I forgot to emphasize that the "logarithmic mapping" in the image processing application used the _complex logarithm_, so that a circle around the origin in "unwrapped" into a periodic function (in the imaginary coordinate), while circles not enclosing the origin are extremely distorted. Objects not enclosing the origin change size by simple translation in the ln (horizontal) coordinate, while rotations about the origin become translations in the imaginary (vertical) coordinate. At 10:30 AM 12/6/2010, Robert Munafo wrote:

The connection with human vision sensitivity is also related to the use of the logarithmic map to optimize the generation of "fractal zoom" animation videos.

Another Mandelbrot artist with whom I am corresponding separately just suggested to me that the image can be produced at lower precision than needed to resolve every pixel in every frame of a zoom animation. The pixels near the edge will be a bit fuzzy, but this typically happens in compressed video because typical motion compensation algorithms smooth out pixel details in areas that are moving quickly. Since the frame rate naturally blurs high-speed motion, and since the viewer typically focuses on the center (at least when the zoom is an "inward" zoom) this fuzziness isn't noticed.

- Robert

On Mon, Dec 6, 2010 at 13:16, Henry Baker <hbaker1@pipeline.com> wrote: In computer vision/image processing, a logarithmic map was suggested ~30 years ago as a way to focus attention/computational cycles on areas of greatest interest. For example, the fovea of the human eye has the highest density of pixels (as well as all of the color pixels), and the density of pixels falls off pretty quickly the further from the fovea that one goes. By utilizing a logarithmic map, one can theoretically achieve any desired resolution, at the cost of moving the center point -- either mechanically or digitally.

I believe that a number of common geometric objects (circles, lines, etc.) were analyzed by these researchers to understand how they mapped under this logarithmic transformation.

At 02:24 PM 12/5/2010, Robert Munafo wrote:

...

I was working on a page about the Mandelbrot set as seen through an exponential (or logarithmic) coordinate transformation:

http://mrob.com/pub/muency/exponentialmap.html

[...]

-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com