Examples of inverse problem

Section: Inverse Problems

Direct problem

Let

  • $x\in\mathbb{R}^N$ be the signal of interest (clean image, music etc.)
  • $A\in\mathbb{R}^{MN}$ be a linear operator (sensing matrix, mixing matrix, diffusion matrix etc.)
  • $y\in\mathbb{R}^M$ be the (noisy) observed/measured signal
  • $b\in\mathbb{R}^M$ be some noise (assumed to be white gaussian noise)

The direct problem is :

$$ y = Ax + b $$

Inverse problem

The goal of the inverse problem is to estimate the original signal $x$ from the measurement $y$

  • If $M\geq N$ the problem is said (over)-determined
  • If $M<N$ the problem is said under-determined