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Projective GeometryFunctions related to Projective Geometry.
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Functions | |
QVMatrix | ComputeProjectiveHomography (const QList< QPointFMatching > &matchings) |
Obtains a planar homography from a list of point correspondences. | |
QVMatrix | ComputeAffineHomography (const QList< QPointFMatching > &matchings) |
Obtains an affine homography from a list of point correspondences. | |
QVMatrix | ComputeEuclideanHomography (const QList< QPointFMatching > &matchings) |
Obtain an euclidean mapping for a set of inter-image point matchings. | |
QVMatrix | cvFindFundamentalMat (const QList< QPointFMatching > &matchings) |
Obtains the fundamental matrix between two images, using the 8 point algorithm. | |
QPointF | ApplyHomography (const QVMatrix &homography, const QPointF &point) |
Maps a point using an homography. | |
QList< QPointF > | ApplyHomography (const QVMatrix &homography, const QList< QPointF > &sourcePoints) |
Maps a set of points using an homography. | |
QVImage< uChar, 1 > | ApplyHomography (const QVMatrix &homography, const QVImage< uChar, 1 > &image, const int interpolation=IPPI_INTER_CUBIC) |
Performs an homography distortion on an image. | |
QVImage< uChar, 3 > | ApplyHomography (const QVMatrix &homography, const QVImage< uChar, 3 > &image, const int interpolation=IPPI_INTER_CUBIC) |
Performs an homography distortion on an image. | |
double | HomographyTestError (const QVMatrix &homography) |
Function to test if a 3x3 matrix corresponds to an homography. | |
void | GetExtrinsicCameraMatrixFromHomography (const QVMatrix &K, const QVMatrix &H, QVMatrix &M4x4) |
void | GetDirectIntrinsicCameraMatrixFromHomography (const QVMatrix &H, QVMatrix &K) |
Diagonal intrinsic camera matrix calibration. | |
void | CalibrateCameraFromPlanarHomography (const QVMatrix &H, QVMatrix &K, QVMatrix &Rt) |
Diagonal intrinsic camera matrix calibration. | |
void | GetPinholeCameraIntrinsicsFromPlanarHomography (const QVMatrix &H, QVMatrix &K, const int iterations=100, const double maxGradientNorm=1e-3, const double step=0.01, const double tol=1e-4) |
Obtains the intrinsic camera matrix from a planar homography. | |
QList< QVMatrix > | getCanonicalCameraMatricesFromEssentialMatrix (const QVMatrix &E) |
Obtains the canonical matrices corresponding to an essential matrix. | |
QVMatrix | getEssentialMatrixFromCanonicalCameraMatrix (const QVMatrix &P) |
Obtains the essential matrix corresponding to a canonical camera matrix. | |
QVMatrix | getCameraMatrixFrom2D3DPointCorrespondences (const QList< QPointF > &points2d, const QList< QV3DPointF > &points3d) |
Obtains the canonical camera matrix corresponding to a set of 3D point to image point matchings. | |
QVMatrix | refineExtrinsicCameraMatrixWithQRDecomposition (const QVMatrix &P) |
Eliminates errors in the rotation component of a canonical camera matrix using a QR decomposition. | |
QVMatrix | refineExtrinsicCameraMatrixWithPolarDecomposition (const QVMatrix &P) |
Eliminates errors in the rotation component of a canonical camera matrix using a Polar decomposition. | |
QV3DPointF | triangulate3DPointFromNViews (const QList< QPointF > &points, const QList< QVMatrix > &Plist) |
Recovers the location of a 3D point from its projection on several images and their corresponding canonical camera matrices. | |
QV3DPointF | triangulate3DPointFrom2Views (const QPointF &point1, const QVMatrix &P1, const QPointF &point2, const QVMatrix &P2) |
Recovers the location of a 3D point from its projection on two images and their corresponding canonical camera matrices. |
Projective Geometry is based on the principles of perspective. It models the relationships between image points and their corresponding points
in the 3D world. It is the most fundamental theoretical tool for 3D reconstruction, and augmented reality applications.
