding to the orientation of the applied filters. An unwrapping scheme that corrects the sign ambiguities by comparing the local derivatives is then proposed. This can be done, assuming that the phase derivatives do not change abruptly among adjacent areas as occurs with smooth continuous phase maps. The proposed algorithm works fast and is robust against noise, as demonstrated in experimental and simulated data.

Spatially incoherent Fourier digital holography using a rotational shearing interferometer for four-step phase-shifting method is proposed. The previous incoherent Fourier holography using a rotational shearing interferometer [Watanabe and Nomura (Appl. Opt. 54:A18, 2015)] employs the two-step phase-shifting method in the vertical and horizontal polarizations. The reconstructed image contains a large bias term. This paper proposes introduction of two kinds of wave plates in one path of a rotational shearing interferometer for a four-step phase-shifting method. A Fourier hologram is obtained from the four recorded holograms for eliminating the bias term and the twin image. The numerical simulation and the optical experiment demonstrate improvement of the image quality of reconstructed image by the twin image and bias level reduction. Furthermore, the effect of the size of an image sensor on the image quality in rotational shearing interferometer is also investigated by the numerical simulations.

As technologies evolve, there have been high demands for the three-dimensional (3D) shape measurement techniques to posses the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects as well as measuring complex shapes. Generally, the existing techniques can satisfy some of the requirements, but not all of them. This paper presents four practical considerations in fringe projection profilometry (FPP) based 3D shape measurements, along with simple but robust solutions, including gamma correction of digital projection, arbitrary setup of system components, phase unwrapping with multi-frequency fringes, and system calibration with a least-squares inverse approach. The validity and practicability of the FPPbased 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. The presented technique is capable of satisfying the various critical demands in enormous scientific and engineering applications.

As technologies evolve, there have been high demands for the three-dimensional (3D) shape measurement techniques to posses the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects as well as measuring complex shapes. Generally, the existing techniques can satisfy some of the requirements, but not all of them. This paper presents four practical considerations in fringe projection profilometry (FPP) based 3D shape measurements, along with simple but robust solutions, including gamma correction of digital projection, arbitrary setup of system components, phase unwrapping with multi-frequency fringes, and system calibration with a least-squares inverse approach. The validity and practicability of the FPPbased 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. The presented technique is capable of satisfying the various critical demands in enormous scientific and engineering applications.

We have applied the phase unwrapping technique to resolve the phase ambiguity problem arising from complex expressions of scattering parameters, for reflection-only measurement configurations, since, at some instances, only one side of the sample under test is accessible for electromagnetic measurements. We considered two different measurement configurations for testing the applicability of the phase unwrapping technique as: 1) two identical samples with different lengths flushed by a short-circuit termination and 2) one sample shorted by a varying short-circuit termination. For each measurement configuration, the underlying expressions for the reflection scattering parameters are derived. For both cases, we evaluated the suitability of the phase unwrapping technique by considering a highly-dispersive medium (distilled water) as our test sample. We note that continuity of the real part of the complex wavelength is a key issue in the unwrapping technique for (one-port) reflection-only measurements.

The use of Mean-Field theory to unwrap principal phase patterns has been recently proposed. In this paper we generalize the Mean-Field approach to process phase patterns with arbitrary degree of undersampling. The phase unwrapping problem is formulated as that of finding the ground state of a locally constrained, finite size, spin-L Ising model under a non-uniform magnetic field. The optimization problem is solved by the Mean-Field Annealing technique. Synthetic experiments show the effectiveness of the proposed algorithm.

A new, to our knowledge, algorithm for the phase unwrapping ͑PU͒ problem that is based on stochastic relaxation is proposed and analyzed. Unlike regularization schemes previously proposed to handle this problem, our approach dispells the following two assumptions about the solution: a Gaussian model for noise and the magnitude of the true phase-field gradient's being less than everywhere. We formulate PU as a constrained optimization problem for the field of integer multiples of 2, which must be added to the wrapped phase gradient to recover the true phase gradient. By solving the optimization problem using simulated annealing with constraints, one can obtain a consistent solution under difficult conditions resulting from noise and undersampling. Results from synthetic test images are reported.

