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Publication# Redundancy in Non-Orthogonal Transforms

Abstract

Compression efficiency is mainly driven by redundancy of the overcomplete set of functions chosen for the signal decomposition. In Matching Pursuit algorithms for example, the redundancy of the dictionary influences the convergence of the residual energy. The set of functions or dictionary plays a crucial role into the non-orthogonal transform properties, and more particularly in the ability of this transform to compact the signal energy. Redundancy provides an important criteria in the design of dictionaries and quantifies the power of the transform to capture signal features. The size of the dictionary provides a first indicator of the dictionary propertie, but it does not take into account the distribution of the atoms. This paper provides a formulation for the structural redundancy of an overcomplete set of functions. We also compute the structural redundancy factor for random dictionaries and show its implication in the practical context of Matching Pursuit.

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Webster's Dictionary

Webster's Dictionary is any of the English language dictionaries edited in the early 19th century by Noah Webster (1758–1843), an American lexicographer, as well as numerous related or unrelated dictionaries that have adopted the Webster's name in his honor. "Webster's" has since become a genericized trademark in the United States for English dictionaries, and is widely used in dictionary titles. Merriam-Webster is the corporate heir to Noah Webster's original works, which are in the public domain.

Dictionary

A dictionary is a listing of lexemes from the lexicon of one or more specific languages, often arranged alphabetically (or by consonantal root for Semitic languages or radical and stroke for logographic languages), which may include information on definitions, usage, etymologies, pronunciations, translation, etc. It is a lexicographical reference that shows inter-relationships among the data. A broad distinction is made between general and specialized dictionaries.

Matching pursuit

Matching pursuit (MP) is a sparse approximation algorithm which finds the "best matching" projections of multidimensional data onto the span of an over-complete (i.e., redundant) dictionary . The basic idea is to approximately represent a signal from Hilbert space as a weighted sum of finitely many functions (called atoms) taken from . An approximation with atoms has the form where is the th column of the matrix and is the scalar weighting factor (amplitude) for the atom . Normally, not every atom in will be used in this sum.

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