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  • 1.
    Adler, Jonas
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Solving ill-posed inverse problems using iterative deep neural networks2017In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 33, no 12, article id 124007Article in journal (Refereed)
    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. Dong, G.
    et al.
    Patrone, A. R.
    Scherzer, O.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Infinite dimensional optimization models and PDEs for dejittering2015In: 5th International Conference on Scale Space and Variational Methods in Computer Vision, SSVM 2015, Elsevier, 2015, Vol. 9087, p. 678-689Conference paper (Refereed)
    Abstract [en]

    In this paper we do a systematic investigation of continuous methods for pixel, line pixel and line dejittering. The basis for these investigations are the discrete line dejittering algorithm of Nikolova and the partial differential equation of Lenzen et al for pixel dejittering. To put these two different worlds in perspective we find infinite dimensional optimization algorithms linking to the finite dimensional optimization problems and formal flows associated with the infinite dimensional optimization problems. Two different kinds of optimization problems will be considered: Dejittering algorithms for determining the displacement and displacement error correction formulations, which correct the jittered image, without estimating the jitter. As a by-product we find novel variational methods for displacement error regularization and unify them into one family. The second novelty is a comprehensive comparison of the different models for different types of jitter, in terms of efficiency of reconstruction and numerical complexity.

  • 3. Gopinath, A.
    et al.
    Xu, G.
    Ress, D.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Subramaniam, S.
    Bajaj, C.
    Shape-based regularization of electron tomographic reconstruction2012In: IEEE Transactions on Medical Imaging, ISSN 0278-0062, E-ISSN 1558-254X, Vol. 31, no 12, p. 2241-2252Article in journal (Refereed)
    Abstract [en]

    We introduce a tomographic reconstruction method implemented using a shape-based regularization technique. Spatial models of known features in the structure being reconstructed are integrated into the reconstruction process as regularizers. Our regularization scheme is driven locally through shape information obtained from segmentation and compared with a known spatial model. We demonstrated our method on tomography data from digital phantoms, simulated data, and experimental electron tomography (ET) data of virus complexes. Our reconstruction showed reduced blurring and an improvement in the resolution of the reconstructed volume was also measured. This method also produced improved demarcation of spike boundaries in viral membranes when compared with popular techniques like weighted back projection and the algebraic reconstruction technique. Improved ET reconstructions will provide better structure elucidation and improved feature visualization, which can aid in solving key biological issues. Our method can also be generalized to other tomographic modalities.

  • 4. Hahn, S.
    et al.
    Mueller, Y.
    Hofmann, R.
    Moosmann, Julian
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Helfen, L.
    Guigay, J. -P
    van de Kamp, Th
    Baumbach, T.
    Spectral transfer from phase to intensity in Fresnel diffraction2016In: PHYSICAL REVIEW A, ISSN 2469-9926, Vol. 93, no 5, article id 053834Article in journal (Refereed)
    Abstract [en]

    We analyze theoretically and investigate experimentally the transfer of phase to intensity power spectra of spatial frequencies through free-space Fresnel diffraction. Depending on lambda z (where lambda is the wavelength and z is the free-space propagation distance) and the phase-modulation strength S, we demonstrate that for multiscale and broad phase spectra critical behavior transmutes a quasilinear to a nonlinear diffractogram except for low frequencies. On the contrary, a single-scale and broad phase spectrum induces a critical transition in the diffractogram at low frequencies. In both cases, identifying critical behavior encoded in the intensity power spectra is of fundamental interest because it exhibits the limits of perturbative power counting but also guides resolution and contrast optimization in propagation-based, single-distance, phase-contrast imaging, given certain dose and coherence constraints.

  • 5. Norlén, L.
    et al.
    Anwar, J.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Accessing the molecular organization of the stratum corneum using high-resolution electron microscopy and computer simulation2014In: Computational Biophysics of the Skin, Pan Stanford Publishing, 2014, p. 289-330Chapter in book (Other academic)
  • 6. Norlén, L.
    et al.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM. KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Skoglund, U.
    Molecular cryo-electron tomography of vitreous tissue sections: current challenges2009In: Journal of Microscopy, ISSN 0022-2720, E-ISSN 1365-2818, Vol. 235, no 3, p. 293-307Article in journal (Refereed)
    Abstract [en]

    Electron tomography of vitreous tissue sections (tissue TOVIS) allows the study of the three-dimensional structure of molecular complexes in a near-native cellular context. Its usage is, however, limited by an unfortunate combination of noisy and incomplete data, by a technically demanding sample preparation procedure, and by a disposition for specimen degradation during data collection. Here we outline some major challenges as experienced from the application of TOVIS to human skin. We further consider a number of practical measures as well as theoretical approaches for its future development.

