Flexible Segmentation and Smoothing of DT-MRI Fields Through a Customizable Structure Tensor

Schultz, Thomas; Burgeth, Bernhard; Weickert, Joachim

doi:10.1007/11919476_46

Cited by 10 publications

(9 citation statements)

References 16 publications

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“…The matrix-valued extensions inherit the filtering capabilities of their scalar counterparts. It is worth mentioning that the results are in good agreement with the results in [11] and [16]. However, the framework presented here is generic, hence more general, and does not rely on any notion of a potentially parameter-steered structure tensor.…”

Section: Methodssupporting

confidence: 82%

“…Even arbitrary exponents have been considered, [1,17]. Extensions of curvature-based PDEs to matrix fields have been proposed in [11] and more recently in [16], based on generalisations of the so-called structure tensor for scalar images to matrix fields. The research on these structure-tensor concepts has been initiated by [19,7].…”

Section: Introductionmentioning

confidence: 99%

“…For this work we concentrate on the matrix-valued analogs of the singular PDEs (1)-(3) as particularly interesting equations. It is also worth mentioning that in contrast to [11] and [16] our framework does not rely on a notion of structure tensor. Nevertheless, the proposed concept ensures an appropriate and desirable coupling of channels.…”

Section: Introductionmentioning

confidence: 99%

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A Generic Approach to the Filtering of Matrix Fields with Singular PDEs

Burgeth

Didas

Florack

et al.

Lecture Notes in Computer Science

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Abstract. There is an increasing demand to develop image processing tools for the filtering and analysis of matrix-valued data, so-called matrix fields. In the case of scalar-valued images parabolic partial differential equations (PDEs) are widely used to perform filtering and denoising processes. Especially interesting from a theoretical as well as from a practical point of view are PDEs with singular diffusivities describing processes like total variation (TV-)diffusion, mean curvature motion and its generalisation, the so-called self-snakes. In this contribution we propose a generic framework that allows us to find the matrix-valued counterparts of the equations mentioned above. In order to solve these novel matrix-valued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting. Numerical experiments performed on both synthetic and real world data substantiate the effectiveness of our matrix-valued, singular diffusion filters.

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Section: Methodssupporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Generic Approach to the Filtering of Matrix Fields with Singular PDEs

Burgeth

Didas

Florack

et al.

Lecture Notes in Computer Science

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“…, λ n ∈ IR from left to right simply by diag(λ i ), and O(n) stands for the matrix group of orthogonal n × n-matrices. Nonlinear partial differential equations have been employed to process matrix fields in [10] and more recently in [19]. Some extensions of scalar PDEs to matrices proposed in these works rely on generalisations of the so-called structure tensor.…”

Section: Introductionmentioning

confidence: 99%

“…For this work we concentrate on the matrix-valued analogs of the Perona-Malik PDE for a proof-of-concept. It is also worth mentioning that in contrast to [10,19,3] our framework does not rely on a notion of structure tensor. Nevertheless, the proposed concept ensures an appropriate and desirable coupling of channels.…”

Section: Introductionmentioning

confidence: 99%

A generic approach to diffusion filtering of matrix-fields

et al. 2007

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Diffusion tensor magnetic resonance imaging (DT-MRI), is a image acquisition method, that provides matrix-valued data, so-called matrix fields. Hence image processing tools for the filtering and analysis of these data types are in demand. In this artricle we propose a generic framework that allows us to find the matrix-valued counterparts of the Perona-Malik PDEs with various diffusivity functions. To this end we extend the notion of derivatives and associated differential operators to matrix fields of symmetric matrices by adopting an operator-algebraic point of view. In order to solve these novel matrixvalued PDEs successfully we develop truly matrix-valued analogs to numerical solution schemes of the scalar setting. Numerical experiments performed on both synthetic and real world data substantiate the effectiveness of our novel matrix-valued Perona-Malik diffusion filters.

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Mathematical Methods for Signal and Image Analysis and Representation

Florack¹,

Duits²,

Jongbloed³

et al. 2011

Computational Imaging and Vision

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This comprehensive book series embraces state-of-the-art expository works and advanced research monographs on any aspect of this interdisciplinary field.Topics covered by the series fall in the following four main categories:• Imaging Systems and Image Processing • Computer Vision and Image Understanding • Visualization •Applications of Imaging TechnologiesOnly monographs or multi-authored books that have a distinct subject area, that is where each chapter has been invited in order to fulfill this purpose, will be considered for the series. Volume 41 For further volumes: www.springer.com/series/5754 Luc Florack r Remco Duits r Geurt Jongbloed r Marie-Colette van Lieshout r Laurie Davies Editors

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Flexible Segmentation and Smoothing of DT-MRI Fields Through a Customizable Structure Tensor

Cited by 10 publications

References 16 publications

A Generic Approach to the Filtering of Matrix Fields with Singular PDEs

A Generic Approach to the Filtering of Matrix Fields with Singular PDEs

A generic approach to diffusion filtering of matrix-fields

Mathematical Methods for Signal and Image Analysis and Representation

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