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Related Concept Videos

Real-Life Applications of Multiple Integrals01:18

Real-Life Applications of Multiple Integrals

Multiple integrals provide a powerful mathematical framework for calculating physical quantities distributed throughout two- and three-dimensional regions. One important application is the determination of volume in objects with curved geometries, such as storage tanks, pipes, and reservoirs. Cylindrical coordinates are especially useful for systems with rotational symmetry because they simplify the description of circular and paraboloid-shaped regions.Consider a paraboloid-shaped water tank...
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In vector calculus, flux measures the total flow of a vector field through a surface. For a closed surface in three-dimensional space, this means measuring how much of the field passes outward through every point on the boundary. Directly calculating this flux can be difficult when the surface has a complicated or irregular shape. The Divergence Theorem provides a powerful alternative by relating surface flux to behavior inside the enclosed region.The Divergence Theorem states that the outward...
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Triple Integrals in Rectangular Coordinates

Triple integrals provide a method for calculating the accumulated value of a function over a three-dimensional region. Common applications include computing volume, mass, and other physical quantities that vary with position. The fundamental idea is to partition a solid region into small rectangular boxes, evaluate the function at sample points within each box, and sum the contributions. As the partitions become finer, this triple Riemann sum approaches the exact value of the triple integral.In...
Triple Integrals over General Regions01:28

Triple Integrals over General Regions

Triple integrals over general bounded regions extend the concept of double integrals from planar domains to three-dimensional solids. A solid region E in space is commonly enclosed within a rectangular box B, and a continuous function f(x, y, z) is integrated over the region by defining F such that it coincides with f on E and is zero outside the solid. The triple integral is therefore expressed as\begin{equation*}\iiint_E f(x,y,z) dV \end{equation*}The existence of the integral requires that f...
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Related Experiment Video

Updated: Jun 25, 2026

Quantification of Strain in a Porcine Model of Skin Expansion Using Multi-View Stereo and Isogeometric Kinematics
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A new watermarking method for 3D models based on integral invariants.

Yu-Ping Wang1, Shi-Min Hu

  • 1Tsinghua National Laboratory for Information Science and Technology, Tsinghua University, Beijing, P.R. China. wyp02@mails.tsinghua.edu.cn

IEEE Transactions on Visualization and Computer Graphics
|February 27, 2009
PubMed
Summary

This study introduces a novel semi-fragile watermarking technique for 3D model authentication using integral invariants. The method embeds watermarks by altering vertex data, proving robust against common modifications and malicious attacks.

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Published on: September 28, 2018

Area of Science:

  • Computer Science
  • Digital Forensics
  • Computer Graphics

Background:

  • 3D models require robust authentication methods to ensure integrity.
  • Existing watermarking techniques often lack robustness against various manipulations or are fragile.
  • Integral invariants offer a promising approach for geometric feature preservation in 3D models.

Purpose of the Study:

  • To propose a novel semi-fragile watermarking algorithm for 3D model authentication.
  • To enhance the security and integrity verification of digital 3D assets.
  • To develop a method that is robust against common modifications and malicious attacks.

Main Methods:

  • A semi-fragile watermarking algorithm is proposed, embedding a watermark image by modifying integral invariants of 3D model vertices.
  • Vertex and neighbor positions are adjusted to alter integral invariants, embedding the watermark.
  • A new algorithm for computing two types of integral invariants is also presented.

Main Results:

  • The watermarking algorithm demonstrates robustness against rigid transformations, format conversions, and rounding errors.
  • The method effectively detects malicious attacks, including mesh editing and cropping.
  • Watermark extraction involves testing vertices and combining information to recover the watermark image for authentication.

Conclusions:

  • The proposed semi-fragile watermarking algorithm provides a reliable method for 3D model authentication.
  • The technique balances robustness for legitimate use with sensitivity to malicious alterations.
  • The integration of integral invariants offers a secure and effective approach to 3D model integrity verification.