Wednesday, April 17, 2024

Multiple View Geometry In Computer Vision

Retrieve Camera Matrices From Fundamental Matrix

Multiple View Geometry – Lecture 1 (Prof. Daniel Cremers)

If $H$ is a $4\times 4$ matrix representing a projective transformation of 3-space, then the fundamental matrices corresponding to the pairs of camera matrices $$ and $$ are the same.

Therefore, although a pair of camera matrices $$ uniquely determine a fundamental matrix $F=_ M$, the converse is not true. The fundamental matrix determines the pair of camera matrices up to a projective transformation.

However, we can still define a specific canonical form for the pair of camera matrices retrieved from a fundamental matrix, then they are

$$P = \quad P^\prime =\begin\left_\times F \mid e^\prime \right] \\\left_\times F + e^\prime v^T \mid \lambda e^\prime \right]\end$$

where $S$ is a skew-symmetric matrix, $e^\prime$ is the epipole such that $e^ F = 0$. $v$ is any 3-vector, and $\lambda$ is a non-zero scalar.

Chapter : Projective Geometry And Transformations Of 2d

If you are finding it hard to grasp the ideas in this chapter, I suggest goingthrough an introductory text on projective geometry. One book I highlyrecommend is Introduction to Projective Geometry by C.R. Wylie Jr. I alsorecommend that you play with the interative 3D graphs that are part of thesolution set for the book on this blog. Just hit the Viewin GeoGebra link and modify the lines and conics to get a feel for perspectiveprojection. The point $C$ is the center of projection, the image plane is $z =0$ and the object plane is $y = 0$.

Here are quicklinks to the exercise solutions in this chapter.

The Essential Matrix $e$

The essential matrix is the specialization of the fundamental matrix to the case of normalized image coordinates, which is $\hat = K^x = X$, and the $\hat = k^P = $ is called normalized camera matrix.

So for $P = $ and $P^\prime = $ $\Rightarrow E=_\times R=R_\times $

In other words, the essential matrix means the camera calibration matrix $K, K^\prime$ is known, where $E = K^ F K$.


\end$$where $u_3$ is the last column of $U$, and $W = \left$

After reconstructing a space point with the camera matrix, there will be only 1 case that the point is in front of 2 cameras, which means only 1 result is valid.

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The Projective Camera Anatomy

A general projective matrix $P = $ maps world points to image points. The meaning of different part of matrix could be revealed as following.

Some descriptions here.

  • The coordinate $C$ of camera center is in world coordinate frame
  • Column vectors are vanishing point of world axes
  • Since the camera centre is in the principal plane, then $p^C = 0$
  • Point at axis planes maps to the $x$ or $y$ axis of the image, since $p^X = 0$, leading to image coordinate $^T$ or $^T$
  • Principal point is computed by mapping the point at infinity $X = ^T$ to the image $X = M^T$

The camera matrix could be decomposed to a 1) projective transformation from world coordinate frame to camera coordinate frame, 2) a projection from 3D space to an image, and a 3) projective transformation of the images.$$P = \left[\begin1 & 0 & 0 & 0 \\

Depth of points

For a finite camera maps a point to the image plane $w^T = P^T$, then $w$ is the depth of a point $\mathbf$ from the camera centre $C$ in the direction of the principal ray, when the camera matrix is normalized so that $\det M > 0$ and $||m^3|| = 1$.

If the matrix hasnt be normalized, then the depth is$$\text = \fracw}$$

Given a general projective camera $P$, then

Camera centre $C$ for which $PC = 0$ can be obtained by$$X = \det \quad Y = -\det \quad Z = \det \quad T = -\det$$

The camera orientation and internal parameters could be found for finite camera, where $P = = $

Then $\widetilde$ could be got by $-M^p_4$, which doesnt have the scale ambiguity.

Who Can I Call If I Have More Questions

[PDF] Multiple View Geometry in Computer Vision

If you have any addition questions or comments, please don’t hesitate to call our customer service at 1-800-THE-BOOK.

