Analysis of partially occluded objects encoded in digital holograms using the Wigner distribution function

In this thesis, we describe new approaches and methods for reconstructing complex-valued wave fields from digital holograms. We focus on Fresnel holograms recorded in an off-axis geometry, for which operational real-time acquisition setups readily exist. The three main research directions presented are the following. First, we derive the necessary tools to port methods and concepts of wavelet-based approaches to the field of digital holography. This is motivated by the flexibility, the robustness, and the unifying view that such multiresolution procedures have brought to many applications in image processing. In particular, we put emphasis on space-frequency processing and sparse signal representations. Second, we propose to decouple the demodulation from the propagation problem, which are both inherent to digital Fresnel holography. To this end, we derive a method for retrieving the amplitude and phase of the object wave through a local analysis of the hologram's interference fringes. Third, since digital holography reconstruction algorithms involve a number of parametric models, we propose automatic adjustment methods of the corresponding parameters. We start by investigating the Fresnel transform, which plays a central role in both the modeling of the acquisition procedure and the reconstruction of complex wave fields. The study of the properties that are central to wavelet and multiresolution analysis leads us to derive Fresnelets, a new family of wavelet-like bases. Fresnelets permit the analysis of holograms with a good localization in space and frequency, in a way similar to wavelets for images. Since the relevant information in a Fresnel off-axis hologram may be separated both in space and frequency, we propose an approach for selectively retrieving the information in the Fresnelet domain. We show that in certain situations, this approach is superior to others that exclusively rely on the separation in space or frequency. We then derive a least-squares method for the estimation of the object wave's amplitude and phase. The approach, which is reminiscent of phase-shifting techniques, is sufficiently general to be applied in a wide variety of situations, including those dictated by the use of microscopy objectives. Since it is difficult to determine the reconstruction distance manually, we propose an automatic procedure. We take advantage of our separate treatment of the phase retrieval and propagation problems to come up with an algorithm that maximizes a sharpness metric related to the sparsity of the signal's expansion in distance-dependent Fresnelet bases. Based on a simulation study, we suggest a number of guidelines for deciding which algorithm to apply to a given problem. We compare existing and the newly proposed solutions in a wide variety of situations. Our final conclusion is that the proposed methods result in flexible algorithms that are competitive with preexisting ones and superior to them in many cases. Overall, they may be applied in a wide range of experimental situations at a low computational cost.

We propose a task-specific digital holographic capture system for three-dimensional scenes, which can reduce the amount of data sent from the camera system to the receiver and can effectively reconstruct partially occluded objects. The system requires knowledge of the object of interest, but it does not require a priori knowledge of either the occlusion or the distance the object is from the camera. Subwindows of the camera-plane Fresnel field are digitally propagated to reveal different perspectives of the scene, and these are combined to overcome the unknown foreground occlusions. The nature of the occlusions and the effect of subwindows are analyzed thoroughly by using the Wigner distribution function. We demonstrate that a careful combination of reconstructions from subwindows can reveal features that are not apparent in a reconstruction from the whole hologram. We provide results by using optically captured digital holograms of real-world objects and simulated occlusions.

Three-dimensional object recognition using joint fractional Fourier transform correlators with the help of digital Fresnel holography

Digital in-line holography is commonly used to reconstruct 3D images from 2D holograms for microscopic objects. One of the technical challenges that arise in the signal processing stage is removing the twin image that is caused by the phase-conjugate wavefront from the recorded holograms. Twin image removal is typically formulated as a non-linear inverse problem due to the irreversible scattering process when generating the hologram. Recently, end-to-end deep learning-based methods have been utilized to reconstruct the object wavefront (as a surrogate for the 3D structure of the object) directly from a single-shot in-line digital hologram. However, massive data pairs are required to train deep learning models for acceptable reconstruction precision. In contrast to typical image processing problems, well-curated datasets for in-line digital holography does not exist. Also, the trained model highly influenced by the morphological properties of the object and hence can vary for different applications. Therefore, data collection can be prohibitively cumbersome in practice as a major hindrance to using deep learning for digital holography. In this paper, we proposed a novel implementation of autoencoder-based deep learning architecture for single-shot hologram reconstruction solely based on the current sample without the need for massive datasets to train the model. The simulations results demonstrate the superior performance of the proposed method compared to the state of the art single-shot compressive digital in-line hologram reconstruction method.

Phase space optics allows the four-dimensional simultaneous visualization of both space and spatial frequency optical information. The Wigner distribution function (WDF) is a common characterization of the phase space. Compared with the two-dimensional complex amplitude coherent optical field, the WDF can characterize optical field with arbitrary coherent states due to its higher dimensions. It is especially advantageous for the representation of partially coherent optical fields. The WDF is real and may have negative values, which are the result of phase-space interference. In this paper, an improved phase-space retrieval method is demonstrated. First, capture three-dimensional intensity focal stack. Then, phase space tomography (PST) combined with a non-linear iterative projection algorithm is conducted to reconstruct the whole WDF. We further analyzed the effect of the microscopy imaging system, i.e., the illumination aperture and the aperture of objective lens effect.

