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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...
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...
Geometric Mean01:15

Geometric Mean

The mean is a measure of the central tendency of a data set. In some data sets, the data is inherently multiplicative, and the arithmetic mean is not useful. For example, the human population multiplies with time, and so does the credit amount of financial investment, as the interest compounds over successive time intervals.
In cases of multiplicative data, the geometric mean is used for statistical analysis. First, the product of all the elements is taken. Then, if there are n elements in the...
Relative Motion Analysis using Rotating Axes-Problem Solving01:29

Relative Motion Analysis using Rotating Axes-Problem Solving

Consider a crane whose telescopic boom rotates with an angular velocity of 0.04 rad/s and angular acceleration of 0.02 rad/s2. Along with the rotation, the boom also extends linearly with a uniform speed of 5 m/s. The extension of the boom is measured at point D, which is measured with respect to the fixed point C on the other end of the boom. For the given instant, the distance between points C and D is 60 meters.
Here, in order to determine the magnitude of velocity and acceleration for point...
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.
Curvilinear Motion: Rectangular Components01:23

Curvilinear Motion: Rectangular Components

Curvilinear motion characterizes the movement of a particle or object along a curved path, notably evident when envisioning a car navigating a winding road. If the car starts at point A, its position vector is established within a fixed frame of reference, where the ratio of the position vector to its magnitude signifies the unit vector pointing in the position vector's direction.
As the car advances, its position evolves over time. Quantifying the car's velocity involves computing the time...

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Related Experiment Video

Updated: Jun 23, 2026

A Methodology for Capturing Joint Visual Attention Using Mobile Eye-Trackers
12:39

A Methodology for Capturing Joint Visual Attention Using Mobile Eye-Trackers

Published on: January 18, 2020

Geometric video approximation using weighted matching pursuit.

Oscar Divorra Escoda1, Gianluca Monaci, Rosa M Figueras I Ventura

  • 1Signal Processing Institute (ITS), Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland. oscar.divorra@ieee.org

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|April 25, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces weighted matching pursuits (wMP) for advanced video approximation, effectively capturing geometric and motion content. The novel approach enhances video representation and extracts salient components for multimedia analysis.

Related Experiment Videos

Last Updated: Jun 23, 2026

A Methodology for Capturing Joint Visual Attention Using Mobile Eye-Trackers
12:39

A Methodology for Capturing Joint Visual Attention Using Mobile Eye-Trackers

Published on: January 18, 2020

Area of Science:

  • Digital Signal Processing
  • Computer Vision
  • Multimedia Analysis

Background:

  • Geometric multidimensional signal representations are closely linked to signal expansions on redundant dictionaries.
  • Matching Pursuits (MP) is a relevant tool for such expansions, but has limitations.
  • Weighted-MP (wMP) has emerged as an alternative to address MP's shortcomings.

Purpose of the Study:

  • To explore weighted-MP (wMP) as a novel framework for motion-adaptive geometric video approximations.
  • To develop a new algorithm for decomposing video sequences into salient components representing both geometric and motion content.
  • To evaluate the effectiveness of wMP in exploiting spatio-temporal video geometry.

Main Methods:

  • Implementation of a novel algorithm based on two-dimensional weighted-MP (2D wMP) for video decomposition.
  • Decomposition of video sequences into a few salient components capturing geometric and motion information.
  • Experimental coding and analysis of highly geometric video content.

Main Results:

  • The proposed wMP paradigm effectively exploits spatio-temporal video geometry.
  • Two-dimensional weighted-MP (2D wMP) provides improved representation compared to 2D MP.
  • Extracted video components represent salient visual structures effectively.

Conclusions:

  • Weighted-MP offers a promising framework for motion-adaptive geometric video approximation.
  • The novel algorithm successfully decomposes video sequences into meaningful components.
  • These components serve as effective video descriptors for joint audio-video analysis.