Signal Processing/Image Editing

< Signal Processing

As an image is a type of 2-D signal; instead of just time-amplitude pairs that correspond to a voice transmission, consider "time in the X domain"-"time in the Y domain"-amplitude pairs. That is, an X coordinate, a Y coordinate and an amplitude. This will give you a monochromatic image. As such, signal processing tools can be used in editing images.

Singular Value DecompositionEdit

The Singular Value Decomposition is a matrix decomposition (another way to say factorization).



  • U is an m×m unitary matrix over.
  • Σ is an m×n diagonal matrix with nonnegative real numbers s_{n,n} on the diagonal where
  • V*, an n×n unitary matrix. V* is the conjugate transpose (take the complex conjugate of all entries, and then perform a transpose operation on the matrix) of V.

The diagonal entries Σi,i of Σ are known as the singular values of M.

Image CompressionEdit

The nature of the singular values Σ is such that for a certain k,


In order to transmit a 10x10 monochromatic image with 2 values ("on" or "off") it would require a matrix that has 10x10 = 100 entries. Consider the following image.


This image can be represented by the following matrix.


The singular value decomposition of which is




Using the matrix


And corresponding "truncated" versions of U and V* (Use only the first three columns of U and the first three columns of V*), we can find that


A cursory examination of the previous matrix will show that M_{sm} is approximately equal to M. Note that the truncated version of U and V* use 3*10 numbers each. The total number of values needed for this type of storage is 2*30+3 = 63. Which means data usage is reduced by nearly half.

Noise RemovalEdit