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Transformations in mathematics alter the position or orientation of a function’s graph while preserving its fundamental shape. One important type of transformation is the horizontal shift, which involves modifying the input variable within a function’s equation. This operation affects where outputs occur along the horizontal axis but does not alter the function’s overall structure.A horizontal shift is achieved by replacing the input variable x with either x + c or x - c,...
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A function's graph can be modified by changing its position or size without altering its overall shape. These transformations allow the graph to be moved across the coordinate plane while preserving its pattern and structure. One of the most common transformations is shifting, which repositions the graph without distorting it.When the output of a function is adjusted by adding or subtracting a constant, the graph shifts vertically. A positive value moves the graph upward, while a negative value...
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Absolute testing method of shift-rotation based on the influence function.

Xiaochuan Hu, Shuai Zhang, Xi Hou

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    Summary
    This summary is machine-generated.

    This study introduces an optimized shift-rotation method for absolute optical surface testing. The new influence function approach improves accuracy and practicality over traditional Zernike polynomial and pixel-level methods.

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    Area of Science:

    • Optical metrology
    • Adaptive optics
    • Surface characterization

    Background:

    • Absolute testing of optical surfaces is crucial for high-accuracy measurements.
    • Traditional shift-rotation methods using Zernike polynomials suffer from computational complexity and poor high-frequency fitting.
    • Pixel-level spatial frequency methods are computationally intensive and lack practical applicability.

    Purpose of the Study:

    • To present an optimized absolute testing method for optical surfaces using the shift-rotation technique.
    • To enhance the accuracy of mid- and high-frequency surface measurements.
    • To improve the practicality of absolute optical surface testing.

    Main Methods:

    • An optimized shift-rotation method based on the influence function is proposed.
    • The influence function concept from adaptive optics replaces Zernike polynomials.
    • Theoretical simulations and experimental verifications were conducted.

    Main Results:

    • The influence function-based method demonstrates improved calculation accuracy for mid- and high-frequency surfaces.
    • Higher precision in absolute surface reconstruction was achieved compared to traditional methods.
    • Experimental results validated the method's effectiveness in reconstructing test and reference surfaces.

    Conclusions:

    • The proposed shift-rotation method utilizing influence functions offers superior accuracy and practicality for absolute optical surface testing.
    • This method effectively overcomes the limitations of Zernike polynomial and pixel-level approaches.
    • It enables precise reconstruction of optical surfaces, particularly for mid- and high-frequency details.