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Joachim Weickert - One of the best experts on this subject based on the ideXlab platform.

  • A Discrete Theory and Efficient Algorithms for Forward-and-Backward Diffusion Filtering
    Journal of Mathematical Imaging and Vision, 2018
    Co-Authors: Martin Welk, Joachim Weickert, Guy Gilboa
    Abstract:

    Image enhancement with forward-and-backward (FAB) Diffusion lacks a sound theory and is numerically very challenging due to its diffusivities that are negative within a certain gradient range. In our paper, we address both problems. First we establish a comprehensive theory for space-discrete and time-continuous FAB Diffusion processes. It requires approximating the gradient magnitude with a non-standard discretisation. Then, we show that this theory carries over to the fully discrete case, when an explicit time discretisation with a fairly restrictive step-size limit is applied. To come up with more efficient algorithms, we propose three accelerated schemes: (i) an explicit scheme with global time step size adaptation that is also well suited for parallel implementations on GPUs, (ii) a randomised two-pixel scheme that offers optimal adaptivity of the time step size, (iii) a deterministic two-pixel scheme which benefits from less restrictive consistency bounds. Our experiments demonstrate that these algorithms allow speed-ups by up to three orders of magnitude without compromising stability or introducing visual artefacts.

  • 3d coherence enhancing Diffusion Filtering for matrix fields
    Mathematical Methods for Signal and Image Analysis and Representation, 2012
    Co-Authors: Bernhard Burgeth, Stephan Didas, Luis Pizarro, Joachim Weickert
    Abstract:

    Coherence-enhancing Diffusion Filtering is a striking application of the structure tensor concept in image processing. The technique deals with the problem of completion of interrupted lines and enhancement of flow-like features in images. The completion of line-like structures is also a major concern in Diffusion tensor magnetic resonance imaging (DT-MRI). This medical image acquisition technique outputs a 3D matrix field of symmetric (3×3)-matrices, and it helps to visualize, for example, the nerve fibers in brain tissue. As any physical measurement DT-MRI is subjected to errors causing faulty representations of the tissue corrupted by noise and with visually interrupted lines or fibers.

  • properties of higher order nonlinear Diffusion Filtering
    Journal of Mathematical Imaging and Vision, 2009
    Co-Authors: Stephan Didas, Joachim Weickert, Bernhard Burgeth
    Abstract:

    This paper provides a mathematical analysis of higher order variational methods and nonlinear Diffusion Filtering for image denoising. Besides the average grey value, it is shown that higher order Diffusion filters preserve higher moments of the initial data. While a maximum-minimum principle in general does not hold for higher order filters, we derive stability in the 2-norm in the continuous and discrete setting. Considering the filters in terms of forward and backward Diffusion, one can explain how not only the preservation, but also the enhancement of certain features in the given data is possible. Numerical results show the improved denoising capabilities of higher order Filtering compared to the classical methods.

  • a general structure tensor concept and coherence enhancing Diffusion Filtering for matrix fields
    2009
    Co-Authors: Bernhard Burgeth, Stephan Didas, Joachim Weickert
    Abstract:

    Coherence-enhancing Diffusion Filtering is a striking application of the structure tensor concept in image processing. The technique deals with the problem of completion of interrupted lines and enhancement of flow-like features in images. The completion of line-like structures is also a major concern in Diffusion tensor magnetic resonance imaging (DT-MRI). This medical image acquisition technique outputs a 3D matrix field of symmetric 3 × 3-matrices, and it helps to visualize, for example, the nerve fibers in brain tissue. As any physical measurement, DT-MRI is subjected to errors causing faulty representations of the tissue corrupted by noise and with visually interrupted lines or fibers.

