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  • 1.
    Andersson, Kenneth
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Harvard Medical School, Brigham and Women's Hospital, Boston, USA.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Prediction from off-grid samples using continuous normalized convolution2007In: Signal Processing, ISSN 0165-1684, E-ISSN 1872-7557, Vol. 87, no 3, p. 353-365Article in journal (Refereed)
    Abstract [en]

    This paper presents a novel method for performing fast estimation of data samples on a desired output grid from samples on an irregularly sampled grid. The output signal is estimated using integration of signals over a neighbourhood employing a local model of the signal using discrete filters. The strength of the method is demonstrated in motion compensation examples by comparing to traditional techniques.

  • 2.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Park, Hae-Jeong
    Clinical Neuroscience Division, Laboratory of Neuroscience, Boston VA, USA Health Care System-Brockton Division, Department of Psychiatry, Harvard Medical School, Boston, MA, USA.
    Shenton, Martha E.
    Clinical Neuroscience Division, Laboratory of Neuroscience, Boston VA, USA Health Care System-Brockton Division, Department of Psychiatry, Harvard Medical School, Boston, MA, USA.
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Clustering Fiber Traces Using Normalized Cuts2004In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2004: 7th International Conference, Saint-Malo, France, September 26-29, 2004. Proceedings, Part I, Springer Berlin/Heidelberg, 2004, p. 368-375Conference paper (Refereed)
    Abstract [en]

    In this paper we present a framework for unsupervised segmentation of white matter fiber traces obtained from diffusion weighted MRI data. Fiber traces are compared pairwise to create a weighted undirected graph which is partitioned into coherent sets using the normalized cut (Ncut) criterion. A simple and yet effective method for pairwise comparison of fiber traces is presented which in combination with the Ncut criterion is shown to produce plausible segmentations of both synthetic and real fiber trace data. Segmentations are visualized as colored stream-tubes or transformed to a segmentation of voxel space, revealing structures in a way that looks promising for future explorative studies of diffusion weighted MRI data.

  • 3.
    Brun, Anders
    et al.
    Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, USA .
    Park, Hae-Jeong
    Clinical Neuroscience Div., Lab. of Neuroscience, Boston VA Health Care System-Brockton Division, Dep. of Psychiatry, Harvard Medical School, and Surgical Planning Laboratory, Brigham and Women’s Hospital, Harvard Medical School, .
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, USA .
    Coloring of DT-MRI fiber traces using Laplacian Eigenmaps2003In: Computer Aided Systems Theory - EUROCAST 2003: 9th International Workshop on Computer Aided Systems Theory Las Palmas de Gran Canaria, Spain, February 24-28, 2003 Revised Selected Papers / [ed] Roberto Moreno-Díaz and Franz Pichler, Springer Berlin/Heidelberg, 2003, Vol. 2809, p. 518-529Conference paper (Refereed)
    Abstract [en]

    We propose a novel post processing method for visualization of fiber traces from DT-MRI data. Using a recently proposed non-linear dimensionality reduction technique, Laplacian eigenmaps [3], we create a mapping from a set of fiber traces to a low dimensional Euclidean space. Laplacian eigenmaps constructs this mapping so that similar traces are mapped to similar points, given a custom made pairwise similarity measure for fiber traces. We demonstrate that when the low-dimensional space is the RGB color space, this can be used to visualize fiber traces in a way which enhances the perception of fiber bundles and connectivity in the human brain.

  • 4.
    Brun, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Svensson, Björn
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Herberthson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Wrangsjö, Andreas
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Knutsson, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Filtering Vector-Valued Images using Importance Sampling2007In: Proceedings of the {SSBA} Symposium on Image Analysis,2007, 2007Conference paper (Other academic)
  • 5.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Svensson, Björn
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering. Linköping University, The Institute of Technology.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Wrangsjö, Andreas
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Using Importance Sampling for Bayesian Feature Space Filtering2007In: Proceedings of the 15th Scandinavian conference on image analysis / [ed] Kjær Bjarne Ersbøll and Kim Steenstrup Pedersen, Berlin, Heidelberg: Springer-Verlag , 2007, p. 818-827Conference paper (Refereed)
    Abstract [en]

    We present a one-pass framework for filtering vector-valued images and unordered sets of data points in an N-dimensional feature space. It is based on a local Bayesian framework, previously developed for scalar images, where estimates are computed using expectation values and histograms. In this paper we extended this framework to handle N-dimensional data. To avoid the curse of dimensionality, it uses importance sampling instead of histograms to represent probability density functions. In this novel computational framework we are able to efficiently filter both vector-valued images and data, similar to e.g. the well-known bilateral, median and mean shift filters.

