Radial Hilbert transform with the spatially variable half-wave plate

Optical imaging usually suffers from aberrations that are induced by various structures when imaging biological samples. Usually aberrations degrade the imaging system performances by broadening the point spread function (PSF). Unexpectedly we show that in spatially incoherent interferometry like full-filed optical coherence tomography (FFOCT), the system PSF width is almost insensitive to aberrations. Instead of considering the PSF of a classical imaging system such as a microscope, we specifically pay attention to the system PSF of interferometric imaging systems for which an undistorted wavefront from a reference beam interferes with the distorted wavefront of the object beam. By comparing the cases of scanning OCT with spatially coherent illumination, wide-field OCT with spatially coherent illumination and FFOCT with spatially incoherent illumination, we found that in FFOCT with spatially incoherent illumination the system PSF width is almost independent of the aberrations and only its amplitude varies. This is demonstrated by theoretical analysis as well as numerical calculations for different aberrations, and confirmed by experiments with a FFOCT system. It is the first time to the best of our knowledge that such specific merit of incoherent illumination in FFOCT has been demonstrated. Based on this, the signal level is used as metric in our adaptive optics FFOCT system for retinal imaging. Only the main aberrations (defocus and astigmatism) that are dominating in eye are corrected to improve the signal to noise ratio and the high order aberrations are skipped. This would increase the correction speed thus reducing the imaging time.

Tailored optical fields whose polarization and phase are coupled in a non-separable fashion usually referred to as vector beams, have become ubiquitous in a wide variety of research fields. These complex light fields can be generated through a combination of physical optical elements such as spiral phase plates, q-plates and quarter wave plates. However, such "hard-coded" optical devices present some restrictions such as their inability to generate multiple structured light fields simultaneously. This has stimulated a wide range of generation techniques based on computer controlled devices such as spatial light modulators (SLMs). SLMs provide on demand, fast and flexible holographic means to create multiple tailored optical fields simultaneously (multiplexing). Here we present a novel technique that utilizes the superposition principle, enabling digital generation of many vector beams on a single hologram. Unlike other techniques, this method is purely digital and allows the generation of a wide variety of vector beams by simply changing the encoded digital hologram. As a proof of concept we demonstrate simultaneous generation of vector beams with different polarization states on the High Order Poincaré Sphere (HOPS). This novel technique will be of impact in the context of quantum and classical communication, optical micro-manipulation and super resolution microscopy.

The focusing accuracy in reective optical systems, usually expressed in terms of the Point Spread Function (PSF) is chiey determined by two factors: the deviation of the mirror shape from the nominal design and the surface finishing. While the effects of the former are usually well described by the geometrical optics, the latter is diffractive/interferential in nature and determined by a distribution of defects that cover several decades in the lateral scale (from a few millimeters to a few microns). Clearly, reducing the level of scattered light is crucial to improve the focusing of the collected radiation, particularly for astronomical telescopes that aim to detect faint light signals from our Universe. Telescopes are typically arranged in multiple reections configuration and the behavior of the multiply-scattered radiation becomes difficult to predict and control. Also it is difficult to disentangle the effect of surface scattering from the PSF degradation caused by the shape deformation of the optical elements. This paper presents a simple and unifying method for evaluating the contribution of optical surfaces defects to the two-dimensional PSF of a multi-reections system, regardless of the classification of a spectral range as "geometry" or "roughness". This method, entirely based on Huygens-Fresnel principle in the far-field approximation, was already applied in grazing-incidence X-ray mirrors and experimentally validated for a single reection system, accounting for the real surface topography of the optics. In this work we show the extension of this formalism to a double reection system and introducing real microroughness data. The formalism is applied to a MAGIC-I panel mirror that was fully characterized, allowing us to predict the PSF and the validation with real measurements of the double reection ASTRI telescope, a prototype of CTA-SST telescope.

An accurate measurement of the point spread function (PSF) for the extended depth of field (EDoF) cell phone camera with CMOS sensor is very important for image processing and image restoration. But due to the coarse sampling of the PSF by CMOS sensor, the overall system including the imaging subsystem and sampling subsystem is a shift-variant system with respect to the sample-scene phase parameter within sub-pixel range. In this paper, we present the sub-pixel digital algorithm to estimate the overall camera PSF based on the measurement of a high resolution PSF of the imaging lens. The sub-pixel digital algorithm averages the shifted high resolution PSFs of the lens over one active pixel area with the assumption of uniform random distribution of point source location within one active pixel area. Then the averaged high resolution PSF is down sampled onto Bayer plane to obtain the shift-invariant overall system PSF. We applied this shift-invariant PSF for image restoration of blurred images captured with an extended depth of field camera. The processed images are compared with originally captured images. Improvement of image quality is seen.

