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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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Updated: Jun 29, 2026

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Detection of Architectural Distortion in Prior Mammograms via Analysis of Oriented Patterns

Published on: August 30, 2013

Texture image classification based on a pseudo-parabolic diffusion model.

Jardel Vieira¹, Eduardo Abreu², Joao B Florindo²

¹Unidade Acadêmica Especial de Matemática e Tecnologia, Universidade Federal de Goiás, Av. Dr. Lamartine Pinto de Avelar 1120, St. Universitário, 75704-020 Catalão, Goiás Brasil.

Multimedia Tools and Applications

|July 20, 2022

Summary

This summary is machine-generated.

A novel pseudo-parabolic diffusion method enhances texture recognition by using nonlinear filters and local binary patterns. This approach achieves superior classification accuracy on plant species and benchmark datasets compared to existing methods.

Keywords:

Image classification Numerical approximation methods for PDEs Pseudo-parabolic equation Texture recognition

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Area of Science:

Computer Vision
Image Processing
Pattern Recognition

Background:

Texture recognition is crucial for image analysis and pattern identification.
Existing methods often struggle with noise and preserving critical details.
Model-based approaches offer an alternative to computationally intensive learning-based methods.

Purpose of the Study:

To introduce a novel pseudo-parabolic diffusion method for texture recognition.
To develop a discrete pseudo-parabolic differential operator for texture description.
To demonstrate the method's effectiveness in plant species identification and benchmark dataset classification.

Main Methods:

A pseudo-parabolic diffusion process is applied to generate a family of nonlinearly filtered images.
Local binary patterns (LBP) are used to encode each filtered image.
Histograms summarize the LBP features, creating an image feature vector.

Main Results:

The proposed method achieves higher classification accuracy than state-of-the-art techniques on benchmark texture databases.
The approach demonstrates strong performance in identifying plant species from leaf surface images.
The pseudo-parabolic operator effectively smooths noise while preserving essential image discontinuities.

Conclusions:

Pseudo-parabolic models offer a competitive and efficient approach to texture recognition.
The method is particularly advantageous when computational resources and labeled data are limited.
This work highlights the potential of model-based image analysis for robust pattern recognition.