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Sequential linear interpolation of multidimensional functions.

J Z Chan1, J P Allebach, C A Bouman

  • 1Color Savvy Syst. Inc., Springboro, OH.

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|January 1, 1997
PubMed
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We developed sequential linear interpolation (SLI), a novel grid method for approximating complex functions. SLI optimizes grid points for nonlinear regions, reducing errors while maintaining computational efficiency.

Area of Science:

  • Numerical Analysis
  • Computational Mathematics
  • Computer Graphics

Background:

  • Approximating multidimensional nonlinear functions is crucial in many scientific and engineering fields.
  • Conventional uniform grid interpolation methods often struggle with high nonlinearity, leading to significant errors.
  • Efficient approximation techniques are needed to balance accuracy and computational cost.

Purpose of the Study:

  • Introduce a new interpolation approach, sequential linear interpolation (SLI), for approximating multidimensional nonlinear functions.
  • Improve the accuracy of function approximation compared to conventional uniform grid methods.
  • Demonstrate the effectiveness of SLI in a practical application like color printer characterization.

Main Methods:

  • Developed a partially separable grid structure (SLI) that adaptively allocates more grid points to highly nonlinear regions.

Related Experiment Videos

  • Utilized asymptotic analysis, drawing parallels with asymptotic vector quantization (VQ) theory, to determine optimal grid point placement.
  • Derived closed-form expressions for mean squared error (MSE) from asymptotic analysis to guide SLI design.
  • Implemented a postprocessing technique to enhance MSE performance when asymptotic theory assumptions are not met.

    • Main Results:

    • The SLI approach effectively reduces the mean squared error (MSE) between the original and approximated functions.
    • SLI retains significant computational advantages over traditional uniform grid interpolation.
    • Asymptotic analysis provided closed-form MSE expressions crucial for designing optimal SLI structures.
    • Experimental application to color printer characterization showed substantial MSE improvement with appropriately designed SLI.

    Conclusions:

    • Sequential linear interpolation (SLI) offers a superior method for approximating multidimensional nonlinear functions.
    • SLI provides a robust and computationally efficient alternative to conventional interpolation techniques.
    • The adaptive grid allocation and theoretical framework of SLI lead to demonstrably improved accuracy in complex approximation tasks.