Application in Image Compression

SVD image transform has different applications in image processing and analysis. We now consider how it can be used for data compression. First we write a matrix $A$ as

$${\bf A}=[a_{ij}]_{N\times N}=[{\bf a}_1,\cdots,{\bf a}_N]$$

where ${\bf a}_i=[a(1,i),\cdots,a(N,i)]^T$ is the ith column vector of ${\bf A}$. The total amount of energy contained in ${\bf A}$ can be represented by the norm of ${\bf A}$ defined as

$$=$$

$$=$$

Now we consider image compression achieved by using only the first eigenimages of the given image ${\bf A}$:

with error

After compression, the energy (information) contained in is:

$$=$$

$$=$$

$$=$$

Note that here we have used the property that . We see that the total amount of energy (information) contained in the original image $A$ is

and the energy (information) lost (contained in ) is

It is therefore obvious that minimum energy is lost if we range $\lambda_i$'s so that

To find out the compression ratio, consider total degrees of freedom in :

The degrees of freedom in the orthogonal vectors are

The same is true for . Including the degrees of freedom in , we have the total d.o.f.:

and the compression ratio is

An example of using SVD for image compression is available here