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Tensor Voting: Current State, Challenges and New Trends in the Context of Medical Image Analysis

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Visualization and Processing of Higher Order Descriptors for Multi-Valued Data

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

Perceptual organisation techniques aim at mimicking the human visual system for extracting salient information from noisy images. Tensor voting has been one of the most versatile of those methods, with many different applications both in computer vision and medical image analysis. Its strategy consists in propagating local information encoded through tensors by means of perception-inspired rules. Although it has been used for more than a decade, there are still many unsolved theoretical issues that have made it challenging to apply it to more problems, especially in analysis of medical images. The main aim of this chapter is to review the current state of the research in tensor voting, to summarise its present challenges, and to describe the new trends that we foresee will drive the research in this field in the next few years. Also, we discuss extensions of tensor voting that could lead to potential performance improvements and that could make it suitable for further medical applications.

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Notes

  1. 1.

    In this case tensorisation denotes the mapping \(t: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3\times 3}\) with \(t(\mathbf{n}) =\mathbf{ n}\mathbf{n}^{T}\;\forall _{\mathbf{n}\,\in \,\mathbb{R}^{3}}\), also referred to as the dyadic product or the outer product.

  2. 2.

    The saliencies are also referred to as curveness, surfaceness and junctionness.

  3. 3.

    Function \(R_{\mathbf{t}}\) is defined as \(R_{\mathbf{t}}(\alpha,\cdot ): \mathbb{R}^{3\times 3} \rightarrow \mathbb{R}^{3\times 3}\) with \(R_{\mathbf{t}}(\alpha,\mathsf{S}) = Q_{\alpha,\mathbf{t}}\;\mathsf{S}\;Q_{\alpha,\mathbf{t}}^{T}\;\forall _{\mathsf{S}\,\in \,\mathbb{R}^{3\times 3}}\), where \(Q_{\alpha,\mathbf{t}} \in \mathrm{ SO(3)}\) performs a rotation of angle α around axis \(\mathbf{t}\).

  4. 4.

    That is Y-crossings.

  5. 5.

    That is X-crossings.

  6. 6.

    The rank of a tensor is defined as the minimal number of first-order tensors, i.e. vectors, which is needed to represent it as a sum of outer products of these.

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Authors and Affiliations

  1. School of Technology and Health, KTH Flemingsberg, Alfred Nobels Allé 10, 14152, Huddinge, Sweden

    Daniel Jörgens & Rodrigo Moreno

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  1. Daniel Jörgens
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  2. Rodrigo Moreno
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Correspondence to Rodrigo Moreno .

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  1. Linköping University, Norrköping, Sweden

    Ingrid Hotz

  2. Visualization and Medical Image Analysis, University of Bonn, Bonn, Germany

    Thomas Schultz

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Jörgens, D., Moreno, R. (2015). Tensor Voting: Current State, Challenges and New Trends in the Context of Medical Image Analysis. In: Hotz, I., Schultz, T. (eds) Visualization and Processing of Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization. Springer, Cham. https://doi.org/10.1007/978-3-319-15090-1_9

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