돛단배

Projective reconstruction
= Affine upgrade (H_p) + Euclidean upgrade (H_a)
=> Auto-calibration

Kruppa equations

kruppa.pdf

 

Range Imaging
Range Sensor Calibration

Fundamental Matrix

http://en.wikipedia.org/wiki/Fundamental_matrix_(computer_vision)
In computer vision, the fundamental matrix is a matrix of rank 2 which relates corresponding points in stereo images. In epipolar geometry, with homogeneous image coordinates and of corresponding points in a stereo image pair, describes a line (an epipolar line) on which the corresponding point y₂ on the other image must lie. That means, for all pairs of corresponding points holds

Being of rank two and determined only up to scale, the fundamental matrix can be estimated given at least seven point correspondences. Its seven parameters represent the only geometric information about cameras that can be obtained through point correspondences alone.

epipole
epipolar line
epipolar plane

http://en.wikipedia.org/wiki/Epipolar_geometry


RANdom SAmple Consensus
http://en.wikipedia.org/wiki/RANSAC



ref.
http://www.robots.ox.ac.uk/~vgg/hzbook/index.html

http://nonsense.danielwedge.com/2008/10/19/the-fundamental-matrix-song/


An Invitation to 3-D Vision
Yi Ma, Stefano Soatto, Jana Kosecka, Shankar Sastry
Springer Verlag, 2003

Learning Epipolar Geometry
The Java code for this page was created by Sylvain Bougnoux.

image filtering
image warping

LSI (Linear Shift Invariance) - convolution

http://en.wikipedia.org/wiki/Convolution
Convolution is a mathematical operation on two functions f and g, producing a third function that is typically viewed as a modified version of one of the original functions.

http://en.wikipedia.org/wiki/Cross_correlation

http://en.wikipedia.org/wiki/Gaussian_filter
Mathematically, a Gaussian filter modifies the input signal by convolution with a Gaussian function; this transformation is also known as the Weierstrass transform.
Gaussian filters are designed to give no overshoot to a step function input while minimizing the rise and fall time (which leads to the steepest possible slope). This behavior is closely connected to the fact that the Gaussian filter has the minimum possible group delay.

Structure from Stereo

two or more images, calibrated cameras (K(f), R, T known)
http://en.wikipedia.org/wiki/Stereo_vision

ref.
UCR stereographs
http://www.cmp.ucr.edu/site/exhibitions/stereo/
The Art of Stereo Photography
http://www.photostuff.co.uk/stereo.htm
History of Stereo Photography
http://www.rpi.edu/~ruiz/stereo_history/text/historystereog.html
Double Exposure
http://home.centurytel.net/s3dcor/index.html
Stereo Photography
http://www.shortcourses.com/book01/chapter09.htm
3D Photography links
http://www.studyweb.com/links/5243.html
National Stereoscopic Association
http://204.248.144.203/3dLibrary/welcome.html
Books on Stereo Photography
http://userwww.sfsu.edu/~hl/3d.biblio.html

http://en.wikipedia.org/wiki/Epipolar_geometry

http://en.wikipedia.org/wiki/Image_rectification
http://en.wikipedia.org/wiki/Rectification_(geometry)


Computing Rectifying Homographies for Stereo Vision
Charles Loop, Zhengyou Zhang

ComputingRectifyingHomographies.pdf

SSD (Sum of Squared Difference) to choose the baseline 

Structure from Motion

 
http://en.wikipedia.org/wiki/Structure_from_motion
http://en.wikipedia.org/wiki/Motion_field

http://en.wikipedia.org/wiki/Nodal_point#Nodal_points


optical flow, image flow
http://en.wikipedia.org/wiki/Optical_flow

Horn-Schunck Algorithm
http://en.wikipedia.org/wiki/Horn-Schunck_algorithm

Determining Optical Flow

horn_schunck_DeterminingOpticalFlow.pdf

shi_tomasi_GoodFeaturesToTrack.pdf

Single View Metrology

criminisi_reid_zisserman_SingleViewMetrology.pdf

 http://en.wikipedia.org/wiki/VRML
VRML (Virtual Reality Modeling Language, pronounced vermal or by its initials, originally — before 1995 — known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphics, designed particularly with the World Wide Web in mind. It has been superseded by X3D.