Use of mean-field annealing theory is proposed for solving the phase-unwrapping ͑PU͒ problem. PU is formulated as a constrained optimization problem for the field of integer corrections to be added to the wrapped gradient field. A deterministic algorithm is described to provide an approximation of the average of the correction field over the global minima of the cost function. The proposed algorithm can be applied for any choice of the cost function. Using a cost function based on second-order differences, we obtain results close to those from simulated annealing and spend less computational time.

Phase unwrapping, i.e. the retrieval of absolute phases from wrapped, noisy measures, is a tough problem because of the presence of rotational inconsistencies (residues), randomly generated by noise and undersampling on the principal phase gradient field. These inconsistencies prevent the recovery of the absolute phase field by direct integration of the wrapped gradients. In this paper we examine the relative merit of known global approaches and then we present evidence that our approach based on "stochastic annealing" can recover the true phase field also in noisy areas with severe undersampling, where other methods fail. Then, some experiments with local approaches are presented. A fast neural filter has been trained to eliminate close residue couples by joining them in a way which takes into account the local phase information. Performances are about 60-70% of the residues. Finally, other experiments have been aimed at designing an automated method for the determination of weight matrices to use in conjunction with local phase unwrapping algorithms. The method, tested with the minimum cost flow algorithm, gives good performances over both simulated and real data.

ABSTRACTPurposeTo develop a rapid and accurate MRI phase unwrapping technique for challenging phase topographies encountered at high magnetic fields, around metal implants or post-operative cavities, that is sufficiently fast to be applied to large group studies including Quantitative Susceptibility Mapping and functional MRI (with phase-based distortion correction).MethodsThe proposed path-following phase unwrapping algorithm, ROMEO, estimates the coherence of the signal both in space - using MRI magnitude and phase information - and over time, assuming approximately linear temporal phase evolution. This information is combined to form a quality map that guides the unwrapping along a three-dimensional path through the object using a computationally efficient minimum spanning tree algorithm. ROMEO was tested against the two most commonly used exact phase unwrapping methods: PRELUDE and BEST PATH in simulated topographies and at several field strengths: in 3 T and 7 T in vivo human h...

An image processing algorithm based on the Fourier Transform Profilometry (FTP) method for 3D reconstruction purposes is presented. This method uses a global and local analysis for the phase unwrapping stage and obtains better results than using a simple unwrapping algorithm in the normal FTP method. A sinusoidal fringe pattern of known spatial frequency is firstly projected on a reference frame and an image is acquired. Then, the object of the shape to know is placed in front of the reference frame, and the same fringe pattern is projected. Once again another image is acquired. The projected pattern is distorted according to the shape of the object. Later, the modified Fourier Transform Profilometry method is applied to the acquired images. The digitized images contains the (x,y) pixels of the object including the fringe pattern, and the phase difference between the acquired images contains the z (height or depth) information. The novelty in the proposed method comes in the part of the unwrapping algorithm at the moment of obtaining the depth information by using the above mentioned combined analysis.

The advent of interferometric synthetic aperature radar for geophysical studies has resulted in the need for accurate, efficient methods of two-dimensional phase unwrapping. Inference of the lost integral number of cycles in phase measurements is critical for three-pass surface deformation studies as well as topographic mapping and can result in an order of magnitude increase in sensitivity for two-pass deformation analysis. While phase unwrapping algorithms have proliferated over the past ten years, two main approaches are currently in use. Each is most useful only for certain restricted applications. All these algorithms begin with the measured gradient of the phase field, which is subsequently integrated to recover the unwrapped phases. The earliest approaches in interferometric applications incorporated residue identification and cuts to limit the possible integration paths, while a second class using least-squares techniques was developed in the early 1990's. We compare the approaches and find that the residue-cut algorithms are quite accurate but do not produce estimates in regions of moderate phase noise. The least-squares methods yield complete coverage but at the cost of distortion in the recovered phase field. A new synthesis approach, combining the cuts from the first class with a least-squares solution, offers greater spatial coverage with less distortion in many instances.

