Re: [math-fun] Exponential map of the Mandelbrot set, and the "logarithm of a circle"
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
and I ran across the need to describe the shape of an offset circle after its logarithm is taken. To be more precise:
If A is a circle (viewed as a set of points on the complex plane) whose distance from the origin is greater than its radius (i.e. the origin is outside the circle), and if B is the set of points you get by taking the (complex-valued) natural logarithm of each point in A, then what type of shape is B?
Is B an ellipse, some kind of superellipse (using a transcendental function perhaps)? If I need to give it a name, is there any name (like "quasi-ellipse") that doesn't already have some other meaning that would confuse my readers?
-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com
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
participants (2)
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Henry Baker -
Robert Munafo