Everybody knows about white Gaussian noise

White Gaussian noise is famous because it has very nice properties:

- It is easy to generate using pseudorandom numbers
- Each pixel is an independent, identically distributed Normal variable
- The discrete Fourier, Hartley and Cosine transforms are also white Gaussian noise (except for the obvious symmetries)
- In particular, the power spectrum is mostly flat
- Applying a linear filter renders the pixel values non-independent, but they are still Normal and identically distributed.

Some properties of dubious convenience:

- The mean is zero, thus it cannot be directly represented as a positive-valued image
- Worse, the pixel values are not bounded, thus it has a-priori infinite dynamic range.
- When you see it from far away (zooming-out), it disappears.

Statistics of white gaussian noise and its DFT:

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

White gaussian noise blurred by a small gaussian kernel:

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

White gaussian noise blurred by a larger gaussian kernel:

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

White gaussian noise blurred by a Cauchy kernel:

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

White gaussian noise blurred by a Laplace kernel:

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

White gaussian noise blurred by a Disk kernel:

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

White gaussian noise blurred by a Square kernel:

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

When the spectrum of noise decays as a power-law, we say that it is ``colored'' noise. The exponent $\alpha$ of the power law determines its color. The particular case of $\alpha=0$ corresponds to white noise (a flat spectrum).

$\alpha=2$ purple | $\alpha=1$ blue | $\alpha=0$ white |

$\phantom{a}$ | ||

$\alpha=-1$ pink | $\alpha=-2$ brown | $\alpha=-3$ smooth |

Statistics of Pink noise ($\alpha=-1$):

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

Statistics of Brown noise ($\alpha=-2$):

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

Statistics of Smooth noise ($\alpha=-3$):

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

Statistics of Blue noise ($\alpha=1$):

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |

Statistics of Purple noise ($\alpha=2$):

$u(x)$ | histogram of $u$ | $\log|\hat u(\xi)|$ | average spectral profile |