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Related Concept Videos

Spherical Coordinates01:23

Spherical Coordinates

Spherical coordinate systems are preferred over Cartesian, polar, or cylindrical coordinates for systems with spherical symmetry. For example, to describe the surface of a sphere, Cartesian coordinates require all three coordinates. On the other hand, the spherical coordinate system requires only one parameter: the sphere's radius. As a result, the complicated mathematical calculations become simple. Spherical coordinates are used in science and engineering applications like electric and...
Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a uniform...
Gauss's Law: Cylindrical Symmetry01:20

Gauss's Law: Cylindrical Symmetry

A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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.
Radius of Gyration of an Area01:12

Radius of Gyration of an Area

The second moment of area, also known as the moment of inertia of area, is a crucial factor in understanding an object's resistance against bending deformation, or stiffness. To accurately estimate the second moment of area along any axis, one needs to concentrate all areas associated with that object into a thin strip, which should be placed parallel to that particular axis.
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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Updated: Jun 6, 2026

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

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Published on: August 30, 2013

Modeling Bidirectional Texture Functions with Multivariate Spherical Radial Basis Functions.

Yu-Ting Tsai, Kuei-Li Fang, Wen-Chieh Lin

    IEEE Transactions on Pattern Analysis and Machine Intelligence
    |December 8, 2010
    PubMed
    Summary
    This summary is machine-generated.

    This study introduces a new way to represent bidirectional texture functions using multivariate spherical radial basis functions (SRBFs) and optimized parameterization. This method efficiently captures complex material appearances and enables high-quality, real-time rendering.

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    Area of Science:

    • Computer Graphics
    • Material Appearance Modeling

    Background:

    • Bidirectional texture functions (BTFs) are crucial for realistic surface rendering.
    • Traditional BTF representations face challenges with heterogeneous materials and fixed parameterization.

    Purpose of the Study:

    • To develop a novel parametric representation for BTFs.
    • To improve the efficiency and quality of representing heterogeneous material appearances.
    • To enable real-time rendering of complex textures.

    Main Methods:

    • Utilized multivariate spherical radial basis functions (SRBFs) for intrinsic material representation.
    • Developed an optimized parameterization technique integrated with SRBFs.
    • Implemented a hierarchical fitting algorithm for BTFs.

    Main Results:

    • Achieved high-quality approximation of complex surface appearances.
    • Demonstrated efficient real-time rendering performance.
    • Successfully represented heterogeneous materials using a sum-of-products model.

    Conclusions:

    • The proposed parametric representation offers an efficient and intrinsic method for BTFs.
    • Optimized parameterization overcomes limitations of traditional approaches.
    • The technique facilitates high-fidelity and real-time rendering of diverse materials.