  • 7. Quinto, Eric Todd
    et al.
    Ozan, Öktem
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Skoglund, Ulf
    Reply to Wang and Yu: Both electron lambda tomography and interior tomography have their uses2010In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 107, no 22, p. E94-E95Article in journal (Other academic)
  • 8. Rullgard, H.
    et al.
    Ofverstedt, L. -G
    Masich, S.
    Daneholt, B.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Simulation of transmission electron microscope images of biological specimens2011In: Journal of Microscopy, ISSN 0022-2720, E-ISSN 1365-2818, Vol. 243, no 3, p. 234-256Article in journal (Refereed)
    Abstract [en]

    We present a new approach to simulate electron cryo-microscope images of biological specimens. The framework for simulation consists of two parts; the first is a phantom generator that generates a model of a specimen suitable for simulation, the second is a transmission electron microscope simulator. The phantom generator calculates the scattering potential of an atomic structure in aqueous buffer and allows the user to define the distribution of molecules in the simulated image. The simulator includes a well defined electron-specimen interaction model based on the scalar Schrodinger equation, the contrast transfer function for optics, and a noise model that includes shot noise as well as detector noise including detector blurring. To enable optimal performance, the simulation framework also includes a calibration protocol for setting simulation parameters. To test the accuracy of the new framework for simulation, we compare simulated images to experimental images recorded of the Tobacco Mosaic Virus (TMV) in vitreous ice. The simulated and experimental images show good agreement with respect to contrast variations depending on dose and defocus. Furthermore, random fluctuations present in experimental and simulated images exhibit similar statistical properties. The simulator has been designed to provide a platform for development of new instrumentation and image processing procedures in single particle electron microscopy, two-dimensional crystallography and electron tomography with well documented protocols and an open source code into which new improvements and extensions are easily incorporated.

  • 9. Vulovic, Milos
    et al.
    Ravelli, Raimond B. G.
    van Vliet, Lucas J.
    Koster, Abraham J.
    Lazic, Ivan
    Lucken, Uwe
    Rullgård, Hans
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.). KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Rieger, Bernd
    Image formation modeling in cryo-electron microscopy2013In: Journal of Structural Biology, ISSN 1047-8477, E-ISSN 1095-8657, Vol. 183, no 1, p. 19-32Article in journal (Refereed)
    Abstract [en]

    Accurate modeling of image formation in cryo-electron microscopy is an important requirement for quantitative image interpretation and optimization of the data acquisition strategy. Here we present a forward model that accounts for the specimen's scattering properties, microscope optics, and detector response. The specimen interaction potential is calculated with the isolated atom superposition approximation (IASA) and extended with the influences of solvent's dielectric and ionic properties as well as the molecular electrostatic distribution. We account for an effective charge redistribution via the Poisson-Boltzmann approach and find that the IASA-based potential forms the dominant part of the interaction potential, as the contribution of the redistribution is less than 10%. The electron wave is propagated through the specimen by a multislice approach and the influence of the optics is included via the contrast transfer function. We incorporate the detective quantum efficiency of the camera due to the difference between signal and noise transfer characteristics, instead of using only the modulation transfer function. The full model was validated against experimental images of 20S proteasome, hemoglobin, and GroEL. The simulations adequately predict the effects of phase contrast, changes due to the integrated electron flux, thickness, inelastic scattering, detective quantum efficiency and acceleration voltage. We suggest that beam-induced specimen movements are relevant in the experiments whereas the influence of the solvent amorphousness can be neglected. All simulation parameters are based on physical principles and, when necessary, experimentally determined.

  • 10.
    Öktem, Ozan
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
    Mathematics of electron tomography2015In: Handbook of Mathematical Methods in Imaging: Volume 1, Second Edition, Springer, 2015, p. 937-1031Chapter in book (Other academic)
    Abstract [en]

    This survey starts with a brief description of the scientific relevance of electron tomography in life sciences followed by a survey of image formation models. In the latter, the scattering of electrons against a specimen is modeled by the Schrödinger equation, and the image formation model is completed by adding a description of the transmission electron microscope optics and detector. Electron tomography can then be phrased as an inverse scattering problem and attention is now turned to describing mathematical approaches for solving that reconstruction problem. This part starts out by explaining challenges associated with the aforementioned inverse problem, such as the extremely low signalto- noise ratio in the data and the severe ill-posedness due to incomplete data, which naturally brings up the issue of choosing a regularization method for reconstruction. Here, the review surveys both methods that have been developed, as well as pointing to new promising approaches. Some of the regularization methods are also tested on simulated and experimental data. As a final note, this is not a traditional mathematical review in the sense that focus here is on the application to electron tomography rather than on describing mathematical techniques that underly proofs of key theorems.

