Example: Infinitesimal translations

A Laplacian smoothness prior, for example, can be related to an approximate symmetry under infinitesimal translations. Considering the group of $d$-dimensional translations which is generated by the gradient operator $\nabla$, this can be verified by recalling the multidimensional Taylor formula for expansion of $\phi$ at $x$

$${\bf S}(\theta) \phi(x) = e^{ \sum_i \theta_i \nabla_i } \p... ..._i \theta_i \nabla_i\right)^{k}}{k!} \phi(x) = \phi(x+\theta).$$ (218)

Up to first order ${\bf S} \approx 1+\sum_i\theta_i \Delta_i$. Hence, for infinitesimal translations, the error measure of Eq. (213) becomes

$$\frac{1}{2}\sum_i \left(\frac{\phi - (1 + \theta_i {\nabla... ..._i^T \nabla_i \phi ) \!=\!-\frac{1}{2}(\phi,\, \Delta \phi ) ,$$ (219)

assuming vanishing boundary terms and choosing ${\bf K}_S$ = ${\bf I}$. This is the classical Laplacian smoothness term.