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}}</ref><ref>{{cite journal | last1 = Crowley | first1 = J. L. | last2 = Sanderson | first2 = A. C. | year = 1987 | title = Multiple resolution representation and probabilistic matching of 2-D gray-scale shape | url = http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-Sanderson-PAMI87.pdf| journal = IEEE Transactions on Pattern Analysis and Machine Intelligence | volume = 9 | issue = 1| pages = 113–121 | doi = 10.1109/tpami.1987.4767876 | pmid = 21869381 | citeseerx = 10.1.1.1015.9294 | s2cid = 14999508 }}</ref><ref>{{cite journal | last1 = Meer | first1 = P. | last2 = Baugher | first2 = E. S. | last3 = Rosenfeld | first3 = A. | year = 1987 | title = Frequency ___domain analysis and synthesis of image generating kernels | doi = 10.1109/tpami.1987.4767939 | journal = IEEE Transactions on Pattern Analysis and Machine Intelligence | volume = 9 | issue = 4| pages = 512–522 | pmid = 21869409 | s2cid = 5978760 }}</ref> Among the suggestions that have been given, the ''binomial kernels'' arising from the [[binomial coefficient]]s stand out as a particularly useful and theoretically well-founded class.<ref name=Crowley1981/><ref>Lindeberg, Tony, "[http://kth.diva-portal.org/smash/record.jsf?pid=diva2%3A472968&dswid=77 Scale-space for discrete signals]," PAMI(12), No. 3, March 1990, pp. 234-254.</ref><ref>{{cite journal | last1 = Haddad | first1 = R. A. | last2 = Akansu | first2 = A. N. | date = March 1991 | title = A Class of Fast Gaussian Binomial Filters for Speech and Image Processing | url = https://web.njit.edu/~akansu/PAPERS/Haddad-AkansuFastGaussianBinomialFiltersIEEE-TSP-March1991.pdf | journal = IEEE Transactions on Signal Processing | volume = 39 | issue = 3| pages = 723–727| doi = 10.1109/78.80892 | bibcode = 1991ITSP...39..723H }}</ref><ref>Lindeberg, Tony. [http://www.csc.kth.se/~tony/book.html Scale-Space Theory in Computer Vision], Kluwer Academic Publishers, 1994, {{ISBN|0-7923-9418-6}} (see specifically Chapter 2 for an overview of Gaussian and Laplacian image pyramids and Chapter 3 for theory about generalized binomial kernels and discrete Gaussian kernels)</ref><ref name=LinBre03-ScSp/><ref>See the article on [[multi-scale approaches]] for a very brief theoretical statement</ref> Thus, given a two-dimensional image, we may apply the (normalized) binomial filter (1/4, 1/2, 1/4) typically twice or more along each spatial dimension and then subsample the image by a factor of two. This operation may then proceed as many times as desired, leading to a compact and efficient multi-scale representation. If motivated by specific requirements, intermediate scale levels may also be generated where the subsampling stage is sometimes left out, leading to an ''oversampled'' or ''hybrid pyramid''.<ref name=LinBre03-ScSp/> With the increasing computational efficiency of [[CPU]]s available today, it is in some situations also feasible to use wider supported [[Gaussian filter]]s as smoothing kernels in the pyramid generation steps.
===Gaussian pyramid===
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