log-polar coordinates

Image analysis

Already at the end of the 1970s, applications for the discrete spiral coordinate system were given in image analysis. To represent an image in this coordinate system rather than in Cartesian coordinates, gives computational advantages when rotating or zooming in an image. Also, the photo receptors in the retina in the human eye are distributed in a way that has big similarities with the spiral coordinate system.[2] It can also be found in the Mandelbrot fractal (see picture to the right).

Log-polar coordinates can also be used to construct fast methods for the Radon transform and its inverse.

retina-like visual sensor

advantages of polar and log-polar mapping (for visual navigation)

apart from the shape invaiance property to scaling and rotations stems from the considerable data reduction obtained with the non-uniform sampling, a high resolution in the central part of the filed of view, which corresponds to the focus of attention.

bins that are uniform in log-polar space, making the descriptor more sensitive to positions of nearby sample points than to those points farther away.

corresponding points on two similar shapes will tend to have similar shape contexts.

Shape context at a given point on a shape is invariant under translation and scaling, shape contexts are not invariant under arbitrary affine transforms, but the log-polar binning ensures that for small locally affine distortions due to pose change, intra-category variation etc., the change of shape context is correspondingly small.

since sc gathers coarse info from the entire shape, it is relatively insensitive to the occlusion of any particular part.

Each shape context is a log-polar histogram of the coordinates of the rest of the point set measured using the reference point as the origin

via http://users.isr.ist.utl.pt/~alex/Projects/TemplateTracking/logpolar.htm

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