The application of the Permanent Scatterers (PS) Technique in multi-temporal data-sets, namely the identification and exploitation of sparse coherent targets, has shown that it is possible to estimate and remove interferometric phase components due to atmospheric effects and orbital fringes. So far, the application of the PS technique has been focused on the extraction of the motion field of the area of interest. However, it is also known that PS relative elevations can be estimated with sub-meter precision while smooth errors can be removed using a coarse resolution DEM or the data of the Shuttle Radar Topography Mission (SRTM). In this paper, we describe a new approach combining the PS Technique and standard interferometry to improve the quality of InSAR DEM's. ERS Tandem interferograms are exploited to increase the number of coherent pixels, while atmospheric effects are estimated and subtracted by means of the sparse PS grid. Prior information and PS elevation are used to reduce the probability of phase-unwrapping errors. Preliminary results are reported and the key-factors for its successful application (e.g. the number of Tandem acquisitions available, PS density) are discussed.

Structured light scanning works by projecting a supplement of controlled information on the scene: the captured signal is processed to provide a unique label (namely a code) for each observed point, and then proceed to geometrical triangulation. In phase shift profilometry sinusoidal patterns are projected so

We present a new phase unwrapping approach, which allows reconstruction of optically thick objects that are optically thin from at least one viewing angle, by considering the information stored in the object phase maps captured from consecutive angles. Our algorithm combines 1-D phase unwrapping in the angular dimension with conventional 2-D phase unwrapping, to achieve unwrapping of the object from the optically thick perspective. We thus obtain quantitative phase imaging of objects that were previously impossible to image in certain viewing angles. To demonstrate our approach, we present both numerical simulation and experimental results for quantitative phase imaging of biological cells.

Interferometric synthetic aperture radar (InSAR) has been widely used in remote sensing field, which can reflect actual topographic trend or possible surface deformation. Due to the orbit attitude influence, the flat-earth phase usually causes the interferogram dense and difficult to be used in further procedures. Before phase unwrapping, interferogram must be flattened to derive accurate topographic or deformation information. In this paper, analysis of performance of two methods of flat-earth removal is done. First method uses imaging geometry and second method uses precise orbital information. Further, 3-degree, 5-degree and 7-degree polynomials are fitted in the method using precise orbital information. Validation is done both visually and empirically using entropy as the evaluation index.

Twenty five superior candidate plus trees (CPTs) of Melia composita were selected through an intensive survey from Haryana, Punjab and Uttrakhand. The trees were selected on the basis of the different characters of economic importance viz. straightness, clear bole height, self-pruning ability, acute branch angle etc. Among all the plus trees the maximum tree height was reported in MC21 (17.9 m) whereas the minimum was reported in MC3 (6.5 m). Highly significant and positive correlation (r = 0.941) was obtained between the tree height and other morphological characters viz. clear bole height, girth at breast height, crown spread etc. with optimum amount of variability among different progenies for various characters. Further, the progeny performance also provided a huge amount of variability which may be helpful in identifying superior plus trees for further scientific improvements of this versatile tree species.

The use of an electro-optic crystal as phase-stepping device in a digital speckle interferometer is discussed. Phase stepping is introduced by varying the external voltage applied to the crystal. The crystal calibration procedure is outlined and errors are discussed. Experimental results compared to those obtained with a piezo-electrically driven mirror are presented.

A filtering algorithm is proposed for processing images generated by TV holography that contain phase jumps and a high noise level. This algorithm first performs phase unwrapping without removing the noise. After that, it removes the noise by use of a conventional low-pass filter. The new approach allows for using low-pass filters with narrow passbands, leading to a better signal-to-noise ratio in the desired signal. Simulation results are presented and discussed. The new algorithm has been applied successfully under real conditions in a holographic station.

Microwave imaging is recognized as an efficient diagnostic modality for no invasively visualizing dielectric contrasts in non metallic bodies. The usefulness of this modality results from the existing correlation between dielectric properties and quantities of practical relevance for industrial or biomedical applications. At the beginning of the 80's, Supélec developed a 2.45 GHz planar microwave camera, in the 90's the group developed algorithms for quantitative microwave imaging. The purpose of this study is to investigate the capability of these existing materials, or an extended version of it, in terms of quantitative imaging of high contrast inhomogeneous object for application of breast cancer detection.

