4. Variational Image Processing

In the last chapter we have observed, that the maximum a-posteriori estimation yields an optimization problem consisting of the sum of a data term and a regularization term including prior information. In this chapter we are analyzing such a class of problems using variational calculus. We want to minimize a family of functionals \(\operatorname{arg\,min}_{u\in X} E(u;f)\) written as

(4.1)\[ E(u; f) = \frac{1}{2} \int _{\Omega} \left( f(x) - (Au)(x) \right)^2 \, dx + \alpha R(u)\,,\]

where \(\alpha > 0\) is a regularization weight for the regularizer \(R:X\to \R\), and \(A:X\to X\) denotes a linear operator modelling the ‘forward’ problem (e.g. the convolution, for denoising tasks we choose \(A = \operatorname{Id}\)).
We start with some examples for a choice of regularizer.

Example 4.1 (Variational Regularizer)

1. Tikhonov regularization penalizes large \(u\) in the least squares sense

   \[R(u) = \frac{1}{2}\|u\|_2^2\,.\]

2. \(\mathbf{l^1}\)-regularization (LASSO) is sparsity enducing

   \[R(u) = \|u\|_1\,.\]

3. Gradient regularization penalizes the rate of change of a function and makes it smoother

   \[R(u) = \frac{1}{2}\|\nabla u\|_2^2\,.\]

4. Total variation regularization penalizes the gradient in the \(l^1\) sense. It induces discontinuouities (jumps) and piecewise constant functions.

   \[R(u) = \|\nabla u\|_1\,.\]