X-ray crystallography remains the primary technique for discovering the arrangement of a protein’s atoms in space. If this arrangement is given by a vector of atom positions r = (r 1 ,r 2 ,…,r N ), then the resulting structure factor F(r)(S) in direction S is given by where F a ( i ) is the atomic scattering function of a(i) ∈ {“Carbon", “Nitrogen", “Hydrogen",…}, the atom type of the ith atom. F a ( i ) is the Fourier transform of ρ a (i), the spherical electron density for an atom of type a(i) placed at the origin.
{"title":"Adaptive Protein Structure Refinement Using Diffraction Data","authors":"Robert A. Grothe","doi":"10.1364/srs.1998.swa.2","DOIUrl":"https://doi.org/10.1364/srs.1998.swa.2","url":null,"abstract":"X-ray crystallography remains the primary technique for discovering the arrangement of a protein’s atoms in space. If this arrangement is given by a vector of atom positions r = (r\u0000 1\u0000 ,r\u0000 2\u0000 ,…,r\u0000 N\u0000 ), then the resulting structure factor F(r)(S) in direction S is given by where F\u0000 a\u0000 (\u0000 i\u0000 ) is the atomic scattering function of a(i) ∈ {“Carbon\", “Nitrogen\", “Hydrogen\",…}, the atom type of the ith atom. F\u0000 a\u0000 (\u0000 i\u0000 ) is the Fourier transform of ρ\u0000 a\u0000 (i), the spherical electron density for an atom of type a(i) placed at the origin.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131766755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Fiddy, R. McGahan, A. E. Morales-Porras, J. B. Morris
Methods for diffraction tomography, which are both numerically feasible and mathematically rigorous, have required that the scattering object only interact weakly with the incident field1,2. The approximations used include the first-order Born and the Rytov methods, which are rarely appropriate in practice, thus limiting their usefulness. More general methods or "exact" inversion procedures have proved extremely difficult, if not impossible, to implement. Recent developments have been made which do extend the domain of validity of the Born and Rytov approximations to some extent2 these are based on the distorted- wave Born or Rytov approximations. These methods assume that the strongly scattering component of the object is known, and that an unknown perturbation to this satisfies the Born or Rytov approximation. Thus, some a priori information about the scatterer must be acquired and which represents a (strongly scattering) background, against which small fluctuations in permittivity are imaged.
{"title":"Image recovery from far-field data at 10GHz","authors":"M. Fiddy, R. McGahan, A. E. Morales-Porras, J. B. Morris","doi":"10.1364/srs.1998.sthd.2","DOIUrl":"https://doi.org/10.1364/srs.1998.sthd.2","url":null,"abstract":"Methods for diffraction tomography, which are both numerically feasible and mathematically rigorous, have required that the scattering object only interact weakly with the incident field1,2. The approximations used include the first-order Born and the Rytov methods, which are rarely appropriate in practice, thus limiting their usefulness. More general methods or \"exact\" inversion procedures have proved extremely difficult, if not impossible, to implement. Recent developments have been made which do extend the domain of validity of the Born and Rytov approximations to some extent2 these are based on the distorted- wave Born or Rytov approximations. These methods assume that the strongly scattering component of the object is known, and that an unknown perturbation to this satisfies the Born or Rytov approximation. Thus, some a priori information about the scatterer must be acquired and which represents a (strongly scattering) background, against which small fluctuations in permittivity are imaged.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"659 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115122443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A main topic of inverse problems in crystallography appears to have been phase recovery of diffracted fields in diffraction crystallography1,2. In this paper we deal with a problem of detecting yet another type of phase in a structure image (or a lattice fringe image) obtained by the direct observation of a crystal using a high-resolution electron microscope. The phase in our problem represents spatial distortion of lattices in a crystal rather than the phase of X-ray or electron wave fields. We note that a quasi periodic structure of atoms observed in a TEM (transmission electron microscopy) image or in a STM (scanning tunnel microscopy) image bears a close similarity to an optical interferometric fringe pattern having spatial carrier frequencies, where lattice distortion or atom displacement may be regarded as a spatial fringe shift. The interpretation of the distorted lattice image as an interferogram permits us the use of spatial heterodyne technique for highly sensitive detection of the lattice distortion, where a phase change by 2π corresponds to the displacement of an atom by a lattice constant. Based on this interpretation, we propose a crystallographic heterodyne technique for precisely determining the positions of dislocated atoms using the Fourier fringe analysis technique originally developed for optical heterodyne interferometry3,4.
