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Related Concept Videos

Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length, the...
Vector Algebra: Method of Components01:08

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like addition, subtraction, and multiplication. There are two ways to circumvent this algebraic complexity. One way is to draw the vectors to scale, as in navigation, and read approximate vector lengths and angles (directions) from the graphs. The other way is to use the method of components.
In many applications, the magnitudes and directions of...
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...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...

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High-speed Particle Image Velocimetry Near Surfaces
11:59

High-speed Particle Image Velocimetry Near Surfaces

Published on: June 24, 2013

Cost-effective implementation of order-statistics-based vector filters using minimax approximations.

M Emre Celebi1, Hassan A Kingravi, Rastislav Lukac

  • 1Department of Computer Science, Louisiana State University, Shreveport, Louisiana 71115, USA. ecelebi@lsus.edu

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|June 3, 2009
PubMed
Summary

This study speeds up vector filters for color image processing using minimax approximations. The new methods efficiently reduce noise and preserve details, balancing speed and accuracy for critical applications.

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Area of Science:

  • Digital image processing
  • Computer vision
  • Signal processing

Background:

  • Robust order statistics vector filters are effective for color image filtering and enhancement.
  • These filters excel at handling impulsive noise while preserving image details.
  • High computational cost limits the application of these filters in time-critical scenarios.

Purpose of the Study:

  • To introduce techniques for accelerating vector filters.
  • To leverage minimax approximation theory for computational efficiency.
  • To maintain accuracy and image quality while reducing processing time.

Main Methods:

  • Development of minimax approximations for vector filters.
  • Implementation of accelerated vector filtering techniques.
  • Extensive experimental evaluation on diverse color image datasets.

Main Results:

  • Proposed approximations significantly reduce computational requirements.
  • Achieved excellent balance between implementation ease, accuracy, and speed.
  • Demonstrated effectiveness across a large and diverse set of color images.

Conclusions:

  • Minimax approximation techniques offer a viable solution for accelerating vector filters.
  • The proposed methods enable efficient and accurate color image processing.
  • These advancements are suitable for time-critical digital imaging applications.