Holography is truly the key to three dimensions in particle image velocimetry, i.e.?the measurement of all spatial components of the velocity vector - and this over a deep measuring field. Sophisticated instruments have been designed that successfully tackle practical problems such as the low scattering efficiency of particles, the inferior depth resolution or the aberrations and distortions in the reconstruction. Furthermore, efficient strategies are introduced to interrogate the holographic storage and process the huge amount of data towards a final flow field representation. Recently, phase-sensitive metrology, familiar in many fields of experimental mechanics, has been examined for use in particle velocimetry. Suitable methods are holographic and speckle interferometry or the optical processing of data for three-dimensional correlation. While in these techniques the power of optics is unrivalled, the practical advantage of video and digital techniques over photographic recording is obvious. The electronic version of speckle interferometry (ESPI/DSPI) is a well-established method used in laser metrology and has received further exploitation for applications in flow analysis recently. Finally, the state-of-the-art of digital particle holography is reviewed to allow estimates of its future in experimental flow analysis.

Digital holograms are capable of encoding three-dimensional (3D) information of a scene or object. Uniform quantization has been used in the past to compress digital holograms with some success. We have developed nonuniform quantization techniques that operate on the complex-valued hologram data for increased performance. Our digital holograms of 3D objects are captured using phase-shift digital interferometry. The quantized holographic pixels are coded using lossless techniques to achieve compression rates in excess of 40.

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] Airy beam is a self-sustained light beam that propagates in free space along a parabolic trajectory with a constant lateral acceleration. It is a diffraction-free beam with an electric field pattern described by the Airy function containing a main lobe and side lobes. Unlike Gaussian beam that expands after the focus, a practical finite-energy Airy beam keeps its pattern without spreading over a relatively long distance during its propagation. This unique feature makes the Airy beam a great potential of designing optical systems for three-dimensional (3D) imaging. In this work, we develop a new optical 3D imaging methodology using Airy beam. The proposed imaging method has an advantage of increased depth of field (DOF) that is important in biological optical imaging. In the first step, we perform a fundamental research on specific parameters that can affect the properties of the Airy beams. The finite-energy Airy beams are generated by optical Fourier transform using a lens with the incoming plane wave that is truncated by an aperture. Our simulation and experiments showed that by changing the aperture size and the focal length of the lens, the Airy beam pattern is modified. In addition, the DOF, the beam size and the main lobe energy are changed. As a result, Airy beam can be designed for specific imaging applications. Next, we consider the phase in the cubic phase pattern for the phase modulation to generate the Airy beam. This problem is related to the wavelength dependent phase modulation since the phase in cubic phase pattern is designed for a specific wavelength whereas a broadband light source is normally used in optical coherence tomography (OCT). A liquid crystal display (LCD) panel from a projector is used as the SLM. The cubic phase patterns with different gray values displayed on the computer screen provide different phase modulations in the SLM. The experimental and simulated results show that the phase modulation affects the beam shape of the Airy beam. If the maximum phase modulation is larger than 1.7pi, the Airy beam can keep its pattern and the unmodulated Gaussian beam can be neglected. Further, we design a 3D imaging system using the phase-space method based on the Wigner distribution function (WDF). We theoretically build the WDF for the Airy beam and show that the WDF of the Airy beam is shifted in the phase-space as the transverse position of incoming Airy beam changes, and the WDF is tilted as changing the axial position (along the propagation direction). A larger truncating factor can reduce the energy contribution in WDF for the side lobes and modify the shape of the main lobe close to a straight line. In this case, the WDF of the Airy beam is similar to that of the Gaussian beam. However, the DOF of the Airy beam is greatly improved. We perform experimental measurements of WDF using the basic idea of measuring both the space and spatial frequency information in a 4-f system. Our experimental results match the simulation and validate the 3D imaging using Airy beam in the phase space.

The concept of the phase space plays a key role in the analysis of oscillating signals. For a 1-D signal, the coordinates of the 2-D phase space are the observation time and the instant frequency. For measurements of propagating wave fields, the time and instant frequency are linked to the spatial location and wave normal, defining a ray. In this case, the phase space is also termed the ray space. Distributions in the ray space find important applications in the analysis of radio occultation (RO) data because they allow the separation of interfering rays in multipath zones. Examples of such distributions are the spectrogram, Wigner distribution function (WDF), and Kirkwood distribution function (KDF). In this study, we analyze the application of the fractional Fourier transform (FrFT) to the construction of distributions in the ray space. The FrFT implements the phase space rotation. We consider the KDF averaged over the rotation group and demonstrate that it equals the WDF convolved with a smoothing kernel. We give examples of processing simple test signals, for which we evaluate the FrFT, KDF, WDF, and smoothed WDF (SWDF). We analyze the advantages of the SWDF and show examples of its application to the analysis of real RO observations.

