Almost a decade ago I wrote about optimizing zoom animations by reusing the center portion of key frame images. In that post I used a fixed scale factor of 2, because I didn't think about generalizing it. This post here is to rectify that oversight.

So, fix the output video image size to \(W \times H\) with a pixel density (for anti-aliasing) of \(\rho^2\). For example, with \(5 \times 5\) supersampling (25 samples per output pixel), \(\rho = 5\).

Now we want to calculate the size of the key frame images \(H_K \times W_K\) and the scale factor \(R\). Suppose we have calculated an inner / deeper key frame in the zoom animation, then the next outer keyframe only needs the outer border calculated, because we can reuse the center. This means only \( H_K W_K \left(1 - \frac{1}{R^2}\right) \) need to be calculated per keyframe. The number of keyframes decreases as \(R\) increases, it turns out to be proportional to \(\frac{1}{\log R}\) for a fixed total animation zoom depth.

Some simple algebra, shows that \(H_K = R \rho H\) and \(W_K = R \rho W\). Putting this all together means we want to minimize the total number of pixels that need calculating, which is proportional to

\[ \frac{R^2 - 1}{\log R} \]

which decreases to a limit of \(2\) as \(R\) decreases to \(1\). But \(R = 1\) is not possible, as this wouldn't zoom at all; as-close-to-1-as-possible means a ring of pixels 1 pixel thick, at which point you are essentially computing an exponential map.

So if an exponential map is the most efficient way to proceed, how much worse is the key frame interpolation approach? Define the efficiency by \(2 / \frac{R^2 - 1}{\log R}\), then the traditional \(R = 2\) has an efficiency of only 46%, \(R = \frac{4}{3}\) 74%, \(R = \frac{8}{7}\) 87%.

These results are pushing me towards adding an exponential map rendering mode to all of my fractal zoom software, because the efficiency savings are significant. Expect it in (at least the command line mode of) the next release of KF which is most likely to be in early September, and if time allows I'll try to make a cross-platform zoom video assembler that can make use of the exponential map output files.