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  • 1.
    Adler, Jonas
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Solving ill-posed inverse problems using iterative deep neural networks2017Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 33, nr 12, artikkel-id 124007Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We propose a partially learned approach for the solution of ill-posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularisation theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularising functional. The method results in a gradient-like iterative scheme, where the 'gradient' component is learned using a convolutional network that includes the gradients of the data discrepancy and regulariser as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of 512 x 512 pixel images in about 0.4 s using a single graphics processing unit (GPU).

  • 2. de Silva, Vin
    et al.
    Morozov, Dmitriy
    Vejdemo-Johansson, Mikael
    St Andrews University, UK.
    Dualities in persistent (co) homology2011Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 27, nr 12Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of the latter algorithm.

    Fulltekst (pdf)
    fulltext
  • 3.
    Gris, Barbara
    et al.
    Univ Paris, LJLL, Sorbonne Univ, CNRS, F-75005 Paris, France..
    Chen, Chong
    Chinese Acad Sci, Acad Math & Syst Sci, ICMSEC, LSEC, Beijing 100190, Peoples R China..
    Öktem, Ozan
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.).
    Image reconstruction through metamorphosis2020Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 36, nr 2, artikkel-id 025001Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The paper describes a method for reconstructing an image from noisy and indirect observations by registering, via metamorphosis, a template. The image registration part consists of two components, one is a geometric deformation that moves intensities without changing them and the other that changes intensity values. Unlike a registration with only geometrical deformation, this framework gives good results also when intensities of the template are poorly chosen. It also allows for appearance of a new structure. The approach is applicable to general inverse problems in imaging and we prove existence, stability and convergence, which implies that the method is a well-defined regularisation method. We also present several numerical examples from tomography.

  • 4.
    Gustafsson, Björn
    et al.
    KTH, Tidigare Institutioner                               , Matematik.
    He, C. Y.
    Milanfar, P.
    Putinar, M.
    Reconstructing planar domains from their moments2000Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 16, nr 4, s. 1053-1070Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    In many areas of science and engineering it is of interest to find the shape of an object or region from indirect measurements which can actually be distilled into moments of the underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the reconstruction of a class of planar semi-analytic domains from their moments. A part of this class, known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the complex plane, and is therefore of great practical importance. We provide an exact reconstruction algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will be presented. In addition, relations of the present theory to computer-assisted tomography and a geophysical inverse problem will be briefly discussed.

  • 5.
    He, Sailing
    et al.
    KTH, Tidigare Institutioner                               , Teoretisk elektroteknik.
    Zhang, H.
    Romanov, V. G.
    Explicit identification of multiple small breast cancers in an optical mammographic imaging2002Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 18, nr 6, s. 1555-1567Artikkel i tidsskrift (Fagfellevurdert)
  • 6.
    Laptev, Ari
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Shterenberg, R.
    Sukhanov, V.
    Ostensson, J.
    Reflectionless potentials for an ordinary differential operator of order four2006Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 22, nr 1, s. 135-153Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    The aim of this paper is to construct exact formulae for reflectionless potentials for ordinary differential operators of order four. They lead to soliton-type solutions which are well known for one-dimensional Schrodinger operators. Such solitons are solutions of some nonlinear integrable systems which appeared in Gelfand and Dikey 1976 Funct. Anal. Appl. 10 13-29 (in Russian) (see also Hoppe et al 2003 Preprint math-ph/0311011).

  • 7.
    Lenells, Jonatan
    et al.
    University of Cambridge, United Kingdom .
    Fokas, A. S.
    An integrable generalization of the nonlinear Schrödinger equation on the half-line and solitons2009Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 25, nr 11, artikkel-id 115006Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We analyze initial-boundary value problems for an integrable generalization of the nonlinear Schrödinger equation formulated on the half-line. In particular, we investigate the so-called linearizable boundary conditions, which in this case are of Robin type. Furthermore, we use a particular solution to verify explicitly all the steps needed for the solution of a well-posed problem.

