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LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Linear Methods for Image Compression Methods SVD Math 420, Prof. Beezer PCA DCT Aidan Meacham University of Puget Sound Outline LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Methods Preliminaries SVD Color Spaces PCA DCT Lossy vs. Lossless Methods SVD PCA DCT Outline LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Methods Preliminaries SVD Color Spaces PCA DCT Lossy vs. Lossless Methods SVD PCA DCT RGB Color Space LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Methods SVD PCA DCT (cid:73) Intensity and Representation (cid:73) Gamut mapping and Translation (cid:73) Absolute Color Spaces Lossy vs. Lossless Methods LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Methods SVD PCA DCT (cid:73) Lossless Methods - GIF / LZW (cid:73) Usefulness of Lossy Compression (cid:73) Limit - arithmetic, entropy, and LZW coding Outline LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Methods Preliminaries SVD Color Spaces PCA DCT Lossy vs. Lossless Methods SVD PCA DCT SVD LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Methods Definition SVD PCA √ √ √ DCT (cid:73) A is a matrix with singular values σ , σ ,..., σ , 1 2 r where r is the rank of A∗A and σ are eigenvalues of A i (cid:73) Define V = [x |x |...|x ], U = [y |y |...|y ] where 1 2 n 1 2 n {x } is an orthonormal basis of eigenvectors for A∗A i and y = √1 Ax i σi i SVD LinearMethodsfor Image Compression √ AidanMeacham (cid:73) Additionally, s = σ i i (cid:73) Preliminaries ColorSpaces Lossyvs.Lossless s1 Methods 0 SVD s PCA 2 DCT ... S = sr 0 0 ... 0 0| (cid:73) Thus, AV = US A = USV∗ SVD Truncated Form LinearMethodsfor Image Compression AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless r Methods (cid:73) A = (cid:80)s x y∗, where r is the rank of A∗A and the s SVD i i i i PCA i=1 DCT are ordered in decreasing magnitude, s ≥ s ≥ ··· ≥ s 1 2 r (cid:73) For i < r, this neglects the lower weighted singular values (cid:73) Discarding unnecessary singular values and the corresponding columns of U and V decreases the amount of storage necessary to reconstruct the image SVD Example LinearMethodsfor Image Compression Sage AidanMeacham Preliminaries ColorSpaces Lossyvs.Lossless Methods SVD PCA DCT (cid:73) Import image and convert to Sage matrix (cid:73) Perform SVD decomposition (cid:73) Choose number of singular values and reconstruct
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