  • 6.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl_Fredrik
    Haker, S.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    A novel approach to averaging, filtering and interpolation of 3-D object orientation data2004In: Proceedings of the Swedish Symposium on Image Analysis (2004), 2004, p. 5-8Conference paper (Other academic)
  • 7.
    Brun, Anders
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Haker, Steven
    Laboratory of Mathematics in Imaging, Harvard Medical School, Boston, MA, USA.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    A tensor-like representation for averaging, filtering and interpolation of 3D object orientation data2005In: Image Processing, 2005. ICIP 2005. IEEE International Conference on  (Volume:3 ), 2005, p. 1092-1095Conference paper (Refereed)
    Abstract [en]

    Averaging, filtering and interpolation of 3-D object orientation data is important in both computer vision and computer graphics, for instance to smooth estimates of object orientation and interpolate between keyframes in computer animation. In this paper we present a novel framework in which the non-linear nature of these problems is avoided by embedding the manifold of 3-D orientations into a 16-dimensional Euclidean space. Linear operations performed in the new representation can be shown to be rotation invariant, and defining a projection back to the orientation manifold results in optimal estimates with respect to the Euclidean metric. In other words, standard linear filters, interpolators and estimators may be applied to orientation data, without the need for an additional machinery to handle the non-linear nature of the problems. This novel representation also provides a way to express uncertainty in 3-D orientation, analogous to the well known tensor representation for lines and hyperplanes.

  • 8.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Laboratory of Mathematics in Imaging Harvard Medical School, Boston, USA.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Fast manifold learning based on Riemannian normal coordinates2005In: Image Analysis: 14th Scandinavian Conference, SCIA 2005, Joensuu, Finland, June 19-22, 2005. Proceedings / [ed] Heikki Kalviainen, Jussi Parkkinen, Arto Kaarna., Springer Berlin/Heidelberg, 2005, p. 920-Conference paper (Refereed)
    Abstract [en]

    We present a novel method for manifold learning, i.e. identification of the low-dimensional manifold-like structure present in a set of data points in a possibly high-dimensional space. The main idea is derived from the concept of Riemannian normal coordinates. This coordinate system is in a way a generalization of Cartesian coordinates in Euclidean space. We translate this idea to a cloud of data points in order to perform dimension reduction. Our implementation currently uses Dijkstra’s algorithm for shortest paths in graphs and some basic concepts from differential geometry. We expect this approach to open up new possibilities for analysis of e.g. shape in medical imaging and signal processing of manifold-valued signals, where the coordinate system is “learned” from experimental high-dimensional data rather than defined analytically using e.g. models based on Lie-groups.

  • 9.
    Brun, Anders
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Lab of Mathematics in Imaging Harvard Medical School, Boston, USA.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    LOGMAP: Preliminary results using a new method for manifold learning2005In: Symposium on Image Analysis SSBA,2005, 2005, p. 101-105Conference paper (Other academic)
  • 10.
    Eriksson, Stefanie
    et al.
    Lund University, Sweden.
    Lasic, Samo
    CR Dev AB, Sweden.
    Nilsson, Markus
    Lund University, Sweden.
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Harvard University, MA 02215 USA.
    Topgaard, Daniel
    Lund University, Sweden.
    NMR diffusion-encoding with axial symmetry and variable anisotropy: Distinguishing between prolate and oblate microscopic diffusion tensors with unknown orientation distribution2015In: Journal of Chemical Physics, ISSN 0021-9606, E-ISSN 1089-7690, Vol. 142, no 10, p. 104201-Article in journal (Refereed)
    Abstract [en]

    We introduce a nuclear magnetic resonance method for quantifying the shape of axially symmetric microscopic diffusion tensors in terms of a new diffusion anisotropy metric, D-Delta, which has unique values for oblate, spherical, and prolate tensor shapes. The pulse sequence includes a series of equal-amplitude magnetic field gradient pulse pairs, the directions of which are tailored to give an axially symmetric diffusion-encoding tensor b with variable anisotropy b(Delta). Averaging of data acquired for a range of orientations of the symmetry axis of the tensor b renders the method insensitive to the orientation distribution function of the microscopic diffusion tensors. Proof-of-principle experiments are performed on water in polydomain lyotropic liquid crystals with geometries that give rise to microscopic diffusion tensors with oblate, spherical, and prolate shapes. The method could be useful for characterizing the geometry of fluid-filled compartments in porous solids, soft matter, and biological tissues. (C) 2015 Author(s).

