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Related Concept Videos

Graphs of Polar Equations01:17

Graphs of Polar Equations

The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...
Polar Curves01:19

Polar Curves

The spirograph is a versatile tool for visualizing the relationship between geometry and mathematical representation. In particular, it demonstrates how polar coordinates offer an alternative framework for describing curves in comparison to Cartesian coordinates. Instead of specifying a point by its horizontal and vertical displacements (x, y), polar coordinates use a radius r, the distance from the origin, and an angle θ, measured counterclockwise from the polar axis. This system is...
Polar Coordinates01:24

Polar Coordinates

The polar coordinate system offers an alternative to the Cartesian coordinate system for specifying points in a plane, using a distance and an angle instead of x and y coordinates. This system is particularly advantageous in situations involving circular or rotational symmetry, such as in physics or engineering problems involving waves, oscillations, or orbital paths.Defining Polar CoordinatesIn polar coordinates, a point is represented as P(r, ��), where r is the radial distance from a fixed...
Polar Coordinate System01:30

Polar Coordinate System

The polar coordinate system provides a natural way to describe points in the plane when distances and directions are more meaningful than horizontal and vertical displacements. It is especially useful for modeling non-rectangular regions such as circles and spirals, where symmetry about a center point is easier to express than it is in a rectangular grid. A familiar example is a ship’s plan position indicator, which marks detected targets as dots positioned relative to the ship at the display’s...
Polar and Cylindrical Coordinates01:22

Polar and Cylindrical Coordinates

The Cartesian coordinate system is a very convenient tool to use when describing the displacements and velocities of objects and the forces acting on them. However, it becomes cumbersome when we need to describe the rotation of objects. So, when describing rotation, the polar coordinate system is generally used.
Curvilinear Motion: Polar Coordinates01:27

Curvilinear Motion: Polar Coordinates

In polar coordinates, the motion of a particle follows a curvilinear path. The radial coordinate symbolized as 'r,' extends outward from a fixed origin to the particle, while the angular coordinate, 'θ,' measured in radians, represents the counterclockwise angle between a fixed reference line and the radial line connecting the origin to the particle.
The particle's location is described using a unit vector along the radial direction. Deriving the particle's position with respect to time...

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Related Experiment Video

Updated: Jun 24, 2026

Polarization-Sensitive Two-Photon Microscopy for a Label-Free Amyloid Structural Characterization
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Polarization-Sensitive Two-Photon Microscopy for a Label-Free Amyloid Structural Characterization

Published on: September 8, 2023

Cyclic mathematical morphology in polar-logarithmic representation.

Miguel Angel Luengo-Oroz1, Jesús Angulo

  • 1Biomedical Image Technologies Lab of ETSI Telecomunicación, Universidad Politécnica de Madrid, 28040 Madrid, Spain. maluengo@die.upm.es

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|April 2, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces cyclic morphology, applying mathematical morphology operators in transformed image domains. This method enables analysis of round objects using regular structuring elements in transformed coordinates, simplifying image processing for pattern recognition tasks.

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Polarization-Sensitive Two-Photon Microscopy for a Label-Free Amyloid Structural Characterization
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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns
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Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Area of Science:

  • Image analysis and pattern recognition
  • Mathematical morphology
  • Geometric transformations

Background:

  • Traditional mathematical morphology often struggles with analyzing objects with varying scales and orientations.
  • Geometric transformations can alter image representations, potentially simplifying complex analysis tasks.
  • Analyzing round or origin-centered objects presents unique challenges in standard image processing.

Purpose of the Study:

  • To develop and evaluate a novel approach called cyclic morphology for image analysis.
  • To demonstrate the equivalence between using regular structuring elements in a transformed domain and deformed elements in the original domain.
  • To showcase the effectiveness of cyclic morphology, particularly with polar-logarithmic coordinate transformations, for specific pattern recognition applications.

Main Methods:

  • Applying mathematical morphology operators within a geometric image transformation framework.
  • Utilizing a polar-logarithmic coordinate transformation for image representation.
  • Testing the cyclic morphology approach on pattern recognition tasks, including erythrocyte shape analysis and iris texture description.

Main Results:

  • Processing images with regular structuring elements in the transformed domain is equivalent to using deformed structuring elements in the original image.
  • The polar-logarithmic coordinate transformation yields effective results for analyzing roughly origin-centered round objects.
  • Cyclic morphology proved beneficial in the pattern recognition examples of erythrocyte shape analysis and multiscale iris texture description.

Conclusions:

  • Geometric transformations combined with mathematical morphology offer a powerful tool for image analysis.
  • Cyclic morphology, especially with polar-logarithmic coordinates, enhances the analysis of specific object shapes like round features.
  • The proposed method demonstrates significant potential for applications in medical imaging and biometric pattern recognition.