Inverse Problem: examples

Section: Inverse Problems

Denoising

$A=\mathrm{Id}_N$

$$ y = x + n $$

Deconvolution

$A$ is a Toeplitz matrix, or convolution matrix ($M\leq N$)

$$ y = Ax + n = h*x + n $$ $$ \hat y = \hat h \hat x + \hat n $$

kernel

blured

blured

Inpainting

$A$ is a binary matrix ($a_{ij} = 1$ if the pixel is measured, $a_{ij} = 0$ if the pixel is not measured)

lena

lena masked

Compressive sensing

$A$ is a random matrix (with $M<N$)