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Isomorphism of projective spaces in geometry
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces
Homography
Relation of two images with software
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
Homography_(computer_vision)
Topics referred to by the same term
A homography may refer to homography, a type of isomorphism of projective spaces, homography (computer vision), a mapping relating perspective images of
Homography_(disambiguation)
Mathematical formulation of vector pairs used in physics (rigid body dynamics)
eight-dimensional space of dual quaternions. This 3-flat F represents space, and the homography constructed, restricted to F, is a screw displacement of space. Let a
Screw_theory
In projective geometry, a bijection between projective spaces that preserves collinearity
of collineations are projective linear transformations (also known as homographies) and automorphic collineations. For projective spaces coming from a linear
Collineation
Mathematical operation
skew-symmetric matrices and special orthogonal matrices. The transform is a homography used in real analysis, complex analysis, and quaternionic analysis. In
Cayley_transform
Distinct words with identical written forms
A homograph (from the Greek: ὁμός, homós 'same' and γράφω, gráphō 'write') is a word that shares the same written form as another word but has a different
Homograph
Projective construction in ring theory
of A. The projective line P1(A) is equipped with a group of homographies. The homographies are expressed through use of the matrix ring over A and its
Projective_line_over_a_ring
Geometric concept of a 2D space with "points at infinity" adjoined
}+k+1\right)(k+1)k^{3}(k-1)^{2}} different homographies. Every collineation is the product of a homography and an automorphism of GF(pm), and there are
Projective_plane
Combining multiple photographic images with overlapping fields of view
erroneous measurements, or simply incorrect data. For the problem of homography estimation, RANSAC works by trying to fit several models using some of
Image_stitching
Möbius transformation generalized to rings other than the complex numbers
transformation is the restriction to the field of a projective transformation or homography of the projective line. When a, b, c, d are integers (or, more generally
Linear fractional transformation
Linear_fractional_transformation
Function theory with quaternion variable
classical application of quaternions to space mapping. In terms of a homography, the rotation is expressed [ q : 1 ] ( u 0 0 u ) = [ q u : u ] ∼ [ u −
Quaternionic_analysis
Generating high-resolution video frames from given low-resolution ones
Branch to integrate aligned frames Some methods align frames by calculated homography between frames. TGA (Temporal Group Attention) divide input frames to
Video_super-resolution
Invariant in projective geometry
not certain in a ring). One approach to cross ratio interprets it as a homography that takes three designated points to 0, 1, and ∞. Under restrictions
Cross-ratio
Group of real 2×2 matrices with unit determinant
in light of the general theory of SL(2, R). Elements of PSL(2, R) are homographies on the real projective line R ∪ {∞}: [ x , 1 ] ↦ [ x , 1 ] ( a c b
SL2(R)
Name list
spelling of this given name in different languages is a case of interlingual homography. Similar cases occur in English given names (Albert, Bertrand, Christine
José
Association of one output to each input
function extension that is encountered when studying homographies of the real line. A homography is a function h ( x ) = a x + b c x + d {\displaystyle
Function_(mathematics)
Diacritic used in French orthography
circumflex avoids possible homography with other words containing "u": sur ~ sûr(e)(s) (from seür → sëur): The homography with the preposition sur, "on"
Circumflex_in_French
In projective geometry, points that define coordinates
of K. As homographies of P(V) are induced by linear endomorphisms of V, it follows that, given two frames, there is exactly one homography mapping the
Projective_frame
Mathematical metric in geometry
second homographies takes a to 1, thus normalizing an arbitrary a in the interval. The composed homographies are called the cross ratio homography of p
Cayley–Klein_metric
New Zealand applied linguist
English in New Zealand Secondary Schools Word Meaning in Academic English: Homography in the Academic Word List Teaching Communicative and Interactive Vocabulary
Paul_Nation
Four-dimensional number system
Cambridge University Press. ISBN 978-0-521-48022-2. Du Val, Patrick (1964). Homographies, quaternions, and rotations. Oxford mathematical monographs. Clarendon
Quaternion
Methodological basis for 3D CAD/CAM solid modeling and image rendering
known as 3D projection, affine transformation, or projective transform (homography). Rendering an image this way is difficult to achieve with hidden surface/edge
Ray_casting
History of reform efforts in French orthography
French alphabet in replacement of consonant I and U, fixing many cases of homography. uil → vil (vile) Many changes were introduced in the sixth edition of
Reforms_of_French_orthography
\{\mathbf {x_{i}} \leftrightarrow \mathbf {x_{i}} '\}} . We wish to find a homography H ^ {\displaystyle {\hat {\mathbf {H} }}} and pairs of perfectly matched
Reprojection_error
Rational function of the form (az + b)/(cz + d)
are named in honor of August Ferdinand Möbius; they are an example of homographies, linear fractional transformations, bilinear transformations, and spin