A Kelvin probe force microscopy (KPFM) image is sometimes difficult to interpret because it is a blurred representation of the true surface potential (SP) distribution of the materials under test. The reason for the blurring is that KPFM relies on the detection of electrostatic force, which is a long-range force compared to other surface forces. Usually, KPFM imaging model is described as the convolution of the true SP distribution of the sample with an intrinsic point spread function (PSF) of the measurement system. To restore the true SP signals from the blurred ones, the intrinsic PSF of the system is needed. In this work, we present a way to experimentally calibrate the PSF of the KPFM system. Taking the actual probe shape and experimental parameters into consideration, this calibration method leads to a more accurate PSF than the ones obtained from simulations. Moreover, a nonlinear reconstruction algorithm based on total variation (TV) regularization is applied to KPFM measurement to reverse the blurring caused by PSF during KPFM imaging process; as a result, noises are reduced and the fidelity of SP signals is improved.

During spatial remote sensing imaging procedure, combined degradation factors conduce to Gaussian image blurring. The precondition of restoring the degraded image is to estimate point spread function (PSF) of the imaging system as precisely as possible. Because the depredating processes are quite complex, the transfer function of the degraded system is often completely or partly unknown, which makes it quite difficult to identify the precise PSF. Considering the similarity between the quantum process and imaging process in the probability and statistics fields, a novel algorithm is proposed by using multilayer feed-forward back-propagation quantum neural network (QBPNN) to estimate PSF of the Gaussian degraded imaging system. Different from the classical artificial neural network (ANN), 2 adjustable parameters of weight connection coefficient and phase coefficient are introduced in its quantum neurons used in learning stage. By establishing different training sets, this estimation method can overcome the limitation in the dependence on initial values and large amount of computation. Test results show that this method can achieve higher precision, faster convergence and stronger generalization ability comparing with the traditional PSF estimation results.

Linear systems theory combined with Fourier transform techniques provides a powerful means to characterize electro-optical systems. We present the description and analysis of a particular scanning IR system whose characteristic Point Spread Function (PSF) is space-variant. This peculiarity of the PSF arises from the scanning geometry; the IR sensor spins as it drops toward the ground. A system impulse response, or Point Spread Function (PSF), is derived from effects including diffraction, motion blur, chromatic aberration and detector area. Other effects may be taken into account by cascading PSF's. Once the system PSF is determined, the output of the system is the convolution of the input irradiance with the PSF. Alternatively, the output is the familiar inverse Fourier transform of the product of transforms. However, the space-variance of the PSF makes the latter option impossible and the former difficult. The convolution progresses in a spiral manner, with the PSF width changing continuously. The method is tested using Fortran programs and a digital image processor. The digital images produced show expected results, revealing a marked increase in resolution as the sensor nears the ground.

Traditional imaging methods often separate the two technical procedures of optical system design and imaging processing, where the lens subsystem is firstly optimized using an optical measure of performance and the image processing subsystem is optimized subsequently, resulting in the failures in efficient cooperation between the optical and digital elements and missing of the optimal solution. Therefore, an innovative approach is presented to combine optical system design and image processing together to obtain high resolution images from the final system. In this paper, the theoretical foundations for this computing optical design method is analyzed by considering the modulation transfer functions (MTF) and aberrations. According to the wave aberration theory and Zernicke formula, the relationship between the point spread function (PSF) and a certain monochromatic aberration can be obtained. Fourier transformation is taken on these PSFs and then modulus value calculation is adopted to get the MTFs. By analyzing the MTFs of idealized optical systems, the influence of each monochromatic aberration on the image can be concluded. Note that some of the MTFs values have zeros and others do not. Considering that imaging processing algorithm cannot recover the information lost at such zeros, thus some aberrations can be easily corrected through imaging processing algorithm, while others not. Aimed at a single monochromatic aberrations respectively, the optical system with single lens is designed in ZEMAX software and the relevant imaging processing algorithm is adopted. From the comparison among the final images, the theoretical foundations for the computing optical design method considering the MTF and aberrations can be verified. The compensation for aberration of imaging processing can be utilized to reduce the pressure of optical system design. The computing optical design method provides a new approach to simplify structure and reduce cost, as well as to gain high resolution images simultaneously, especially for designing certain complex optical system, which shows promising perspective of industrial application.