  • 11.
    Forsberg, Daniel
    et al.
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Sectra Imtec, Linkoping, Sweden.
    Rathi, Yogesh
    Harvard Medical School, Boston, MA, USA.
    Bouix, Sylvain
    Harvard Medical School, Boston, MA, USA.
    Wassermann, Demian
    Harvard Medical School, Boston, MA, USA.
    Knutsson, Hans
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Harvard Medical School, Boston, MA, USA.
    Improving Registration Using Multi-channel Diffeomorphic Demons Combined with Certainty Maps2011In: Multimodal Brain Image Analysis: First International Workshop, MBIA 2011, Held in Conjunction with MICCAI 2011, Toronto, Canada, September 18, 2011. Proceedings, Springer Berlin/Heidelberg, 2011, Vol. 7012/2011, p. 19-26Conference paper (Refereed)
    Abstract [en]

    The number of available imaging modalities increases both in clinical practice and in clinical studies. Even though data from multiple modalities might be available, image registration is typically only performed using data from a single modality. In this paper, we propose using certainty maps together with multi-channel diffeomorphic demons in order to improve both accuracy and robustness when performing image registration. The proposed method is evaluated using DTI data, multiple region overlap measures and a fiber bundle similarity metric.

  • 12.
    Knutsson, Hans
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Faculty of Science & Engineering.
    An Iterated Complex Matrix Approach for Simulation and Analysis of Diffusion MRI Processes2015In: MEDICAL IMAGE COMPUTING AND COMPUTER-ASSISTED INTERVENTION - MICCAI 2015, PT I, SPRINGER INT PUBLISHING AG , 2015, Vol. 9349, p. 61-68Conference paper (Refereed)
    Abstract [en]

    We present a novel approach to investigate the properties of diffusion weighted magnetic resonance imaging (dMRI). The process of restricted diffusion of spin particles in the presence of a magnetic field is simulated by an iterated complex matrix multiplication approach. The approach is based on first principles and provides a flexible, transparent and fast simulation tool. The experiments carried out reveals fundamental features of the dMRI process. A particularly interesting observation is that the induced speed of the local spatial spin angle rate of change is highly shift variant. Hence, the encoding basis functions are not the complex exponentials associated with the Fourier transform as commonly assumed. Thus, reconstructing the signal using the inverse Fourier transform leads to large compartment estimation errors, which is demonstrated in a number of 1D and 2D examples. In accordance with previous investigations the compartment size is under-estimated. More interestingly, however, we show that the estimated shape is likely to be far from the true shape using state of the art clinical MRI scanners.

  • 13.
    Knutsson, Hans
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Harvard Medical School, Laboratory of Mathematics in Imaging (LMI).
    An Information Theoretic Approach to Optimal Q-space Sampling2014In: ISMRM-ESMRMB 2014, 2014Conference paper (Other academic)
  • 14.
    Knutsson, Hans
    et al.
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Harvard School of Medicin.
    Charged Containers for Optimal 3D Q-space Sampling2013In: Proceedings of the International Society for Magnetic Resonance in Medicine annual meeting (ISMRM'13), International Society for Magnetic Resonance in Medicine ( ISMRM ) , 2013Conference paper (Other academic)
    Abstract [en]

    Conclusions: We have presented a novel method for generating evenly distributed samples in a part of q-space that can be pre- specified in a general way. We have demonstrated the feasibility for two shapes, a sphere and a cube. The results are interesting from several points of view. There is a market tendency for the samples to group in shells indicating that the present work may provide a preferable alternative to recently proposed shell-interaction schemes [9]. The distributions attained for the cube case are far from Cartesian, this may be an advantage in a sparse reconstruction, e.g. compressed sensing, setting.

  • 15.
    Knutsson, Hans
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Harvard Medical School, USA .
    From Expected Propagator Distribution to Optimal Q-space Sample Metric2014In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014: 17th International Conference, Boston, MA, USA, September 14-18, 2014, Proceedings, Part III / [ed] Polina Golland, Nobuhiko Hata, Christian Barillot, Joachim Hornegger, Robert Howe, Springer, 2014, p. 217-224Conference paper (Refereed)
    Abstract [en]

    We present a novel approach to determine a local q-space metric that is optimal from an information theoretic perspective with respect to the expected signal statistics. It should be noted that the approach does not attempt to optimize the quality of a pre-defined mathematical representation, the estimator. In contrast, our suggestion aims at obtaining the maximum amount of information without enforcing a particular feature representation.

    Results for three significantly different average propagator distributions are presented. The results show that the optimal q-space metric has a strong dependence on the assumed distribution in the targeted tissue. In many practical cases educated guesses can be made regarding the average propagator distribution present. In such cases the presented analysis can produce a metric that is optimal with respect to this distribution. The metric will be different at different q-space locations and is defined by the amount of additional information that is obtained when adding a second sample at a given offset from a first sample. The intention is to use the obtained metric as a guide for the generation of specific efficient q-space sample distributions for the targeted tissue.

