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Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
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Bayesian Inference for Neighborhood Filters With Application in Denoising.

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    This study introduces a unified Bayesian framework for image denoising filters. It enables direct reasoning and parameter estimation for filters like Yaroslavsky and bilateral, improving image quality efficiently.

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    Area of Science:

    • Computer Vision
    • Image Processing
    • Statistical Modeling

    Background:

    • Range-weighted neighborhood filters are popular for edge preservation but lack direct property reasoning and parameter estimation.
    • Previous methods required indirect connections to other models for analysis.

    Purpose of the Study:

    • To introduce a unified empirical Bayesian framework for direct reasoning and parameter estimation of neighborhood filters.
    • To develop an efficient algorithm for estimating the essential parameter, range variance.

    Main Methods:

    • A neighborhood noise model was proposed for inferring Yaroslavsky, bilateral, and modified non-local means filters using joint maximum a posteriori and maximum likelihood estimation.
    • Range variance estimation was achieved via model fitting to an empirical distribution of a chi scale mixture variable.
    • An expectation-maximization and quasi-Newton optimization algorithm was devised for efficient model fitting.

    Main Results:

    • The proposed framework successfully fits noisy images and efficiently estimates range variance.
    • Extensive experiments demonstrated the framework's effectiveness across various configurations (kernel functions, filter types, noise types).
    • A recursive fitting and filtering scheme improved color-image denoising quality.

    Conclusions:

    • The unified empirical Bayesian framework provides a direct and efficient method for neighborhood filter analysis and parameter estimation.
    • This approach significantly enhances image denoising performance, particularly for color images.