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In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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    Area of Science:

    • Computer Vision
    • Image Processing
    • Machine Learning

    Background:

    • Kernel-based filters (e.g., bilateral filter, nonlocal means) are widely used in image processing.
    • Variational, Bayesian, and proximal formulations offer powerful frameworks for regularized inverse problems.
    • These formulations often lead to complex optimization problems requiring iterative solutions.

    Purpose of the Study:

    • To establish a connection between kernel-based filtering and variational Bayesian formulations.
    • To demonstrate how local adaptive filters can approximate solutions to global optimization problems.
    • To provide a unified framework for interpreting local filters within variational methods.

    Main Methods:

    • Connecting kernel-based filtering with variational Bayesian regularized least squares.
    • Utilizing proximal operators to analyze optimization problems.
    • Deriving one-shot, locally adaptive kernelized solutions.

    Main Results:

    • A novel link is established between kernel-based filters and variational Bayesian frameworks.
    • Locally adaptive filters are shown to approximate global optimization solutions efficiently.
    • A method is presented to derive kernelized solutions in a single step using local operations.

    Conclusions:

    • The work bridges local filtering techniques with advanced variational methods.
    • This approach offers a powerful and efficient way to solve complex image processing problems.
    • It enables a new interpretation of existing local filters within a broader theoretical context.