  • 16.
    Knutsson, Hans
    et al.
    Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, The Institute of Technology.
    Local Frequency1995In: Signal Processing for Computer Vision / [ed] Gösta H. Grandlund and Hans Knutsson, Dordrecht: Kluwer , 1995, Vol. 2749, p. 279-295Chapter in book (Refereed)
    Abstract [en]

    This chapter deals with the estimation of local frequency and bandwidth. Local frequency is an important concept which provides an indication of the appropriate range of scales for subsequent analysis. A number of one-dimensional and two-dimensional examples of local frequency and bandwidth estimation are given.

  • 17.
    Knutsson, Hans
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA.
    Monomial Phase: A Matrix Representation of Local Phase2014In: Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data / [ed] Carl-Fredrik Westin, Anna Vilanova, Bernhard Burgeth, Springer, 2014, p. 37-73Chapter in book (Other academic)
    Abstract [en]

    Local phase is a powerful concept which has been successfully used in many image processing applications. For multidimensional signals the concept of phase is complex and there is no consensus on the precise meaning of phase. It is, however, accepted by all that a measure of phase implicitly carries a directional reference. We present a novel matrix representation of multidimensional phase that has a number of advantages. In contrast to previously suggested phase representations it is shown to be globally isometric for the simple signal class. The proposed phase estimation approach uses spherically separable monomial filter of orders 0, 1 and 2 which extends naturally to N dimensions. For 2-dimensional simple signals the representation has the topology of a Klein bottle. For 1-dimensional signals the new phase representation reduces to the original definition of amplitude and phase for analytic signals. Traditional phase estimation using quadrature filter pairs is based on the analytic signal concept and requires a pre-defined filter direction. The new monomial local phase representation removes this requirement by implicitly incorporating local orientation. We continue to define a phase matrix product which retains the structure of the phase matrix representation. The conjugate product gives a phase difference matrix in a manner similar to the complex conjugate product of complex numbers. Two motion estimation examples are given to demonstrate the advantages of this approach.

  • 18.
    Knutsson, Hans
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics.
    Andersson, Mats
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Representing local structure using tensors II2011In: Proceedings of the 17th Scandinavian conference on Image analysis / [ed] Anders Heyden, Fredrik Kahl, Springer, 2011, p. 545-556Conference paper (Refereed)
    Abstract [en]

    Estimation of local spatial structure has a long history and numerous analysis tools have been developed. A concept that is widely recognized as fundamental in the analysis is the structure tensor. However, precisely what it is taken to mean varies within the research community. We present a new method for structure tensor estimation which is a generalization of many of it's predecessors. The method uses filter sets having Fourier directional responses being monomials of the normalized frequency vector, one odd order sub-set and one even order sub-set. It is shown that such filter sets allow for a particularly simple way of attaining phase invariant, positive semi-definite, local structure tensor estimates. We continue to compare a number of known structure tensor algorithms by formulating them in monomial filter set terms. In conclusion we show how higher order tensors can be estimated using a generalization of the same simple formulation.

  • 19.
    Knutsson, Hans
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Westin, Carl-Fredrik
    Linköping University, The Institute of Technology. Linköping University, Department of Biomedical Engineering, Medical Informatics.
    Andersson, Mats
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Structure Tensor Estimation: Introducing Monomial Quadrature Filter Sets2012In: New Developments in the Visualization and Processing of Tensor Fields / [ed] David H. Laidlaw, Anna Vilanova, Springer, 2012, p. 3-28Chapter in book (Other academic)
    Abstract [en]

       "Bringing together key researchers in disciplines ranging from visualization and image processing to applications in structural mechanics, fluid dynamics, elastography, and numerical mathematics, the workshop that generated this edited volume was the third in the successful Dagstuhl series. Its aim, reflected in the quality and relevance of the papers presented, was to foster collaboration and fresh lines of inquiry in the analysis and visualization of tensor fields, which offer a concise model for numerous physical phenomena. Despite their utility, there remains a dearth of methods for studying all but the simplest ones, a shortage the workshops aim to address. Documenting the latest progress and open research questions in tensor field analysis, the chapters reflect the excitement and inspiration generated  by this latest Dagstuhl workshop, held in July 2009. The topics they address range from applications of the analysis of tensor fields to purer research into their mathematical and analytical properties. They show how cooperation and the sharing of ideas and data between those engaged in pure and applied research can open new vistas in the study of tensor fields."--Publisher's website.