  • locally analytic schemes a link between Diffusion Filtering and wavelet shrinkage
    Applied and Computational Harmonic Analysis, 2008
    Co-Authors: Martin Welk, Gabriele Steidl, Joachim Weickert
    Abstract:

    Abstract We study a class of numerical schemes for nonlinear Diffusion Filtering that offers insights on the design of novel wavelet shrinkage rules for isotropic and anisotropic image enhancement. These schemes utilise analytical or semi-analytical solutions to dynamical systems that result from space-discrete nonlinear Diffusion Filtering on minimalistic images with 2 × 2 pixels. We call them locally analytic schemes (LAS) and locally semi-analytic schemes (LSAS), respectively. They can be motivated from discrete energy functionals, offer sharp edges due to their locality, are very simple to implement because of their explicit nature, and enjoy unconditional absolute stability. They are applicable to singular nonlinear Diffusion filters such as TV flow, to bounded nonlinear Diffusion filters of Perona–Malik type, and to tensor-driven anisotropic methods such as edge-enhancing or coherence-enhancing Diffusion Filtering. The fact that these schemes use processes within 2 × 2 -pixel blocks allows to connect them to shift-invariant Haar wavelet shrinkage on a single scale. This interpretation leads to novel shrinkage rules for two- and higher-dimensional images that are scalar-, vector- or tensor-valued. Unlike classical shrinkage strategies they employ a Diffusion-inspired coupling of the wavelet channels that guarantees an approximation with an excellent degree of rotation invariance. By extending these schemes from a single scale to a multi-scale setting, we end up at hybrid methods that demonstrate the possibility to realise the effects of the most sophisticated Diffusion filters within a fairly simplistic wavelet setting that requires only Haar wavelets in conjunction with coupled shrinkage rules.

Alexei A Samsonov - One of the best experts on this subject based on the ideXlab platform.

  • noise adaptive nonlinear Diffusion Filtering of mr images with spatially varying noise levels
    Magnetic Resonance in Medicine, 2004
    Co-Authors: Alexei A Samsonov, Christopher R Johnson
    Abstract:

    Anisotropic Diffusion Filtering is widely used for MR image enhancement. However, the anisotropic filter is nonoptimal for MR images with spatially varying noise levels, such as images reconstructed from sensitivity-encoded data and intensity inhomogeneity-corrected images. In this work, a new method for Filtering MR images with spatially varying noise levels is presented. In the new method, a priori information regarding the image noise level spatial distribution is utilized for the local adjustment of the anisotropic Diffusion filter. Our new method was validated and compared with the standard filter on simulated and real MRI data. The noise-adaptive method was demonstrated to outperform the standard anisotropic Diffusion filter in both image error reduction and image signal-to-noise ratio (SNR) improvement. The method was also applied to inhomogeneity-corrected and sensitivity encoding (SENSE) images. The new filter was shown to improve segmentation of MR brain images with spatially varying noise levels. Magn Reson Med 52:798 – 806, 2004. © 2004 Wiley-Liss, Inc.

  • noise adaptive nonlinear Diffusion Filtering of mr images with spatially varying noise levels
    Magnetic Resonance in Medicine, 2004
    Co-Authors: Alexei A Samsonov, Christopher R Johnson
    Abstract:

    Anisotropic Diffusion Filtering is widely used for MR image enhancement. However, the anisotropic filter is nonoptimal for MR images with spatially varying noise levels, such as images reconstructed from sensitivity-encoded data and intensity inhomogeneity-corrected images. In this work, a new method for Filtering MR images with spatially varying noise levels is presented. In the new method, a priori information regarding the image noise level spatial distribution is utilized for the local adjustment of the anisotropic Diffusion filter. Our new method was validated and compared with the standard filter on simulated and real MRI data. The noise-adaptive method was demonstrated to outperform the standard anisotropic Diffusion filter in both image error reduction and image signal-to-noise ratio (SNR) improvement. The method was also applied to inhomogeneity-corrected and sensitivity encoding (SENSE) images. The new filter was shown to improve segmentation of MR brain images with spatially varying noise levels.

Christopher R Johnson - One of the best experts on this subject based on the ideXlab platform.

