Singular Value Decompostion application to Image Compression

The singular value decomposition (SVD) is an important matrix factorization technique that provides a numerically stable matrix decomposition that can be used for a variety of purposes such as to obtain low-rank approximations to matrices and to perform pseudo-inverses of non-square matrices to find the solution of a system of equations $$\text{Ax = b}$$.

Generally, we are interested in analyzing a large data set $X \in \mathbb{C}^{n\times m}$.

$$ \mathbf{X}=\left[\begin{array}{cccc} \mid & \mid & & \mid \\ \mathbf{x}_1 & \mathbf{x}_2 & \cdots & \mathbf{x}_m \\ \mid & \mid & & \mid \end{array}\right] $$

For many systems $$ n \gg m $$, resulting in a tall-skinny matrix, as opposed to a short-fat matrix when $$ n \ll m $$. The SVD is a unique matrix decomposition that exists for every complex valued matrix $$ X \in \mathbb{C}^{n\times m} $$:

$$ \mathrm{X} = \mathrm{U\Sigma V}^* $$

where $$ U \in \mathbb{C}^{n\times n} $$ and $$ V \in \mathbb{C}^{m\times m} $$ are unitary matrices with orthonormal columns, and $$ \mathrm{\Sigma} \in \mathbb{C}^{n\times m} $$ is a matrix with real, non-negative entries on the diagonal and zeros off the diagonal.

When $$ n \geq m $$, the matrix $$ \mathrm{\Sigma} $$ has at most m non-zero elements on the diagonal, and may be written as $ \boldsymbol{\Sigma}=\left[\begin{array}{l} \hat{\Sigma} \ 0 \end{array}\right] \text {. } $ Therefore, it is possible to exactly represent X using the economy SVD: $$

\mathrm{X} = \mathrm{U\Sigma V}^* = \left[\begin{array}{ll} \hat{U} & \hat{U}^{\perp} \end{array}\right]\left[\begin{array}{c} \hat{\Sigma} \ 0 \end{array}\right] \mathrm{V}^* = \hat{U}\hat{\Sigma}\hat{V}^* $$

We consder the image of a guy with beautiful landscape behind as shown below This image has 8192 by 5461 pixels.

Original	Gray

We apply the svd code on this image:

• install necessary files in requirements.txt
• run the svd.py
• The plots below show truncation of the columns to when r = 5, 10, 20

r =5	r =10	r=20

Plot the diagonal singular values:

• Run the scd_analysis.py code
  

The result

Singular values	Cummulatve Singular Values

Test for around 90% of the cummulative energy where r = 1200. we get:

The above steps can be done in the matlab file. Just by running SVD_1.m

.That's all. Thank You