  • 20.
    Lundberg, Jonas
    et al.
    Linköping University, Department of Science and Technology, Media and Information Technology. Linköping University, Faculty of Science & Engineering.
    Westin, Carl-Fredrik
    Linköping University, Department of Science and Technology, Media and Information Technology. Linköping University, Faculty of Science & Engineering.
    Arvola, Mattias
    Linköping University, Department of Computer and Information Science, Human-Centered systems. Linköping University, Faculty of Arts and Sciences.
    Holmlid, Stefan
    Linköping University, Department of Computer and Information Science, Human-Centered systems. Linköping University, Faculty of Science & Engineering.
    Josefsson, Billy
    Luftfartsverket, Norrköping, Sweden.
    Cognitive work analysis and conceptual designing for unmanned air traffic management in cities2018Conference paper (Refereed)
    Abstract [en]

    Cognitive Work Analysis (CWA) is an appropriate approach in high-stakes domains, such as Air Traffic Management (ATM). It provides focus on human expert performance in regular as well as contingency situations. However, CWA is not suitable for the design of a first-of-a-kind system, since there is nothing to analyze before the start of the design process. In 2017, unmanned traffic management (UTM) for intense drone traffic in cities was such a system. Making things worse, the UTM system has to be in place before the traffic, since it provides basic safety. In this paper we present conceptual designing as a bootstrapping approach to CWA for UTM as a first-of-a-kind system.

  • 21.
    Nordin, Teresa
    et al.
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Zsigmond, Peter
    Department of Neurosurgery and Department of Clinical and Experimental Medicine, Linköping University Hospital.
    Pujol, Sonja
    Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, USA, Surgical Planning Laboratory, Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, USA.
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering. Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, USA.
    Wårdell, Karin
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Computer models in deep brain stimulation based on diffusion MRI2018Conference paper (Refereed)
  • 22.
    Nordin, Teresa
    et al.
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Zsigmond, Peter
    Department of Neurosurgery and Department of Clinical and Experimental Medicine, Linköping University Hospital, SE.
    Pujol, Sonja
    Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, USA, Surgical Planning Laboratory, Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, USA.
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering. Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, USA.
    Wårdell, Karin
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Deep brain stimulation: Patient-specific electrical field simulation2018Conference paper (Refereed)
  • 23.
    Sigfridsson, Andreas
    et al.
    Linköping University, Center for Medical Image Science and Visualization, CMIV. Linköping University, Department of Medicine and Health Sciences, Clinical Physiology . Linköping University, Faculty of Health Sciences. Östergötlands Läns Landsting, Heart Centre, Department of Clinical Physiology.
    Estepar, R.
    E.T.S.I. Telecomunicaci´on, University of Valladolid, Spain.
    Wigström, Lars
    Linköping University, Center for Medical Image Science and Visualization, CMIV. Linköping University, Department of Medicine and Health Sciences, Clinical Physiology . Linköping University, Faculty of Health Sciences.
    Alberola, C.
    E.T.S.I. Telecomunicaci´on, University of Valladolid, Spain.
    Westin, C-F.
    Brigham and Women’s Hospital, Harvard Medical School, Boston.
    Diffusion tensor visualization using random field correlation and volume rendering2003Conference paper (Refereed)
    Abstract [en]

    The visualization of diffusion tensor fields remains a challenging topic. A representation based on volume rendering of a scalar field is presented. The method uses the tensor to correlate a noise field in the direction of greater diffusivity while preserving the high frequency components of the noise field in transversal diffusion directions.

  • 24.
    Sjölund, Jens
    et al.
    Linköping University, Department of Biomedical Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology. Elekta Instrument AB, Sweden.
    Nilsson, Markus
    MR Physics, Lund University, Sweden.
    Topgaard, Daniel
    Physical Chemistry, Lund University, Sweden.
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, The Institute of Technology.
    Constrained optimization of gradient waveforms for generalized diffusion encoding2015In: Journal of magnetic resonance, ISSN 1090-7807, E-ISSN 1096-0856, Vol. 261, p. 157-168Article in journal (Refereed)
    Abstract [en]

    Diffusion MRI is a useful probe of tissue structure. The prototypical diffusion encoding sequence, the single pulsed field gradient, has recently been challenged with the introduction of more general gradient waveforms. Out of these, we focus on q-space trajecory imaging, which generalizes the scalar b-value to a tensor valued property. To take full advantage of its capabilities, it is imperative to respect the constraints imposed by the hardware, while at the same time maximizing the diffusion encoding strength. We formulate this as a constrained optimization problem that accomodates constraints on maximum gradient amplitude, slew rate, coil heating and positioning of radiofrequency pulses. The power of this approach is demonstrated by a comparison with previous work on optimization of isotropic diffusion sequences, showing possible gains in diffusion weighting or in heat dissipation, which in turn means increased signal or reduced scan-times.

