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Orthogonal trajectories describe the geometric relationship between two families of curves that intersect each other at right angles. One illustrative case involves a family of parabolas that open sideways along the x-axis. These curves share a common shape but differ by a scaling parameter, resulting in a set of curves that all pass through the origin and widen at different rates.Determining Orthogonal TrajectoriesTo identify the orthogonal trajectories for these parabolas, the first step...
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Orthogonal basis for the optical transfer function.

Chelo Ferreira, José L López, Rafael Navarro

    Applied Optics
    |December 14, 2016
    PubMed
    Summary

    We developed orthogonal functions to represent optical transfer functions (OTFs), incorporating the perfect OTF as the first function. This method accurately expands OTFs, with accuracy depending on image quality and the number of basis functions used.

    Area of Science:

    • Optics
    • Image Science
    • Applied Mathematics

    Background:

    • Optical Transfer Functions (OTFs) are crucial for characterizing imaging system performance.
    • Representing complex OTFs efficiently is a persistent challenge in optical engineering.
    • Existing methods may lack a unified theoretical framework for diverse OTF types.

    Purpose of the Study:

    • To introduce novel systems of orthogonal functions for representing optical transfer functions (OTFs).
    • To establish a rigorous theoretical framework for OTF expansion using orthogonal functions.
    • To explore the efficiency and applicability of the proposed method for various imaging scenarios.

    Main Methods:

    • Development of orthogonal function systems (q_n) tailored for OTF representation.

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  • Inclusion of the diffraction-limited OTF (OTF_perfect) as the initial basis function (q_0).
  • Application of a theoretical framework involving variable changes to established orthogonal systems (Legendre polynomials, spherical harmonics).

    • Main Results:

    • Accurate linear expansion of rotationally symmetric OTFs achieved with approximately 10 basis functions.
    • The number of required basis functions for accurate OTF representation is dependent on image quality.
    • Increased complexity and number of basis functions are needed for poorer image quality scenarios.

    Conclusions:

    • The proposed orthogonal function systems offer an effective method for representing optical transfer functions.
    • The efficiency of the method is influenced by image quality, potentially requiring more basis functions for lower quality images.
    • Potential applications include the development of new image quality metrics and advanced optical system analysis.