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    Area of Science:

    • Optical Metrology
    • Signal Processing
    • Computational Physics

    Background:

    • Interferometry generates phase data crucial for measurements.
    • Existing methods for phase estimation can be sensitive to noise and require complex processing.
    • Direct phase estimation from single, noisy patterns remains a challenge.

    Purpose of the Study:

    • To introduce a novel algorithm for direct phase estimation from single noisy interferometric patterns.
    • To generalize smoothing spline interpolation for implicitly defined data.
    • To demonstrate the algorithm's applicability in classical and digital speckle pattern interferometry (DSPI).

    Main Methods:

    • Developed the implicit smoothing spline (ISS) algorithm.
    • Derived the necessary mathematical equations for ISS.
    • Applied ISS to direct continuous phase estimation problems.

    Main Results:

    • The ISS algorithm provides accurate direct phase estimation.
    • Numerical illustrations confirm the high quality of results obtained with ISS.
    • The method is effective for both classical interferometry and DSPI.

    Conclusions:

    • The implicit smoothing spline (ISS) method offers a robust approach for direct phase estimation.
    • ISS is a valuable tool for analyzing noisy interferometric data.
    • This technique enhances phase measurement capabilities in optical metrology.