Integral image resizing

Question

I'm trying to find an approach for integral image resizing. I found out that I can do it with a bilinear interpolation method, but with this approach I can only resizing by the factor which is a power of two.

I found a paper (Speeding Up Object Detection Fast Resizing in the Integral Image Domain) which describes a formula which allows to resize arbitrarily:

$$II_r(x,y) \approx \frac{1}{4a^2}\space \cdot bilinear(II, (2ax + b, 2ay+b))$$

$$2a \space - resizing\space factor$$ $$b \space - resizing\space filter\space offset$$

I would like to ask about explanation of this formula. I assume that I have to find a correct value of resizing filter offset but I don't know how to do this.

Answer 1

The formula says that the resized (smaller) image pixel at (x,y) is a bilinear approximation of the pixel at location (2ax+b, 2ay+b). Ignoring b for the moment, you can see that the scale factor (or "resizing factor") is 2a. So for example, if a=1.13, the pixel at 1,1 would come from a (bilinear) estimate of the pixel at (2.26,2.26).

As the paper mentions (the link the OP gave is no longer valid, it can be found at Speeding Up Object Dection...), the variable b is the filter phase, -a<b<a. It allows for a slight variation in exactly where in the image you will grab the downscaled image. Think of it as a global shift in the entire image of a small amount (up to 1.13 pixels in the example of a=1.13).