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Affine Shape Alignment Using Covariant Gaussian Densities: A Direct Solution

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Abstract

We propose a novel approach for the estimation of 2D affine transformations aligning a planar shape and its distorted observation. The exact transformation is obtained as a least-squares solution of a linear system of equations constructed by fitting Gaussian densities to the shapes which preserve the effect of the unknown transformation. In the case of compound shapes, we also propose a robust and efficient numerical scheme achieving near real-time performance. The method has been tested on synthetic as well as on real images. Its robustness in the case of segmentation errors, missing data, and modelling error has also been demonstrated. The proposed method does not require point correspondences nor the solution of complex optimization problems, has linear time complexity and provides an exact solution regardless of the magnitude of deformation.

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Notes

  1. For a given set of point coordinates \(\{\mathbf{x}_i\}_{i=1}^n\) the sample mean vector is calculated as \(\varvec{\mu }=\frac{1}{n}\sum _{i=1}^n\mathbf{x}_i\) and the covariance matrix is given by \(\varvec{\varSigma }=\frac{1}{n}\mathbf{X}\mathbf{X}^T\), where \(\mathbf{X}=\begin{bmatrix} \mathbf{x}_1-\varvec{\mu }&\cdots&\mathbf{x}_n-\varvec{\mu } \end{bmatrix}\).

  2. The float arithmetic is inherently the most precise in the \([-1,1]\) interval. Therefore, in order to increase the numerical stability, we transform all the point coordinates into that interval, hence the power of those numbers will also remain in the \([-1,1]\) interval.

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Acknowledgments

This work has been partially supported by the Hungarian Scientific Research Fund—OTKA K75637, a PhD Fellowship of the University of Szeged, Hungary and by the European Union and the State of Hungary, co-financed by the European Social Fund through Project TAMOP–4.2.4.A/2-11-1-2012–0001 National Excellence Program.

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  1. Department of Image Processing and Computer Graphics, University of Szeged, P.O. Box 652, Szeged, 6701, Hungary

    Csaba Domokos & Zoltan Kato

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  1. Csaba Domokos
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  2. Zoltan Kato
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Correspondence to Zoltan Kato.

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Domokos, C., Kato, Z. Affine Shape Alignment Using Covariant Gaussian Densities: A Direct Solution. J Math Imaging Vis 51, 385–399 (2015). https://doi.org/10.1007/s10851-014-0530-3

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