Fundamental principles of image processing

Idea

• An image is a
  • matrix $I_{ij}\ \Rightarrow$ Use linear algebra
  • function of two variables $f(x,y)\ \Rightarrow$ Use calculus
• An image can be decscribed in terms of frequencies
  • Low frequencies $\leftrightarrow$ background, overall shape
  • High frequencies $\leftrightarrow$ details, edges, noise etc.

Linear algebra in image processing $\Rightarrow$ Fourier analysis

• A image that we look at is given in its "standard" Euclidean basis.
• Find new basis where different features (shapes, edges, noise) are represented as basis vectors.
• Look at an image in its standard basis, but manipulate the image in a different basis.

  

Figure: High pass filtering of an image. Here coordinates for high frequency basis vectors (located towards the boundaries in the Fourier representation) are emphasized.