## Curvature-driven PDE methods for matrix-valued images (2004)

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Citations: | 13 - 4 self |

### BibTeX

@TECHREPORT{Feddern04curvature-drivenpde,

author = {Christian Feddern and Joachim Weickert and Bernhard Burgeth and Martin Welk},

title = {Curvature-driven PDE methods for matrix-valued images},

institution = {},

year = {2004}

}

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### Abstract

Abstract. Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edge-like structures in tensor fields, we first generalise Di Zenzo’s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts. Keywords: DT-MRI, denoising, segmentation, edge detection, structure tensor, mean curvature motion, selfsnakes, active contours