  • 25.
    Westin, Carl-Fredrik
    Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, The Institute of Technology.
    Vector and Tensor Field Filtering1995In: Signal Processing for Computer Vision / [ed] Gösta H. Granlund and Hans Knutsson, Dordrecht: Kluwer , 1995, Vol. 2749, p. 343-365Chapter in book (Refereed)
    Abstract [en]

    This chapter discusses techniques for processing of higher order data such as vector and tensor fields. As abstraction implies a more complex descriptor, developing methods for processing of higher order data is an essential part of any hierarchical or layered approach to vision. The chapter focuses on models for extracting local symmetries and discontinuities in higher order fields.

  • 26.
    Westin, Carl-Fredrik
    et al.
    Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, The Institute of Technology.
    Knutsson, Hans
    Linköping University, Department of Electrical Engineering, Computer Vision. Linköping University, The Institute of Technology.
    Representation and Averaging1995In: Signal Processing for Computer Vision / [ed] Gösta H. Grandlund and Hans Knutsson, Dordrecht: Kluwer , 1995, Vol. 2749, p. 297-308Chapter in book (Refereed)
    Abstract [en]

    This chapter considers what the important properties are for an information representation to behave well in various transformations. There is an extended discussion on the necessity to separate between class membership and certainty of a signal.

  • 27.
    Westin, Carl-Fredrik
    et al.
    Brigham and Women’s Hospital, Harvard Medical School, Boston, US .
    Knutsson, Hans
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Representation and Estimation of Tensor-Pairs2012In: New Developments in the Visualization and Processing of Tensor Fields: Part V / [ed] David H. Laidlaw, Anna Vilanova, Springer Berlin/Heidelberg, 2012, p. 267-280Chapter in book (Refereed)
    Abstract [en]

    Over the years, several powerful models have been developed to represent specific elementary signal patterns, e.g. locally linear and planar structures. However, in real world problems there is often a need for handling more than one elementary pattern simultaneously. The straightforward approach of adaptive model selection has proven to be difficult and fragile. At the core of this problem is the vicious intractable search space created by having to simultaneously select models and corresponding samples. This calls for higher order models where multiple patterns are represented as one more complex pattern. In this work, we illustrate the advantages of this approach on data that has bi-modal tensor-valued distributions.The method uses first and second order invariants as a representation, and an eigenvector based solution for recovering the elementary tensor components. We show that this method allows estimation of the two tensors that best represent a given tensor distribution. This distribution can for example be samples from a local neighborhood. A bi-modal distribution will produce the two tensors corresponding to the peaks of the distribution. In addition, numbers indicating the amount of samples belonging to each sub distribution are produced. We demonstrate the potential of the approach by processing a number of simple tensor image examples. The results clearly show that new valuable information regarding the local tensor structure is revealed.

  • 28.
    Westin, Carl-Fredrik
    et al.
    Brigham and Women’s Hospital, Harvard Medical School, Boston.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Kikinis, Ron
    Harvard Medical School, Boston.
    Adaptive Image Filtering2000In: Handbook of Medical Imaging: Processing and Analysis Management (Biomedical Engineering) / [ed] Isaac Bankman, Academic Press , 2000, , p. 901Chapter in book (Other academic)
  • 29.
    Westin, Carl-Fredrik
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Faculty of Science & Engineering. Brigham and Women's Hospital, Harvard Medical School, Boston, MA, USA.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Faculty of Science & Engineering.
    Pasternak, Ofer
    Brigham and Women's Hospital, Harvard Medical School, Boston, MA, USA.
    Szczepankiewicz, Filip
    Department of Medical Radiation Physics, Lund University, Lund, Sweden.
    Özarslan, Evren
    Linköping University, Department of Biomedical Engineering. Linköping University, Faculty of Science & Engineering. Brigham and Women's Hospital, Harvard Medical School, Boston, MA, USA; Department of Physics, Bogazici University, Istanbul, Turkey.
    van Westen, Danielle
    Department of Diagnostic Radiology, Lund University, Lund, Sweden.
    Mattisson, Cecilia
    Clinical Sciences, Psychiatry, Lund University, Lund, Sweden.
    Bogren, Mats
    Clinical Sciences, Psychiatry, Lund University, Lund, Sweden.
    O'Donnell, Lauren J
    Brigham and Women's Hospital, Harvard Medical School, Boston, MA, USA.
    Kubicki, Marek
    Brigham and Women's Hospital, Harvard Medical School, Boston, MA, USA.
    Topgaard, Daniel
    Division of Physical Chemistry, Department of Chemistry, Lund University, Lund, Sweden.
    Nilsson, Markus
    Lund University Bioimaging Center, Lund University, Lund, Sweden.
    Q-space trajectory imaging for multidimensional diffusion MRI of the human brain2016In: NeuroImage, ISSN 1053-8119, E-ISSN 1095-9572, Vol. 135, p. 345-362Article in journal (Refereed)
    Abstract [en]

