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Piecewise defined functions are mathematical models where different expressions define a function over distinct intervals of the domain. These functions are useful for representing systems with varying behaviors depending on input values.For example, the function:  uses a linear rule for inputs less than or equal to –1 and a quadratic rule for values greater than –1. Although it has two formulas, it still defines a single function.Another common type is the absolute value...
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Flat belts are crucial in many industrial applications as they help transmit power from one pulley to another. The concept of forces and moments is used to determine the maximum moment on a pulley. For instance, consider a flat belt that wraps around two pulleys, A and B, with radii of 30 cm and 10 cm, respectively. The angle between the belt and the horizontal is 20 degrees at the pulleys. As pulley B rotates clockwise and drives pulley A, tension T2 is caused at one end of the belt, while...
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Piecewise Flat Embedding for Image Segmentation.

Chaowei Fang, Zicheng Liao, Yizhou Yu

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    We introduce Piecewise Flat Embedding (PFE), a novel method for image segmentation. PFE enhances region identifiability by promoting sparse representations, significantly improving segmentation accuracy on benchmark datasets.

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    Area of Science:

    • Computer Vision
    • Image Processing
    • Machine Learning

    Background:

    • Image segmentation is crucial for computer vision tasks.
    • Existing methods struggle with accurately identifying distinct regions and boundaries.
    • Sparse signal recovery offers a promising approach for improved image representation.

    Purpose of the Study:

    • To introduce a new multi-dimensional nonlinear embedding, Piecewise Flat Embedding (PFE), for enhanced image segmentation.
    • To develop a method that promotes sparse solutions for image representation, improving region identifiability.
    • To demonstrate the efficacy of PFE in improving existing image segmentation frameworks.

    Main Methods:

    • Piecewise Flat Embedding (PFE) formulated as a variant of Laplacian Eigenmap embedding.
    • Incorporation of an L1,p regularization term (0
    • Development of a two-stage numerical algorithm using Bregman iterations for L1,1-regularized PFE, generalized via iterative reweighting for L1,p regularization.

    Main Results:

    • PFE suppresses slowly varying signals, yielding higher region identifiability.
    • Integration of PFE into clustering and contour detection segmentation frameworks.
    • Significant improvements in segmentation accuracy observed on BSDS500, MSRC, Stanford Background Dataset, and PASCAL Context datasets.

    Conclusions:

    • PFE provides a powerful new embedding for image segmentation and semantic analysis.
    • The proposed algorithms efficiently compute sparse solutions for PFE.
    • PFE demonstrates broad applicability and effectiveness across diverse image segmentation tasks and datasets.