This FY03 final report on Wet Etch Figuring involves a 2D thermal tool. Its purpose is to flatten (0.3 to 1 mm thickness) sheets of glass faster thus cheaper than conventional sub aperture tools. An array of resistors on a circuit board was used to heat acid over the glass Optical Path Difference (OPD) thick spots and at times this heating extended over the most of the glass aperture. Where the acid is heated on the glass it dissolves faster. A self-referencing interferometer measured the glass thickness, its design taking advantage of the parallel nature and thinness of these glass sheets. This measurement is used in close loop control of the heating patterns of the circuit board thus glass and acid. Only the glass and acid were to be moved to make the tool logistically simple to use in mass production. A set of 4-circuit board, covering 80x80-cm aperture was ordered, but only one 40x40-cm board was put together and tested for this report. The interferometer measurement of glass OPD was slower than needed on some glass profiles. Sometimes the interference fringes were too fine to resolve which would alias the sign of the glass thickness profile. This also caused the phase unwrapping code (FLYNN) to struggle thus run slowly at times taking hours, for a 10 inch square area. We did extensive work to improve the speed of this code. We tried many different phase unwrapping codes. Eventually running (FLYNN) on a farm of networked computers. Most of the work reported here is therefore limited to a 10-inch square aperture. Researched into fabricating a better interferometer lens from Plexiglas so to have less of the scattered light issues of Fresnel lens groves near field scattering patterns, this set the Nyquest limit. There was also a problem with the initial concept of wetting the 1737 glass on its bottom side with acid. The wetted 1737 glass developed an Achromatic AR coating, spoiling the reflection needed to see glass thickness interference fringes. In response to this dilemma, the acid wetting was moved to the top of the glass and the thermal diffusion layer under the glass replaced with rubber. The bottom side of the glass was then protected from etching with a vacuum grease layer on the glass, though many other ways were tried and need to be further developed. We switched our mission away from flattening the OPD of 1737 glass, as the Eyeglass project no longer wanted glass. We turned our attention to the NIF Borofloat Debris shields.

3 . The images were obtained at two different echo times (TE:s) using a dual echo, multi slice, spoiled, fast gradient echo pulse sequence. The first echo was obtained using TE1 = 2.3 ms with the water and fat signals out of phase, and the second using TE2 = 4.6 ms with the water and fat in phase. The repetition

In this work an efficient parallel implementation of the Chirp Scaling Algorithm (CSA) for Synthetic Aperture Radar (SAR) processing is presented. The architecture selected for the implementation is General Purpose Graphic Processing Unit (GPGPU), as it is well suited for scientific applications and real time implementation of algorithms. The analysis of a first implementation led to several improvements which resulted in an important final speedup. Details of the issues found are explained, and the performance improvement of their correction explicitly shown.

As technologies evolve, there have been high demands for the three-dimensional (3D) shape measurement techniques to posses the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects as well as measuring complex shapes. Generally, the existing techniques can satisfy some of the requirements, but not all of them. This paper presents four practical considerations in fringe projection profilometry (FPP) based 3D shape measurements, along with simple but robust solutions, including gamma correction of digital projection, arbitrary setup of system components, phase unwrapping with multi-frequency fringes, and system calibration with a least-squares inverse approach. The validity and practicability of the FPPbased 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. The presented technique is capable of satisfying the various critical demands in enormous scientific and engineering applications.

3 . The images were obtained at two different echo times (TE:s) using a dual echo, multi slice, spoiled, fast gradient echo pulse sequence. The first echo was obtained using TE1 = 2.3 ms with the water and fat signals out of phase, and the second using TE2 = 4.6 ms with the water and fat in phase. The repetition

Phase unwrapping is a procedure that allows 3D representation for surfaces when the data is acquired using interferometry based techniques. An interferogram contains phase difference values wrapped onto a fixed range of angles 0-360 degrees. In order to compute surface heights and generate an elevation model, the interferogram have to be unwrapped. This paper presents two 2D phase unwrapping algorithms: the zigzag approach and the least-squares method, and shows an implementation of them on a mobile device based on Symbian S60 platform. Also, filtering methods (i.e mean filter and vector filter) for preprocessing the interferogram are discussed and implemented. The implementation was tested using SAR (Synthetic Aperture Radar) interferograms. Based on the results, it can be concluded that mobile devices offer enough processing and graphical power and development options, in order to be a good choice to test some new signal processing algorithms.

