# Log Polar Transform

### Overview

The log-polar transform warps an image onto new axes, angle () and log of distance (). The angle and distance are measured in relation to the centre of the image, .

### Forward coordinate transformation

Using the centre of the image as the axis reference, pixel coordinates are given by

the angle is

and, with a base of , the log of the distance is

The base, chosen so the diameter of the transformed image is the same as that of the input image, is

where is the minimum dimension (probably height) of the input image.

### Reverse coordinate transformation

When warping images, it is often impractical to use the forward transform. Since we use discrete coordinates (i.e. integer x and y values), more than one input coordinate may map to the same output coordinate. Even worse, not every output coordinate will be covered.

One solution is to calculate the irregular grid of coordinates obtained by transforming each input coordinate (without discretising). Then, the grid is resampled (using interpolation) at the required output positions.

An easier (and sometimes less computationally intensive) method is to reverse the process. For each output coordinate, the transformation is applied in reverse, to obtain a coordinate in (or outside) the input image. Using interpolation, a value is determined.

This can be done if, neglecting the effect of discretisation, the transformation function is bijective (one-to-one correspondence, and all input and output coordinates are mapped).

Given and , we would now like to find and . First, we calculate the distance from the centre:

whereafter we can determine and as

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