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Related Concept Videos

Derivatives of Inverse Trigonometric Functions01:30

Derivatives of Inverse Trigonometric Functions

A ship tracking an approaching aircraft relies on geometric measurements to find out the aircraft’s position relative to the observer. By measuring the slant distance to the aircraft and the angle of elevation, the horizontal and vertical components of the distance can be obtained using trigonometric relationships. This geometric approach provides a basis for analyzing how the observed angle changes as the aircraft moves closer to the ship.To examine the mathematical behavior of the angle of...
Multivariable Functions and Higher Derivatives01:30

Multivariable Functions and Higher Derivatives

A multivariable function assigns a single output value to each ordered set of independent inputs, thereby defining a surface in three-dimensional space. For a function f(x, y), each point (x, y) corresponds to a height z = f(x, y). This geometric interpretation allows systematic analysis of how the output varies as multiple variables change simultaneously. Such functions frequently arise in physical models and optimization problems, where system behavior depends on several interacting...
Second Derivatives and the Shape of a Graph01:29

Second Derivatives and the Shape of a Graph

The second derivative of a function provides essential information about a graph's curvature and how it changes over an interval. It helps determine whether a function is concave upward or concave downward and identifies points where the curvature changes. These properties are fundamental in analyzing real-world scenarios, such as changes in road elevation, population growth, and economic trends.A function f(x) is considered concave upward on an interval if its graph lies above all its tangent...
Directional Derivatives01:26

Directional Derivatives

In multivariable calculus, partial derivatives describe how a function changes when movement is restricted to a single coordinate direction. For a surface represented by a function of two variables, one partial derivative measures the slope in the x-direction, while the other measures the slope in the y-direction. Although these quantities are useful for analyzing local behavior, most physical motion does not occur strictly parallel to the coordinate axes. Applications such as fluid flow, heat...
Higher Derivatives01:29

Higher Derivatives

In calculus, higher-order derivatives extend the idea of differentiation beyond the first derivative to capture successive rates of change. These derivatives provide detailed information about the behavior of functions and have important applications in both mathematics and physics. To illustrate these concepts, consider the example function\begin{equation*}f(x) = x^3 - x\end{equation*}which serves as a useful case study for exploring higher derivatives.The first derivative represents the slope...
Derivatives of Vector Functions01:17

Derivatives of Vector Functions

A vector-valued function describes position as a function of time. For example, in Cartesian coordinates, the position of a car moving along a curved road can be written as\begin{equation*}\textbf{r}(t)=\langle x(t),y(t),z(t)\rangle\end{equation*}Secant Vector and Average Velocity:This secant vector captures the overall change in position during the interval and provides a crude estimate of the direction of motion.At time t, the car is at point P, with position r(t). After a short interval h,...

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Related Experiment Video

Updated: Jun 19, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Boolean derivatives with application to edge detection for imaging systems.

Sos S Agaian1, Karen A Panetta, Shahan C Nercessian

  • 1Department of Electrical Engineering, University of Texas/San Antonio, San Antonio, TX 78249, USA. sos.agaian@utsa.edu

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
|November 4, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces Boolean derivatives for image processing. These derivatives enhance edge detection in both binary and grayscale images, offering competitive results with novel parameter optimization.

Related Experiment Videos

Last Updated: Jun 19, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
13:44

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Area of Science:

  • Computer Vision
  • Image Processing
  • Digital Image Analysis

Background:

  • Partial derivatives of Boolean functions (PDBFs) are a foundational concept.
  • Existing edge detection methods have limitations in accuracy and parameter tuning.

Purpose of the Study:

  • To introduce Boolean derivatives as a novel concept fusing PDBFs.
  • To develop efficient algorithms for calculating Boolean derivatives.
  • To apply Boolean derivatives to edge detection in binary and grayscale images and introduce a new parameter optimization measure.

Main Methods:

  • Developed three efficient algorithms for calculating PDBFs.
  • Extended Boolean derivatives for edge detection in grayscale images.
  • Introduced a new measure for automatic parameter selection in binarization.
  • Utilized computer simulations with synthetic and natural images for validation.

Main Results:

  • Boolean derivatives effectively identify edge pixels in binary images.
  • The new edge detection algorithm for grayscale images achieves competitive results.
  • The novel measure automates parameter selection for binarization effectively.
  • Quantitative comparisons using image recovery and Pratt's figure of merit demonstrate effectiveness.

Conclusions:

  • Boolean derivatives offer a powerful tool for image analysis, particularly edge detection.
  • The proposed edge detection algorithm provides a competitive alternative to traditional methods.
  • Automated parameter selection enhances the practical applicability of the binarization procedure.