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| doi = 10.1109/TPAMI.1984.4767500
| citeseerx = 10.1.1.161.3102
}}</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 }}</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 }}</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://www.nada.kth.se/~tony/abstracts/Lin90-PAMI.html Scale-space for discrete signals]," PAMI(12), No. 3, March 1990, pp. 234-254.</ref><ref>Lindeberg, Tony. [http://www.nada.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>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 support [[Gaussian filter]]s as smoothing kernels in the pyramid generation steps.
===Gaussian pyramid===
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===Steerable pyramid===
A steerable pyramid, developed by [[Eero Simoncelli|Simoncelli]] and others, is an implementation of a multi-scale, multi-orientation [[band-pass filter]] bank used for applications including [[image compression]], [[texture synthesis]], and [[Outline of object recognition|object recognition]]. It can be thought of as an orientation selective version of a Laplacian pyramid, in which a bank of [[steerable filter]]s are used at each level of the pyramid instead of a single Laplacian or [[Gaussian filter]].<ref>{{Cite web |first=Eero |last=Simoncelli |url=http://www.cns.nyu.edu/~eero/STEERPYR/ |title=The Steerable Pyramid |publisher=cns.nyu.edu }}</ref><ref>{{Cite web |first1=Roberto |last1=Manduchi |first2=Pietro |last2=Perona |first3=Doug |last3=Shy |title=Efficient Deformable Filter Banks |url=http://www.vision.caltech.edu/publications/ManduchiPeronaShy_efficient_deformable.pdf |publisher=[[California Institute of Technology]]/[[University of Padua]] |year=1997 }} <br />Also in {{Cite journal |journal=Transactions on Signal Processing |title=Efficient Deformable Filter Banks |volume=46 |issue=4 |pages=1168–1173 |year=1998 |doi=10.1109/78.668570|last1=Manduchi |first1=R. |last2=Perona |first2=P. |last3=Shy |first3=D. |bibcode=1998ITSP...46.1168M |citeseerx=10.1.1.5.3102 }}</ref><ref>Stanley A. Klein ; Thom Carney ; Lauren Barghout-Stein and Christopher W. Tyler
"Seven models of masking", Proc. SPIE 3016, Human Vision and Electronic Imaging II, 13 (June 3, 1997); {{DOI|10.1117/12.274510}}</ref>
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