Babel function

Background

The Babel function measures the maximum total coherence between a fixed atom and a collection of other atoms in a dictionary.

Definition and Formulation

The Babel function (also known as cumulative coherence) of a dictionary ${\boldsymbol {A}}$ with normalized columns is a real-valued function that is defined as

$\mu (p)=\max _{|\lambda |=p}\{\max _{j\notin \lambda }\{\sum _{i\in \lambda }{|{\boldsymbol {a}}_{i}^{\boldsymbol {T}}{\boldsymbol {a}}_{j}|}\}\}$

where ${\boldsymbol {a}}_{k}$ are the columns (atoms) of the dictionary ${\boldsymbol {A}}$