Approximate invariance

Prior terms can often be related to the assumption of approximate invariances or approximate symmetries. A Laplacian smoothness functional, for example, measures the deviation from translational symmetry under infinitesimal translations.

Consider for example a linear mapping

$$\phi \rightarrow \tilde \phi = {\bf S} \phi ,$$ (208)

given by the operator ${\bf S}$. To compare $\phi$ with $\tilde \phi$ we define a (semi-)distance defined by choosing a positive (semi-)definite ${\bf K}_S$, and use as error measure

$$\frac{1}{2}\Big( (\phi-{\bf S}\phi),\, {\bf K}_S(\phi-{\bf S}\phi) \Big) =\frac{1}{2}\Big( \phi,\,{\bf K} \phi\Big) .$$ (209)

Here

$${\bf K} = ({\bf I}-{\bf S})^T {\bf K}_S({\bf I}-{\bf S})$$ (210)

is positive semi-definite if ${\bf K}_S$ is. Conversely, every positive semi-definite K can be written ${\bf K}$ = ${\bf W}^T {\bf W}$ and is thus of form (210) with ${\bf S}$ = ${\bf I}-{\bf W}$ and ${\bf K}_S = {\bf I}$. Including terms of the form of (210) in the error functional forces $\phi$ to be similar to $\tilde \phi$.

A special case are mappings leaving the norm invariant

$$(\phi,\, \phi) = ({\bf S} \phi, {\bf S} \phi) = (\phi, \,{\bf S}^T {\bf S} \phi) .$$ (211)

For real $\phi$ and $\tilde \phi$ i.e., $({\bf S}\phi)$ = $({\bf S}\phi)^*$, this requires ${\bf S}^T = {\bf S}^{-1}$ and ${\bf S}^* = {\bf S}$. Thus, in that case ${\bf S}$ has to be an orthogonal matrix $\in O(N)$ and can be written

$${\bf S}(\theta) = e^{\bf A} = e^{\sum_i \theta_i {\bf A}_i}... ...}^\infty \frac{1}{k!}\left(\sum_i \theta_i{\bf A}_i\right)^k ,$$ (212)

with antisymmetric ${\bf A} = - {\bf A}^T$ and real parameters $\theta_i$. Selecting a set of (generators) ${\bf A}_i$ the matrices obtained be varying the parameters $\theta_i$ form a Lie group. Up to first order the expansion of the exponential function reads ${\bf S} \approx 1+\sum_i\nabla_i {\bf A}_i$. Thus, we can define an error measure with respect to an infinitesimal transformation by

$$\frac{1}{2} \sum_i \left(\frac{\phi - (1 + \theta_i {\bf A... ...{1}{2}(\phi,\, \sum_i {\bf A}_i^T {\bf K}_S {\bf A}_i \phi ) .$$ (213)