{"url":"/method/laplacian-pyramid","slug":"laplacian-pyramid","name":"Laplacian Pyramid","full_name":"Laplacian Pyramid","full_name_withheld":false,"description_markdown":"A **Laplacian Pyramid** is a linear invertible image representation consisting of a set of band-pass\r\nimages spaced an octave apart, plus a low-frequency residual. Formally, let $d\\left(.\\right)$ be a downsampling operation that blurs and decimates a $j \\times j$ image $I$ so that $d\\left(I\\right)$ is a new image of size $\\frac{j}{2} \\times \\frac{j}{2}$. Also, let $u\\left(.\\right)$ be an upsampling operator which smooths and expands $I$ to be twice the size, so $u\\left(I\\right)$ is a new image of size $2j \\times 2j$. We first build a Gaussian pyramid $G\\left(I\\right) = \\left[I\\_{0}, I\\_{1}, \\dots, I\\_{K}\\right]$, where\r\n$I\\_{0} = I$ and $I\\_{k}$ is $k$ repeated application of $d\\left(.\\right)$ to $I$. $K$ is the number of levels in the pyramid selected so that the final level has a minimal spatial extent ($\\leq 8 \\times 8$ pixels).\r\n\r\nThe coefficients $h\\_{k}$ at each level $k$ of the Laplacian pyramid $L\\left(I\\right)$ are constructed by taking the difference between adjacent levels in the Gaussian pyramid, upsampling the smaller one with $u\\left(.\\right)$ so that the sizes are compatible:\r\n\r\n$$ h\\_{k} = \\mathcal{L}\\_{k}\\left(I\\right) = G\\_{k}\\left(I\\right) − u\\left(G\\_{k+1}\\left(I\\right)\\right) = I\\_{k} − u\\left(I\\_{k+1}\\right) $$\r\n\r\nIntuitively, each level captures the image structure present at a particular scale. The final level of the\r\nLaplacian pyramid $h\\_{K}$ is not a difference image, but a low-frequency residual equal to the final\r\nGaussian pyramid level, i.e. $h\\_{K} = I\\_{K}$. Reconstruction from a Laplacian pyramid coefficients\r\n$\\left[h\\_{1}, \\dots, h\\_{K}\\right]$ is performed using the backward recurrence:\r\n\r\n$$ I\\_{k} = u\\left(I\\_{k+1}\\right) + h\\_{k} $$\r\n\r\nwhich is started with $I\\_{K} = h\\_{K}$ and the reconstructed image being $I = I\\_{o}$. In other words, starting at the coarsest level, we repeatedly upsample and add the difference image h at the next finer level until we return to the full-resolution image.\r\nSource: [LAPGAN](https://paperswithcode.com/method/lapgan)\r\n\r\nImage : [Design of FIR Filters for Fast Multiscale Directional Filter Banks](https://www.researchgate.net/figure/Relationship-between-Gaussian-and-Laplacian-Pyramids_fig2_275038450)","description_state":"present","introduced_year":1983,"introduced_by":{"title":null,"paper":null,"first_author":null,"n_authors":0,"url_abs":null,"archive_paper_url":null},"source":{"url":null,"title":null,"url_on_a_paper_host":false},"code_snippet_url":null,"code_snippet_url_on_a_code_host":false,"categories":[{"area":"Computer Vision","area_id":"computer-vision","collection":"Image 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