For a multi-scale keypoints detector, local extremas are found between the scales at EACH octave. The scale-space is implemented as follows: Each octave is downsampled by half from the first image in the previous octave. In each octave the images are smoothed with a gaussian kernel s times (s in the number of levels in each octave). So, the scale space is constructed by o ocatves with s levels = o*s images at different sizes and different smoothing levels.
Say, I have keypoints at the last octave (the smallest image), in scale s. Approximating the maximas (for better accuracy) gives (x',y',s').
My question is:
How are these values (x',y',s') converted to the original size image ? (surely, the location (x','y) of an image that was downsampled is different than the point location in the real image)