{"title":"Crystallographic Heterodyne Phase Detection Technique for Highly-Sensitive Lattice-Distortion Measurements","authors":"M. Takeda, J. Suzuki","doi":"10.1364/srs.1995.rtub4","DOIUrl":"https://doi.org/10.1364/srs.1995.rtub4","url":null,"abstract":"A main topic of inverse problems in crystallography appears to have been phase recovery of diffracted fields in diffraction crystallography1,2. In this paper we deal with a problem of detecting yet another type of phase in a structure image (or a lattice fringe image) obtained by the direct observation of a crystal using a high-resolution electron microscope. The phase in our problem represents spatial distortion of lattices in a crystal rather than the phase of X-ray or electron wave fields. We note that a quasi periodic structure of atoms observed in a TEM (transmission electron microscopy) image or in a STM (scanning tunnel microscopy) image bears a close similarity to an optical interferometric fringe pattern having spatial carrier frequencies, where lattice distortion or atom displacement may be regarded as a spatial fringe shift. The interpretation of the distorted lattice image as an interferogram permits us the use of spatial heterodyne technique for highly sensitive detection of the lattice distortion, where a phase change by 2π corresponds to the displacement of an atom by a lattice constant. Based on this interpretation, we propose a crystallographic heterodyne technique for precisely determining the positions of dislocated atoms using the Fourier fringe analysis technique originally developed for optical heterodyne interferometry3,4.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134013207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cleaning the distortions of an unknown image due to an unknown spatially invariant point spread function (PSF) is a problem of particular interest for optical systems, like telescopes, wherein the radiation from the object passes trough a medium in turbulent motion. The aim of this paper is to present a stable method, based on the approach presented in [1] for the case of coherent radiation, which is able to estimate both the optical path disturbance (OPD) and the unknown object from noisy and incoherent multiple images, in absence of a reference source. The presented technique, at variance of other existing methods, [2-3], is able to take explicitly into account the measurement errors and noise without requiring any hypothesis on their statistical nature. The stability of the method is ensured by exploiting both a physical constraint on the OPD and a stabilizing functional.
{"title":"A Method for Image Restoration and Wavefront Sensing by Using Phase Diversity","authors":"O. Bucci, A. Capozzoli, G. D'Elia","doi":"10.1364/srs.1998.stuc.1","DOIUrl":"https://doi.org/10.1364/srs.1998.stuc.1","url":null,"abstract":"Cleaning the distortions of an unknown image due to an unknown\u0000 spatially invariant point spread function (PSF) is a problem of\u0000 particular interest for optical systems, like telescopes, wherein the\u0000 radiation from the object passes trough a medium in turbulent motion.\u0000 The aim of this paper is to present a stable method, based on the\u0000 approach presented in [1] for the case of coherent radiation, which is\u0000 able to estimate both the optical path disturbance (OPD) and the\u0000 unknown object from noisy and incoherent multiple images, in absence\u0000 of a reference source. The presented technique, at variance of other\u0000 existing methods, [2-3], is able to take explicitly into account the\u0000 measurement errors and noise without requiring any hypothesis on their\u0000 statistical nature. The stability of the method is ensured by\u0000 exploiting both a physical constraint on the OPD and a stabilizing\u0000 functional.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132657552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
X-ray crystallography is a technique for determining the structures of molecules [1,2]. It involves irradiating a crystalline specimen of the molecule with a monochromatic beam of x-rays and measuring the resulting diffraction pattern. The complex amplitude of the diffracted x-rays is equal to the Fourier transform of the electron density in the crystalline specimen, and only the intensity, but not the phase, of the diffracted x-rays can be measured. Reconstruction of the electron density therefore constitutes a phase problem.
