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I'm working on a product that has a visual transition. I've found that applying a simple filter that results in an exponential decay (starting fast, then tapering off) is pleasing in one direction. The problem is that going the other direction with the same filter looks bad. The filter I'm using is of the form:

$ Y_\text{next} = Y_\text{current} + \alpha\left(\text{target}-Y_\text{current}\right) $

What I basically need is the reverse behavior, where the transition starts out slow and then speeds up. Imagine playing the exponential decay I currently see "in reverse". I'm stumped as to the formulation, however. Note that I needn't have one formula to cover all cases, as I can switch the behavior depending on direction and/or current value.

This feels like a recurrence-type problem, but my familiarity with that topic is woefully out of date.

Thanks in advance.

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    Neal: I will try something like what you're proposing. I also will be investigating some simple rate limiting based on absolute value. Not elegant, but this is a real device (with a real ship date!), and being a perception thing it needn't be a perfect transition. If I can achieve something close it will probably look OK.2012-03-13

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