This module uses the concept of projection camera matrices. A projection camera matrix models the linear relationship between the location 3D points and their corresponding pixels in a certain image. The projection camera matrix contains thus information about the intrinsic camera parameters, and the location and orientation of the camera, in the 3D space.
Using homogeneous notation (where is a 3D point, and
is its corresponding projection location in the image) the following equation holds:
Matrix is decomposed in the intrinsic (
) and the extrinsic (
) camera matrices, with the following expression:
Another convention used in this module is the canonical camera matrix. It is equivalent to the extrinsic matrix of a camera matrix:
The essential matrix relates the normalized locations of corresponding points at two different images:
These normalized locations are obtained eliminating the deformations induced by the intrinsic camera parameters in the coordinates of the pixels.
QVImage<uChar, 3> ApplyHomography | ( | const QVMatrix & | homography, | |
const QVImage< uChar, 3 > & | image, | |||
const int | interpolation = IPPI_INTER_CUBIC | |||
) |
Performs an homography distortion on an image.
The homography is represented as a matrix. These matrices can be obtained using methods like ComputeEuclideanHomography, ComputeProjectiveHomography, or ComputeAffineHomography.
This function takes a three channel image, and an homography matrix as inputs. Each point in the input image is mapped to its location in the resulting image
using the homography
as follows:
homography | The homography transformation matrix | |
image | The input image to distort | |
interpolation | Type of interpolation. Possible values for this parameter are:
|
Definition at line 397 of file qvprojective.cpp.
QVImage<uChar, 1> ApplyHomography | ( | const QVMatrix & | homography, | |
const QVImage< uChar, 1 > & | image, | |||
const int | interpolation = IPPI_INTER_CUBIC | |||
) |
Performs an homography distortion on an image.
The homography is represented as a matrix. These matrices can be obtained using methods like ComputeEuclideanHomography, ComputeProjectiveHomography, or ComputeAffineHomography.
This function takes a single channel image, and an homography matrix as inputs. Each point in the input image is mapped to its location in the resulting image
using the homography
as follows:
homography | The homography transformation matrix | |
image | The input image to distort | |
interpolation | Type of interpolation. Possible values for this parameter are:
|
Definition at line 390 of file qvprojective.cpp.
QList<QPointF> ApplyHomography | ( | const QVMatrix & | homography, | |
const QList< QPointF > & | sourcePoints | |||
) |
Maps a set of points using an homography.
This is an overloaded version of the ApplyHomography(const QVMatrix &, const QPointF &) function provided by convenience. This funcion takes a list of points from the 2D plane, and an homography matrix as inputs. The output will be a list of points obtained by mapping the points from the input list, using the homography.
homography | The homography transformation matrix | |
sourcePoints | Points to apply the homography transformation |
Definition at line 382 of file qvprojective.cpp.
QPointF ApplyHomography | ( | const QVMatrix & | homography, | |
const QPointF & | point | |||
) |
Maps a point using an homography.
This function maps a point in the 2D plane, using a planar homography. The homography is represented as a
matrix. These matrices can be obtained using methods like ComputeEuclideanHomography, ComputeProjectiveHomography, or ComputeAffineHomography from a set of point correspondences between the original 2D plane, and the mapped plane.
The output of the function is a point which holds the following equation:
homography | The homography transformation matrix | |
point | Point to apply the homography transformation |
Definition at line 371 of file qvprojective.cpp.
Referenced by ApplyHomography().
Diagonal intrinsic camera matrix calibration.
This function obtains a diagonal intrinsic camera matrix, consisting on the focal distance only. This matrix is such that
Where is a rotation matrix. This matrix is obtained with a minimization process, so its result is more precise than that obtained with GetDirectIntrinsicCameraMatrixFromHomography function.
