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Mathematical definition and analysis of the retinex algorithm.
Edoardo Provenzi1, Luca De Carli, Alessandro Rizzi
1Dipartimento di Tecnologie dell'Informazione, Università di Milano, Via Bramante 65, Crema (CR), Italy. provenzi@dti.unimi.it
Summary
This study provides a mathematical formula to analyze the Retinex algorithm, explaining its behavior and iterations. The formula offers a unified approach for various Retinex implementations, validated by image tests.
Area of Science:
- Computer Vision
- Image Processing
- Computational Mathematics
Background:
- The Retinex algorithm, developed by Land and McCann, is fundamental for image processing tasks like color constancy.
- Existing analyses often lack a unified mathematical framework to explain diverse Retinex implementations.
Purpose of the Study:
- To provide a detailed mathematical analysis of the original Retinex algorithm.
- To develop an analytic formula describing Retinex algorithm behavior.
- To establish a common mathematical ground for various Retinex implementations.
Main Methods:
- Developing an analytic formula to model the Retinex algorithm's behavior.
- Examining multiple Retinex versions, including those with and without thresholding.
- Mathematically analyzing recursive iterations and predicting behavior based on the number of paths.
Main Results:
- An analytic formula accurately describes the behavior of the Retinex algorithm.
- The formula allows prediction of algorithm performance with varying parameters, such as the number of paths.
- Mathematical analysis confirmed the validity of the proposed framework through image tests.
Conclusions:
- The presented mathematical framework offers a unified understanding of the Retinex algorithm.
- The analytic formula facilitates the analysis and comparison of different Retinex implementations.
- This work validates the mathematical approach through empirical testing on image data.