A computational camera uses a combination of optics and processing to produce images that cannot be captured with traditional cameras. Over the last decade, a range of computational cameras have been proposed, which use various optics designs to encode and use computation to decode useful visual information. What is often missing, however, is the quantitative analysis of the relation between camera design and the captured visual information, and little systematic work has been done to evaluate and optimize these computational camera designs. While the optics of computational cameras may be quite complicated, many of them can be effectively characterized by their point spread functions (PSFs): the intensity distribution on an image sensor as a response to a point light source in a scene. This thesis explores the techniques to characterize, evaluate and optimize computational cameras via PSF engineering for computer vision tasks, including image recovery, 3D reconstruction, and image refocusing. I first explore PSF engineering techniques to recover image details from blurry images. Image blur is a problem commonly seen in photography due to a number of reasons, including defocus, lens aberration, diffraction, atmospheric turbulence, object motion, and etc. Image blur is often formulated as a convolution of the latent blur-free image and a PSF, and deconvolution (or deblurring) techniques have to be used to recover image details from a blurry region. Here, I propose a comprehensive framework of PSF evaluation for the purpose of image deblurring, in which the effects of image noise, deblurring algorithm, and the structure of natural images are all taken into account. In defocus blur, the shape of defocus PSF is largely determined by the aperture pattern of the camera lens. By using an evaluation criterion derived from the comprehensive framework, I optimize the aperture pattern to preserve many more image details when defocus occurs. Both through simulations and experiments, I demonstrate the significant improvement gained by using optimized aperture patterns. While defocus blur causes a loss in image detail, it also encodes depth information of scenes in images. A typical depth from defocus (DFD) technique computes depth from two or more images that are captured with circular aperture lenses of different focus settings. Circular apertures produce circular defocus PSFs. In this thesis, I show that the use of a circular aperture severely restricts the accuracy of DFD, and propose a comprehensive framework of PSF evaluation for depth recovery. With this framework, we can derive a criterion for evaluating a pair of apertures with respect to the precision of depth recovery. This criterion is optimized using a genetic algorithm and gradient descent search to arrive at a pair of high resolution apertures. The two coded apertures are found to complement each other in the scene frequencies they preserve. With this property it becomes possible to not only recover depth with greater fidelity but also to obtain a high quality all-focused image from the two defocused images. While depth recovery can significantly benefit from optimized aperture patterns, its overall performance is rigidly limited by the lens aperture's physical size. To transcend this limitation, I propose a novel depth recovery technique using an optical diffuser - referred to as depth from diffusion (DFDiff). I show that DFDiff is analogous to conventional DFD, in which the scatter angle of the diffuser determines the system's effective aperture. High precision depth estimation can be achieved by choosing a proper diffuser and no longer requires the large lenses that DFD requires. Even a consumer camera with a low-end small lens can be used to do high-precision depth estimation when coupled with an optical diffuser. In my detailed analysis of the image formation properties of a DFDiff system, I show a number of examples demonstrating greater precision in depth estimation when using DFDiff. While the finite depth of field (DOF) of a lens camera leads to defocus blur, it also produces artistic visual experience. Many of today's displays are interactive in nature, which opens up a possibility for new kind of visual representations. Users could, for example, interactively refocus images to different depths, so that they can experience the artistic narrow DOF images while simultaneously making available the image detail for the entire image. To enable image refocusing, one typical approach is to capture the entire light field. But this method has the drawback of a significant sacrifice in spatial resolution due to the dimensionality gap: the captured information (light field) is 4D, while the required information (focal stack) is only 3D.In this thesis, I present an imaging system that directly captures focal stacks by a sweeping focal plane. First, I describe how to synchronize focus sweeping with image capturing so that the summed DOF of a focal stack efficiently covers the entire depth range. Then, I take a customized algorithm to enable a seamless refocusing experience, even in textureless regions or with moving objects. Prototype cameras are presented to capture real space-time focal stacks. There is also an interactive refocusing viewer available online at www.focalsweep.com.

In a previous work we have demonstrated a novel numerical model for the point spread function (PSF) of an optical system that can efficiently model both experimental measurements and lens design simulations of the PSF. The novelty lies in the portability and the parameterization of this model, which allows for completely new ways to validate optical systems, which is especially interesting for mass production optics like in the automotive industry, but also for ophtalmology. The numerical basis for this model is a non-linear regression of the PSF with an artificial neural network (ANN). In this work we examine two important aspects of this model: the spatial resolution and the accuracy of the model. Measurement and simulation of a PSF can have a much higher resolution then the typical pixel size used in current camera sensors, especially those for the automotive industry. We discuss the influence this has on on the topology of the ANN and the final application where the modeled PSF is actually used. Another important influence on the accuracy of the trained ANN is the error metric which is used during training. The PSF is a distinctly non-linear function, which varies strongly over field and defocus, but nonetheless exhibits strong symmetries and spatial relations. Therefore we examine different distance and similarity measures and discuss its influence on the modeling performance of the ANN.