Möbius_transformation
Four-sided polygon
Quadrangle (geography) Homography - Any quadrilateral can be transformed into another quadrilateral by a projective transformation (homography) Sometimes called
Quadrilateral
Sacred Islamic relic
shoulder-blade;[citation needed] also, محمد خاتمی by coincidential near-homography, the name of Mohammad Amir Khatam. Tavernier, Jean-Baptiste. "Nouvelle
Seal_of_Muhammad
Construction in group theory
called projective linear transformation, projective transformation or homography. If V is the n-dimensional vector space over a field F, namely V = Fn
Projective_linear_group
Model of the extended complex plane plus a point at infinity
be written as a composition of these. The Möbius transformations are homographies on the complex projective line. In projective coordinates, the transformation
Riemann_sphere
Branch of mathematics
specified, and studied in terms of linear maps. This is also the case of homographies and Möbius transformations when considered as transformations of a projective
Linear_algebra
Completion of the usual space with "points at infinity"
Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term homography, which
Projective_space
Special class of linear fractional transformations
algebra B can be extended to P(B) through the projective line over a ring. Homographies on P(B) include translations: U ( q : 1 ) ( 1 0 t 1 ) = U ( q + t : 1
Special conformal transformation
Special_conformal_transformation
Non-Euclidean geometry
+x_{n}^{2}))^{2}}}} . The isometry to the half-space model can be realised by a homography sending a point of the unit sphere to infinity. Hyperboloid model: In
Hyperbolic_space
Sinitic language spoken in Central Asia
heard as [ɯ˞] in Kyrgyzstan. With х-, -уй is respelled as -уэй to avoid homography with the Russian-language vulgarity. This is the same treatment as in
Dungan_language
Indian American physician and computer scientist
PMID 7547154. Kundu, Shinjini (2012). "Light field compression using homography and 2D warping". 2012 IEEE International Conference on Acoustics, Speech
Shinjini_Kundu
Commune in Nouvelle-Aquitaine, France
Aquitaine), southwestern France. It is located north of Bordeaux. The homography with Bruges (Belgium) is purely coincidental. The place-name comes from
Bruges,_Gironde
Design technique
geometry Engineering drawing Exploded-view drawing Homogeneous coordinates Homography Map projection (including cylindrical projection) Multiview projection
3D_projection
Algorithm to solve systems of equations
scene and their projection onto the image plane of a pinhole camera, and homographies. An ordinary system of linear equations x k = A y k {\displaystyle \mathbf
Direct_linear_transformation
Computer vision geometry concept
normalized image coordinates can be transformed by means of an arbitrary 2D homography. This includes 2D translations and rotations as well as scaling (isotropic
Camera_matrix
Study of angle-preserving transformations
circle inversion, is anticonformal. In this case a homography is conformal while an anti-homography is anticonformal. The (n − 1)-sphere with equation
Inversive_geometry
Geometric transformation that preserves lines but not angles nor the origin
applications of affine transformations Bent function Flat (geometry) Homography Multilinear polynomial Berger 1987, p. 38. Samuel 1988, p. 11. Snapper
Affine_transformation
Property of points all lying on a single line
collineations. In projective geometry these linear mappings are called homographies and are just one type of collineation. In any triangle the following
Collinearity
Word with different meaning in different languages
word dividers, etc. Language portal Linguistics portal Heterography and homography – Distinct words with identical written formsPages displaying short descriptions
Interlingual_homograph
Image distortion caused by projection
where the projector must be mounted outside the frame of the screen. Homography (computer vision) Perspective control Introduction to stereo microscopy
Keystone_effect
Comparison of Scandinavian languages
consonants before inflexional endings beginning in a consonant (unless homography needs to be avoided), but not before derivational endings and in compounding
Comparison of Danish, Norwegian and Swedish
Comparison_of_Danish,_Norwegian_and_Swedish
Computer vision technique for optical flow estimation
optical flow algorithm Python example using the Lucas-Kanade tracker for homography matching MATLAB quick example of Lucas-Kanade method to show optical flow
Lucas–Kanade_method
Transformation process to project images
problem Epipolar geometry Geographic information system Georeferencing Homography Image registration Photogrammetry Point set registration Rubbersheeting
Image_rectification
Process of estimating the parameters of a pinhole camera model
{\displaystyle R} and T {\displaystyle T} calibration parameters. Assume we have a homography H {\displaystyle {\textbf {H}}} that maps points x π {\displaystyle x_{\pi
Camera_resectioning
Embedding a topological space into a compact space as a dense subset
conformal group of spacetime described in Associative Composition Algebra/Homographies at Wikibooks Roubíček, T. (1997). Relaxation in Optimization Theory and
Compactification (mathematics)
Compactification_(mathematics)
Robot control by visual feedback
a homography, (essentially 3D information) giving an axis of rotation and the angle (by computing the eigenvalues and eigenvectors of the homography).