{"title":"Phase Retrieval for Icosahedral Particles","authors":"R. Millane, W. Stroud","doi":"10.1364/srs.1995.rtub3","DOIUrl":"https://doi.org/10.1364/srs.1995.rtub3","url":null,"abstract":"X-ray crystallography is a technique for determining the structures of molecules [1,2]. It involves irradiating a crystalline specimen of the molecule with a monochromatic beam of x-rays and measuring the resulting diffraction pattern. The complex amplitude of the diffracted x-rays is equal to the Fourier transform of the electron density in the crystalline specimen, and only the intensity, but not the phase, of the diffracted x-rays can be measured. Reconstruction of the electron density therefore constitutes a phase problem.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"41 1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129454702","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Planar-integrated free-space optics1,2 is a promising scheme to integrate large scale optical systems into a thick transparent substrate of plane-parallel geometry. For this approach all optical elements are located on the surfaces of the substrate (Fig. 1). Light signals propagate along a folded optical axis. All elements of the entire optical system form a two dimensional structure. This makes planar-integrated optics compatible with micro-electronics and allows, for instance, the fabrication of the system as a single diffractive optical surface relief by use of standard photo-lithographic techniques.
{"title":"Numerical optimization of diffractive optical elements for planar-integrated free-space optics","authors":"M. Testorf, M. Fiddy","doi":"10.1364/srs.1998.swb.3","DOIUrl":"https://doi.org/10.1364/srs.1998.swb.3","url":null,"abstract":"Planar-integrated free-space optics1,2 is a promising scheme to integrate large scale optical systems into a thick transparent substrate of plane-parallel geometry. For this approach all optical elements are located on the surfaces of the substrate (Fig. 1). Light signals propagate along a folded optical axis. All elements of the entire optical system form a two dimensional structure. This makes planar-integrated optics compatible with micro-electronics and allows, for instance, the fabrication of the system as a single diffractive optical surface relief by use of standard photo-lithographic techniques.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129750238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a massively parallel implementation of Richardson-Lucy or maximum likelihood restoration with a spatially-variant point spread function (PSF). Richardson-Lucy iterates involve the computation of sums of the form: where O(x' q ) is the incident optical field estimate at discrete source location x' q , I(x q ) is the measured discrete image at discrete field location x q , and P(x q , x' q ) is the discrete PSF – the probability that a photon from source region x' q is incident on the detector at field region x q . In general P is a function of source and field coordinates, and the computational burden of Eq. 1 is intractably large.
{"title":"Massively Parallel Spatially-Variant Maximum Likelihood Image Restoration","authors":"A. Boden, D. Redding, R. Hanisch, J. Mo","doi":"10.1364/srs.1995.rwb3","DOIUrl":"https://doi.org/10.1364/srs.1995.rwb3","url":null,"abstract":"We consider a massively parallel implementation of Richardson-Lucy or maximum likelihood restoration with a spatially-variant point spread function (PSF). Richardson-Lucy iterates involve the computation of sums of the form: where O(x'\u0000 q\u0000 ) is the incident optical field estimate at discrete source location x'\u0000 q\u0000 , I(x\u0000 q\u0000 ) is the measured discrete image at discrete field location x\u0000 q\u0000 , and P(x\u0000 q\u0000 , x'\u0000 q\u0000 ) is the discrete PSF – the probability that a photon from source region x'\u0000 q\u0000 is incident on the detector at field region x\u0000 q\u0000 . In general P is a function of source and field coordinates, and the computational burden of Eq. 1 is intractably large.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130138319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ground-based imaging of astronomical objects typically requires some form of post-processing to realize the full information content in the recorded image. The recorded image can be expressed as which is the standard expression for incoherent imaging in the absence of noise. i(r→) is the measured target, o(r→) is the true object distribution, p(r→) represents the point spread function (PSF) of the optical system, and * denotes the convolution operation. Thus, when the PSF is known, inversion of this expression will yield the object distribution. However, in many cases, the PSF is either poorly determined or unknown. Thus standard deconvolution techniques cannot be applied.