Definition at line 443 of file qvprojective.cpp.
QVMatrix ComputeAffineHomography | ( | const QList< QPointFMatching > & | matchings | ) |
Obtains an affine homography from a list of point correspondences.
This function obtains the homography matrix which most closely maps the source points to their destination, in the input point matching list. This homography matrix will represent an affine transformation.
The function returns the matrix corresponding to the planar homography, from a list of three or more point correspondences between the location of those points at the source plane and their location in the destination plane.
Usage:
QList< QPair<QPointF, QPointF> > matchings; matchings.append(QPair<QPointF, QPointF>(QPointF(-171,109), QPointF(-100,+100))); matchings.append(QPair<QPointF, QPointF>(QPointF(-120,31), QPointF(-100,-100))); matchings.append(QPair<QPointF, QPointF>(QPointF(117,53), QPointF(+100,-100))); const QVMatrix M = ComputeAffineHomography(matchings);
Any point from the 2D plane can be mapped to another point in the plane
with an affine matrix
using the following C++ code:
QPointF q = M * QVVector::homogeneousCoordinates(p);
Or by using the ApplyHomography functions.
matchings | list of point matchings from the original location, to the destination location. |
Definition at line 153 of file qvprojective.cpp.
QVMatrix ComputeEuclideanHomography | ( | const QList< QPointFMatching > & | matchings | ) |
Obtain an euclidean mapping for a set of inter-image point matchings.
This function obtains an euclidean mapping between the source an destination locations of a set of point matchings. The mapping is returned as a matrix which can be multiplied to the source location of each mapping point in homogeneous coordinates, obtaining a location close to the destination location from the matching.
Definition at line 201 of file qvprojective.cpp.
QVMatrix ComputeProjectiveHomography | ( | const QList< QPointFMatching > & | matchings | ) |
Obtains a planar homography from a list of point correspondences.
This function obtains the homography matrix which most closely maps the source points to their corresponding destination, in the input point matching list. This homography matrix will represent a projective transformation.
The function returns the matrix corresponding to the planar homography, from a list of four or more point correspondences between the location of those points at the source plane and their location in the destination plane.
Projective homography matrices can be used to obtain a frontal view of any planar figure appearing in an image. The following code and images shows this effect:
Image on the left is the original picture obtained with a pin-hole camera. Image on the right was obtained aplying a rectification homography to the pixels in the original image. This rectification homography is obtained using the function ComputeProjectiveHomography with the four point correspondences between the points in the images featured with yellow dots:
QList< QPair<QPointF, QPointF> > matchings; matchings.append(QPair<QPointF, QPointF>(QPointF(-171,109), QPointF(-100,+100))); matchings.append(QPair<QPointF, QPointF>(QPointF(-120,31), QPointF(-100,-100))); matchings.append(QPair<QPointF, QPointF>(QPointF(117,53), QPointF(+100,-100))); matchings.append(QPair<QPointF, QPointF>(QPointF(11,115), QPointF(+100,+100))); const QVMatrix H = ComputeProjectiveHomography(matchings);
The obtained matrix holds the following equation:
Also for each point matching contained in the matchings list, the following equation holds.
or
In C++ source code:
QPair<QPointF, QPointF> matching; foreach(matching, matchings) { const QVVector p_0 = QVVector::homogeneousCoordinates(matching.first), p_1 = QVVector::homogeneousCoordinates(matching.second); std::cout << "p1 x H p0 =" << (p_1 ^ H*p_0) << std::endl; }
The printed values are close to 0.
Any point from the 2D plane can be mapped to another point in the plane
with an homography matrix
using the following C++ code:
QPointF q = H * QVVector::homogeneousCoordinates(p);
Or by using the ApplyHomography functions.
matchings | list of point matchings from the original location, to the destination location. |
Definition at line 93 of file qvprojective.cpp.