3-D Metrology
http://research.microsoft.com/~antcrim/

single view reconstruction

http://en.wikipedia.org/wiki/Ground_truth

radial distortion
http://en.wikipedia.org/wiki/Distortion_(optics)

A point in the image is a ray in projective space.

=> All points on the ray are equivalent.

http://en.wikipedia.org/wiki/Grassman_algebra


vanishing point
http://mathdl.maa.org/mathDL/1//?pa=content&sa=viewDocument&nodeId=477&bodyId=598

vanishing line

eigen value
eigen vector

matrix rank

symmetric matrix
http://en.wikipedia.org/wiki/Symmetric_matrix

eigenvalue decomposition
http://en.wikipedia.org/wiki/Spectral_theorem

singular value decomposition
http://en.wikipedia.org/wiki/Singular_value_decomposition

linear equations
- inhomogeneous equations
- homogeneous equations - unconstrained / constrained



triangular decompostion:
QR, LL, LU decomposition

QR decomposition
http://en.wikipedia.org/wiki/QR_decomposition
- householder transformation
http://en.wikipedia.org/wiki/Householder_transformation
- Givens rotation
http://en.wikipedia.org/wiki/Givens_rotation
- Gram-Schmidt transformation
http://en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process

Cholesky LL decomposition
http://en.wikipedia.org/wiki/Cholesky_decomposition

LU decomposition
ref. http://rkb.home.cern.ch/rkb/AN16pp/node160.html#SECTION0001600000000000000000
http://en.wikipedia.org/wiki/LU_decomposition



nonlinear optimization

http://en.wikipedia.org/wiki/Newton%27s_method
http://en.wikipedia.org/wiki/Newton%27s_method_in_optimization

http://en.wikipedia.org/wiki/Linear_least_squares

cf. nonlinear least squares
http://en.wikipedia.org/wiki/Non-linear_least_squares

http://en.wikipedia.org/wiki/Pinhole_camera
Cameras using small apertures, and the human eye in bright light both act like a pinhole camera. The smaller the hole, the sharper the image, but the dimmer the projected image. Optimally, the size of the aperture, should be 1/100 or less of the distance between it and the screen.

http://en.wikipedia.org/wiki/Projection_matrix
a projection is a linear transformation P from a vector space to itself such that P2 = P. It leaves its image unchanged. Though abstract, this definition of "projection" formalizes and generalizes the idea of graphical projection.

http://en.wikipedia.org/wiki/Orthographic_projection
Increasing the focal length and distance of the camera to infinity in a perspective projection results in an orthographic projection. It is a form of parallel projection, where the view direction is orthogonal to the projection plane.

http://en.wikipedia.org/wiki/Homogeneous_coordinate
The homogeneous coordinates of a point of projective space of dimension n are usually written as (x : y : z : ... : w), a row vector of length n + 1, other than (0 : 0 : 0 : ... : 0). Two sets of coordinates that are proportional denote the same point of projective space: for any non-zero scalar c from the underlying field K, (cx : cy : cz : ... : cw) denotes the same point. Therefore this system of coordinates can be explained as follows: if the projective space is constructed from a vector space V of dimension n + 1, introduce coordinates in V by choosing a basis, and use these in P(V), the equivalence classes of proportional non-zero vectors in V.

http://en.wikipedia.org/wiki/Affine_transformation
In geometry, an affine transformation or affine map or an affinity (from the Latin, affinis, "connected with") between two vector spaces (strictly speaking, two affine spaces) consists of a linear transformation followed by a translation:
             
