image pyramid

An image pyramid is a stack of copies of the same image at progressively coarser resolutions, the full-size image at the bottom, then a half-size version, a quarter-size, and so on, like a pyramid narrowing toward the top. The reason to build one is that different things are visible at different scales: a face is obvious in a small thumbnail but individual pores need full resolution. Processing coarse-to-fine, find the rough answer cheaply on a tiny image, then refine it on larger ones, makes many tasks faster and more robust, and gives algorithms a handle on scale.

There are two main kinds. The Gaussian pyramid is built by repeatedly Gaussian-blurring an image and then downsampling it by a factor of two; each level G(l+1) = downsample(blur(G(l))). The blur is essential, it removes high frequencies that would otherwise alias into ugly artifacts when you throw away pixels. The Laplacian pyramid stores instead the detail lost at each step: each level L(l) = G(l) minus an upsampled-and-blurred version of G(l+1), a band-pass image capturing the frequencies present at that scale but not the next. The Laplacian pyramid is invertible, you can reconstruct the original exactly by adding the levels back up, which is why it works as a multi-scale decomposition for editing and compression.

Image pyramids are everywhere. The Burt-Adelson Laplacian-pyramid blend seamlessly merges two photos (the classic apple-orange splice). Coarse-to-fine optical flow tracks large motions by estimating them first on small images. SIFT builds a scale space of Gaussian-blurred octaves and finds keypoints as extrema of the Difference of Gaussians across scales, giving scale-invariant features. In deep learning the Feature Pyramid Network (FPN) fuses multi-resolution CNN feature maps so detectors can find both large and small objects. The cost is modest: a full pyramid adds only about one-third more storage than the original image.