Shape analysis is the automatic analysis of geometric shapes.
Once the objects are given, either by modeling, by scanning or by extracting shape from 2D or 3D images, they have to be simplified in order for a comparison to be achieved.
The simplified representation is often called a shape descriptor (which is what we relate to fingerprints or a signature). These simplified representations try to carry most of the important information, while being easier to handle, to store and to compare than the shapes directly.
A complete shape descriptor is a representation that can be used to completely reconstruct the original object (for example the medial axis transform).
Shape preserving approximations are used in the design of curves or surfaces, in order to predict or control their “shape” by checking the shape of the control points, i.e. the vertices of a given polygonal arc or maybe a polyhedral surface.
Shape preserving approximations usually find applications in Computer Aided Design, as well as in Computer Graphics and in the Design of Fonts. On the other hand, in Approximation Theory, the construction of approximation operators that mimic the “shape” of the function to be approximated is required in many applications.
The key ingredient to construct shape preserving approximations and operators is the variation diminishing property. In mathematics, the variation diminishing property of certain mathematical objects involves diminishing the number of changes in sign (positive to negative or vice versa.)
In order to implement that, an algorithm is needed. For example, the Chaikin’s algorithm is an efficient algorithm to generate quadratic spline curves starting from a
set of control points. Chaikin’s algorithm is a simple example of a wide
class of algorithms, usually called corner cutting algorithms. The algorithm cuts iteratively the corners of a polygonal arc {(Pk)^m−1, k ∈ Z} converging in the limit
to a smooth curve.
More informations about the Chaikin’s algorithm can be found here.
Shape preserving can be related also with image scaling. There are different algorithm with which it’s possible to succesfully scale an image. More informations about those algorithms can be found here.
Rambling things 