The point spread function (PSF) of an optical system is a function of both the amplitude and phase of the electric field in the exit pupil. Interferometric testing of optics yields information regarding variations in phase, but not in amplitude. Typically the PSF, encircled energy function (EEF) and Strehl ratio are calculated from the phase information alone, implicitly assuming that the amplitude is uniform across the pupil. For incoherent, unvignetted imaging systems this assumption is generally valid. Coherent optical systems are a different case due to the Gaussian irradiance profile of lasers. The optics in such systems are non-uniformly illuminated, the amplitude function in the pupil being characterized by a Gaussian or truncated Gaussian profile. This effect of apodization is examined for aberrated and unaberrated beams. Analytic forms and numerical results are presented, as are measurements of laser diode based optical systems. In general, little effect from apodization is seen on parameters which are typically of interest (e.g., Strehl, FWHM of PSF, 80% encircled energy) except for cases of pupil irradiance decrease of greater than half from center to edge.

3-D optical fluorescent microscopy becomes now an efficient tool for volume investigation of living biological samples. Developments in instrumentation have permit to beat off the conventional Abbe limit, in any case the recorded image can be described by the convolution equation between the original object and the Point Spread Function (PSF) of the acquisition system. If the goal is 3-D quantitative analysis, whether you improve the instrument capabilities, or (and) you restore the data. These last is until now the main task in our laboratory. Based on the knowledge of the optical Transfer Function of the microscope, deconvolution algorithms were adapted to automatic determine the regularisation threshold in order to give less subjective and more reproducible results. The PSF represents the properties of the image acquisition system; we have proposed the use of statistical tools and Zernike moments to describe a 3-D system PSF and to quantify the variation of the PSF. This first step toward standardization is helpful to define an acquisition protocol optimizing exploitation of the microscope depending on the studied biological sample. We have pointed out that automating the choice of the regularization level; if it facilitates the use, it also greatly improves the reliability of the measurements. Furthermore, to increase the quality and the repeatability of quantitative measurements a pre-filtering of images improves the stability of deconvolution process. In the same way, the PSF pre-filtering stabilizes the deconvolution process. We have shown that Zernike polynomials can be used to reconstruct experimental PSF, preserving system characteristics and removing the noise contained in the PSF. Fluorescent microscopes suffer from limitations; photobleaching and phototoxicity effects, or influence of the sample optical properties to 3-D observation. Amplitude and phase of the object can be reached with optical tomography based on a combination of microholography with a tomographic illumination. So indices cartography of the specimen can be obtained, and combined with fluorescence information it will open new possibilities in 3-D optical microscopy.

The distribution of the ray density of the spot diagram formed in the image plane is called the geometric point spread function (PSF). It plays an important role in the image formation theory, since it describes the impulse response of an optical system to a source point. However, the literature contains very few techniques for deriving the PSF of optical systems. Accordingly, this study presents a method based on an irradiance model for computing the geometric PSF of an optical system by considering the energy conservation along a single light ray. It is shown that the proposed method obtains a reliable and accurate estimate of the PSF and enables the computation of the centroid and root-mean-square radius of the focus spot on the image plane. In addition, compared to existing ray-counting methods presented in the literature, in which the quality of the PSF solution depends on the number of rays traced and the grid size used to mesh the image plane, the proposed irradiance-based method requires just one tracing operation. Overall, the results presented in this study confirm that the proposed method provides an ideal solution for calculating the merit function and modulation transfer function of an optical system.

Self-referenced speckle holography (SRSH) is a post-detection turbulence-compensation technique for obtaining diffraction limited imagery from ground based telescopes degraded by atmospheric turbulence. In SRSH image plane information is used together with wave front distortion information to reconstruct an estimate of the object spectrum. The wave front distortion information is obtained from a wave front sensor (WFS) in the pupil plane of the telescope. The data from the WFS is used in a post-processing environment to estimate the point spread function (PSF) of the combined telescope and atmosphere. The PSF is then used to obtain an estimate of the object via deconvolution. We present the results of a detailed performance analysis of SRSH. Performance is quantified in terms of a 'system transfer function' and a 'system point spread function'. The results show how the performance of the technique is dependent on the WFS sampling intervals and shot noise. The results also indicate how the technique, for a given set of design parameters, responds to changing seeing conditions. For subaperture dimensions on the order of a Fried coherence cell size ro and adequate light levels, SRSH boosts the high spatial frequencies (those near the diffraction limit of the telescope) to nearly 0.6. The 'system strehl ratio' approaches 0.6.

In an incoherent imaging system using the liquid crystal active lens, a point spread function can be controlled electrically. Since the liquid crystal active lens is a lens attached to an electrically addressable liquid crystal phase modulator, and the point spread function is an intensity distribution of the Fourier transform of the modulator's phase distribution. In this study, an electro-optical learning system is employed to produce a desirable point spread function by optimizing the phase distribution. Th phase distribution is controlled by a computer and the point spread function of the optical system is captured and fed into the computer. The point spread function is evaluated by the computer to optimize the phase distribution. The optimization method is based on the simulated annealing. The experimentally obtained point spread functions are closer to the desired ones compared to those generated only using a computer, becuase the electro-optical system learns the characteristics of the optical system during the optimization process. The performance of the electro-optical system is discussed.