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

1. Download the application zip-file
2. 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.
CameraCalibrationFromImagedSquaresAssertingZeroSkew[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. 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.

RANSAC

RANSAC[x, fittingfn, distfn, degenfn, s, t, feedback, maxDataTrials, maxTrials]

is a general RANSAC implementation. The last three arguments are optional.

File System

LoadImagesInDirectory[dir]

returns all images in a directory.

Miscelaneous

BressenhamPoints[A,B]

gives a list of the pixel-points on the line segment between A and B.