Introduction to the general framework

Linearization is the key to design a feedback strategy between recognition and reconstruction problems. Linearization is a way of approximating a function at a given point using a linear representation of the function. Fibrations allow to integrate the linearization of transformations and functions defined over objects and the linearization of functionals defined over procedures.

Recognition

In this framework, features can be represented as the elements of a fiber bundle

1. Feature detectors outputs are represented in the base space $\mathbf{B}$ of the fiber bundle
2. Feature descriptors are represented as the fiber $\mathbf{F}$ of the fiber bundle, which can be a vectorial fiber bundle (in the general case), a principal fiber bundle (using transformation groups) or a fibration (when dealing with deformations).

Reconstruction

Some remarks about the modeling of the information:

• Linearized information about the objects can be modeled using plane arrangements in arbitrary dimension (hyperplanes)
• The elements of a dual $\mathbf{V^*}$ of a vector space $\mathbf{V}$ defined over a field $\mathbf{F}$ are called covectors or one-forms. In fact, they are linear functionals of the form

• A field is just a set $ \mathbf{F} $ which is a commutative group with respect to multiplication and addition where the additive identity $0$ has no multiplicative inverse. Any field can be used as the scalars of a vector space for linear algebra.
• A plane is the result of the linearization of an object but it can also be represented as a one-form. For instance, the one-form $[1,1,1] \in \mathbf{R}^3$ defines an infinite number of parallel planes over the field $\mathbf{F}$ such as $x + y + z = 0$. See the example at Wolfram Alpha
• A differential one-form is linked to the numeric evaluation of the fields (escalar, vectorial or tensorial) modeling the changes in the surface of the model. In theory, these one-forms can have arbitrary dimension defined as the fiber of a fiber bundle.
• Advantage: a one-form can model objects, transformations and deformations, including functionals defined over them