Based on the Collins integral formula and the Hermite–Gaussian expansion of a Lorentz function, an analytical expression for the Wigner distribution function (WDF) of a Lorentz–Gauss vortex beam passing through a paraxial ABCD optical system is derived. The WDF properties of a Lorentz–Gauss vortex beam propagating in free space are also demonstrated. During propagation, the WDF pattern shrinks in the direction of the spatial frequency variables and elongates in the direction of the transverse spatial coordinates. With given the transverse spatial coordinates or the spatial frequency variables, the evolvement of the WDF of a Lorentz–Gauss vortex beam during propagation is examined in detail. To show the WDF application, the second-order moments of the WDF for a Lorentz–Gauss vortex beam in free space are also presented. This research exhibits the propagation properties of a Lorentz–Gauss vortex beam from the phase space.

A gray-level intensity image can be employed as a host image for hiding a watermark image to protect information security. Past research works demonstrate that a gray-level hidden image can be embedded into the host image by a digital phase-only holography method. However, the fidelity of retrieved watermark image from the host image is not very satisfactory and the host image quality is degraded due to the insertion of external data bits, especially when observers focus on the saliency regions in the host image. To address this problem, we propose a steganography method based on digital holography and the saliency map of the host image. First, we calculate the hidden capacity (number of bits to be replaced) for each host image pixel based on the weighted sum of pixel intensity and saliency value. Next, a multilevel phase-only digital hologram of the watermark image will be calculated by the Gerchberg–Saxton method under the constraint of the hidden capacity of host image. Finally, we embedded the multilevel phase-only digital hologram into the host image by replacing a corresponding number of bits in each pixel. In this way, the host image can preserve good image fidelity for its saliency regions even if we hide a large amount of digital hologram data into the host image. The experimental results show that the quality of retrieved watermark image from the host image and the quality of saliency regions in the watermarked host image, in our proposed scheme, are superior to the state-of-the-art works reported.

On the similarities between the Wigner distribution function in classical and quantum optics

Compression is essential for efficient storage and transmission of three-dimensional (3D) digital holograms. The inherent speckle content in holographic data causes lossless compression techniques, such as Huffman and Burrows-Wheeler (BW), to perform poorly. Therefore, the combination of lossy quantisation followed by lossless compression is essential for effective compression of digital holograms. Our complex-valued digital holograms of 3D real-world objects were captured using phase-shift interferometry (PSI). Quantisation reduces the number of different real and imaginary values required to describe each hologram. Traditional data compression techniques can then be applied to the hologram to actually reduce its size. Since our data has a nonuniform distribution, the uniform quantisation technique does not perform optimally. We require nonuniform quantisation, since in a histogram representation our data is denser around the origin (low amplitudes), thus requiring more cluster centres, and sparser away from the origin (high amplitudes). By nonuniformly positioning the cluster centres to match the fact that there is a higher probability that the pixel will have a low amplitude value, the cluster centres can be used more efficiently. Nonuniform quantisation results in cluster centres that are adapted to the exact statistics of the input data. We analyse a number of iterative (k-means clustering, Kohonen competitive neural network, SOM, and annealed Hopfield neural network), and non-iterative (companding, histogram, and optimal) nonuniform quantisation techniques. We discuss the strengths and weaknesses of each technique and highlight important factors that must be considered when choosing between iterative and non-iterative nonuniform quantisation. We measure the degradation due to lossy quantisation in the reconstruction domain, using the normalised rms (NRMS) metric.

High-speed optical reconstruction of 3D-scenes can be achieved using digital holography with binary digital micromirror devices (DMD) or a ferroelectric spatial light modulator (fSLM). There are many algorithms for binarizing digital holograms. The most common are methods based on global and local thresholding and error diffusion techniques. In addition, hologram binarization is used in optical encryption, data compression, beam shaping, 3D-displays, nanofabrication, materials characterization, etc. This paper proposes an adaptive binarization method based on a combination of local threshold processing, hologram division into blocks, and error diffusion procedure (the LDE method). The method is applied for binarization of optically recorded and computer-generated digital holograms of flat objects and three-dimensional scenes. The quality of reconstructed images was compared with different methods of error diffusion and thresholding. Image reconstruction quality was up to 22% higher by various metrics than that one for standard binarization methods. The optical hologram reconstruction using DMD confirms the results of the numerical simulations.

A modified aperture-synthesis method is reported to improve the resolution of a reconstructed image in digital holography by a single hologram. A series of sub-holograms incoherently overlapped on a CCD are recorded as a single aperture-synthesized digital hologram. The angular division multiplexing is introduced to both the object and the reference path for holographic recording. The final super-resolved image is obtained by synthesizing the several reconstructed images. In the experiment, a two-dimensional transparent USAF resolution test target is used. The result demonstrates that the resolution of the reconstructed image is improved from 16.00 lp/mm to 22.64 lp/mm in the horizontal direction.