Differential SAR interferometry (DInSAR) has proven to be a processing approach that is wellsuited to precisely identifying large-scale land deformation patterns. This is useful for many environmental monitoring applications, but the speckle noise and temporal decorrelation present in SAR images presents particular challenges in processing SAR images. This research focuses on the phase unwrapping problem, proposing two new approaches: Polynomial-Based Region-Growing Phase Unwrapping (PBRGPU), which expands upon the traditional region-growing approach to phase unwrapping; and Path-Based Least-Squares Phase Unwrapping (PBLSPU), which extends the leastsquares phase unwrapping models in a path-based framework. Both algorithms were tested using simulated data and interferograms generated from RADARSAT-2 data. Both approaches significantly reduced the root mean square error compared to the algorithms they build from, and achieved a similar level of performance to the commonly-used SNAPHU algorithm without the need for masking low coherence areas. iii Acknowledgements I would like to express appreciation to York University, NSERC, the Canadian Space Agency, and PCI Geomatics for providing financial support for this research. In addition, I would like to thank the Canadian Space Agency for providing the SAR data used in this research and PCI Geomatics for the use of their software and technical support services. I would like to thank my supervisor, Professor Jianguo Wang, and my supervisory committee member, Professor Baoxin Hu, for the support they have provided throughout this research and through all my studies. I would like to extend my thanks to Professor Mojgan Jadidi and Professor Juejiao Fu for having served on my M.A.Sc. defense committee. I would also like to thank my family for supporting and encouraging me through my studies.

A generic reconstruction algorithm is proposed, which is obtained by using a phase-shifting technique with an arbitrary number of phase shifts between intensity measurements. The generic algorithm entirely describes the structure of known phase-shifting algorithms and permits the construction of new ones with an arbitrary number of phase shifts.

The bulk air cooling and refrigeration system at Impala Platinum No. 16 Shaft is unique in that it was designed from the beginning for the long-term phased expansion necessary for the mine to satisfy its life-of-mine cooling requirements. Most of the civil structures were constructed upfront to be equipped when additional cooling capacity became necessary. The Phase A refrigeration system produced cooled downcast air at a flow rate of 666kg/s and temperature of 13.1°C. At the final phase with the introduction of ice thermal storage, the refrigeration system produced ultra-cold downcast air at a flow rate of 1000kg/s and a temperature of 4.4°C. The refrigeration output, after the final phase was commissioned in April 2022, is 42.0MWR. The integrated ice thermal storage system provides a range of opportunities in operating the refrigeration system; colder air temperatures sent to the shaft are enabled by using near-zerodegree water produced by melting ice to cool the air, flexible timing of cooling capacity by melting ice when needed, and refrigeration capacity amplification by storing excess cooling as ice at night which can be melted during the hottest part of the day. This paper describes the thermodynamic design, construction, and performance assessment of the new ice storage system at Impala 16 No. Shaft. Furthermore, this paper compares the design and performance of an R134a ice-making chiller to an equivalent ammonia machine as well as the decision matrix used to proceed with the installation of an R134a refrigeration machine at Impala No. 16 Shaft. Aspects of comparison include global warming potential, flammability, toxicity, corrosiveness of the refrigerant, full load efficiency, refrigerant supply costs, first cost basis as well as operating cost competitiveness of the refrigeration machines during their lifetime.

