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A vanishing point is a point on the
image plane In 3D computer graphics, the image plane is that plane in the world which is identified with the plane of the display monitor used to view the image that is being rendered. It is also referred to as screen space. If one makes the analogy of taki ...
of a
perspective drawing Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation, ...
where the two-dimensional
perspective projection Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate representation ...
s of mutually
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster o ...
lines in three-dimensional space appear to converge. When the set of parallel lines is
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', ⟂. It c ...
to a
picture plane In painting, photography, graphical perspective and descriptive geometry, a picture plane is an image plane located between the "eye point" (or ''oculus'') and the object being viewed and is usually coextensive to the material surface of the ...
, the construction is known as one-point perspective, and their vanishing point corresponds to the
oculus Oculus (a term from Latin ''oculus'', meaning 'eye'), may refer to the following Architecture * Oculus (architecture), a circular opening in the centre of a dome or in a wall Arts, entertainment, and media * ''Oculus'' (film), a 2013 American s ...
, or "eye point", from which the image should be viewed for correct perspective geometry. Kirsti Andersen (2007) ''Geometry of an Art'', p. xxx, Springer, Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points. Italian
humanist Humanism is a philosophical stance that emphasizes the individual and social potential and agency of human beings. It considers human beings the starting point for serious moral and philosophical inquiry. The meaning of the term "human ...
polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...
and architect
Leon Battista Alberti Leon Battista Alberti (; 14 February 1404 – 25 April 1472) was an Italian Renaissance humanist author, artist, architect, poet, priest, linguist, philosopher, and cryptographer; he epitomised the nature of those identified now as polymaths. H ...
first introduced the concept in his treatise on perspective in art, '' De pictura'', written in 1435.


Vector notation

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.B. Caprile, V. Torr

"Using Vanishing Points for Camera Calibration", International Journal of Computer Vision, Volume 4, Issue 2, pp. 127-139, March 1990
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 (set theory), 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 Humphry Ditton (29 May 1675 – 15 October 1715) was an English mathematician. He was the author of several influential works. Life Ditton was born on 29 May 1675 in Salisbury, the only son of Humphry Ditton, gentleman and ardent nonconformist, ...
called the result the "main and Great Proposition".
Brook Taylor Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. Life and work 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. She notes, in terms of
projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, ...
, the vanishing point is the image of the
point at infinity In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line. In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. ...
associated with , as the
sightline In architecture, sightlines are a particularly important consideration in the design of civic structures, such as a stage, arena, or monument. They determine the configuration of such items as theater and stadium design, road junction layout ...
from through the vanishing point is parallel to .


Vanishing line

As a vanishing point originates in a line, so a vanishing line originates in a plane that is not parallel to the picture . Given the eye point , and the plane parallel to and lying on , then the vanishing line of is . For example, when is the ground plane and is the horizon plane, then the vanishing line of is the horizon line . To put it simply, the vanishing line of some plane, say , is obtained by the intersection of the image plane with another plane, say , parallel to the plane of interest (), passing through the camera center. For different sets of lines parallel to this plane , their respective vanishing points will lie on this vanishing line. The horizon line is a theoretical line that represents the eye level of the observer. If the object is below the horizon line, its lines angle up to the horizon line. If the object is above, they slope down.


Properties

1. Projections of two sets of parallel lines lying in some plane appear to converge, i.e. the vanishing point associated with that pair, on a horizon line, or vanishing line formed by the intersection of the image plane with the plane parallel to and passing through the pinhole. Proof: Consider the ground plane , as which is, for the sake of simplicity, orthogonal to the image plane. Also, consider a line that lies in the plane , which is defined by the equation . Using perspective pinhole projections, a point on projected on the image plane will have coordinates defined as, : : This is the parametric representation of the image of the line with as the parameter. When it stops at the point on the axis of the image plane. This is the vanishing point corresponding to all parallel lines with slope in the plane . All vanishing points associated with different lines with different slopes belonging to plane will lie on the axis, which in this case is the horizon line. 2. Let , , and be three mutually orthogonal straight lines in space and , , be the three corresponding vanishing points respectively. If we know the coordinates of one of these points, say , and the direction of a straight line on the image plane, which passes through a second point, say , we can compute the coordinates of both and 3. Let , , and be three mutually orthogonal straight lines in space and , , be the three corresponding vanishing points respectively. The orthocenter of the triangle with vertices in the three vanishing points is the intersection of the optical axis and the image plane.


