Pyramid or 'pyramid representation' is a type of multiscale signal 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 representation and multiresolution analysis.
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There are two main types of pyramids; lowpass pyramids and bandpass pyramids. A lowpass pyramid is generated by first smoothing the image with an appropriate smoothing filter and then subsampling the smoothed image, usually by a factor of two along each coordinate direction. As this process proceeds, the result will be a set of gradually more smoothed images, where in addition the spatial sampling density decreases level by level. If illustrated graphically, this multiscale representation will look like a pyramid, from which the name has been obtained. A bandpass pyramid is obtained by forming the difference between adjacent levels in a pyramid, where in addition some kind of interpolation is performed between representations at adjacent levels of resolution, to enable the computation of pixelwise differences.
A variety of different smoothing kernels have proposed for generating pyramids.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}^{[6]} Among the suggestions that have been given, the binomial kernels arising from the binomial coefficients stand out as a particularly useful and theoretically wellfounded class.^{[2]}^{[7]}^{[8]}^{[9]} Thus, given a twodimensional 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 multiscale 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 CPUs available today, it is in some situations also feasible to use wider support Gaussian filters as smoothing kernels in the pyramid generation steps.
In the early days of computer vision, pyramids were used as the main type of multiscale representation for computing multiscale image features from realworld 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 multiscale 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 scalespace representation.^{[10]}^{[11]}^{[12]}
