ImgExample

Image Processing Example

A color-indexed or RGB digital image is essentially a big matrix (or three), and as such provides an excellent example of some of the properties of the SVD. We can make use of the ordering of the singular values to reduce the data needed to describe the most 'important' aspects of the system. Specifically, we can write the following SVD version of the Spectral Decomposition:

M = d_{1} \vec{U_{1}} \otimes \vec{V_{1}} + d_{2} \vec{U_{2}} \otimes \vec{V_{2}} + . . . + d_{n} \vec{U_{n}} \otimes \vec{V_{n}}

where

d_{i}

\vec{U_{i}}

, and

\vec{V_{i}}

are the i^{th} Singular Value, Left Singular Vector (column of

U

), and Right Singular Vector (row of

V

), respectively.

Just as with the eigenvalues in a Spectral Decomposition, the first few Singular Values will be the largest, and thus these terms will make the largest 'contributions' to

M

. In fact, the matrix constructed by adding the first

k

such terms is the rank-k matrix closest to

M

, in terms of least-squares distance between elements. This provides a simple compression or feature extraction technique.

The following image is the blue channel from a 650 \times 600 RGB image. We convert the elements to floats, contruct the SVD, and then recreate the image using the approximation above for various values of

k

. Note that for, say,

k = 10

, we are using (at most)

k * 600 * 2 + 1 = 1201

numbers, rather then the original

600 * 650 = 390,000

, a significant compression.