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Monomial Phase: A Matrix Representation of Local Phase

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

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

Abstract

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.

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Notes

  1. 1.

    Some researchers may prefer to express Eq. (1) as: \(\;f_{\!\!_{\mathcal{H}}}(x)\, =\, (f\, {\ast}\, g)(x);\;\;\;\; g(x) = \frac{\,-1\ } {\pi x}\). We will, however, continue to use the notation of Bracewell, i.e. \(\;f_{\!\!_{\mathcal{H}}}(x) = f(x)\, {\ast}\, g(x)\), no confusion should arise from this.

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Acknowledgements

The authors would like to thank Mats Andersson, Michael Felsberg, Gustaf Johansson and Jens Sjölund for valuable discussions and proof reading, and Anders Brun for demonstrating the Klein bottle phase structure of oriented patches using his LogMap manifold learning algorithm. We also gratefully acknowledge the support from the Swedish Research Council grants 2011–5176, 2012–3682 and NIH grants R01MH074794, P41RR013218, and P41EB015902.

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Authors and Affiliations

  1. Department of Biomedical Engineering, Center for Medical Image Science and Visualization (CMIV), Linköping University, Linköping, Sweden

    Hans Knutsson

  2. Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA

    Carl-Fredrik Westin

  3. Department of Biomedical Engineering, Linköping University, Linköping, Sweden

    Carl-Fredrik Westin

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  1. Hans Knutsson
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  2. Carl-Fredrik Westin
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Correspondence to Hans Knutsson .

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Editors and Affiliations

  1. Dept. of Radiology, Harvard Medical School, Brigham, and Women's Hospital, Boston, Massachusetts, USA

    Carl-Fredrik Westin

  2. Technical University of Delft, Dept of Computer Graphics and Visualization, Delft, The Netherlands

    Anna Vilanova

  3. Universität des Saarlandes, Dept. of Mathematics, Saarbrücken, Germany

    Bernhard Burgeth

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Knutsson, H., Westin, CF. (2014). Monomial Phase: A Matrix Representation of Local Phase. In: Westin, CF., Vilanova, A., Burgeth, B. (eds) Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54301-2_3

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