    This work describes a new diffusion MR framework for imaging and modeling of microstructure that we call q-space trajectory imaging (QTI). The QTI framework consists of two parts: encoding and modeling. First we propose q-space trajectory encoding, which uses time-varying gradients to probe a trajectory in q-space, in contrast to traditional pulsed field gradient sequences that attempt to probe a point in q-space. Then we propose a microstructure model, the diffusion tensor distribution (DTD) model, which takes advantage of additional information provided by QTI to estimate a distributional model over diffusion tensors. We show that the QTI framework enables microstructure modeling that is not possible with the traditional pulsed gradient encoding as introduced by Stejskal and Tanner. In our analysis of QTI, we find that the well-known scalar b-value naturally extends to a tensor-valued entity, i.e., a diffusion measurement tensor, which we call the b-tensor. We show that b-tensors of rank 2 or 3 enable estimation of the mean and covariance of the DTD model in terms of a second order tensor (the diffusion tensor) and a fourth order tensor. The QTI framework has been designed to improve discrimination of the sizes, shapes, and orientations of diffusion microenvironments within tissue. We derive rotationally invariant scalar quantities describing intuitive microstructural features including size, shape, and orientation coherence measures. To demonstrate the feasibility of QTI on a clinical scanner, we performed a small pilot study comparing a group of five healthy controls with five patients with schizophrenia. The parameter maps derived from QTI were compared between the groups, and 9 out of the 14 parameters investigated showed differences between groups. The ability to measure and model the distribution of diffusion tensors, rather than a quantity that has already been averaged within a voxel, has the potential to provide a powerful paradigm for the study of complex tissue architecture.

  • 30.
    Westin, Carl-Fredrik
    et al.
    Harvard Medical School.
    Martin-Fernandez, Marcos
    University ov Valladolid.
    Alberola-Lopez, Carlos
    University ov Valladolid.
    Ruiz-Alzola, Juan
    University of Las Palmas de Gran Canaria.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Tensor Field Regularization Using Normalized Convolution and Markov Random Fields in a Bayesian Framework2006In: Visualization and Image Processing of Tensor Fields / [ed] Joachim Weickert, Hans Hagen, Springer Berlin/Heidelberg, 2006, p. 381-398Chapter in book (Refereed)
    Abstract [en]

    This chapter presents two techniques for regularization of tensor fields. We first present a nonlinear filtering technique based on normalized convolution, a general method for filtering missing and uncertain data. We describe how the signal certainty function can be constructed to depend on locally derived certainty information and further combined with a spatially dependent certainty field. This results in reduced mixing between regions of different signal characteristics, and increased robustness to outliers, compared to the standard approach of normalized convolution using only a spatial certainty field. We contrast this deterministic approach with a stochastic technique based on a multivariate Gaussian signal model in a Bayesian framework. This method uses a Markov random field approach with a 3D neighborhood system for modeling spatial interactions between the tensors locally. Experiments both on synthetic and real data are presented. The driving tensor application for this work throughout the chapter is the filtering of diffusion tensor MRI data.

  • 31.
    Westin, Carl-Fredrik
    et al.
    Harvard Medical School.
    Nilsson, M.
    Lund University.
    Pasternak, O.
    Harvard Medical School.
    Knutsson, Hans
    Linköping University, Center for Medical Image Science and Visualization (CMIV). Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Diffusion tensors from double-PFG of the human brain2013In: ISMRM 2013, The International Society for Magnetic Resonance in Medicine , 2013Conference paper (Other academic)
  • 32.
    Westin, Carl-Fredrik
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Harvard Medical School, Laboratory of Mathematics in Imaging (LMI).
    Nilsson, Markus
    Lund University, Sweden.
    Szczepankiewicz, Filip
    Lund University, Sweden.
    Pasternak, Ofer
    Harvard Medical School.
    Ozarslan, Evren
    Harvard Medical School.
    Topgaard, Daniel
    Lund University, Sweden.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    In-vivo diffusion q-space trajectory imaging2014In: ISMRM 2014, 2014Conference paper (Other academic)
  • 33.
    Westin, Carl-Fredrik
    et al.
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Brigham and Women’s Hospital and Harvard Medical School, Boston, MA, USA .
    Szczepankiewicz, Filip
    Lund University, Sweden.
    Pasternak, Ofer
    Brigham and Women’s Hospital and Harvard Medical School, Boston, MA, USA.
    Özarslan, Evren
    Brigham and Women’s Hospital and Harvard Medical School, Boston, MA, USA.
    Topgaard, Daniel
    Lund University, Sweden.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Nilsson, Markus
    Lund University, Sweden.
    Measurement Tensors in Diffusion MRI: Generalizing the Concept of Diffusion Encoding2014In: Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014: 17th International Conference, Boston, MA, USA, September 14-18, 2014, Proceedings, Part III, Springer, 2014, p. 209-216Conference paper (Refereed)
    Abstract [en]

