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    Area of Science:

    • Harmonic analysis
    • Spherical geometry
    • Signal processing

    Background:

    • Haar-type tight framelets offer efficient signal representation.
    • Spherical data analysis presents unique challenges due to curvature.
    • Hierarchical partitions are crucial for localized analysis on complex domains.

    Purpose of the Study:

    • To develop a general framework for Haar-type tight framelets on compact sets.
    • To construct area-regular spherical Haar tight framelets with directionality.
    • To evaluate the denoising effectiveness of these framelets and a CNN model.

    Main Methods:

    • Construction of a general theoretical framework for Haar-type tight framelets.
    • Development of an area-regular hierarchical partition on the two spheres.
    • Implementation of spherical Haar tight framelets and a convolutional neural network (CNN).

    Main Results:

    • Successful construction of area-regular spherical Haar tight framelets.
    • Demonstrated effectiveness of framelets in denoising experiments.
    • Proposed CNN model outperformed traditional thresholding methods.

    Conclusions:

    • Area-regular spherical Haar tight framelets are effective for signal denoising.
    • The proposed CNN model offers robust and generalizable spherical signal denoising.
    • This work advances signal processing techniques on spherical domains.