  • 8. Romanov, V. G.
    et al.
    He, Sailing
    KTH, Tidigare Institutioner                               , Teoretisk elektroteknik.
    Some uniqueness theorems for mammography-related time-domain inverse problems for the diffusion equation2000Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 16, nr 2, s. 447-459Artikkel i tidsskrift (Fagfellevurdert)
  • 9.
    Strömberg, Jan-Olov
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Andersson, Joel
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    An identity for triplets of double Hilbert transforms, with applications to the attenuated Radon transform2012Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 28, nr 12, s. 125007-Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We consider an elementary identity for double singular integrals in the plane and show that one can apply this to deduce inversion and product formulae for the Hilbert transform and inversion formulae for the affine and weighted Radon transforms. We will be able to allow many of the previously known weights for which there is an inversion formula for the weighted Radon transform and also pose some new conditions on which weights that can be used.

  • 10.
    Zickert, Gustav
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Maretzke, Simon
    Univ Gottingen, Inst Numer & Appl Math, Gottingen, Germany..
    Cryogenic electron tomography reconstructions from phaseless data2018Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 34, nr 12, artikkel-id 124001Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We perform three-dimensional (3D) reconstructions from simulated and real cryogenic electron tomography (cryo-ET) data. Our reconstructions are based on a nonlinear and phaseless forward model very reminiscent of a commonly used model for phase contrast x-ray tomography.

  • 11.
    Öktem, Ozan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Chen, Chong
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.). Chinese Academy of Sciences, China.
    Onur Domaniç, N.
    Ravikumar, P.
    Bajaj, C.
    Shape-based image reconstruction using linearized deformations2017Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 33, nr 3, artikkel-id 035004Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We introduce a reconstruction framework that can account for shape related prior information in imaging-related inverse problems. It is a variational scheme that uses a shape functional, whose definition is based on deformable template machinery from computational anatomy. We prove existence and, as a proof of concept, we apply the proposed shape-based reconstruction to 2D tomography with very sparse and/or highly noisy measurements.

  • 12.
    Öktem, Ozan
    et al.
    Sidec AB, Kista, Sweden.
    Fanelli, Duccio
    University of Florens.
    Electron tomography: A short overview with an emphasis on the absorption potential model for the forward problem2008Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 24, nr 1, s. 013001-Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    This review of the development and current status of electron tomography deals mainly with the mathematical and algorithmic aspects. After a very brief description of the role of electron tomography in structural biology, we turn our attention to the derivation of the forward operator. Starting from the Schrodinger equation, the electron - specimen interaction is modelled as a diffraction tomography problem and the picture is completed by adding a description of the optical system of the transmission electron microscope. The first- order Born approximation enables one to explicitly express the intensity for any finite wavenumber in terms of the propagation operator acting on the specimen convolved with a point spread function, derived from the optics in the transmission electron microscope. Next, we focus on the difficulties that cause the reconstruction problem to be quite challenging. Special emphasis is put on explaining the extremely low signal- to- noise ratio in the data combined with the incomplete data problems, which lead to severe ill- posedness. The next step is to derive the standard phase contrast model used in the electron tomography community. The above- mentioned expression for the intensity generalizes the standard phase contrast model which can be obtained by replacing the propagation operator by its high- energy limit, the x- ray transform, as the wavenumber tends to infinity. The importance of more carefully including the wave nature of the electron - specimen interaction is supported by performing an asymptotic analysis of the intensity as the wavenumber tends to infinity. Next we provide an overview of the various reconstruction methods that have been employed in electron tomography and we conclude by mentioning a number of open problems. Besides providing an introduction to electron tomography written in the 'language of inverse problems', the authors hope to raise interest among experts in integral geometry and regularization theory for the mathematical and algorithmic difficulties that are encountered in electron tomography.

  • 13.
    Öktem, Ozan
    et al.
    KTH, Skolan för teknikvetenskap (SCI), Matematik (Inst.), Matematik (Avd.).
    Rullgård, Hans
    Stockholm University.
    Skoglund, Ulf
    Karolinska Institutet.
    A component-wise iterated relative entropy regularization method with updated prior and regularization parameter2007Inngår i: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 23, nr 5, s. 2121-2139Artikkel i tidsskrift (Fagfellevurdert)
    Abstract [en]

    We present a componentwise iterated relative entropy regularization method (COMET) where the prior and regularization parameter could be updated in the iterates. Such a reconstruction method could be useful for multicomponent inverse problems, such as the one occurring in electron tomography. The paper also contains a brief introduction to regularization theory with emphasis on variational based regularization methods, and we rigorously prove that the tolerance-based entropy reconstruction method that occurs in the COMET iterates is a regularization method. We conclude by showing examples of COMET applied to electron tomography data.

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