Visual_servoing
Orthography of the Italian language
although it may still appear in contexts where ambiguity might arise from homography. For example, it can be used to differentiate words such as geni ("genes"
Italian_orthography
Geometric axis of rotation and translation
it to a parallel rotation axis. Such a conjugation, using quaternion homographies, produces the appropriate screw axis to express the given spatial displacement
Screw_axis
German mathematician (1862 – 1930)
field of kinematics, in particular exhibiting an element of E(3) as a homography of dual quaternions. Study's use of abstract algebra was noted in A History
Eduard_Study
Concept in geometry
Horton Conway and Derek A. Smith ISBN 978-1-56881-134-5 Patrick Du Val, Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon
Coxeter_element
Concept in mathematics
of an associative ring with 1, see projective line over a ring and the homography group GL(2,A). From the topological point of view the quaternionic projective
Quaternionic_projective_space
Feature detection algorithm in computer vision
correspondences are then used to find m candidate matching images for each image. Homographies between pairs of images are then computed using RANSAC and a probabilistic
Scale-invariant feature transform
Scale-invariant_feature_transform
Israeli computer scientist
1281–1283. doi:10.1109/TPAMI.2013.79. Eshel, Ran; Moses, Yael (2008). "Homography based multiple camera detection and tracking of people in a dense crowd"
Yael_Moses
Position and orientation of an object in an image
world coordinates of a point viewed by both cameras. Gesture recognition Homography (computer vision) Camera calibration Structure from motion Essential matrix
Pose_(computer_vision)
1879, p.283) homography 1. An isomorphism between projective spaces induced by an isomorphism of vector spaces. 2. An axis of homography is a line associated
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Line with a point at infinity added
infinity may be represented as [1 : 0]. Quite generally, the group of homographies with coefficients in K acts on the projective line P1(K). This group
Projective_line
Traditional Chinese fivefold deity
its Shang versions, was alternately represented as a cross potent ☩, in homography with 巫; wū ("shaman"). The Chinese concept equivalent of the Indian mandala
Wufang_Shangdi
for example, bundle adjustment method) we wish to define rectifying homography H {\displaystyle H} such that { P j H , H − 1 X J } {\displaystyle \left\{P^{j}H
Camera_auto-calibration
Process of determining spatial characteristics of objects
object recognition Articulated body pose estimation Camera calibration Homography (computer vision) Trifocal tensor Pose estimation Javier Barandiaran (28
3D_pose_estimation
in the English literature one often finds "path curve" or "pathcurve". Homography Felix Klein and Sophus Lie: Ueber diejenigen ebenen Curven... in Mathematische
W-curve
{\displaystyle I(.)} is the indicator function, H m {\displaystyle H_{m}} is the homography transformation from I 0 {\displaystyle I_{0}} to I m {\displaystyle I_{m}}
Rank_SIFT
Matrix in computer version
{P}}'{\textbf {X}}\end{aligned}}} Say we transform space by a general homography matrix H 4 × 4 {\displaystyle {\textbf {H}}_{4\times 4}} such that X 0
Fundamental matrix (computer vision)
Fundamental_matrix_(computer_vision)
Locating a moving object by analyzing frames of a video
objects, the motion model is a 2D transformation (affine transformation or homography) of an image of the object (e.g. the initial frame). When the target is
Video_tracking
Bavarian mathematician and surveyor (1862–1951)
von Brill. Finsterwalder observed that Rudolf Sturm's analysis of the "homography problem" (1869) can be used to solve the problem of 3D-reconstruction
Sebastian_Finsterwalder
Real numbers with an added point at infinity
real projective line. The projectivities are described algebraically as homographies, since the real numbers form a ring, according to the general construction