    In traditional diffusion MRI, short pulsed field gradients (PFG) are used for the diffusion encoding. The standard Stejskal-Tanner sequence uses one single pair of such gradients, known as single-PFG (sPFG). In this work we describe how trajectories in q-space can be used for diffusion encoding. We discuss how such encoding enables the extension of the well-known scalar b-value to a tensor-valued entity we call the diffusion measurement tensor. The new measurements contain information about higher order diffusion propagator covariances not present in sPFG. As an example analysis, we use this new information to estimate a Gaussian distribution over diffusion tensors in each voxel, described by its mean (a diffusion tensor) and its covariance (a 4th order tensor). © 2014 Springer International Publishing.

  • 34.
    Westin, C-F.
    et al.
    Harvard Medical School.
    Nilsson, M.
    Lund University.
    Pasternak, Ofer
    Harvard Medical School.
    Topgaard, D.
    Lund University.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology.
    Rotationally invariant gradient schemes for diffusion MRI2012In: Proceedings of the ISMRM (2012), 2012Conference paper (Other academic)
  • 35.
    Özarslan, Evren
    et al.
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Yolcu, Cem
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering. Laboratory for Mathematics in Imaging, Department of Radiology, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA.
    Influence of the Size and Curvedness of Neural Projections on the Orientationally Averaged Diffusion MR Signal2018In: Frontiers in Physics, E-ISSN 2296-424X, Vol. 6, p. 1-10, article id 17Article in journal (Refereed)
    Abstract [en]

    Neuronal and glial projections can be envisioned to be tubes of infinitesimal diameter as far as diffusion magnetic resonance (MR) measurements via clinical scanners are concerned. Recent experimental studies indicate that the decay of the orientationally-averaged signal in white-matter may be characterized by the power-law, Ē(q) ∝ q−1, where q is the wavenumber determined by the parameters of the pulsed field gradient measurements. One particular study by McKinnon et al. [1] reports a distinctively faster decay in gray-matter. Here, we assess the role of the size and curvature of the neurites and glial arborizations in these experimental findings. To this end, we studied the signal decay for diffusion along general curves at all three temporal regimes of the traditional pulsed field gradient measurements. We show that for curvy projections, employment of longer pulse durations leads to a disappearance of the q−1 decay, while such decay is robust when narrow gradient pulses are used. Thus, in clinical acquisitions, the lack of such a decay for a fibrous specimen can be seen as indicative of fibers that are curved. We note that the above discussion is valid for an intermediate range of q-values as the true asymptotic behavior of the signal decay is Ē(q) ∝ q−4 for narrow pulses (through Debye-Porod law) or steeper for longer pulses. This study is expected to provide insights for interpreting the diffusion-weighted images of the central nervous system and aid in the design of acquisition strategies.

  • 36.
    Özarslan, Evren
    et al.
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Yolcu, Cem
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.
    Westin, Carl-Fredrik
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering. Laboratory for Mathematics in Imaging, Department of Radiology, Harvard Medical School, Brigham and Women's Hospital, Boston, MA, United States.
    Knutsson, Hans
    Linköping University, Department of Biomedical Engineering, Division of Biomedical Engineering. Linköping University, Faculty of Science & Engineering. Linköping University, Center for Medical Image Science and Visualization (CMIV).
    Effective Potential for Magnetic Resonance Measurements of Restricted Diffusion2017In: Frontiers in Physics, E-ISSN 2296-424X, Vol. 5, article id 68Article in journal (Refereed)
    Abstract [en]

    The signature of diffusive motion on the NMR signal has been exploited to characterize the mesoscopic structure of specimens in numerous applications. For compartmentalized specimens comprising isolated subdomains, a representation of individual pores is necessary for describing restricted diffusion within them. When gradient waveforms with long pulse durations are employed, a quadratic potential profile is identified as an effective energy landscape for restricted diffusion. The dependence of the stochastic effective force on the center-of-mass position is indeed found to be approximately linear (Hookean) for restricted diffusion even when the walls are sticky. We outline the theoretical basis and practical advantages of our picture involving effective potentials.

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