  • noise adaptive nonlinear Diffusion Filtering of mr images with spatially varying noise levels
    Magnetic Resonance in Medicine, 2004
    Co-Authors: Alexei A Samsonov, Christopher R Johnson
    Abstract:

    Anisotropic Diffusion Filtering is widely used for MR image enhancement. However, the anisotropic filter is nonoptimal for MR images with spatially varying noise levels, such as images reconstructed from sensitivity-encoded data and intensity inhomogeneity-corrected images. In this work, a new method for Filtering MR images with spatially varying noise levels is presented. In the new method, a priori information regarding the image noise level spatial distribution is utilized for the local adjustment of the anisotropic Diffusion filter. Our new method was validated and compared with the standard filter on simulated and real MRI data. The noise-adaptive method was demonstrated to outperform the standard anisotropic Diffusion filter in both image error reduction and image signal-to-noise ratio (SNR) improvement. The method was also applied to inhomogeneity-corrected and sensitivity encoding (SENSE) images. The new filter was shown to improve segmentation of MR brain images with spatially varying noise levels. Magn Reson Med 52:798 – 806, 2004. © 2004 Wiley-Liss, Inc.

  • noise adaptive nonlinear Diffusion Filtering of mr images with spatially varying noise levels
    Magnetic Resonance in Medicine, 2004
    Co-Authors: Alexei A Samsonov, Christopher R Johnson
    Abstract:

    Anisotropic Diffusion Filtering is widely used for MR image enhancement. However, the anisotropic filter is nonoptimal for MR images with spatially varying noise levels, such as images reconstructed from sensitivity-encoded data and intensity inhomogeneity-corrected images. In this work, a new method for Filtering MR images with spatially varying noise levels is presented. In the new method, a priori information regarding the image noise level spatial distribution is utilized for the local adjustment of the anisotropic Diffusion filter. Our new method was validated and compared with the standard filter on simulated and real MRI data. The noise-adaptive method was demonstrated to outperform the standard anisotropic Diffusion filter in both image error reduction and image signal-to-noise ratio (SNR) improvement. The method was also applied to inhomogeneity-corrected and sensitivity encoding (SENSE) images. The new filter was shown to improve segmentation of MR brain images with spatially varying noise levels.

Pavel Mrazek - One of the best experts on this subject based on the ideXlab platform.

  • selection of optimal stopping time for nonlinear Diffusion Filtering
    International Journal of Computer Vision, 2003
    Co-Authors: Pavel Mrazek, Mirko Navara
    Abstract:

    We develop a novel time-selection strategy for iterative image restoration techniques: the stopping time is chosen so that the correlation of signal and noise in the filtered image is minimized. The new method is applicable to any images where the noise to be removed is uncorrelated with the signal, under the assumptions that the filter used is suitable for the given type of data, and that neither the additive noise nor the Filtering procedure alter the average gray values no other knowledge (e.g. the noise variance, training data etc.) is needed. We analyse the theoretical properties of the method, then test the performance of our time estimation procedure experimentally, and demonstrate that it yields near-optimal results for a wide range of noise levels and for various Filtering methods.

  • selection of optimal stopping time for nonlinear Diffusion Filtering
    Lecture Notes in Computer Science, 2001
    Co-Authors: Pavel Mrazek
    Abstract:

    We develop a novel time-selection strategy for iterative image restoration techniques: the stopping time is chosen so that the correlation of signal and noise in the filtered image is minimised. The new method is applicable to any images where the noise to be removed is uncorrelated with the signal; no other knowledge (e.g. the noise variance, training data etc.) is needed. We test the performance of our time estimation procedure experimentally, and demonstrate that it yields near-optimal results for a wide range of noise levels and for various Filtering methods.

Santiago Ajafernandez - One of the best experts on this subject based on the ideXlab platform.

  • noise driven anisotropic Diffusion Filtering of mri
    IEEE Transactions on Image Processing, 2009
    Co-Authors: Karl Krissian, Santiago Ajafernandez
    Abstract:

    A new Filtering method to remove Rician noise from magnetic resonance images is presented. This filter relies on a robust estimation of the standard deviation of the noise and combines local linear minimum mean square error filters and partial differential equations for MRI, as the speckle reducing anisotropic Diffusion did for ultrasound images. The parameters of the filter are automatically chosen from the estimated noise. This property improves the convergence rate of the Diffusion while preserving contours, leading to more robust and intuitive Filtering. The partial derivative equation of the filter is extended to a new matrix Diffusion filter which allows a coherent Diffusion based on the local structure of the image and on the corresponding oriented local standard deviations. This new filter combines volumetric, planar, and linear components of the local image structure. The numerical scheme is explained and visual and quantitative results on simulated and real data sets are presented. In the experiments, the new filter leads to the best results.