We have a brilliant Shiny application demonstrating Image Compression with Singular Value Decomposition. One could get a sense ofm SVD quickly from it: https://yihui.shinyapps.io/imgsvd/.

ImgSVD made by Nan Xiao, who is Data Analyst Intern at SupStat and Yihui Xie who is Software Engineer at RStudio, also advisory Data Scientist at SupStat Inc.

The interface looks like the following:

Singular Value Decomposition is a complex technique. It is a matrix factorization and is widely used in dimensionality reduction. Its result is an approximation of a matrix, and it is flexible because we can determine the degree of the approximation by the parameter **k**. Besides, image compression is also focusing on approximation. We can use this method for image compression if we regard an image as a matrix. Following is a demonstration of the result of our algorithm.

Original image is

When we set k=1

When we set k=5

When we set k=10

When we set k=20

When we set k=100

When k=1, we can only see the color of the original picture. When k=5, the scheme of the scene is vaguely recognizable. As k keeps increasing, the picture becomes clearer. When k=100, it is almost the same as the original one. SVD is a computation expensive algorithm, therefore the function svd() in R cannot fulfill our requirement. So instead, we use the function svds() from package rARPACK which will only calculate the first k terms. The code for computing is as follows:

https://gist.github.com/casunlight/9628504

**References:**Image Compression with the SVD in RFoundations of Data ScienceSVD on Wikipedia