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Monomial Phase: A Matrix Representation of Local Phase
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Linköping University, Center for Medical Image Science and Visualization (CMIV).ORCID iD: 0000-0002-9091-4724
Linköping University, Department of Biomedical Engineering, Medical Informatics. Linköping University, The Institute of Technology. Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA.
2014 (English)In: Visualization and Processing of Tensors and Higher Order Descriptors for Multi-Valued Data / [ed] Carl-Fredrik Westin, Anna Vilanova, Bernhard Burgeth, Springer, 2014, 37-73 p.Chapter in book (Other academic)
Abstract [en]

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.

Place, publisher, year, edition, pages
Springer, 2014. 37-73 p.
, Mathematics and Visualization, ISSN 1612-3786
Keyword [en]
Mathematics, Computer vision, Computer graphics, Differential equations, partial, Visualization, Global differential geometryPartial Differential Equations, Differential Geometry, Computer Imaging, Vision, Pattern Recognition and Graphics, Theoretical, Mathematical and Computational Physics
National Category
Computer Vision and Robotics (Autonomous Systems)
URN: urn:nbn:se:liu:diva-95762DOI: 10.1007/978-3-642-54301-2_3ISBN: 978-3-642-54300-5 (print)ISBN: 978-3-642-54301-2 (online)OAI: diva2:637593
Swedish Research CouncilLinnaeus research environment CADICSNIH (National Institute of Health)
Available from: 2013-07-19 Created: 2013-07-19 Last updated: 2015-08-19Bibliographically approved

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Knutsson, HansWestin, Carl-Fredrik
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Medical InformaticsThe Institute of TechnologyCenter for Medical Image Science and Visualization (CMIV)
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