elements, $S \to G$. For a Galilean geometry $\Gamma_{n+1}$ the frame field is a mapping $\mathbb{R}^n \to \Gamma_{n+1}$. A frame field can be conceptualized by its action on an arbitrary coordinate system for the tangent space of the base space. For $\Gamma_{n+1}$ we can e.g. attach a Galilean ON-system at each point.
Definition 2. A $\Gamma_{n+1}$ coordinate system is an affine coordinate system where n vectors lies in the spatial part. A $\Gamma_{n+1}$ ON-system is a $\Gamma_{n+1}$ coordinate system s.t. the spatial part consists of n dimensional ON-coordinate system and the remaining base vector has unit temporal length.
The property of being a $\Gamma_{n+1}$ ON-system is a Galilean invariant. In the sequel we will use the coordinate system view of frame fields as we find it easier to visualise.
The main idea of Cartans theory about moving frames is to put a frame at each point that is connected to the local structure of the sub-manifold or the function in an invariant way. In this way we get a frame field.
For a function $f$ defined on $S$, all expressions over mixed derivatives w.r.t. the Cartan frame at a certain point are by construction geometrical invariants. This class of invariants are called differential invariants.
On sub-manifolds, we can find the local geometrical structure from how the frame field varies in the local neighborhood.
Let $i$ be any (global) frame and $e$ a frame connected to the local structure s.t. $e = Ai$, where the attitude transformation $A \in G$ is a function of position. The local variation of $e$ can be described in an invariant way in terms of $e$,
where the one-form (see [3]) $C(A)$ is called the connection matrix. In a certain sense, the connection matrix contains all geometric information there is.
Scalar invariants can be generated by contracting the coefficients in the connection matrix on the vectors in the Cartan frame, $c_{ij}e_k$. A useful property of the connection matrix is,
which is a direct consequence of the definition.
The level-sets $f^{-1}(c)$ of smooth scalar functions $f$ are sub-manifolds, the geometric structure of those, the level-set invariants, are invariant w.r.t. the group of constant monotonic transformations $g \circ f$, $g: \mathbb{R} \to \mathbb{R}$, $g' > 0$.
4 Image Geometry
Now we will study Galilean differential geometry of moving images using Cartan frames. Image spaces can be considered being trivial fiber bundle $S \otimes I$, where $S$ is the base space and the fiber $I$ is log intensity [14]. Most of the time we will discuss the image geometry in terms of an arbitrary section of the fiber bundle i.e. functions $f: S \to I$. We will start by reviewing differential geometry for