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Linear Approximations01:23

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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...
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Approximation errors and model reduction in three-dimensional diffuse optical tomography.

Ville Kolehmainen1, Martin Schweiger, Ilkka Nissilä

  • 1Department of Physics, University of Kuopio, P.O. Box 1627, 70211 Kuopio, Finland. Ville.Kolehmainen@uku.fi

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|October 3, 2009
PubMed
Summary
This summary is machine-generated.

Model reduction in diffuse optical tomography (DOT) is improved using an approximation error model. This method compensates for errors from coarse meshes and truncated domains, enabling more efficient DOT computations.

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Area of Science:

  • Biomedical optics
  • Computational modeling
  • Inverse problems

Background:

  • Diffuse optical tomography (DOT) often requires model reduction due to computational constraints.
  • Conventional DOT methods face limitations with coarse meshes and truncated domains.
  • Modeling errors significantly impact DOT accuracy.

Purpose of the Study:

  • To introduce and evaluate an approximation error model for diffuse optical tomography.
  • To compensate for modeling errors arising from domain truncation and coarse meshes in DOT.
  • To assess the feasibility of using less restrictive computational models in DOT.

Main Methods:

  • Application of a (Bayesian) approximation error model.
  • Compensation for modeling errors in the forward problem of DOT.
  • Testing with a three-dimensional numerical example and experimental data.

Main Results:

  • The approximation error model effectively compensates for modeling errors.
  • Enables the use of coarser meshes and smaller computation domains than conventional methods.
  • Demonstrates improved feasibility for computationally intensive DOT models.

Conclusions:

  • The approximation error model enhances the practicality of diffuse optical tomography.
  • Allows for more computationally efficient DOT by relaxing mesh and domain constraints.
  • Offers a viable solution for overcoming computational limitations in DOT.