Structure tensor: Difference between revisions

Revision as of 20:46, 15 October 2022

Revision as of 02:33, 19 October 2022

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In particular, if the ellipsoid is stretched along one axis only, like a cigar (that is, if <math>\lambda_1</math> is much larger than both <math>\lambda_2</math> and <math>\lambda_3</math>), it means that the gradient in the window is predominantly aligned with the direction <math>e_1</math>, so that the [[isosurface]]s of <math>I</math> tend to be flat and perpendicular to that vector. This situation occurs, for instance, when ''p'' lies on a thin plate-like feature, or on the smooth boundary between two regions with contrasting values.

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| [[File:STsurfel.png|thumb|180px|The structure tensor ellipsoid of a surface-like neighborhood ("[[surfel]]"), where <math>\lambda_1 >\!> \lambda_2 \approx \lambda_3</math>.]]||[[File:StepPlane3D.png|thumb|180px|A 3D window straddling a smooth boundary surface between two uniform regions of a 3D image.]]||[[File:StepPlane3DST.png|thumb|180px|The corresponding structure tensor ellipsoid.]]

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If the ellipsoid is flattened in one direction only, like a pancake (that is, if <math>\lambda_3</math> is much smaller than both <math>\lambda_1</math> and <math>\lambda_2</math>), it means that the gradient directions are spread out but perpendicular to <math>e_3</math>; so that the isosurfaces tend to be like tubes parallel to that vector. This situation occurs, for instance, when ''p'' lies on a thin line-like feature, or on a sharp corner of the boundary between two regions with contrasting values.

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| [[File:STcurvel.png|thumb|180px|The structure tensor of a line-like neighborhood ("curvel"), where <math>\lambda_1 \approx \lambda_2 >\!> \lambda_3</math>.]]||[[File:curve3D.png|thumb|180px|A 3D window straddling a line-like feature of a 3D image.]]||[[File:curve3DST.png|thumb|180px|The corresponding structure tensor ellipsoid.]]

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Finally, if the ellipsoid is roughly spherical (that is, if <math>\lambda_1\approx\lambda_2\approx\lambda_3</math>), it means that the gradient directions in the window are more or less evenly distributed, with no marked preference; so that the function <math>I</math> is mostly isotropic in that neighborhood. This happens, for instance, when the function has [[spherical symmetry]] in the neighborhood of ''p''. In particular, if the ellipsoid degenerates to a point (that is, if the three eigenvalues are zero), it means that <math>I</math> is constant (has zero gradient) within the window.

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| [[File:STball.png|thumb|180px|The structure tensor in an isotropic neighborhood, where <math>\lambda_1\approx\lambda_2\approx\lambda_3</math>.]]||[[File:Sphere3D.png|thumb|180px|A 3D window containing a spherical feature of a 3D image.]]||[[File:Sphere3DST.png|thumb|180px|The corresponding structure tensor ellipsoid.]]

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==The multi-scale structure tensor==