QVMatrix cvFindFundamentalMat | ( | const QList< QPointFMatching > & | matchings | ) |
Obtains the fundamental matrix between two images, using the 8 point algorithm.
This function performs point normalization to robustly obtain the F matrix.
matchings | list of 8 point matchings |
Definition at line 264 of file qvprojective.cpp.
Referenced by cvFindFundamentalMat().
QVMatrix getCameraMatrixFrom2D3DPointCorrespondences | ( | const QList< QPointF > & | points2d, | |
const QList< QV3DPointF > & | points3d | |||
) |
Obtains the canonical camera matrix corresponding to a set of 3D point to image point matchings.
The following formula models the relation between a set of points from the 3D world and their projections
at an image:
This function uses a DLT to obtain the camera matrix from a given set of 3D points and their corresponding image points.
points2d | List containing the points from the image. | |
points3d | List containing the 3D points. |
Definition at line 557 of file qvprojective.cpp.
Obtains the canonical matrices corresponding to an essential matrix.
See section 9.6.2 from Multiple View Geometry in Computer Vision.
E | The input essential matrix. |
Definition at line 534 of file qvprojective.cpp.
Diagonal intrinsic camera matrix calibration.
This function obtains a diagonal intrinsic camera matrix, consisting on the focal distance only. This matrix is such that
Where is a rotation matrix. This function returns a direct approximation for the
matrix.
Definition at line 427 of file qvprojective.cpp.
Referenced by CalibrateCameraFromPlanarHomography().
Obtains the essential matrix corresponding to a canonical camera matrix.
This method estimates the essential matrix of a two view scenario, provided the second camera matrix , considering the first camera matrix as the identity:
See section 9.6.1 from Multiple View Geometry in Computer Vision.
P | The input canonical matrix. |
Definition at line 552 of file qvprojective.cpp.
void GetPinholeCameraIntrinsicsFromPlanarHomography | ( | const QVMatrix & | H, | |
QVMatrix & | K, | |||
const int | iterations = 100 , |
|||
const double | maxGradientNorm = 1e-3 , |
|||
const double | step = 0.01 , |
|||
const double | tol = 1e-4 | |||
) |
Obtains the intrinsic camera matrix from a planar homography.
This functions obtains the intrinsic calibration matrix corresponding to a simple pinhole camera model. The intrinsic camera matrix has only one free parameter, related to the focal distance of the camera:
This function should receive a planar homography corresponding to the mapping of a set of know template points, to an image frame captured with the camera containing a view of that template.
The following is an example of a full intrinsic camera calibration, knowing a set of point matchings between the template image and the input image read from the camera:
[...] QList< QPointFMatching > matchings; matchings.append(QPointFMatching(QPointF(-171,109), QPointF(-100,+100))); matchings.append(QPointFMatching(QPointF(-120,31), QPointF(-100,-100))); matchings.append(QPointFMatching(QPointF(117,53), QPointF(+100,-100))); matchings.append(QPointFMatching(QPointF(11,115), QPointF(+100,+100))); QVMatrix H, K; H = ComputeProjectiveHomography(matchings); GetPinholeCameraIntrinsicsFromPlanarHomography(H, K); [...]
For further understanding of the planar homography calibration process, see ComputeProjectiveHomography function documentation.
H | Input planar homography. | |
K | Matrix to store the intrinsic camera matrix | |
iterations | Cumber of iterations to perform camera calibration | |
maxGradientNorm | Minimal value of the gradient size (norm 2) to stop the minimization when reached. | |
step | Corresponds to parameter step for the gsl_multimin_fdfminimizer_set function. | |
tol | Corresponds to parameter tol for the gsl_multimin_fdfminimizer_set function. |
Definition at line 500 of file qvprojective.cpp.
double HomographyTestError | ( | const QVMatrix & | homography | ) |
Function to test if a 3x3 matrix corresponds to an homography.