Colored fringe pattern approach has been proposed to overcome the limitations of the sequential phase-shifting fringe pattern profilometry for dynamic measurements. However, the use of colored fringe patterns leads to a mayor problem, the color cross-talk, which has been considered to be equivalent to reconstructing the phase map from the fringe patterns with unknown phase shifts. Therefore, phase error is introduced when conventional phase-shifting algorithms with fixed phase shift values are used to retrieve the phase. To overcome the cross-talk issue, we propose the use of a fringe pattern normalization algorithm to achieve the intensity modulation balance among the different color channels and then the use of an advanced iterative algorithm to retrieve the phase. Simulations and experimental results show that the proposed procedure can significantly reduce the influence of the color cross-talk.

As technologies evolve, there have been high demands for the three-dimensional (3D) shape measurement techniques to posses the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects as well as measuring complex shapes. Generally, the existing techniques can satisfy some of the requirements, but not all of them. This paper presents four practical considerations in fringe projection profilometry (FPP) based 3D shape measurements, along with simple but robust solutions, including gamma correction of digital projection, arbitrary setup of system components, phase unwrapping with multi-frequency fringes, and system calibration with a least-squares inverse approach. The validity and practicability of the FPPbased 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. The presented technique is capable of satisfying the various critical demands in enormous scientific and engineering applications.

Two-dimensional (2-D) fringe pattern analysis algorithms have been developed for several decades. They impel the frontier interferometric techniques such as synthetic aperture radar, interferometric synthetic aperture sonar, magnetic resonance imaging, diffraction tomography and optical measurement being widely used in more and more measurement applications. Due to the factors such as noises caused by acquisition mechanism and speckle effect, subsampling, and long discontinuities, the modern 2-D fringe analysis algorithms for phase denoising and unwrapping are complicated and usually time consuming. The windowed Fourier transform (WFT) based algorithms including the windowed Fourier filtering (WFF) and the windowed Fourier ridges (WFR) are among the effective fringe analysis algorithms. Along with quality-guided phase unwrapping algorithm, the WFT based algorithms can successfully reconstruct the phase by removing the noises and effectively identify the long discontinuities in the fringe patterns. Despite the merits of the WFR algorithm, a small phase bias is produced by the WFR algorithm for phase estimate. In this dissertation, phase compensation methods are proposed for the WFR algorithm, which is denoted as the WFRC algorithm. Then the theoretical MSEs for local frequency and phase estimates by the WFRC algorithm are derived in the first order perturbation perspective and compared with the corresponding Cramér-Rao bounds to show the efficiency of the proposed algorithm. Another drawback of the WFT-based algorithms is their long computation time, prohibiting them from real-School of Computer Engineering XVI | P a g e time applications. However, the WFT-based algorithms are highly parallelizable, which implies the potential of being efficiently accelerated by the modern parallel computing hardware. Thus in this thesis we explore using parallel hardware to accelerate the WFTbased algorithms and propose a high-speed heterogeneous system based on multicore CPU and graphics processing units for phase denoising and unwrapping using the two algorithms. The system can be easily integrated into a fringe analysis system.

As technologies evolve, there have been high demands for the three-dimensional (3D) shape measurement techniques to posses the following combined technical features: high accuracy, fast speed, easy implementation, capability of measuring multiple objects as well as measuring complex shapes. Generally, the existing techniques can satisfy some of the requirements, but not all of them. This paper presents four practical considerations in fringe projection profilometry (FPP) based 3D shape measurements, along with simple but robust solutions, including gamma correction of digital projection, arbitrary setup of system components, phase unwrapping with multi-frequency fringes, and system calibration with a least-squares inverse approach. The validity and practicability of the FPPbased 3D shape measurement technique using the four corresponding technical approaches have been verified by experiments. The presented technique is capable of satisfying the various critical demands in enormous scientific and engineering applications.

Rather than regarding the phase singularities as obstacles or nuisances in phase unwrapping, we explore new possibilities of making use of the phase singularities in optical metrology. Instead of intensity correlation techniques used in conventional speckle metrology, we propose a new technique of displacement measurement that makes use of the density of phase singularities in the complex analytic signal of the speckle pattern, which is generated by Hilbert filtering. Experimental results and theoretical analysis are presented that demonstrate the validity of the proposed technique.

