Fast Convolution
نویسنده
چکیده
We present a very simple and fast algorithm to compute the convolution of an arbitrary sequence x with a sequence of a specific type, a. The sequence a is any linear combination of polynomials, exponentials and trigonometric terms. The number of steps for computing the convolution depends on a certain complexity of a and not on its length, thus making it feasible to convolve a sequence with very large kernels fast. Computing the convolution (correlation, filtering) of a sequence x together with a fixed sequence a is one of the ubiquitous operations in graphics, image and signal processing. Often the sequence a is a polynomial, exponential or trigonometric function sampled at discrete points or a piecewise sum of such terms, such as, splines, or else the Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Journal of WSCG, Vol.11, No.1., ISSN 1213-6972 WSCG(92)2003, February 3-7, 2003, Plzen, Czech Republic. Copyright UNION Agency (96) Science Press sequence can can be well approximated with a few such terms. The computation of these convolutions is usually computed straight from the definition taking O(|x||a|) time or using a more complicated FFT based O(|x| log |a|) time algorithm. Here we present a simple and fast algorithm to compute the convolution of x1, x2, . . . , xn with am, am−1, . . . , a1, namely y1, y2, . . . , yn−m where yi = ∑m k=1 akxi+k−1. The number of steps of the algorithm depends on a measure of complexity of a and not on m, its length. The number of steps to compute the convolution is O(dn) (m < n) where the sequence a satisfies a linear homogeneous equation, (LHE), ∑d i=0 βiar+i = 0 (where the β do not depend on r), or equivalently, ar = ∑d i=1 αiar+i. For d smaller than log |m| this is faster and much simpler than using FFT. Examples of such sequences are; • polynomials of degree d−1, ai = ∑d−1 j=0 λj i j , the LHE is ∑d j=0(−1) ( d j ) ai+j = 0, this is of complexity d. • ai = βλ, the LHE is λai−ai+1 = 0, this is of complexity 2. • ai = λ ∑d−1 j=0 αji j , the LHE is ∑d j=0(−1) ( d
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تاریخ انتشار 2003