A basic problem in computer vision is to understand the structure of a real world scene. This book covers relevant geometric principles and how to represent objects algebraically so they can be computed and applied. Recent major developments in the theory and practice of scene reconstruction are described in detail in a unified framework. Richard Hartley and Andrew Zisserman provide comprehensive background material and explain how to apply the methods and implement the algorithms. First Edition HB : 0-521-62304-9

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Recover Affine Properties From Image

Affine rectification: given an image which is the projective transformation of real world plane , computes the image which is the affine transformation of the real world plane, where the parallel line will looks parallel

This could be done by mapping the vanishing line $^T$ back to line at infinity $^T$ with$$$$

in order to compute the vanishing line, there are several options

  • Use 2 pairs of line parallel in real world to compute 2 intersections. Then 2 intersections lay on the line at infinity
  • Use 3 points whose length ratio are known in a line to compute the vanishing point by 1D projective transformation, and found another vanishing point from another line. Then 2 vanishing points lay on the line at infinity.
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    Multiple View Geometry – Lecture 3 (Prof. Daniel Cremers)

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    Multiple View Geometry Series

    A very condensed series on multiple view geometry, heavily inspired by the excellent Hartley and Zisserman book, but with the aim of being more direct on some key topics, and hopefully more accessible for the novice reader

    This Multiple View Geometry Series in broken down into the following sub-series, closely following some of the chapters in the Hartley-Zisserman book.

    In this series, the basics of projective geometry are introduced, laying the foundation for subsequent chapters.

    Direct Linear Transformation Algorithm

    DLT transforms the $x_i^\prime = Hx_i, i = 1, 2, \dots, n$ relationship to $Ah = 0$, where $A$ is a $3n \times 9$ or a $2n \times 9$ matrix.

    Minimal solution

    If $n = 4$, then the $h$ could be solved by Gaussian Elimination.

    Over-determined solution

    If $n > 4$, which means the equations are over-determined, then the $h$ could be solved by minimizing the algebraic distance $||Ah||$. The solution is the unit singular vector corresponding to the smallest singular value of matrix $A$. e.g. If $A=UDV^T$ with D diagonal with positive diagonal entries, arranged in descending order down the diagonal, then $h$ is the last column of $V$.

    Usually before DLT algorithm, a data normalization step is required for less well conditioned problem, such as DLT computation of the fundamental matrix or the trifocal tensor. The steps are as follows

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    Recover Similarity Properties From Image

    Metric rectification: computes the image which is the similarity transformation of the real world plane, where the parallel line will looks parallel, and tangential lines are tangential

    This could be done by transforming the circular points to their canonical position $^T$. And dual conic $C_^$ includes these 2 points.

    Recover similarity properties from projective images

    • The dual conic $C_^ = U \left U^T$, and $U$ could achieve the metric rectification.
    • Choose 2 lines $l, m$ which should be orthogonal in real world, then $l^T C_^ m = 0$ could have 1 constraint. Since $C_^$ has 5 DoF, then we need 5 orthogonal pairs.

    Recover similarity properties from images after affine rectification

    The orthogonal pair serves the constraint of $C_^$ as$$l^T \left m = 0$$Since $K$ is a up to scale symmetric matrix, then it only has 2 DoF. So 2 orthogonal pair could solve it.

    Vanishing Points And Lines

    Figure 14.11 from Multiple View Geometry in Computer Vision
    Vanishing lines
    Definition the intersection $v=Kd$ of the image plane with a ray through the camera centre which parallel to the world line with direction $d$ in 3D space intersections on the image with a plane parallel to the scene plane through the camera centre
    How to get from image intersecting world parallel lines in images – 2 vanishing points from two sets of lines parallel to the plane – 3 coplanar equally spaced parallel lines $l = \left^T \right)l_1 + 2\left^T \right)l_2$
    Properties Vanishing point of a line parallel to a plane lies on the vanishing line of the plane – All parallel 3D planes interest in the vanishing lines at $\pi_$ – A plane with vanishing line $l$ has orientation $n = K^Tl$ in the cameras Euclidean coordinate frame

    Example of vanishing points and lines from an image

    Using 2 paris of vanishing points in 2 calibrated cameras image could compute the relative camera rotation. Each pair contributes 2 constraints $d_i^ = Rd_i$, where $d_i = K^ v_i / ||K^ v_i||$ is the direction of the vanishing point $v_i$ in the camera frame

    Relationship with conic $w$

    • The vanishing points of lines with perpendicular directions satisfy $v_1^Twv_2 = 0$
    • If a line is perpendicular to a plane then their respective vanishing point $v$ and vanishing line $l$ are related by $l = wv$
    • The vanishing lines of two perpendicular planes satisfy $l_1^T w^ l_2 = 0$

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    An Index To The Worked Solutions

    After searching in vain for solutions to the exercises in this book, I decidedto start documenting my solutions with the hope that it might provideencouragement to others like me on the path of self-study.

    On the note of self-study, I would like to provide some feedback to those whohave just begun or are contemplating using this book to learn computer vision.Firstly, if you dont know this already , multiple view geometry isjust one, albeit major, facet of computer vision, not the whole of it. Youmight want to first explore computer vision breadth-wise before you decide tocommit to this one particular area.

