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Tangent to a Curve01:30

Tangent to a Curve

The graph of a function where each output is the square of the input creates a smooth curve that bends upward, becoming steeper as one moves further from the center. At any chosen position along this curve, the curve reaches a certain height depending on the input value. This position can be a reference for analyzing how the curve behaves in its immediate vicinity.To understand the change in the curve near a particular position, imagine selecting another point slightly ahead along the curve.
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Horizontal curves are essential in highway and railroad design, ensuring smooth and safe transitions between straight path segments, or tangents. These curves allow vehicles to maintain speed without abrupt changes, minimizing accidents and improving travel efficiency.A horizontal curve is typically defined by its geometric relationship to two tangents that meet at an intersection point (P.I.), where a simple curve is introduced to connect them. The back tangent refers to the initial tangent...
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Quantifying Fibrillar Collagen Organization with Curvelet Transform-Based Tools
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Published on: November 11, 2020

Tangent bundle curve completion with locally connected parallel networks.

Guy Ben-Yosef1, Ohad Ben-Shahar

  • 1Computer Science Department and Zlotowski Center for Neuroscience, Ben-Gurion University, Beer-Sheva 84105, Israel. guybeny@cs.bgu.ac.il

Neural Computation
|September 14, 2012
PubMed
Summary
This summary is machine-generated.

This study presents a neural network model for how the brain completes visual curves, even with missing information. The research explores the biological plausibility of representing these curves as least energetic mathematical paths.

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Area of Science:

  • Computational Neuroscience
  • Visual Perception
  • Artificial Neural Networks

Background:

  • The visual system generates completed curves to fill in missing visual information, such as occlusions.
  • Existing theories suggest these curves emerge from activation patterns in the primary visual cortex.
  • Prior models represented these patterns as least energetic 3D curves in an abstract cortical space.

Purpose of the Study:

  • To propose a theory for the cortical representation and computation of visually completed curves.
  • To assess the biological plausibility of existing theories on curve completion.
  • To bridge physiological findings on curve completion with shape theory.

Main Methods:

  • Developed a neural architecture implementing the theory using locally connected parallel networks.
  • Simulated curve completion in natural and synthetic scenes using the proposed model.
  • Analyzed the model's ability to represent and compute visually completed curves.

Main Results:

  • Demonstrated a biologically plausible neural architecture for visual curve completion.
  • Presented simulations showing successful completion of curves in various scenes.
  • Identified observations and predictions related to curve completion from the theoretical model.

Conclusions:

  • The proposed theory and neural architecture offer a framework for understanding cortical computation of completed curves.
  • The model provides a bridge between neurophysiological data and computational theories of shape perception.
  • Further research can explore the model's predictions in more complex visual scenarios.