Projectively extended real line
Projectively_extended_real_line
projectivity (projective transformation, projective collineation and homography are synonyms). There are several results concerning projectivities and
Perspectivity
Projective line over the real numbers
automorphisms of a real projective line are called projective transformations, homographies, or linear fractional transformations. They form the projective linear
Real_projective_line
The Laguerre transformations or axial homographies are an analogue of Möbius transformations over the dual numbers. When studying these transformations
Laguerre_transformations
Hungarian mathematician
groupe commutative", Comptes rendus 210:288. 1940: "Sur le group des homographies et des antihomographies d’une variable complexe", Commentarii Mathematici
Béla_Kerékjártó
Overview of and topical guide to computer vision
correction Anisotropic diffusion (Perona–Malik equation) Affine transform Homography (computer vision) Hough transform Radon transform Walsh–Hadamard transform
Outline_of_computer_vision
{\displaystyle x_{A}=H\cdot x_{B}} . For a pair of two images related through a homography matrix H {\displaystyle H} , two points, x a {\displaystyle \mathbf {x_{a}}
Harris_affine_region_detector
affine spaces: bijective affine transformation. For projective spaces: homography. In fact, Bourbaki stipulates two additional features. First, several
Equivalent definitions of mathematical structures
Equivalent_definitions_of_mathematical_structures
Mathematical operation modeling parallel resistors
the homographies on the projective line are represented by 2 x 2 matrices M(2,F), and the field operations (+ and ×) are extended to homographies. Each
Parallel_(operator)
Creation of a 3D model from a set of images
in the projective stratum is a series of projective transformations (a homography), in the affine stratum is a series of affine transformations, and in
3D reconstruction from multiple images
3D_reconstruction_from_multiple_images
Interactive geometry software
curve; Dilation, generic affinity, inversion, projective application, homography and harmonic homology; A hyperbola with given asymptotes; Bézier curves
Kig_(software)
geometry dissertation on two general principles of science, duality and homography.] (in French) (2nd ed.). Paris (École polytechnique): Gauthiers-Villars
List of polyhedral stellations
List_of_polyhedral_stellations
to satisfy either an epipolar constraint for a general rigid-body or a homography constraint for a planar object. Planar motion in a sequence is the motion
Rigid_motion_segmentation
MacKay Bravais Lattices in Four-dimensional Space 1964 Patrick du Val, Homographies, quaternions and rotations, quaternion-based 4D point groups 1975 Jan
Point groups in four dimensions
Point_groups_in_four_dimensions
_{b}A)}}} where A is the locally linearized affine transformation of the homography between the two images, and μ a ∩ ( A T μ b A ) {\displaystyle \mu _{a}\cap
Kadir–Brady_saliency_detector
Group representation of a Riemann surface
L 2 ( R ) {\displaystyle \mathrm {PSL} _{2}(\mathbb {R} )} acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free
Fuchsian_model
Topics referred to by the same term
algebraic geometry Linear fractional transformation, an (invertible) homography based on a ratio of linear polynomials Linearly normal (AKA 1-normal)
Linear_(disambiguation)
Blob detection technique
eight-point algorithm. This algorithm can be tested here (Epipolar or homography geometry constrained matches): WBS Image Matcher The MSER algorithm has
Maximally stable extremal regions
Maximally_stable_extremal_regions
British mathematician (1903–1987)
hyperelliptic, Canadian Journal of Mathematics 4: 204–221. MR 0048090 1964: Homographies, quaternions and rotations, Oxford Mathematical Monographs, Clarendon
Patrick_du_Val
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