The whole and the parts: the MDL principle and the a-contrario framework

TL;DR

This paper investigates the theoretical connections between the MDL principle and the a-contrario framework, demonstrating their similarities and conditions for equivalence across various data analysis scenarios.

Contribution

It reveals the shared concepts and tools between MDL and a-contrario methods, and formalizes conditions under which they are equivalent.

Findings

Both methods share common concepts and tools.

They yield similar formulations in practical applications.

Conditions for formal equivalence are established.

Abstract

This work explores the connections between the Minimum Description Length (MDL) principle as developed by Rissanen, and the a-contrario framework for structure detection proposed by Desolneux, Moisan and Morel. The MDL principle focuses on the best interpretation for the whole data while the a-contrario approach concentrates on detecting parts of the data with anomalous statistics. Although framed in different theoretical formalisms, we show that both methodologies share many common concepts and tools in their machinery and yield very similar formulations in a number of interesting scenarios ranging from simple toy examples to practical applications such as polygonal approximation of curves and line segment detection in images. We also formulate the conditions under which both approaches are formally equivalent.