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Reprojection Error
The reprojection error is a geometric error corresponding to the image distance between a projected point and a measured one. It is used to quantify how closely an estimate of a 3D point \hat recreates the point's true projection \mathbf. More precisely, let \mathbf be the projection matrix of a camera and \hat be the image projection of \hat, i.e. \hat=\mathbf \, \hat. The reprojection error of \hat is given by d(\mathbf, \, \hat), where d(\mathbf, \, \hat) denotes the Euclidean distance between the image points represented by vectors \mathbf and \hat. Minimizing the reprojection error can be used for estimating the error from point correspondences between two images. Suppose we are given 2D to 2D point imperfect correspondences \. We wish to find a homography In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camera Matrix
In computer vision a camera matrix or (camera) projection matrix is a 3 \times 4 matrix which describes the mapping of a pinhole camera from 3D points in the world to 2D points in an image. Let \mathbf be a representation of a 3D point in homogeneous coordinates (a 4-dimensional vector), and let \mathbf be a representation of the image of this point in the pinhole camera (a 3-dimensional vector). Then the following relation holds : \mathbf \sim \mathbf \, \mathbf where \mathbf is the camera matrix and the \, \sim sign implies that the left and right hand sides are equal except for a multiplication by a non-zero scalar k \neq 0: : \mathbf = k \, \mathbf \, \mathbf . Since the camera matrix \mathbf is involved in the mapping between elements of two projective spaces, it too can be regarded as a projective element. This means that it has only 11 degrees of freedom since any multiplication by a non-zero scalar results in an equivalent camera matrix. Derivation Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pinhole Camera
A pinhole camera is a simple camera without a lens but with a tiny aperture (the so-called ''pinhole'')—effectively a light-proof box with a small hole in one side. Light from a scene passes through the aperture and projects an inverted image on the opposite side of the box, which is known as the camera obscura effect. The size of the images depends on the distance between the object and the pinhole. History Camera obscura The camera obscura or pinhole image is a natural optical phenomenon. Early known descriptions are found in the Chinese Mozi writings (circa 500 BCE) and the Aristotelian ''Problems'' (circa 300 BCE – 600 CE). Ibn al-Haytham (965–1039), an Arab physicist also known as Alhazen, described the camera obscura effect. Over the centuries others started to experiment with it, mainly in dark rooms with a small opening in shutters, mostly to study the nature of light and to safely watch solar eclipses. Giambattista Della Porta wrote in 1558 in his Magia Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century. The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homography (computer Vision)
In the field of computer vision, any two images of the same planar surface in space are related by a homography (assuming a pinhole camera model). This has many practical applications, such as image rectification, image registration, or camera motion—rotation and translation—between two images. Once camera resectioning has been done from an estimated homography matrix, this information may be used for navigation, or to insert models of 3D objects into an image or video, so that they are rendered with the correct perspective and appear to have been part of the original scene (see Augmented reality). 3D plane to plane equation We have two cameras ''a'' and ''b'', looking at points P_i in a plane. Passing from the projection ^bp_i=\left(^bu_i;^bv_i;1\right) of P_i in ''b'' to the projection ^ap_i=\left(^au_i;^av_i;1\right) of P_i in ''a'': : ^ap_i = \fracK_a \cdot H_ \cdot K_b^ \cdot ^bp_i where ^az_i and ^bz_i are the z coordinates of P in each camera frame and where the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
