A library for functions related to multiple view geometry in Mathematica.
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mathematicacomputervision's Introduction
MathematicaComputerVision
A library for functions related to multiple view geometry in Mathematica.
Author: Thijs Vogels
Supervisor: Dr. Richard van den Doel
Roosevelt University College, Middelburg, Netherlands
Installation
Download the application zip-file
Go to File>Install in Mathematica and select the zip.
Documentation
The application documentation will be integrated in the native Mathematica docs.
Available functions
Utilities
Hgc[x]
converts a list x representing a point in n-dimensional space
to homogeneous coordinates by appending a 1.
Nc[x]
converts a list x representing a point in homogeneous coordinates
to non-homogeneous coordinates by stripping the last element and dividing by it.
If the point lies on infintiy, a warning is thrown.
Intersect[l,m]
returns the intersection the homogeneous 2d lines l and m.
LineThrough[p,q]
returns the line joining the homogeneous 2d points p and q.
LineDirection[l]
returns the direction of the line l.
RQ[M]
is a variant of the QR Decomposition such that A=R.Q, and R is upper triangular and Q orthogonal.
Rxyz[α,β,γ]
gives a 3x3 rotation matrix that first translates a in the x-direction, then b in the y direction and c in the z-direction.
Homographies
NormalizePoints[pts]
normalizes a set of non-homogeneous or non-infinite homogeneous 2d points such that:
* their centroid is in the origin
* their mean distance from the origin is sqrt(2)
Returns the normalized set of points and the transformation that transforms a set of homogeneous points to the normalized ones (for denormalization purposes)
NormalizePoints3D[pts]
normalizes a set of non-homogeneous or non-infinite homogeneous 3d points such that:
* their centroid is in the origin
* their mean distance from the origin is sqrt(3)
Returns the normalized set of points and the transformation that transforms a set of homogeneous points to the normalized ones (for denormalization purposes)
Homography2D[x,y]
calculates a homography between two sets of homogeneous 2D points x and y using the Direct Linear Transformation algorithm (Hartley&Zisserman, p.89). For optimal results the Gold Standard algorithm should be used.
Camera Matrix
CameraMatrixFromCorrespondences[corr]
gives the camera matrix for >=6 {world,pixel} correspondences (homogeneous coordinates) using a DLT method.
DecomposeCamera[P]
decomposes the camera into {K,R,t} such that P = K.R.[I | -t].
DrawCamera[P]
gives a 3D object showing the camera orientation. To be used inside a Graphics3D environment. It uses the img for its dimensions.
DrawCamera[P,size] does the same with a given size.
DrawCamera[P,size,img] let's you specify an image used to find correct the width and height of the screen.
Camera Calibration
Based on images of three black squares on a contrasting background.
SeparateSquares[img,nsquares]
takes an image of n black squares (on a constrasting background) and returns n white images of those squares on a black background.
It defaults to 3 squares.
CornerInQuadrangleImage[square]
calculates the four corner points in an image of a quadrangle by first finding the edges and intersecting those.
Returns the points in non-homogeneous coordinates.
CameraCalibrationFromImagedSquares[squares]
returns the internal camera parameters K from the (non-homogeneous) corner points of (at least) three imaged squares. It identifies the image of the absolute conic, and uses a Cholesky decomposition to derive K from omega.
returns the internal camera parameters K from the (non-homogeneous) corner points of (at least) three imaged squares. It identifies the image of the absolute conic, and uses a Cholesky decomposition to derive K from omega. It asserts zero skew.
Fundamental Matrix
FFromCorrespondences[corr]
gives a fundamental from a list of point correspondences using a DLT algorithm.
EFromFK[F,K]
gives the essential matrix corresponding to a fundamental matrix F and a calibration matrix K.
DecomposeE[E]
returns four possible second camera matrices that form a couple with [I|0] based on an essential matrix E.
Scene Reconstruction
Triangulate[corr,P1,P2]
gives a DLT-estimate of the 3D-location of corresponding image point correlations in two images and the two camera matrices.
Image Manipulation
ImageCoordinateToDataPoint[p,img]
converts image coordinates (origin left-bottom) to a corresponding data point (origin left-top)
ImageColorAtCoordinate[p,img]
returns the color at coordinate p (origin left-bottom)
LinePointsInImage[l,img]
gives the two extreme points on an image of a homogeneous line (snapped to pixel-positions)
NOT TESTED for lines that lie completely outside the image
LineInImage[l,img]
takes the points from LinePointsInImage and makes it a Graphics primitive.
Least squares fitting
FitConic[points]
fits a conic through at least 5 homogeneous points using linear minimization.
FitLine[points]
gives a least squares fit of a line to a set of points.