# Image compression

What is the relevance between change of basis with linear algebra ? Matrix is the coordinate based description of linear transformation.

If you have an image that is $latex {512 \times 512}&fg=000000$ pixels and each is a gray scale coded, the representing this data is very expensive. Instead one way to compress this data is to choose the right basis. If you break down the matrix in to$latex {8\times 8}&fg=000000$ block,then you can represent the data in $latex {\bf R^{64}}&fg=000000$. If you can find a way to represent this 64 dimensional vector in such a way that only the first two or three components are important, then one can compress the data in such a way that you only have to store 3 or 4 coefficients , i.e each vector in $latex {\bf R^{64}}&fg=000000$ can be written as

$latex \displaystyle y = c_1 w_1 + c_2 w_2 + c_3 w_3 \; , where \{ w_1, w_2, \ldots \} \text{ is a basis } &fg=000000$

JPEG uses to work with Fourier basis but they have changed to Wavelet basis.

Two matrices are similar if $latex {B = M^{-1}\;A\;M}&fg=000000$

A good basis is one that has fast inversion and good compression.

One can also talk about change of basis for a transformation.

I don’t know the number of times I have watched strang lectures. They always have something new to offer, a newer perspective to see things.