The central projection transform can be employed to extract invariant features by combining contour-based and region-based methods. However, the central projection transform only considers the accumulation of the pixels along the radial direction. Consequently, information along the radial direction is inevitably lost. In this paper, we propose the Mellin central projection transform to extract affine invariant features. The radial factor introduced by the Mellin transform, makes up for the loss of information along the radial direction by the central projection transform. The Mellin central projection transform can convert any object into a closed curve as a central projection transform, so the central projection transform is only a special case of the Mellin central projection transform. We prove that closed curves extracted from the original image and the affine transformed image by the Mellin central projection transform satisfy the same affine transform relationship. A method is provided for the extraction of affine invariants by employing the area of closed curves derived by the Mellin central projection transform. Experiments have been conducted on some printed Chinese characters and the results establish the invariance and robustness of the extracted features.

We introduce a new affinely invariant structure on smooth surfaces in ℝ3 by defining a family of reflections in all points of the surface. We show that the bifurcation set of this family has a special structure at ‘ points’, which are not detected by the flat geometry of the surface. These points (without an associated structure on the surface) have also arisen in the study of the centre symmetry set; using our technique we are able to explain how the points are created and annihilated in a generic family of surfaces. We also present the bifurcation set in a global setting.

Let $K\,\subset \,{{\mathbb{R}}^{n+1}}$ be a convex body of class ${{C}^{2}}$ with everywhere positive Gauss curvature. We show that there exists a positive number $\delta \left( K \right)$ such that for any $\delta \,\in \,\left( 0,\,\delta \left( K \right) \right)$ we have $\text{Vol}\left( {{K}_{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}_{\delta }} \right)}^{*}} \right)\,\ge \,\text{Vol}\left( K \right)\,\cdot \,\text{Vol}\left( {{K}^{*}} \right)\,\ge \,\text{Vol}\left( {{K}^{\delta }} \right)\,\cdot \,\text{Vol}\left( {{\left( {{K}^{\delta }} \right)}^{*}} \right)$, where ${{K}_{\delta }}$, ${{K}^{\delta }}$ and ${{K}^{*}}$ stand for the convex floating body, the illumination body, and the polar of $K$, respectively. We derive a few consequences of these inequalities.