  • 11.
    Öktem, Ozan
    Sidec, Kista, Sweden.
    Reconstruction methods in electron tomography2008In: Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT) / [ed] Y. Censor, Jiang M., and Louis A. K., Springer Berlin/Heidelberg, 2008, p. 289-320Chapter in book (Refereed)
    Abstract [en]

    Already in 1968 one recognized that the transmission electron micro- scope could be used in a tomographic setting as a tool for structure determination of macromolecules. However, its usage in mainstream structural biology has been limited and one reason is the devastating combination of noisy data and incomplete data problems that leads to severe ill-posedness of the inverse problem. Despite these issues, the importance of electron tomography is beginning to increase, espe- cially in drug discovery. This review begins with a brief introduction to the model for image formation, i.e. the forward operator. Next, we state the difficulties and review some of the various attempts at overcoming those in solving the inverse problem.

  • 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 problem2008In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 24, no 1, p. 013001-Article in journal (Refereed)
    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.
    Sidec, Kista, Sweden.
    Quinto, Eric Todd
    Tufts University.
    Inversion of the X-ray transform from limited angle parallel beam region of interest data with applications to electron tomography2007In: Proceedings in Applied Mathematics and Mechanics: PAMM, ISSN 1617-7061, E-ISSN 1617-7061, Vol. 7, no 1, p. 1050301-1050302Article in journal (Refereed)
    Abstract [en]

    We present a new local tomographic algorithm applicable to electron microscopy tomography. Our algorithm applies to the standard data acquisition method, single-axis tilting, as well as for more arbitrary acquisition methods. Using microlocal analysis we put the reconstructions in a mathematical context, explaining which singularities are stably visible from the limited data given by the data collection protocol in the electron microscope.

  • 14.
    Öktem, Ozan
    et al.
    Sidec Technologies, Kista, Sweden.
    Quinto, Eric Todd
    Tufts University.
    Local tomography in electron microscopy2008In: SIAM Journal on Applied Mathematics, ISSN 0036-1399, E-ISSN 1095-712X, Vol. 68, no 5, p. 1282-1303Article in journal (Refereed)
    Abstract [en]

    We present a new local tomographic algorithm applicable to electron microscope tomography. Our algorithm applies to the standard data acquisition method, single-axis tilting, as well as to more arbitrary acquisition methods including double axis and conical tilt. Using microlocal analysis we put the reconstructions in a mathematical context, explaining which singularities are stably visible from the limited data given by the data collection protocol in the electron microscope. Finally, we provide reconstructions of real specimens from electron tomography data.

  • 15.
    Öktem, Ozan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.). KTH, School of Engineering Sciences (SCI), Centres, Center for Industrial and Applied Mathematics, CIAM.
    Quinto, Eric Todd
    Tufts University.
    Skoglund, Ulf
    Okinawa Institute of Science and Technology.
    Electron Lambda-tomography2009In: Proceedings of the National Academy of Sciences of the United States of America, ISSN 0027-8424, E-ISSN 1091-6490, Vol. 106, no 51, p. 21842-21847Article in journal (Refereed)
    Abstract [en]

    Filtered back-projection and weighted back-projection have long been the methods of choice within the electron microscopy com- munity for reconstructing the structure of macromolecular assem- blies from electron tomography data. Here, we describe electron lambda-tomography, a reconstruction method that enjoys the ben- efits of the above mentioned methods, namely speed and ease of implementation, but also addresses some of their shortcomings. In particular, compared to these standard methods, electron lambda- tomography is less sensitive to artifacts that come from structures outside the region that is being reconstructed, and it can sharpen boundaries.

  • 16.
    Öktem, Ozan
    et al.
    KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
    Rullgård, Hans
    Stockholm University.
    Skoglund, Ulf
    Karolinska Institutet.
    A component-wise iterated relative entropy regularization method with updated prior and regularization parameter2007In: Inverse Problems, ISSN 0266-5611, E-ISSN 1361-6420, Vol. 23, no 5, p. 2121-2139Article in journal (Refereed)
    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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