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Perspective-n-Point
Perspective-''n''-Point is the problem of estimating the pose of a calibrated camera given a set of 3D points in the world and their corresponding 2D projections in the image. The camera pose consists of 6 degrees-of-freedom (DOF) which are made up of the rotation (roll, pitch, and yaw) and 3D translation of the camera with respect to the world. This problem originates from camera calibration and has many applications in computer vision and other areas, including 3D pose estimation, robotics and augmented reality. A commonly used solution to the problem exists for called P3P, and many solutions are available for the general case of . A solution for exists if feature orientations are available at the two points. Implementations of these solutions are also available in open source software. Problem Specification Definition Given a set of 3D points in a world reference frame and their corresponding 2D image projections as well as the calibrated intrinsic camera parameters, det ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Camera Resectioning
Camera resectioning is the process of estimating the parameters of a pinhole camera model approximating the camera that produced a given photograph or video; it determines which incoming light ray is associated with each pixel on the resulting image. Basically, the process determines the pose of the pinhole camera. Usually, the camera parameters are represented in a 3 × 4 projection matrix called the ''camera matrix''. The extrinsic parameters define the camera '' pose'' (position and orientation) while the intrinsic parameters specify the camera image format (focal length, pixel size, and image origin). This process is often called geometric camera calibration or simply camera calibration, although that term may also refer to photometric camera calibration or be restricted for the estimation of the intrinsic parameters only. Exterior orientation and interior orientation refer to the determination of only the extrinsic and intrinsic parameters, respectively. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Pose Estimation
3D pose estimation is a process of predicting the transformation of an object from a user-defined reference pose, given an image or a 3D scan. It arises in computer vision or robotics where the pose or transformation of an object can be used for alignment of a computer-aided design models, identification, grasping, or manipulation of the object. The image data from which the pose of an object is determined can be either a single image, a stereo image pair, or an image sequence where, typically, the camera is moving with a known velocity. The objects which are considered can be rather general, including a living being or body parts, e.g., a head or hands. The methods which are used for determining the pose of an object, however, are usually specific for a class of objects and cannot generally be expected to work well for other types of objects. From an uncalibrated 2D camera It is possible to estimate the 3D rotation and translation of a 3D object from a single 2D photo, if an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degrees Of Freedom
Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or other physical processes are possible. This relates to the philosophical concept to the extent that people may be considered to have as much freedom as they are physically able to exercise. Applications Statistics In statistics, degrees of freedom refers to the number of variables in a statistic calculation that can vary. It can be calculated by subtracting the number of estimated parameters from the total number of values in the sample. For example, a sample variance calculation based on n samples will have n-1 degrees of freedom, because sample variance is calculated using the sample mean as an estimate of the actual mean. Mathematics In mathematics, this notion is formalized as the dimension of a manifold or an algebraic variety. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Linear Transformation
Direct linear transformation (DLT) is an algorithm which solves a set of variables from a set of similarity relations: : \mathbf_ \propto \mathbf \, \mathbf_ for \, k = 1, \ldots, N where \mathbf_ and \mathbf_ are known vectors, \, \propto denotes equality up to an unknown scalar multiplication, and \mathbf is a matrix (or linear transformation) which contains the unknowns to be solved. This type of relation appears frequently in projective geometry. Practical examples include the relation between 3D points in a scene and their projection onto the image plane of a pinhole camera, and homographies. Introduction An ordinary system of linear equations : \mathbf_ = \mathbf \, \mathbf_ for \, k = 1, \ldots, N can be solved, for example, by rewriting it as a matrix equation \mathbf = \mathbf \, \mathbf where matrices \mathbf and \mathbf contain the vectors \mathbf_ and \mathbf_ in their respective columns. Given that there exists a unique ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RANSAC
Random sample consensus (RANSAC) is an iterative method to estimate parameters of a mathematical model from a set of observed data that contains outliers, when outliers are to be accorded no influence on the values of the estimates. Therefore, it also can be interpreted as an outlier detection method. It is a non-deterministic algorithm in the sense that it produces a reasonable result only with a certain probability, with this probability increasing as more iterations are allowed. The algorithm was first published by Fischler and Bolles at SRI International in 1981. They used RANSAC to solve the Location Determination Problem (LDP), where the goal is to determine the points in the space that project onto an image into a set of landmarks with known locations. RANSAC uses repeated random sub-sampling. A basic assumption is that the data consists of "inliers", i.e., data whose distribution can be explained by some set of model parameters, though may be subject to noise, and "outlier ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
OpenCV
OpenCV (''Open Source Computer Vision Library'') is a library of programming functions mainly aimed at real-time computer vision. Originally developed by Intel, it was later supported by Willow Garage then Itseez (which was later acquired by Intel). The library is cross-platform and free for use under the open-source Apache 2 License. Starting with 2011, OpenCV features GPU acceleration for real-time operations. History Officially launched in 1999 the OpenCV project was initially an Intel Research initiative to advance CPU-intensive applications, part of a series of projects including real-time ray tracing and 3D display walls. The main contributors to the project included a number of optimization experts in Intel Russia, as well as Intel's Performance Library Team. In the early days of OpenCV, the goals of the project were described as: * Advance vision research by providing not only open but also optimized code for basic vision infrastructure. No more reinventing the wheel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (linear Algebra)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: :\ker(L) = \left\ . Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows that the image of is isomorphic to the quotient of by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value Decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is related to the polar decomposition. Specifically, the singular value decomposition of an \ m \times n\ complex matrix is a factorization of the form \ \mathbf = \mathbf\ , where is an \ m \times m\ complex unitary matrix, \ \mathbf\ is an \ m \times n\ rectangular diagonal matrix with non-negative real numbers on the diagonal, is an n \times n complex unitary matrix, and \ \mathbf\ is the conjugate transpose of . Such decomposition always exists for any complex matrix. If is real, then and can be guaranteed to be real orthogonal matrices; in such contexts, the SVD is often denoted \ \mathbf^\mathsf\ . The diagonal entries \ \sigma_i = \Sigma_\ of \ \mathbf\ are uniquely determined by and are known as the singular values of . The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Newton Algorithm
The Gauss–Newton algorithm is used to solve non-linear least squares problems, which is equivalent to minimizing a sum of squared function values. It is an extension of Newton's method for finding a minimum of a non-linear function. Since a sum of squares must be nonnegative, the algorithm can be viewed as using Newton's method to iteratively approximate zeroes of the components of the sum, and thus minimizing the sum. It has the advantage that second derivatives, which can be challenging to compute, are not required. Non-linear least squares problems arise, for instance, in non-linear regression, where parameters in a model are sought such that the model is in good agreement with available observations. The method is named after the mathematicians Carl Friedrich Gauss and Isaac Newton, and first appeared in Gauss' 1809 work ''Theoria motus corporum coelestium in sectionibus conicis solem ambientum''. Description Given m functions \textbf = (r_1, \ldots, r_m) (often called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
