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Related Concept Videos

Partial Fractions01:28

Partial Fractions

225
A partial fraction is a component of a rational expression represented as the sum of simpler fractions. When a rational function is expressed as a ratio of two polynomials, it can often be decomposed into a sum of fractions whose denominators are simpler polynomials, typically linear or irreducible quadratic factors. This process is called partial fraction decomposition, and it is used to simplify complex expressions for integration, solving equations, or analysis.Partial fraction decomposition...
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Mixtures of Gases: Dalton's Law of Partial Pressures and Mole Fractions03:03

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Unless individual gases chemically react with each other, the individual gases in a mixture of gases do not affect each other’s pressure. Each gas in a mixture exerts the same pressure that it would exert if it were present alone in the container. The pressure exerted by each individual gas in a mixture is called its partial pressure.
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Integration of Rational Functions Using Partial Fractions01:29

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Rational functions are expressions written as the ratio of two polynomials, and their integrals are evaluated by simplifying the integrand into manageable parts. These functions are classified as proper or improper based on the degrees of the numerator and denominator.A rational function is proper when the degree of the numerator is less than the degree of the denominator. In this case, partial fraction decomposition is used to rewrite the function as a sum of simpler rational terms. The...
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Inverse z-Transform by Partial Fraction Expansion01:20

Inverse z-Transform by Partial Fraction Expansion

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The inverse z-transform is a crucial technique for converting a function from its z-domain representation back to the time domain. One effective method for finding the inverse z-transform is the Partial Fraction Method, which involves decomposing a function into simpler fractions with distinct coefficients. These fractions correspond to known z-transform pairs, facilitating the inverse transformation process.
To begin the process, the poles of the function are identified and the function is...
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Subcellular Fractionation01:32

Subcellular Fractionation

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The homogenate obtained after cell lysis contains various membrane-bound organelles that can be further separated into pure fractions by subcellular fractionation. These isolates are used to study specific cellular components, analyze localized protein activity, and are even employed in diagnostics. Fractionation is typically achieved using centrifugation methods, the most common being density-gradient and differential centrifugation.
Differential Centrifugation
Differential centrifugation is...
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Phase Contrast and Differential Interference Contrast Microscopy01:26

Phase Contrast and Differential Interference Contrast Microscopy

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Phase-Contrast Microscopes
In-phase-contrast microscopes, interference between light directly passing through a cell and light refracted by cellular components is used to create high-contrast, high-resolution images without staining. It is the oldest and simplest type of microscope that creates an image by altering the wavelengths of light rays passing through the specimen. Altered wavelength paths are created using an annular stop in the condenser. The annular stop produces a hollow cone of...
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Lumped-Parameter and Finite Element Modeling of Heart Failure with Preserved Ejection Fraction
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A Fractional-Order Variational Framework for Retinex: Fractional-Order Partial Differential Equation-Based

Yi-Fei Pu, Patrick Siarry, Amitava Chatterjee

    IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
    |July 11, 2018
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    This study introduces a novel fractional-order variational framework for retinex, enhancing image contrast while preserving textures. This approach overcomes limitations of traditional methods, offering superior texture and edge preservation.

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    Area of Science:

    • Image Processing
    • Applied Mathematics
    • Computer Vision

    Background:

    • Traditional contrast enhancement algorithms suffer from artifacts like ringing and staircase effects.
    • Fractional calculus offers unique properties such as long-term memory and nonlocality, beneficial for image processing.
    • Existing methods struggle with preserving intricate textural details during contrast enhancement.

    Purpose of the Study:

    • To introduce a novel fractional-order variational framework for retinex.
    • To address the limitations of integer-order contrast enhancement algorithms.
    • To investigate the hybridization of fractional calculus with multi-scale nonlocal contrast enhancement for texture preservation.

    Main Methods:

    • Formulation of a fractional-order partial differential equation (FPDE) for retinex.
    • Implementation using the fractional-order steepest descent method.
    • Application of restrictive fractional-order optimization and the Courant-Friedrichs-Lewy condition.

    Main Results:

    • The proposed FPDE effectively enhances image contrast.
    • Demonstrated superior preservation of edges and textural details compared to traditional methods.
    • The framework successfully integrates multi-scale nonlocal contrast enhancement with texture preservation.

    Conclusions:

    • The fractional-order variational framework for retinex offers significant advantages over integer-order methods.
    • This novel approach is particularly effective for images with rich textural content.
    • The FPDE formulation provides a robust solution for contrast enhancement with texture preservation.