{"title":"Multiframe Blind Deconvolution for Object and PSF Recovery for Astronomical Imaging","authors":"J. Christou, E. Hege, S. Jefferies","doi":"10.1364/srs.1995.rwa2","DOIUrl":"https://doi.org/10.1364/srs.1995.rwa2","url":null,"abstract":"Ground-based imaging of astronomical objects typically requires some form of post-processing to realize the full information content in the recorded image. The recorded image can be expressed as which is the standard expression for incoherent imaging in the absence of noise. \u0000i(r→) is the measured target, \u0000o(r→) is the true object distribution, \u0000p(r→) represents the point spread function (PSF) of the optical system, and * denotes the convolution operation. Thus, when the PSF is known, inversion of this expression will yield the object distribution. However, in many cases, the PSF is either poorly determined or unknown. Thus standard deconvolution techniques cannot be applied.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"36 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114032325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The inverse problem is particularly important for device and application development. For non-periodic media, approximations can be made which allow estimates of the scattering permittivity distribution to be found. Our own work over the last 15 years [1,2,3] has moved from trying to interpret the most restrictive yet computationally simple of these approximations, the linearizing first Born and Rytov approximations, to the development of techniques which can be applied to both strongly scattering media as well as nonlinear (e.g. χ3) structures. We have also applied these methods to real experimental data as well as simulated cases, as such work has wide ranging uses in fields as diverse as medical and geophyiscal imaging, as well as the design of optical components. Emerging from these studies is a clearer understanding as to how the differential cepstral method (see paper by Morris et al in this volume and [4]) and distorted wave methods can be integrated in order to synthesize structures which are both strongly scattering and which have prescribed optically controllable scattering patterns. The differential cepstral filtering technique processes the function recovered by Fourier inversion of far-field scattering data, recognizing that it represents the product of the permittivity distribution and the total field within the scattering volume. This filtering method can be applied to scattering structures of arbitrarily high permittivity, in principle.
{"title":"Photonic band gap design based on inverse scattering techniques","authors":"D. Pommet, L. Malley, M. Fiddy","doi":"10.1364/srs.1995.rtuc3","DOIUrl":"https://doi.org/10.1364/srs.1995.rtuc3","url":null,"abstract":"The inverse problem is particularly important for device and application development. For non-periodic media, approximations can be made which allow estimates of the scattering permittivity distribution to be found. Our own work over the last 15 years [1,2,3] has moved from trying to interpret the most restrictive yet computationally simple of these approximations, the linearizing first Born and Rytov approximations, to the development of techniques which can be applied to both strongly scattering media as well as nonlinear (e.g. χ3) structures. We have also applied these methods to real experimental data as well as simulated cases, as such work has wide ranging uses in fields as diverse as medical and geophyiscal imaging, as well as the design of optical components. Emerging from these studies is a clearer understanding as to how the differential cepstral method (see paper by Morris et al in this volume and [4]) and distorted wave methods can be integrated in order to synthesize structures which are both strongly scattering and which have prescribed optically controllable scattering patterns. The differential cepstral filtering technique processes the function recovered by Fourier inversion of far-field scattering data, recognizing that it represents the product of the permittivity distribution and the total field within the scattering volume. This filtering method can be applied to scattering structures of arbitrarily high permittivity, in principle.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121960109","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In optical imaging systems that image coherently illuminated diffuse scattering objects, the complex field in both the object plane and the measurement plane of the object are described by a circularly complex random process1 with a spatial correlation function for the phase of the field given by2, where k o is the complex coherence factor3 in the measurement plane, and x→ and x→ are arbitrary positions in the measurement plane. The parameters γ and Ω depend solely on the complex coherence factor and so equation (1) depends only on the object’s ensemble statistics and does not model branch cut effects.
{"title":"Branch cut effects in optimally estimating a coherent diffracted field","authors":"W. Arrasmith","doi":"10.1364/srs.1998.swb.6","DOIUrl":"https://doi.org/10.1364/srs.1998.swb.6","url":null,"abstract":"In optical imaging systems that image coherently illuminated diffuse scattering objects, the complex field in both the object plane and the measurement plane of the object are described by a circularly complex random process1 with a spatial correlation function for the phase of the field given by2, where k\u0000 o\u0000 is the complex coherence factor3 in the measurement plane, and \u0000x→ and \u0000x→ are arbitrary positions in the measurement plane. The parameters γ and Ω depend solely on the complex coherence factor and so equation (1) depends only on the object’s ensemble statistics and does not model branch cut effects.","PeriodicalId":184407,"journal":{"name":"Signal Recovery and Synthesis","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121979594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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