In the finite-dimensional case each affine transformation is given by a matrix A and a vector b, satisfying certain properties described below.
Physically, an affine transformation is one that preserves

In general, an affine transform is composed of zero or more linear transformations (rotation, scaling or shear) and a translation (or "shift"). Several linear transformations can be combined into a single matrix, thus the general formula given above is still applicable.

http://en.wikipedia.org/wiki/Projective_space
The idea of a projective space relates to perspective, more precisely to the way an eye or a camera projects a 3D scene to a 2D image. All points which lie on a projection line (i.e. a "line-of-sight"), intersecting with the focal point of the camera, are projected onto a common image point. In this case the vector space is R3 with the camera focal point at the origin and the projective space corresponds to the image points.
( For example, in the standard geometry for the plane two lines always intersect at a point except when the lines are parallel. In a projective representation of lines and points, however, such an intersection point exists even for parallel lines, and it can be computed in the same way as other intersection points. )

http://en.wikipedia.org/wiki/Projective_geometry
Projective geometry is the most general and least restrictive in the hierarchy of fundamental geometries, i.e. Euclidean - metric (similarity) - affine - projective. It is an intrinsically non-metrical geometry, whose facts are independent of any metric structure. Under the projective transformations, the incidence structure and the cross-ratio are preserved. It is a non-Euclidean geometry. In particular, it formalizes one of the central principles of perspective art: that parallel lines meet at infinity and therefore are to be drawn that way. In essence, a projective geometry may be thought of as an extension of Euclidean geometry in which the "direction" of each line is subsumed within the line as an extra "point", and in which a "horizon" of directions corresponding to coplanar lines is regarded as a "line". Thus, two parallel lines will meet on a horizon line in virtue of their possessing the same direction.

http://en.wikipedia.org/wiki/Camera_resectioning
When a camera is used, light from the environment is focused on an image plane and captured. This process reduces the dimensions of the data taken in by the camera from three to two (light from a 3D scene is stored on a 2D image). Each pixel on the image plane therefore corresponds to a shaft of light from the original scene. Camera resectioning determines which incoming light is associated with each pixel on the resulting image.

http://en.wikipedia.org/wiki/Camera_matrix
Since the camera matrix  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.

http://en.wikipedia.org/wiki/Euclidean_space
An n-dimensional space with notions of distance and angle that obey the Euclidean relationships is called an n-dimensional Euclidean space.
An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.

http://en.wikipedia.org/wiki/Euclidean_transformation
The Euclidean group E(n) is a subgroup of the affine group for n dimensions, and in such a way as to respect the semidirect product structure of both groups.
1. by a pair (A, b), with A an n×n orthogonal matrix, and b a real column vector of size n; or
2. by a single square matrix of size n + 1, as explained for the affine group.
 

http://en.wikipedia.org/wiki/Orthogonal_matrix



http://en.wikipedia.org/wiki/Singular_value_decomposition

where U is an m-by-m unitary matrix over K, the matrix Σ is m-by-n diagonal matrix with nonnegative numbers on the diagonal, and V* denotes the conjugate transpose of V, an n-by-n unitary matrix over K. Such a factorization is called a singular-value decomposition of M.

• The matrix V thus contains a set of orthonormal "input" or "analysing" basis vector directions for M
• The matrix U contains a set of orthonormal "output" basis vector directions for M
• The matrix Σ contains the singular values, which can be thought of as scalar "gain controls" by which each corresponding input is multiplied to give a corresponding output.

A common convention is to order the values Σ_i,i in non-increasing fashion. In this case, the diagonal matrix Σ is uniquely determined by M (though the matrices U and V are not).


http://en.wikipedia.org/wiki/Total_least_squares
total least squares = errors in variables = rigorous least squares = orthogonal regression

http://en.wikipedia.org/wiki/RQ_decomposition#Relation_to_RQ_decomposition