Every matrix which holds a perspective deformation from one plane to another should validate some constrains. The most important is that its two first columns should be perpendicular, and with a similar size, because for the matrix to correspond to an homography, they should be contained in a base of a rotated coordinate system. Given the following homography matrix:
The error value returned by this function for it will be:
That error is a measure of the difference of sizes between the norm of the two column vectors of the homography, corresponding to the two first columns of the rotation matrix, and their dot product. When both values are close to zero, the matrix corresponds to an homography, and it won't otherwise.
A good method to prove that a matrix corresponds to an homography using this function, can be done testing the return value with a threshold of aproximately 0.3. If the return value of this function for a matrix is lower than this threshold, that matrix is likely to correspond to an homography, and is not likely to correspond otherwise.
homography | a possible homography matrix. |
Definition at line 405 of file qvprojective.cpp.
Eliminates errors in the rotation component of a canonical camera matrix using a Polar decomposition.
Given a matrix P* = [R*|t*] which approximates a canonical camera matrix, this function obtains the closest valid canonical matrix . The rotation matrix of this valid canonical matrix must be an orthogonal matrix. Thus it must fullfill:
This function uses a correcting square 3x3 matrix E-1 that satisfies the following equation:
This matrix E-1 also satisfies that the matrix R is the rotation matrix closest to R* in a certain sense. Using the polar decomposition of R*, the function obtains the E which corrects it to the closest valid rotation matrix R regarding the Frobenius norm .
The polar decomposition of the transpose of matrix R* is the following:
Where U is a rotation matrix and D is a positive-semidefinite Hermitian matrix. The matrix DT is used as the matrix E.
Definition at line 637 of file qvprojective.cpp.
Referenced by QVEuclideanMapping3::QVEuclideanMapping3().
Eliminates errors in the rotation component of a canonical camera matrix using a QR decomposition.
Function refineExtrinsicCameraMatrixWithPolarDecomposition(const QVMatrix &) obtains a valid canonical camera matrix P = [R|t] from a given initial approximation P* = [R*|t*], using the Polar decomposition.
This function obtains a computationally more efficient approximation of the P matrix using a QR decomposition. The rotation matrix R of the resulting P is not the closest rotation matrix to the initial R* regarding the Frobenius norm .
Despite of that, this function can be used when a faster and slightly less accurate version of the best P is wanted.
Definition at line 621 of file qvprojective.cpp.
QV3DPointF triangulate3DPointFrom2Views | ( | const QPointF & | point1, | |
const QVMatrix & | P1, | |||
const QPointF & | point2, | |||
const QVMatrix & | P2 | |||
) |
Recovers the location of a 3D point from its projection on two images and their corresponding canonical camera matrices.
Given the projection formula:
The location of the 3D point can be triangulated with its projection on two images.
This function triangulates the location of a 3D point from the locations of two image projections
,
and the corresponding canonical camera matrices for these images
,
.
The method used is described at section 12.2 from Multiple View Geometry in Computer Vision.
point1 | The projected location of the 3D point in the first image. | |
P1 | The camera matrix for the first image. | |
point2 | The projected location of the 3D point in the second image. | |
P2 | The camera matrix for the second image. |
Definition at line 664 of file qvprojective.cpp.
QV3DPointF triangulate3DPointFromNViews | ( | const QList< QPointF > & | points, | |
const QList< QVMatrix > & | Plist | |||
) |
Recovers the location of a 3D point from its projection on several images and their corresponding canonical camera matrices.
Given the projection formula:
The location of the 3D point can be triangulated with its projection on two or more images.
This function triangulates the location of a 3D point from the locations of two or more image projections
and the corresponding canonical camera matrices for these images
.
The method used is described at section 12.2 from Multiple View Geometry in Computer Vision.
points | List containing the points from the image. | |
Plist | List containing the canonical camera matrices. |
Definition at line 685 of file qvprojective.cpp.