Log-polar mapping made easy


Start with an image

and a sensor like this one composed of 64x32 photoreceptors of different size (64 of them for each of the 32 concentric circles). The sensor has two peculiarities:
  1. It has a polar structure: Taking the center as the origin the position of each photosite can be represented using the angle (theta) and the distance from the origin.
  2. The distance between neighboring photosites increases linearly with eccentricity and is smallest in the center of the structure where eccentricity is zero.
Now imagine to superimpose the sensor on the image and take just one value for each photoreceptor (this can be the average computed over the photoreceptor's area).
The result is a space variant image whose resolution is highest in the center and decreases with eccentricity.
The polar representation of the sensor can be mapped into a cartesian one in the way shown here for just a part of the image.
The overall 64x32 matrix of pixels looks like this. This is called log-polar representation because a generic pixels P(r,theta) in the original image is mapped into a cartesian space at coordinates log(r),theta).

This is how the rim of the left spectacle maps in log-polar space