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Related Concept Videos

Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Divergence Theorem in 3D Space01:20

Divergence Theorem in 3D Space

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Extended Versions of Green’s Theorem01:27

Extended Versions of Green’s Theorem

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Gauss's Law: Problem-Solving01:10

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Related Experiment Videos

Gerchberg's extrapolation algorithm in two dimensions.

R J Marks Ii

    Applied Optics
    |March 25, 2010
    PubMed
    Summary
    This summary is machine-generated.

    This study generalizes Gerchberg

    Related Experiment Videos

    Area of Science:

    • Signal processing
    • Image reconstruction
    • Computational imaging

    Background:

    • Bandlimited signals present challenges in signal and image processing.
    • Gerchberg's algorithm offers a method for extrapolating 1-D bandlimited signals.
    • Extending these techniques to 2-D is crucial for advanced imaging applications.

    Purpose of the Study:

    • To generalize Gerchberg's 1-D iterative extrapolation algorithm to two dimensions.
    • To explore two distinct 2-D extrapolation approaches.
    • To analyze the impact of known spectral information on extrapolation accuracy.

    Main Methods:

    • Developed two distinct 2-D generalizations of Gerchberg's algorithm.
    • One method requires full spectral pupil knowledge; the other uses pupil projections.
    • Investigated the use of known image portions in the extrapolation process.

    Main Results:

    • The first 2-D method requires complete spectral pupil information.
    • The second 2-D method utilizes only 1-D projections (vertical and horizontal).
    • For real low-pass bandlimited images, this simplifies to knowing maximum spatial frequencies.
    • The second algorithm, when discretized, yields a closed-form solution.

    Conclusions:

    • Two effective 2-D generalizations of Gerchberg's algorithm are presented.
    • The second method offers a more practical approach by requiring less spectral information.
    • The discrete formulation of the second algorithm provides a computationally efficient solution for image extrapolation.