187,692 research outputs found

    Phase Retrieval with Background Information: Decreased References and Efficient Methods

    Full text link
    Fourier phase retrieval(PR) is a severely ill-posed inverse problem that arises in various applications. To guarantee a unique solution and relieve the dependence on the initialization, background information can be exploited as a structural priors. However, the requirement for the background information may be challenging when moving to the high-resolution imaging. At the same time, the previously proposed projected gradient descent(PGD) method also demands much background information. In this paper, we present an improved theoretical result about the demand for the background information, along with two Douglas Rachford(DR) based methods. Analytically, we demonstrate that the background required to ensure a unique solution can be decreased by nearly 1/21/2 for the 2-D signals compared to the 1-D signals. By generalizing the results into dd-dimension, we show that the length of the background information more than (2d+1dāˆ’1)(2^{\frac{d+1}{d}}-1) folds of the signal is sufficient to ensure the uniqueness. At the same time, we also analyze the stability and robustness of the model when measurements and background information are corrupted by the noise. Furthermore, two methods called Background Douglas-Rachford (BDR) and Convex Background Douglas-Rachford (CBDR) are proposed. BDR which is a kind of non-convex method is proven to have the local R-linear convergence rate under mild assumptions. Instead, CBDR method uses the techniques of convexification and can be proven to own a global convergence guarantee as long as the background information is sufficient. To support this, a new property called F-RIP is established. We test the performance of the proposed methods through simulations as well as real experimental measurements, and demonstrate that they achieve a higher recovery rate with less background information compared to the PGD method

    Broadband Phase Retrieval for Image-Based Wavefront Sensing

    Get PDF
    A focus-diverse phase-retrieval algorithm has been shown to perform adequately for the purpose of image-based wavefront sensing when (1) broadband light (typically spanning the visible spectrum) is used in forming the images by use of an optical system under test and (2) the assumption of monochromaticity is applied to the broadband image data. Heretofore, it had been assumed that in order to obtain adequate performance, it is necessary to use narrowband or monochromatic light. Some background information, including definitions of terms and a brief description of pertinent aspects of image-based phase retrieval, is prerequisite to a meaningful summary of the present development. Phase retrieval is a general term used in optics to denote estimation of optical imperfections or aberrations of an optical system under test. The term image-based wavefront sensing refers to a general class of algorithms that recover optical phase information, and phase-retrieval algorithms constitute a subset of this class. In phase retrieval, one utilizes the measured response of the optical system under test to produce a phase estimate. The optical response of the system is defined as the image of a point-source object, which could be a star or a laboratory point source. The phase-retrieval problem is characterized as image-based in the sense that a charge-coupled-device camera, preferably of scientific imaging quality, is used to collect image data where the optical system would normally form an image. In a variant of phase retrieval, denoted phase-diverse phase retrieval [which can include focus-diverse phase retrieval (in which various defocus planes are used)], an additional known aberration (or an equivalent diversity function) is superimposed as an aid in estimating unknown aberrations by use of an image-based wavefront-sensing algorithm. Image-based phase-retrieval differs from such other wavefront-sensing methods, such as interferometry, shearing interferometry, curvature wavefront sensing, and Shack-Hartmann sensing, all of which entail disadvantages in comparison with image-based methods. The main disadvantages of these non-image based methods are complexity of test equipment and the need for a wavefront reference

    Frames and Phase Retrieval

    Get PDF
    Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as X-ray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phase-retrievable frames as a way of measuring a frame\u27s redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable frames can be of any lengths between 2d - 1 and d(d + 1)=2, inclusive. The complex Hilbert space case remains open. In chapter 4, we investigate phase-retrievability by studying maximal phase-retrievable subspaces with respect to a given frame. These maximal PR-subspaces can have different dimensions. We are able to identify the ones with the largest dimension and this can be considered as a generalization of the characterization of real phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace then |supp(x)| ā‰„ k for every nonzero vector x 2 M. Moreover, if 1 ā‰¤ k \u3c [(d + 1)=2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x Ļµ M such that |supp(x)| = k|. Chapter 5 is devoted to investigating phase-retrievable operator-valued frames. We obtain some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vector-valued frames, fusion frames, and frames of Hermitian matrices. Finally, in Chapter 6, we consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering of the phase-retrieval problem. We prove that every irreducible projective representation of a finite abelian group admits a frame vector with the maximal span property. All such vectors can be explicitly characterized. These generalize some of the recent results about phase-retrieval with Gabor (or STFT) measurements

    The Impact of Digitisation Projects on the Work of Local Historians

    Get PDF
    Background: the methodology of historical research depends on a commitment to reporting and record keeping based on investigation of primary and secondary sources. This dependence on finding and assessing the reliability of information sources presents particular challenges to designers of information retrieval software intended to support historical research. Aim: this paper reports the results of a questionnaire distributed as the first phase of a research project intended to assess the information needs of historians working with original and/or digitised primary resources. We have been particularly interested in the work of ā€œlocalā€ historians working with archives of local newspapers and with image collections that are in the process of being digitised. The prim ary aim at this stage was to discover whether these historians have a preference between original or digitised resources and to understand the reasons behind any preferences. In the longer term we aim to apply what we have learned to the development of a model of information seeking behaviour that can be used to guide the design of information retrieval applications supporting local historical research. Method: an online survey was distributed to historians in 43 universities in the UK. Results: The results show that historians make an interesting distinction between their preference for working with original documents and the greater ā€œusefulnessā€ of digitised resources. Conclusions: Historical research presents a number of unique challenges. Further research is needed into the unique aspects of the local historianā€™s information seeking behaviour. An important focus for this research will be the design of archival information retrieval systems that address these unique requirements

    A non-linear optimal estimation inverse method for radio occultation measurements of temperature, humidity and surface pressure

    Get PDF
    An optimal estimation inverse method is presented which can be used to retrieve simultaneously vertical profiles of temperature and specific humidity, in addition to surface pressure, from satellite-to-satellite radio occultation observations of the Earth's atmosphere. The method is a non-linear, maximum {\it a posteriori} technique which can accommodate most aspects of the real radio occultation problem and is found to be stable and to converge rapidly in most cases. The optimal estimation inverse method has two distinct advantages over the analytic inverse method in that it accounts for some of the effects of horizontal gradients and is able to retrieve optimally temperature and humidity simultaneously from the observations. It is also able to account for observation noise and other sources of error. Combined, these advantages ensure a realistic retrieval of atmospheric quantities. A complete error analysis emerges naturally from the optimal estimation theory, allowing a full characterisation of the solution. Using this analysis a quality control scheme is implemented which allows anomalous retrieval conditions to be recognised and removed, thus preventing gross retrieval errors. The inverse method presented in this paper has been implemented for bending angle measurements derived from GPS/MET radio occultation observations of the Earth. Preliminary results from simulated data suggest that these observations have the potential to improve NWP model analyses significantly throughout their vertical range.Comment: 18 (jgr journal) pages, 7 figure
    • ā€¦
    corecore