Publications:Symmetry derivatives of Gaussians illustrated by cross tracking


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Title Symmetry derivatives of Gaussians illustrated by cross tracking
Author Josef Bigun and Tomas Bigun
Year 2001
PublicationType Report
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Abstract We propose a family of complex differential operators, symmetry derivatives, for pattern recognition in images. We present three theorems on their properties as applied to Gaussians. These show that all orders of symmetry derivatives of Gaussians yield compact expressions obtained by replacing the original differential polynomial with an ordinary polynomial. Just like Gaussians, the symmetry derivatives of Gaussians are (form) invariant to Fourier transform, that is they are rescaled versions of the original. As a result, the symmetry derivatives of Gaussians are closed under the convolution operator, i.e. they map on a member of the family when convolved with each other. Since Gaussians are utilized extensively in image processing, the revealed properties have practical consequences, e.g. when designing filters and filtering schemes that are unbiased w.r.t. orientation (isotropic). A use of these results is illustrated by an application: tracking the cross markers in long image sequences from vehicle crash tests. The implementation and the results of this application are discussed in terms of the theorems presented, along with conclusions.