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Related Concept Videos

Parametric Surfaces01:30

Parametric Surfaces

A parametric surface in three-dimensional space is defined through a vector-valued function\begin{equation*}\mathbf{r}(u, v) = x(u, v)\mathbf{i} + y(u, v)\mathbf{j} + z(u, v)\mathbf{k}\end{equation*}where u and v are parameters within a specified domain D in the uv-plane. The functions x(u, v), y(u, v), and z(u, v) define the coordinates of points on the surface. As u and v vary over D, the position vector r(u, v) traces a continuous surface in space. This parametric representation is essential...
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Oriented Surfaces

A surface is called orientable if a consistent choice of unit normal vector can be made at every point on the surface. A thin soap film stretched across a wire loop provides a familiar example. The film separates the air on one side from the air on the other, so one side can be selected as positive and the opposite side as negative. Once this choice is made, a unit normal vector can be assigned smoothly across the entire surface.At each point on the soap film, a unit normal vector points...
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Understanding the movement of air masses is fundamental to meteorological analysis and atmospheric modeling. A key component in this process is quantifying the total mass of air that flows into or out of a defined region over a specified period of time. This is achieved by evaluating the mass flux across a boundary surface, a conceptual tool that simplifies the complex dynamics of atmospheric systems.To begin, an imaginary boundary surface S is introduced, enclosing the region of interest. The...
Tangent Planes to a Parametric Surface01:22

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A tangent plane provides a linear approximation to a curved surface at a specific point, capturing the local behavior of the surface. It can be understood as the plane that just touches the surface at that point and is defined by the tangent directions of curves lying on the surface. These tangent directions arise naturally when the surface is described parametrically, allowing systematic construction of the plane.For a surface expressed in parametric form, the position of any point is...
Tangent Planes to Surfaces01:19

Tangent Planes to Surfaces

In multivariable calculus, the concept of a tangent plane plays a central role in approximating curved surfaces. When dealing with a surface defined by a function of two variables, such as z = f(x, y), the tangent plane at a given point provides the best linear approximation to the surface near that point. This local linearization allows complex, nonlinear geometries to be treated using simpler, planar models.The construction of the tangent plane involves taking vertical slices of the surface...
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Quadric Surfaces

Quadric surfaces are three-dimensional surfaces characterized by second-degree equations in the variables x, y, and z. These surfaces are smooth and continuous, and specific combinations of squared and linear terms define their shapes. The main types of quadric surfaces include ellipsoids, cones, paraboloids, and hyperboloids. Each type exhibits distinct geometric features depending on how the variables are arranged and related within the equation.Ellipsoids are closed surfaces formed when all...

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Reconstruction of 3-Dimensional Histology Volume and its Application to Study Mouse Mammary Glands
10:59

Reconstruction of 3-Dimensional Histology Volume and its Application to Study Mouse Mammary Glands

Published on: July 26, 2014

Markov random field surface reconstruction.

Rasmus R Paulsen1, Jakob Andreas Baerentzen, Rasmus Larsen

  • 1Informatics and Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark. rrp@imm.dtu.dk

IEEE Transactions on Visualization and Computer Graphics
|May 15, 2010
PubMed
Summary
This summary is machine-generated.

This study introduces a novel implicit surface reconstruction method using Markov Random Field regularization for distance fields. The approach enhances accuracy and hole-closing capabilities for 3D scanning applications.

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Area of Science:

  • Computer Vision
  • Computer Graphics
  • Computational Geometry

Background:

  • Implicit surface reconstruction is crucial for 3D modeling from sensor data.
  • Existing methods face challenges in integrating prior knowledge and sensor data effectively.
  • Accurate reconstruction, especially with noisy or incomplete data, remains an active research area.

Purpose of the Study:

  • To propose a novel method for implicit surface reconstruction.
  • To leverage Markov Random Field (MRF) regularization for distance fields.
  • To improve the accuracy and hole-closing capabilities of surface reconstruction algorithms.

Main Methods:

  • Adaptation of Markov Random Field regularization for distance fields.
  • Orthogonal integration of surface priors and data observation models.
  • Development of local models incorporating scene-specific knowledge and scanner properties.
  • Optimization using conjugate gradients, sparse Cholesky factorization, and multiscale iterative schemes.

Main Results:

  • Demonstrated performance on scanned human head datasets.
  • Achieved comparable or superior accuracy to state-of-the-art methods.
  • Showcased enhanced ability to close holes in reconstructed surfaces.
  • Validated the effectiveness of MRF regularization in distance field-based reconstruction.

Conclusions:

  • The proposed MRF-regularized distance field method offers a robust approach to implicit surface reconstruction.
  • The method effectively integrates diverse knowledge sources for improved 3D model generation.
  • This technique presents a significant advancement for applications requiring high-fidelity surface reconstruction from 3D data.