    Secondly, I found this book to be more of a compendium of research papers onmultiple view geometry rather than an introductory textbook for beginners inthe field. So if youre starting from scratch like me, I strongly recommendbecoming conversant with projective geometry, probability & statistics, linearalgebra, calculus, optimization and some image processing, before attemptingthe material, to get the most out of it.

    Finally, if you put in the work, you will find the material to be rewarding.Through this book, Ive been able to learn things that I didnt even know werepossible. It has literally broadened my horizons .

    Multiple View Geometry In Computer Vision

    This book has beencited by the following publications. This list is generated based on data provided by CrossRef.

    Multimodal 3-D tracking and event detection via the particle filter

    • , Australian National University, Canberra,, University of Oxford
    • Publisher: Cambridge University Press
    • Online publication date: January 2011
    • Print publication year: 2004
    • Email your librarian or administrator to recommend adding this book to your organisation’s collection.

      Multiple View Geometry in Computer Vision

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    Projective Space And Transformation

    Projective space $\mathbf^2$ consist of the normal points in $\mathbf^2$, whose homogeneous coordinate is $^T, x_3 \neq 0$, and the ideal points $^T$.

    All ideal points lay on the same line, the line at infinity: $^T$

    Projective transformation is an invertible mapping $h$ from $\mathbf^2$ to itself such that points $x_1, x_2, x_3$ lie on the same line if and only if $h, h, h$ do. It could be represents by

    4 point pairs not in same line concurrency, collinearity, order of contact matrix up to scale

    Affine transformation maps points in infinity to points in infinity, while projective transformation maps points in infinity to normal points.

    Projective transformation can be decomposed into a chain of transformations, where each matrix in the chain represents a transformation higher in the hierarchy than the previous one.$$\left\left\left =\left$$where $v \neq 0$ and K an upper-triangular matrix normalized as $\det K = 1$.

    Actions Of Projective Cameras

    Multiple View Geometry – Lecture 9 (Prof. Daniel Cremers)

    On plane

    For points in the plane $\pi$ which lays in the world XY-plane, its image point is$$x = \left$$then this mapping is can be defined by a $3\times 3$ matrix with the rank of 3.

    It indicates that the mapping between a world plane and an image is a general planar homography .

    On lines

    • A line in 3-space projects to a line in the image
    • The set of points in space which map to a line in the image is a plane $P^Tl$ in space defined by the camera centre and image line $l$.

    On conics

    A conic $C$ back-projects to a cone. A cone is a degenerate quadric, i.e. the $4 \times 4$ matrix representing the quadric does not have full rank.$$$$where $C$ is the conic, and $Q_$ is the cone quadric.

    The cone vertex, in this case the camera centre, is the null-vector of the quadric matrix, since $Q_C = P^TC = 0$.

    On smooth surfaces

    The contour generator of a surface are the set of points $X$ on the surface $S$ at which rays from camera centre are tangent to the surface, thus it only depends on the relative position of the camera centre and the surface.

    The apparent contour is defined by the intersection of the image plane with the rays to the contour generator, so it depends on the position of the image plane.

    On quadrics

    A quadric is a smooth surface and so its outline curve is given by points where the back-projected rays are tangent to the quadric surface.

    For general quadric, its contour generator is a conic, and its apparent conic is also a conic.

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    Entities From Calibration Matrix

    Define rays $d = K^x$ $x$ is the image point, $d$ is the direction of ray in the cameras Euclidean coordinate frame who start from camera centre and pass through the image point. $K$ here is an affine transformation
    Measure angles $\cos \theta = \fracK^) x_2}K^) x_1} \sqrtK^) x_2}}$ Angle between the 2 rays. Thus, a calibrated camera whose calibration matrix is known is a direction sensor like a 2D protractor
    Define plane $n = K^Tl$ The image line $l$ and the camera centre defines the plane. Its direction $n$ is under the cameras Euclidean coordinate frame.
    Define image of points at infinity $x=KR ^T = KRd$ Property: the mapping between plane at infinity $\pi_$ and an image is given by the planar homography. Also this is only related to calibration matrix and camera rotation, but independent of the position of the camera.
    Define image of absolute conic $\beginw & = ^ I ^ \\ & = ^ = K^K^\end$ Is the image of absolute conic $\Omega_ = I$. Therefore, once the conic $w$ can be identified in an image, then $K$ is also determined by Cholesky factorization.

    Multiple View Geometry In Computer Vision Second Edition

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