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Related Concept Videos

Level Curves and Contour Maps01:22

Level Curves and Contour Maps

Level curves and contour maps provide a way to visualize functions of two variables on a two-dimensional plane. A useful example is a topographic map, where curved lines represent locations that share the same elevation. In mathematics, these curves are called level curves or contour lines. Each contour line corresponds to points in the domain where the function has a constant value. For a function of two variables written as z = f(x,y), a level curve is defined by the equation f(x,y) = k,...
Interpretations of Partial Derivatives01:14

Interpretations of Partial Derivatives

A surface defined by a function of two variables can be visualized as a vast, uneven terrain, where each point is identified using Cartesian coordinates. The elevation of the terrain at any point is determined by a function that assigns a height value to every pair of horizontal coordinates. This representation allows the surface to be studied in terms of how its height varies across different directions.At a specific point on this terrain, understanding how the height changes requires...
Linear Approximations01:23

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...
Topographic Surveying and Contours01:29

Topographic Surveying and Contours

Topographic surveying is critical for documenting the Earth's surface, focusing on capturing elevations, slopes, and natural and man-made features. It is essential in construction planning, water resource management, and land-use analysis. The primary outcome of such surveys is a topographic map, which uses contour lines to visually represent the shape and slope of the terrain, providing valuable insights into the landscape's characteristics.Contour lines are fundamental to understanding the...
Tangent Planes to Surfaces01:19

Tangent Planes to Surfaces

In multivariable calculus, the concept of a tangent plane plays a central role in approximating curved surfaces. When dealing with a surface defined by a function of two variables, such as z = f(x, y), the tangent plane at a given point provides the best linear approximation to the surface near that point. This local linearization allows complex, nonlinear geometries to be treated using simpler, planar models.The construction of the tangent plane involves taking vertical slices of the surface...
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Plotting of Topographic Maps

Topographic maps represent the Earth's surface features using contour lines, which connect points of equal elevation to create a two-dimensional representation of three-dimensional terrain. Creating a topographic map requires a systematic approach.Begin by plotting a scaled grid and marking intersections corresponding to the survey's elevation data points. Assign elevation values at these intersections to build the base map. Next, determine contour levels using a consistent contour interval,...

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Local determinants of contour interpolation.

Marianne Maertens1, Robert Shapley

  • 1Otto-von-Guericke University Magdeburg, Experimental Psychology, USA. marianne.maertens@ovgu.de

Journal of Vision
|January 17, 2009
PubMed
Summary

Subjective contour perception relies on the geometric arrangement of visual elements, not just the amount of visible information. This finding highlights the role of localized features in how we perceive object shapes.

Area of Science:

  • * Visual perception
  • * Cognitive neuroscience
  • * Computational vision

Background:

  • * Objects are perceived as whole despite incomplete retinal images.
  • * Subjective contours are visual illusions where contours are perceived without physical presence.
  • * Previous research explored factors influencing subjective contour strength, including contour support and inducer properties.

Purpose of the Study:

  • * To investigate whether perceptual precision of subjective interpolation depends on visible information proportion or geometrical arrangement.
  • * To compare Varin-type subjective shapes with Kanizsa-type figures regarding stimulus information.
  • * To determine if perceptual precision is influenced by contour support or inducer line endings' geometry.

Main Methods:

  • * Utilized Varin-type subjective shapes with partially occluded concentric arcs as inducers.

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  • * Employed a probe localization task to measure performance around subjective boundaries.
  • * Assessed sensitivity via the just noticeable position difference for probes inside and outside the subjective contour.

    • Main Results:

    • * Perceptual precision critically depended on the geometric arrangement of line endings in Varin figures.
    • * Subjective contour strength was determined by the number and separation of inducers' line endings, not primarily contour support.
    • * Sensitivity varied significantly based on the geometrical configuration of the visual stimuli.

    Conclusions:

    • * The geometrical arrangement of line endings is a key determinant of subjective contour strength.
    • * Neuronal mechanisms sensitive to localized 2D features are vital for perceived object shape.
    • * Findings challenge the primacy of contour support in subjective contour formation, emphasizing geometric factors.