An Introduction to Epipolar Geometry
Understanding the intuition behind 3D to 2D mappings, the Fundamental Matrix with a demo
In this post we will take a look at how Camera Projections work. A demo at the end will illustrate the important segments of the theory. Prerequisites to understand the material are available in the Readings & References section below.
Epipolar Geometry ^{1} is the intrinsic projective geometry between two views. See A Primer on Projective Geometry for a concise introduction. This knowledge becomes an interesting piece in the puzzle of estimating the 3D geometry of a given image projection and the estimated 3D model can then be applied to a myriad of meaningful realworld problems.
To get there, we will first establish a theoretical framework on how projections are formed and realize some interesting properties.
The Pinhole Camera Model
We will start with a very simple camera where the centre of projection is at the origin of the Euclidean Coordinate System. Let $Z = f$ be the plane of projection, more simply the plane where the points in the 3D world space are projected. It is also called the image plane or the focal plane.
$C$ is the camera centre and $p$ is the principal point. By virtue of similar triangles, we can establish a relation between a point in 3D space at $(X,Y,Z)$ to a point on the focal plane at $(x^\prime,y^\prime)$, a distance of focal length $f$. This has been visualized in the figure above on the right.
$\frac{f}{Z} = \frac{y}{Y} \leftrightarrow y = \frac{fY}{Z}$A similar analysis (think of viewing the $zx$ plane down the $y$ axis) gives us
$\frac{f}{Z} = \frac{x}{X} \leftrightarrow x = \frac{fX}{Z}$and we equivalently define a mapping (by dropping the $z$ coordinate)
$(X,Y,Z) \mapsto (x,y) \leftrightarrow ({fX}/{Z},{fY}/{Z})$More formally, this can be defined in terms of homogeneous coordinates and a matrix transform as
$\begin{pmatrix}X \\ Y \\ Z \\ 1 \end{pmatrix} \mapsto \begin{pmatrix}fX \\ fY \\ Z \end{pmatrix} = \begin{bmatrix}f & 0 & 0 & 0 \\ 0 & f & 0 & 0 \\ 0 & 0 & 1 & 0 \end{bmatrix} \begin{pmatrix}X \\ Y \\ Z \\ 1 \end{pmatrix}$ $x = PX$This $3 \times 4$ matrix $P$ is known as the Camera Projection Matrix. To adjust for the reference frame in the image plane  origin at corner v/s origin at the center, we add a translation component to the left $3 \times 3$ matrix of $P$.
$K = \begin{bmatrix}f & 0 & p_x \\ 0 & f & p_y \\ 0 & 0 & 1 \end{bmatrix}$and this is called the Camera Calibration Matrix. For added generality to allow for nonsquare pixels we multiple the matrix with an extra factor of $diag(m_x,m_y,1)$ where $m$ is the pixels per unit distance. Thereâ€™s another possibility of distortion due to the $x$ and $y$ axes not being perpendicular and this is denoted by $s$. The final matrix then becomes 
$K = \begin{bmatrix}\alpha_x & s & x_0 \\ 0 & \alpha_y & y_0 \\ 0 & 0 & 1 \end{bmatrix}$In most cases, the camera coordinates will be with respect to a different coordinate system where it is not at the origin in the world coordinate system. And as we had seen in an earlier discussion on projective geometry, Euclidean transformations consist of a rotation and a translation component. Hence, any coordinate can be shifted from the world coordinate space to the camera coordinate space (where the camera is the origin) with the equation 
$X_\text{camera} = R(X_\text{world}  C)$where $R$ is the rotation matrix and $C$ represents a translation with respect to the camera center to make it the origin. Combining the camera calibration matrix and the coordinate frame shifting above, we can draw a mapping between the point in world coordinate system $X$ and the point $x$ in the image plane, represented by
$x = KR \begin{bmatrix}I \vert C\end{bmatrix}X$ $P = KR \begin{bmatrix}I \vert C\end{bmatrix}$Parameters in $K$ are called the camera intrinsics (they never change) and the remaining parameters $R$ and $C$ are called the camera extrinsics. In literature, these terms will come often and now we know how these are constructed.
