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Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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Advanced Self-Healing Asphalt Reinforced by Graphene Structures: An Atomistic Insight

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Machine-Learning-Driven Stochastic Modeling Method for 3D Asphalt Mixture Reconstruction from 2D Images.

Jiayu Zhang¹, Liang Huang¹

¹School of Civil Engineering, Zhengzhou University, Zhengzhou 450001, China.

Materials (Basel, Switzerland)

|August 28, 2025

Summary

This summary is machine-generated.

We developed a low-cost method for 3D asphalt mixture reconstruction using image analysis and stochastic modeling. This approach accurately captures spatial structures and material properties for better asphalt behavior understanding.

Keywords:

3D model asphalt mixture multiple-point statistics segment anything model

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Area of Science:

Materials Science
Civil Engineering
Computational Modeling

Background:

Accurate 3D models are crucial for understanding asphalt mixture behavior and the relationship between spatial structure and physical properties.
Existing methods for 3D reconstruction can be costly and data-intensive.

Purpose of the Study:

To develop a low-cost and data-efficient framework for creating 3D asphalt mixture models.
To integrate foundational segmentation and stochastic modeling for enhanced reconstruction.

Main Methods:

Utilized smartphone photography and image quilting for 2D image capture.
Employed the Segment Anything Model (SAM) for high-quality segmentation of aggregates and asphalt binder.
Applied Multiple-Point Statistics (MPS) with Nearest Neighbor Simulation (NNSIM) for efficient 3D model construction.
Introduced a probability aggregation framework for calculating 3D conditional probabilities.

Main Results:

Successfully reconstructed 3D asphalt mixture models from 2D images.
Validated reconstruction quality using two-point correlation functions, distance analysis, and grain size distribution.
Demonstrated preservation of spatial patterns and representation of uncertainty in material production.

Conclusions:

The proposed method offers a cost-effective and efficient approach to 3D asphalt mixture reconstruction.
The framework accurately represents the complex spatial structures and properties of asphalt materials.
This technique aids in understanding asphalt behavior and optimizing material production.