Indexing term: Speech synthesis A new phase unwrapping algorithm for phase interpolation in voiced speech synthesis is proposed. Computer simulation results show that the new algorithm outperforms the original phase unwrapping method proposed by Griffin and Lim in terms of the number of instances of wrongly unwrapped phase.

This study proposed an improved center array-sequencing phase unwrapping (ICASPU) algorithm. 2% agarose phantom dopped with 0.6mM/l MnCl2 was used with clinical 1.5T MRI system and commercial knee coil. Obtained k-space data(raw-data) was transmitted to PC and reconstructed into phase image with MATLAB software. Previous center array-sequence phase unwrapping algorithm wascompared with proposed ICASP algorithm using second order regression analysis. As a result, we found that the amount of error on proposed ICASPU method is less about 5 times than that of previous CASPU method. In this study, we exploit improved Center array-sequence phase unwrapping algorithm and expect to apply to images including phase informations.

Quantitative assessment of tissue microstructure is important in studying human brain diseases and disorders. Ultra-high field magnetic resonance imaging (MRI) data obtained using a multi-echo gradient echo sequence have been shown to contain information on myelin, axonal, and extracellular compartments in tissue. Quantitative assessment of water fraction, relaxation time (T 2 *), and frequency shift using multicompartment models has been shown to be useful in studying white matter properties via specific tissue parameters. It remains unclear how tissue parameters vary with model selection based on 7T multiple echo time gradient-recalled echo (GRE) MRI data. We applied existing signal compartment models to the corpus callosum and investigated whether a three-compartment model can be reduced to two compartments and still resolve white matter parameters [i.e., myelin water fraction (MWF) and g-ratio]. We show that MWF should be computed using a three-compartment model in the corpus callosum, and the g-ratios obtained using three compartment models are consistent with previous reports. We provide results for other parameters, such as signal compartment frequency shifts.

Using computer simulations we examine the ranges of validity of the first Born and first Rytov approximations employed in diffraction tomography. To that end we apply the filtered backpropagation ͑FBP͒ algorithm in conjunction with the first Born approximation and the hybrid FBP algorithm in conjunction with the first Rytov approximation. We find that the range of validity of the first Born approximation is approximately 3 times smaller than that of the first Rytov approximation and that the range of validity of each approximation can be expressed in terms of the product of the refractive-index difference between the object and the background and the size of the object. Also, we establish precise criteria for the validity of diffraction tomography within each of these two approximations. For the first Rytov approximation the validity of the hybrid FBP algorithm is found to be limited by phase-unwrapping problems.

Using computer simulations we examine the ranges of validity of the first Born and first Rytov approximations employed in diffraction tomography. To that end we apply the filtered backpropagation ͑FBP͒ algorithm in conjunction with the first Born approximation and the hybrid FBP algorithm in conjunction with the first Rytov approximation. We find that the range of validity of the first Born approximation is approximately 3 times smaller than that of the first Rytov approximation and that the range of validity of each approximation can be expressed in terms of the product of the refractive-index difference between the object and the background and the size of the object. Also, we establish precise criteria for the validity of diffraction tomography within each of these two approximations. For the first Rytov approximation the validity of the hybrid FBP algorithm is found to be limited by phase-unwrapping problems.

Diffraction tomography reconstructions of objects from limited transmitted field data sets are discussed together with theoretical analyses and results of numerical experiments. It is shown that limited data sets, representing only a small part of the complete data sets, can be used for reconstructions in diffraction tomography with satisfactory accuracy. We also find that, in diffraction tomography based on the hybrid filtered backpropagation and the first-Rytov approximation, the use of limited data sets can provide a larger range of validity than the use of complete data sets, the reason being that limited data sets pose less-severe phase-unwrapping problems.

The instantaneous frequency is the barycenter of the instantaneous spectrum, and it may provide high-resolution information about the anelastic absorption. This seismic attribute is normally computed by the derivative of the instantaneous phase, but is spiky and unstable when the phase unwrapping process fails. We present a new direct approach that does not require any phase unwrapping, so reducing the related numerical instabilities and user-defined parameters. We validate this method with synthetic seismic data and a case history from Antarctica offshore. This new tool can improve the detection of fractures and fluids for reservoir characterization and monitoring.