Avanishing point is apoint on theimage plane of aperspective rendering where the two-dimensionalperspective projections ofparallel lines in three-dimensional space appear to converge. When the set of parallel lines isperpendicular to apicture plane, the construction is known as one-point perspective, and their vanishing point corresponds to theoculus, or "eye point", from which the image should be viewed for correct perspective geometry.[1] Traditional linear drawings use objects with one to three sets of parallels, defining one to three vanishing points.
Italianhumanistpolymath and architectLeon Battista Alberti first introduced the concept in his treatise on perspective in art,De pictura, written in 1435.[2] Straightrailroad tracks are a familiar modern example.[3]
The vanishing point may also be referred to as the "direction point", as lines having the same directional vector, sayD, will have the same vanishing point. Mathematically, letq ≡ (x,y,f) be a point lying on the image plane, wheref is the focal length (of the camera associated with the image), and letvq ≡ (x/h,y/h,f/h) be the unit vector associated withq, whereh =√x2 +y2 +f2. If we consider a straight line in spaceS with the unit vectorns ≡ (nx,ny,nz) and its vanishing pointvs, the unit vector associated withvs is equal tons, assuming both point towards the image plane.[4]
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 thex,y, andz axes and therefore lines parallel to these axes intersect, resulting in three different vanishing points.
Thevanishing point theorem is the principal theorem in the science of perspective. It says that the image in a picture planeπ of a lineL in space, not parallel to the picture, is determined by itsintersection withπ 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, andHumphry Ditton called the result the "main and Great Proposition".[5]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 historianKirsti Andersen compiled these observations.[1]: 244–6 She notes, in terms ofprojective geometry, the vanishing point is the image of thepoint at infinity associated withL, as thesightline fromO through the vanishing point is parallel toL.
As a vanishing point originates in a line, so a vanishing line originates in a planeα that is not parallel to the pictureπ. Given the eye pointO, andβ the plane parallel toα and lying onO, then thevanishing line ofα isβ ∩π. For example, whenα is the ground plane andβ is the horizon plane, then the vanishing line ofα is thehorizon lineβ ∩π.
To put it simply, the vanishing line of some plane, sayα, is obtained by the intersection of the image plane with another plane, sayβ, parallel to the plane of interest (α), passing through the camera center. For different sets of lines parallel to this planeα, 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.
1. Projections of two sets of parallel lines lying in some planeπA appear to converge, i.e. the vanishing point associated with that pair, on a horizon line, or vanishing lineH formed by the intersection of the image plane with the plane parallel toπA and passing through the pinhole. Proof: Consider the ground planeπ, asy =c which is, for the sake of simplicity, orthogonal to the image plane. Also, consider a lineL that lies in the planeπ, which is defined by the equationax +bz = d.Using perspective pinhole projections, a point onL projected on the image plane will have coordinates defined as,
This is the parametric representation of the imageL′ of the lineL withz as the parameter. Whenz → −∞ it stops at the point(x′,y′) = (−fb/a,0) on thex′ axis of the image plane. This is the vanishing point corresponding to all parallel lines with slope−b/a in the planeπ. All vanishing points associated with different lines with different slopes belonging to planeπ will lie on thex′ axis, which in this case is the horizon line.
2. LetA,B, andC be three mutually orthogonal straight lines in space andvA ≡ (xA,yA,f),vB ≡ (xB,yB,f),vC ≡ (xC,yC,f) be the three corresponding vanishing points respectively. If we know the coordinates of one of these points, sayvA, and the direction of a straight line on the image plane, which passes through a second point, sayvB, we can compute the coordinates of bothvB andvC[4]
3. LetA,B, andC be three mutually orthogonal straight lines in space andvA ≡ (xA,yA,f),vB ≡ (xB,yB,f),vC ≡ (xC,yC,f) 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.[4]
Acurvilinear perspective 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.
Areverse perspective is a drawing with vanishing points that are placed outside the painting with the illusion that they are "in front of" the painting.
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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 theaccumulation step, the image is mapped onto a bounded space called the accumulator space. The accumulator space is partitioned into units called cells. Barnard[6] assumed this space to be aGaussian 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 theHough 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 thesearch 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.
| 1. | The width of the side street, W is computed from the known widths of the adjacent shops. |
| 2. | As avanishing point, V is visible, the width of only one shop is needed. |
