Image Compression using SVD

Source code notebook Author Update time

This demonstration shows how to work with color channels to explore image compression using the Singular Value Decomposition (SVD).

using Images, TestImages
using LinearAlgebra

img = float.(testimage("mandrill"))
channels = channelview(img)

function rank_approx(F::SVD, k)
    U, S, V = F
    M = U[:, 1:k] * Diagonal(S[1:k]) * V[:, 1:k]'

For each channel, we do SVD decomposition, and then reconstruct the channel using only the K largest singular values.

The image is compressed because for each channel we only need to save two small matrices and one vector – truncated part of (U, S, V). For example, if the original image is gray image of size (512, 512), and we rebuild the image with $50$ singular values, then we only need to save $2 \times 512 \times 50 + 50$ numbers to rebuild the image, while original image has $512 \times 512$ numbers. Hence this gives us a compression ratio $19.55\%$ if we don't consider the storage type.

# before julia v1.1:
# svdfactors = (svd(channels[1,:,:]), svd(channels[2,:,:]), svd(channels[3,:,:]))
svdfactors = svd.(eachslice(channels; dims=1))
imgs = map((10, 50, 100)) do k
    colorview(RGB, rank_approx.(svdfactors, k)...)

mosaicview(img, imgs...; nrow=1, npad=10)

From left to right: original image, reconstructed images using 10, 50, 100 largest singular values. We can see that $50$ largest singular values are capable of rebuilding a pretty good image.

This page was generated using DemoCards.jl and Literate.jl.