Re: How to compute the addition of millions of functions efficiently and dynamically?
From: lucy (losemind_at_yahoo.com)
Date: 08/13/04
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Date: Fri, 13 Aug 2004 10:19:27 -0700
"networm" <networm8848@yahoo.com> wrote in message
news:cfiq28$c6p$1@news.Stanford.EDU...
> Dear all,
>
> I am headache on the following problem:
>
> I have a bunch of 2D functions, they are the replication of the same 2D
> function at different shift (x, y) locations.
>
> They can be viewed as a 2D impulse function replicated at both X and Y
> dimension, and then convolve with another 2D function. Since impulse
> function convolve with another function leads to that very function. So
now
> after convolution I have a bunch of 2D functions...
>
> These 2D functions can overlap, if the 2D functions have infinite support,
> for example, a Z(x, y)=(sinc(x, y))^2, this function has infinite support.
> Suppose there are 1000x1000 such infinite support 2D functions, shifted at
> different (x,y) locations...
>
> Now I want to compute their sum.
>
> (In Matlab), the simplest way to do this is to write out the complete
> expression and then evaluate it at any point (x, y)...
>
> For example, my
>
> Z_SUM(x, y)=Z(x, y) + Z(x-x1, y-y1)+Z(x-x2, y-y2)+Z(x-x3, y-y3)+ ...
> +Z(x-x1000000, y-y1000000)
>
> I guess no computing system can handle this expression...
>
> I guess the only way is to go point-oriented:
>
> Given a point (x, y), asking for how much it should be for Z_SUM(x, y),
then
> for that particular point, I do 1000000-1 evaluations and additions to get
> that one point Z_SUM(x, y) value...
>
> This is still troublesome.
>
> Now I approximate the infinite support of each function Z by a finite
> support -- chop the decaying tail of the sinc^2 slope at a certain
distance
> from the center of the function.
>
> This leads to error, but saves time. (storage is not a problem, since I
> evaluate one point by one point by querying its value...)
>
> However, this leads to a problem I want to ask for help:
>
> How to decide if a point is within a support of a function?
>
> What I meant is that if a point can be decided to only relate to a few
> functions, then I can evaluate and add those a few functions only.
>
> But how to decide which some functions are relevant to the point?
>
> For rectangular finite support of function Zs, this is a little easier.
>
> But how about for circular-shaped finite support? Doing detecting a point
to
> see if the point is with the circular support needs to take
> (x-xm)^2+(y-ym)^2 and then compare with some threshold, doing this 1000000
> square and comparsion isn't quite faster than doing the 1000000
(sinc(x-xm,
> y-ym))^2 evalution then add directly...
>
> Any depending on sampling density, the above query for one point's value
> needs to be millions of times...
>
> Please advise me on how to do this efficiently?
>
> Are there any similar problems and solutions existing in the field?
>
> Thank you very much and I really appreciate your help!
>
> -Net
>
sorry, this is repeated post. please ignore...
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