Curvilinear and reverse perspective

A
curvilinear perspective Curvilinear perspective, also five-point perspective, is a graphical projection used to draw 3D objects on 2D surfaces. It was formally codified in 1968 by the artists and art historians André Barre and Albert Flocon in the book ''La Perspective c ...
is a drawing with either 4 or 5 vanishing points. In 5-point perspective the vanishing points are mapped into a circle with 4 vanishing points at the cardinal headings N, W, S, E and one at the circle's origin. A reverse perspective is a drawing with vanishing points that are placed outside the painting with the illusion that they are "in front of" the painting. Image:Zentralperspektive.png, Single point perspective projection. Image:Study in Vanishing Perspective.jpg, Single point perspective in photography Image:TwoPointPerspective.png, Double point perspective projection. File:Entrega de las llaves a San Pedro (Perugino).jpg,
Pietro Perugino Pietro Perugino (, ; – 1523), born Pietro Vannucci, was an Italian Renaissance painter of the Umbrian school, who developed some of the qualities that found classic expression in the High Renaissance. Raphael was his most famous pupil. E ...
's use of perspective in the '' Delivery of the Keys''
fresco Fresco (plural ''frescos'' or ''frescoes'') is a technique of mural painting executed upon freshly laid ("wet") lime plaster. Water is used as the vehicle for the dry-powder pigment to merge with the plaster, and with the setting of the plast ...
at the
Sistine Chapel The Sistine Chapel (; la, Sacellum Sixtinum; it, Cappella Sistina ) is a chapel in the Apostolic Palace, the official residence of the pope in Vatican City. Originally known as the ''Cappella Magna'' ('Great Chapel'), the chapel takes its nam ...
(1481–82) helped bring the
Renaissance The Renaissance ( , ) , from , with the same meanings. is a period in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ide ...
to Rome.


Detection

Several methods for vanishing point detection make use of the line segments detected in images. Other techniques involve considering the intensity gradients of the image pixels directly. There are significantly large numbers of vanishing points present in an image. Therefore, the aim is to detect the vanishing points that correspond to the principal directions of a scene. This is generally achieved in two steps. The first step, called the accumulation step, as the name suggests, clusters the line segments with the assumption that a cluster will have a common vanishing point. The next step finds the principal clusters present in the scene and therefore it is called the search step. In the accumulation step, the image is mapped onto a bounded space called the accumulator space. The accumulator space is partitioned into units called cells. Barnard 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 The Hough transform is a feature extraction technique used in image analysis, computer vision, and digital image processing. The purpose of the technique is to find imperfect instances of objects within a certain class of shapes by a voting proce ...
, 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

# 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.D. Liebowitz and A. Zisserman "Metric Rectification for perspective images of planes" ,IEEE Conf. Computer Vision and Pattern Recognition, June 1998, Santa Barbara, CA, pp. 482 -488 #
3D reconstruction In computer vision and computer graphics, 3D reconstruction is the process of capturing the shape and appearance of real objects. This process can be accomplished either by active or passive methods. If the model is allowed to change its shape ...
: 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 CoelhoC. Coelho, M. Straforani, M. Campani " Using Geometrical Rules and a priori Knowledge for the Understanding of Indoor Scenes" Proceedings BMVC90, pp. 229–234 Oxford, September 1990. 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 Object detection is a computer technology related to computer vision and image processing that deals with detecting instances of semantic objects of a certain class (such as humans, buildings, or cars) in digital images and videos. Well-researched ...
.


See also

* Graphical projection


References

{{Reflist


External links


Vanishing point detection by three different proposed algorithmsVanishing point detection for images and videos using open CV
Brief explanation of the rationale with an easy example Perspective projection Geometry in computer vision