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'''Pyramid''' or ''''pyramid representation'''' is a type of [[Scale model|multi-scale]] [[Signal (information theory)|signal]] [[Knowledge representation|representation]] developed by the [[computer vision]], [[image processing]] and [[signal processing]] communities, in which a signal or an image is subject to repeated smoothing and subsampling. Historically, pyramid representation is a predecessor to [[scale-space|scale-space representationspace]] representation and [[multiresolution analysis]].
==Pyramid generation==
==Pyramid generation kernels==
A variety of different smoothing kernels have proposed for generating pyramids (.<ref>Burt, P.J. "Fast filter transforms for image processing", Computer Vision, Graphics and Image Processing, vol 16, pages 20-51, 1981;.</ref><ref name=Crowley1981>Crowley, 1981;James "A representation for visual information", PhD thesis, Carnegie-Mellon University, Robotics Institute, Pittsburgh, Pennsylvania 1981.</ref><ref>Burt, Peter and Adelson, 1983;Ted, "The Laplacian Pyramid as a Compact Image Code", IEEE Trans. Communications, 9:4, 532–540, 1983.</ref><ref>Crowley, J. and Parker, 1984;A.C, "A Representation for Shape Based on Peaks and Ridges in the Difference of Low Pass Transform", IEEE Transactions on PAMI, 6(2), pp 156-170, March 1984.</ref><ref>Crowley, J. L. and Sanderson, A. C. "[http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-Sanderson-PAMI87.pdf Multiple resolution representation and probabilistic matching of 2-D gray-scale shape]", IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(1), pp 113-121, 1987;.</ref><ref>P. Meer, etE. alS. Baugher and A. Rosenfeld "Frequency ___domain analysis and synthesis of image generating kernels", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 9, pages 512-522, 1987).</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(Crowley12), 1981;No. Lindeberg3, 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), (seeISBN 0-7923-9418-6</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 motivatived 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''. 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.
==Applications of pyramids==
In the early days of computer vision, pyramids were used as the main type of multi-scale representation for computing multi-scale image [[feature detection (computer vision)|features]] from real-world image data. Today, this role has been taken over by [[scale- space]] representation, motivated by the more solid theoretical foundation, the ability to decouple the subsampling stage from the multi-scale representation, the more powerful tools for theoretical analysis as well as the ability to compute a representation at ''any'' desired scale, thus avoiding the algorithmic problems of relating image representations at different resolution. Nevertheless, pyramids are still frequently used for expressing computationally efficient approximations to scale-space representation.<ref>Crowley, (J, Riff O. [http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-ScaleSpace03.pdf Fast computation of scale normalised Gaussian receptive fields], Proc. Scale-Space'03, Isle of Skye, Scotland, Springer Lecture Notes in Computer Science, volume 2695, 2003.</ref><ref>Lindeberg, T. and Bretzner, 2003;L. Crowley[http://www.nada.kth.se/cvap/abstracts/cvap279.html andReal-time Riffscale 2003;selection in hybrid multi-scale representations], Proc. Scale-Space'03, Isle of Skye, Scotland, Springer Lecture Notes in Computer Science, volume 2695, pages 148-163, 2003.</ref><ref>Lowe, D. G., “[http://citeseer.ist.psu.edu/lowe04distinctive.html Distinctive image features from scale-invariant keypoints]”, International Journal of Computer Vision, 60, 2, pp. 91-110, 2004).</ref>
==References==
<references/>
* Burt, P.J. "Fast filter transforms for image processing", Computer Vision, Graphics and Image Processing, vol 16, pages 20-51, 1981.
* Burt, Peter and Adelson, Ted, "The Laplacian Pyramid as a Compact Image Code", IEEE Trans. Communications, 9:4, 532–540, 1983.
* Crowley, James "A representation for visual information", PhD thesis, Carnegie-Mellon University, Robotics Institute, Pittsburgh, Pennsylvania 1981.
* Crowley, J. and Parker, A.C, "A Representation for Shape Based on Peaks and Ridges in the Difference of Low Pass Transform", IEEE Transactions on PAMI, 6(2), pp 156-170, March 1984.
*[http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-Sanderson-PAMI87.pdf Crowley, J. L. and Sanderson, A. C. "Multiple resolution representation and probabilistic matching of 2-D gray-scale shape", IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(1), pp 113-121, 1987.]
*[http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-ScaleSpace03.pdf Crowley, J, Riff O: Fast computation of scale normalised Gaussian receptive fields, Proc. Scale-Space'03, Isle of Skye, Scotland, Springer Lecture Notes in Computer Science, volume 2695, 2003.]
*[http://www.nada.kth.se/~tony/abstracts/Lin90-PAMI.html Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254.]
*[http://www.nada.kth.se/~tony/book.html Lindeberg, Tony, Scale-Space Theory in Computer Vision, Kluwer Academic Publishers, 1994], ISBN 0-7923-9418-6
*[http://www.nada.kth.se/cvap/abstracts/cvap279.html Lindeberg, T. and Bretzner, L.: Real-time scale selection in hybrid multi-scale representations, Proc. Scale-Space'03, Isle of Skye, Scotland, Springer Lecture Notes in Computer Science, volume 2695, pages 148-163, 2003.]
*[http://citeseer.ist.psu.edu/lowe04distinctive.html Lowe, D. G., “Distinctive image features from scale-invariant keypoints”, International Journal of Computer Vision, 60, 2, pp. 91-110, 2004.]
* P. Meer, E. S. Baugher and A. Rosenfeld "Frequency ___domain analysis and synthesis of image generating kernels", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 9, pages 512-522, 1987.
==See also==
*[[scale-Scale space implementation]]
*[[scale-space implementation]]
*[[multiresolution analysis]]
*[[feature detection (computer vision)]]
[[Category:Image processing]]
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