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Kendall's Tau Test01:16

Kendall's Tau Test

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Kendall's tau test, also known as the Kendall rank coefficient test, is a nonparametric method for assessing association between two variables. This test is particularly useful for identifying significant correlations when the distributions of the sample and population are unknown. Developed in 1938 by the British statistician Sir Maurice George Kendall, the tau coefficient (denoted as τ) serves as a rank correlation coefficient, with values ranging from -1 to +1.
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A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis...
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Gauss's Law: Planar Symmetry01:27

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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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The measures of central tendency calculated from a data set may not reveal much about its intrinsic distribution. If a plot is made of the data set’s values, the mean and the median may not only differ, but also the plot may have more values on one side of the central tendencies. Such a data set is said to be skewed towards that side.
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Kendall's Coefficient of Concordance (W), also known as Kendall's W, is a non-parametric statistical measure used to assess the agreement or concordance between multiple raters or judges when they rank a set of items. It is often used when you have ordinal data (ranks) and you want to see if there is consistency or consensus among the raters. It is widely applied in research areas such as psychology, medicine, and social sciences, where multiple judges are asked to rank or rate subjects...
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Gauss's Law: Spherical Symmetry01:26

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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Morphology-Based Distinction Between Healthy and Pathological Cells Utilizing Fourier Transforms and Self-Organizing Maps
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Caricature Expression Extrapolation Based on Kendall Shape Space Theory.

Na Liu, Dan Zhang, Xudong Ru

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    |March 31, 2021
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    Summary
    This summary is machine-generated.

    This study introduces a new method for facial expression extrapolation in caricatures using a 3-D model within the Kendall shape space. The technique effectively generates exaggerated expressions, enhancing digital content creation.

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    Area of Science:

    • Computer Vision
    • Computer Graphics
    • Geometric Deep Learning

    Background:

    • Facial expression editing is crucial for digital media, but extrapolation (exaggerating beyond original expressions) is underexplored.
    • Current 2-D caricature editing primarily uses interpolation, limiting creative possibilities.

    Purpose of the Study:

    • To propose a novel method for 2-D caricature facial expression extrapolation.
    • To introduce a 3-D model representation within the Kendall shape space to handle rigid transformations.

    Main Methods:

    • Utilizing the Kendall shape space with a novel 3-D model representation to remove translation, scaling, and rotation.
    • Reconstructing a 3-D model from a 2-D caricature and extrapolating expressions using exponential mapping in Riemannian space.
    • Generating exaggerated caricature expressions based on the extrapolated 3-D facial model.

    Main Results:

    • The method effectively and automatically extrapolates facial expressions in caricatures with high consistency and fidelity.
    • Generated 3-D facial models with diverse expressions, expanding the FaceWarehouse database.
    • Demonstrated robustness to facial poses and avoidance of complex 3-D caricature training sets, unlike deep learning methods.

    Conclusions:

    • The proposed Kendall shape space method offers a robust and efficient approach for facial expression extrapolation in 2D caricatures.
    • This technique provides a valuable tool for digital content creation, particularly in film and gaming.
    • The method's reliance on standard datasets and avoidance of complex training data makes it a practical alternative to deep learning approaches.