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A lowpass pyramid is made by smoothing the image with an appropriate smoothing filter and then subsampling the smoothed image, usually by a factor of 2 along each coordinate direction. The resulting image is then subjected to the same procedure, and the cycle is repeated multiple times. Each cycle of this process results in a smaller image with increased smoothing, but with decreased spatial sampling density (that is, decreased image resolution). If illustrated graphically, the entire multi-scale representation will look like a pyramid, with the original image on the bottom and each cycle's resulting smaller image stacked one atop the other.
A bandpass pyramid is made by forming the difference between images at adjacent levels in the pyramid and performing some kind of image interpolation between adjacent levels of resolution, to enable computation of pixelwise differences..<ref>
E.H. Andelson and C.H. Anderson and J.R. Bergen and P.J. Burt and J.M. Ogden.
[http://persci.mit.edu/pub_pdfs/RCA84.pdf "Pyramid methods in image processing"].
==Pyramid generation kernels==
A variety of different smoothing [[Kernel_Kernel (image_processingimage processing)|kernels]] have been proposed for generating pyramids.<ref>{{Cite journal
| last1 = Burt | first1 = P. J.
| doi = 10.1016/0146-664X(81)90092-7
| pages = 20–51
|date=May 1981
}}</ref><ref name=Crowley1981>{{Cite paperjournal |last=Crowley |first=James L. |title=A representation for visual information |publisher=Carnegie-Mellon University, Robotics Institute |date=November 1981 |id=tech. report CMU-RI-TR-82-07 |url=http://www.ri.cmu.edu/publication_view.html?pub_id=37}}</ref><ref>{{cite journal | last1 = Burt, | first1 = Peter and| last2 = Adelson, | first2 = Ted, "[| year = 1983 | title = The Laplacian Pyramid as a Compact Image Code | url = http://web.mit.edu/persci/people/adelson/pub_pdfs/pyramid83.pdf| Thejournal Laplacian= PyramidIEEE asTrans. aCommunications Compact| Imagevolume Code]",= IEEE9 Trans.| Communications,issue = 9:4,| pages = 532–540, 1983.}}</ref><ref>{{Cite journal
| last1 = Crowley | first1 = J. L.
| last2 = Parker | first2 = A. C.
| pmid = 21869180
| doi = 10.1109/TPAMI.1984.4767500
}}</ref><ref>{{cite journal | last1 = Crowley, | first1 = J. L. and| 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| Multiplejournal resolution representation and probabilistic matching of 2-D gray-scale shape]",= IEEE Transactions on Pattern Analysis and Machine Intelligence, | volume = 9( | issue = 1),| pppages 113-121,= 1987.113–121 }}</ref><ref>{{cite journal | last1 = Meer | first1 = P. Meer,| last2 = Baugher | first2 = E. S. Baugher| andlast3 = Rosenfeld | first3 = A. Rosenfeld| year = 1987 | title = "Frequency ___domain analysis and synthesis of image generating kernels", | url = | journal = IEEE Transactions on Pattern Analysis and Machine Intelligence, vol| volume = 9, | issue = | pages 512-522,= 1987.512–522 }}</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===
===Steerable pyramid===
A steerable pyramid 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 of [[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 |format=PDF |publisher=[[California Institute of Technology]]/[[University of Padua]] |year=1997 }} <br />Also in {{Cite journal |publisher=[[Institute of Electrical and Electronics Engineers|IEEE]] |journal=Transactions on Signal Processing |title=Efficient Deformable Filter Banks |volume=46 |issue=4 |pages=1168–1173 |year=1998 |doi=10.1109/78.668570}}</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; http://dx.doi.org/{{DOI|10.1117/12.274510}}</ref>
==Applications of pyramids==
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