thumb|A photo demonstrating a vanishing point at the end of the railroad.

A vanishing point is a point on the image plane of a perspective rendering where the two-dimensional perspective projections of parallel lines in three-dimensional space appear to converge. When the set of parallel lines is perpendicular to a picture plane, the construction is known as one-point perspective, and their vanishing point corresponds to the oculus, or "eye point", from which the image should be viewed for correct perspective geometry. Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points.

Italian humanist polymath and architect Leon Battista Alberti first introduced the concept in his treatise on perspective in art, De pictura, written in 1435. Straight railroad tracks are a familiar modern example.

Vector notation

left|thumb|289px|A 2D construction of perspective viewing, showing the formation of a vanishing point

The vanishing point may also be referred to as the "direction point", as lines having the same directional vector, say D, will have the same vanishing point. Mathematically, let be a point lying on the image plane, where is the focal length (of the camera associated with the image), and let be the unit vector associated with , where . If we consider a straight line in space with the unit vector and its vanishing point , the unit vector associated with is equal to , assuming both point towards the image plane.

When the image plane is parallel to two world-coordinate axes, lines parallel to the axis that is cut by this image plane will have images that meet at a single vanishing point. Lines parallel to the other two axes will not form vanishing points as they are parallel to the image plane. This is one-point perspective. Similarly, when the image plane intersects two world-coordinate axes, lines parallel to those planes will meet form two vanishing points in the picture plane. This is called two-point perspective. In three-point perspective the image plane intersects the , , and axes and therefore lines parallel to these axes intersect, resulting in three different vanishing points.

Theorem

The vanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture plane of a line in space, not parallel to the picture, is determined by its intersection with and its vanishing point. Some authors have used the phrase, "the image of a line includes its vanishing point". Guidobaldo del Monte gave several verifications, and Humphry Ditton called the result the "main and Great Proposition". Brook Taylor wrote the first book in English on perspective in 1714, which introduced the term "vanishing point" and was the first to fully explain the geometry of multipoint perspective, and historian Kirsti Andersen compiled these observations. assumed this space to be a Gaussian sphere centered on the optical center of the camera as an accumulator space. A line segment on the image corresponds to a great circle on this sphere, and the vanishing point in the image is mapped to a point. The Gaussian sphere has accumulator cells that increase when a great circle passes through them, i.e. in the image a line segment intersects the vanishing point. Several modifications have been made since, but one of the most efficient techniques was using the Hough Transform, mapping the parameters of the line segment to the bounded space. Cascaded Hough Transforms have been applied for multiple vanishing points.

The process of mapping from the image to the bounded spaces causes the loss of the actual distances between line segments and points.

In the search step, the accumulator cell with the maximum number of line segments passing through it is found. This is followed by removal of those line segments, and the search step is repeated until this count goes below a certain threshold. As more computing power is now available, points corresponding to two or three mutually orthogonal directions can be found.

Applications

  1. Camera calibration: The vanishing points of an image contain important information for camera calibration. Various calibration techniques have been introduced using the properties of vanishing points to find intrinsic and extrinsic calibration parameters.
  2. 3D reconstruction: A man-made environment has two main characteristics – several lines in the scene are parallel, and a number of edges present are orthogonal. Vanishing points aid in comprehending the environment. Using sets of parallel lines in the plane, the orientation of the plane can be calculated using vanishing points. Torre and Coelho performed extensive investigation in the use of vanishing points to implement a full system. With the assumption that the environment consists of objects with only parallel or perpendicular sides, also called Lego-land, using vanishing points constructed in a single image of the scene they recovered the 3D geometry of the scene. Similar ideas are also used in the field of robotics, mainly in navigation and autonomous vehicles, and in areas concerned with object detection.

See also

  • Graphical projection

References

  • Vanishing point detection by three different proposed algorithms
  • Vanishing point detection for images and videos using open CV
  • A tutorial covering many examples of linear perspective
  • Trigonometric Calculation of Vanishing Points Brief explanation of the rationale with an easy example