Matching

Extraction of global invariants of shapes
Constructed from centered moments:
$\mu_{i\,j}\:=\:\iint_S (x-\bar{x}_S)^i (y-\bar{y}_S)^j\,dx\,dy$ Inertia matrix $I_S = \begin{pmatrix}\mu_{2\,0} & \mu_{1\,1} \\ \mu_{1\,1} & \mu_{0\,2}\end{pmatrix}$ is not invariant to rotation: $I_{R_\theta S}\:=\: R_\theta S R_{-\theta}$ Thus $\det\,I_S$ and tr$\,I_S$ are invariants for rotation
For similarity, divide by $\mu_{0,0}=\vert S\vert$ to the right power Invariants
Rotation: $\vert S\vert$, tr$\,I_S$ and $\det{I_S}$
Similarity: tr$\,I_S\:/ \vert S\vert^2$ and $\det{I_S}\:/ \vert S\vert^4$
Others: cf [Reiss]
Correspondences
$S_1\leftrightarrow S_2$ if their invariants are equal (up to a certain tolerance)

Summer School on Mathematical Problems in Image Processing, Trieste, Italy, September 4th-18th 2000