Epipolar Geometry
Now that we have established all the camera parameters, let us consider two camera centers $C$ and $C^\prime$ with both image planes shown in the figure below. These cameras project a point $X$ in the world coordinate system to $x$ and $x^\prime$ in respective image planes. The plane formed by all these points is known as the epipolar plane.
The line joining the two camera centers is known as the baseline and intersects both planes at $e$ and $e^\prime$. These points are known as the epipoles, visualized on the right. We are aiming to establish and constraint between $x$ and $x^\prime$. If one observes the figure on the right above, the baseline ($CC^\prime$) and the ray backprojected from $x$ ($CX$) form the epipolar plane $\pi$. When $\pi$ intersects with the image plane on the right, it forms the line $l^\prime$ which is known as the epipolar line corresponding to $x$. A search for the point correspondence of $x$ denoted by $x^\prime$ is now constrained to this epipolar line which comes as a direct consequence of the way we constructed $\pi$.
If we rotate the epipolar plane with baseline as the axis, we get a pencil of planes and all these planes will still intersect at the epipoles $e$ and $e^\prime$. This is a very interesting result!
The Fundamental Matrix
Weâ€™ve established some basics for epipolar geometry and realized how images are projected by a given camera matrix $P$. Now we will give a more formal treatment to the problem both algebraically and geometrically and arrive at a very important matrix known as the Fundamental Matrix (literally!).
Construction via the Epipolar Plane
In a previous discussion (see Readings), we had seen Projectivities which are mappings from point in one plane to another. By the above construction of the epipolar plane, we can assert that a similar matrix exists, known as the Planar Homography $H_\pi$.
$x^\prime = H_\pi x$Construction via Camera Matrices
In another interesting result, Fundamental Matrix does not need the epipolar plane to be necessarily defined. Let us see how it directly relates to the camera matrices we described above.
If one remembers the alternate definition of epiline corresponding to point $x$, it was the image of the ray backprojected to the world coordinate $X$ in the second camera view. The solution to the equation
$x = PX$is given by a family of solutions in the form of
$X(\lambda) = P^+x + \lambda C$To find the epipolar line $l^\prime$ for point $x$, we know from the above discussion that both the epipole $e^\prime$ and the point correspondence $x^\prime$ lie on the epiline. The intersection of these (from our projective geometry primer) gives us the line,
$l^\prime = e^\prime \times x^\prime = [e^\prime]_\times x$ $l^\prime = [e^\prime]_\times H_\pi x = Fx$where $F = [e^\prime]_\times H_\pi$ is known as the Fundamental Matrix, a matrix of rank 2. In plain words, $F$ represents a mapping of the projective plane $\mathbb{P}^2$ of the first image to a pencil of epipolar lines $\mathbb{P}^1$ through $e^\prime$.
Note that the notation $[e]_\times$ is a skewsymmetric matrix equivalent for the cross product.
$[e]_\times = \begin{bmatrix} 0 & e_3 & e_2 \\ e_3 & 0 & e_1 \\ e_2 & e_1 & 0 \end{bmatrix}$where $P^+$ is the pseudoinverse of $P$, giving $PP^+ = I$ and $C$ is the rightnull space of $P$ commonly called the camera center. This has its own interesting little derivation and out of scope here.
We have two known points $C$ and $P^+x$ on the ray which can be projected onto the second image plane via a second camera matrix $P^\prime$ and the epiline is the crossproduct of these two projected points.
$l^\prime = (P^\prime C) \times (P^\prime P^+ x)$We have just arrived at an alternate representation of $l^\prime = e^\prime \times x^\prime$ and hence we realize that epipole $e^\prime$ is actually the image of the other camera center. Again we see that
$l^\prime = [e^\prime]_\times (P^\prime P^+ x) = Fx$where $F = [e^\prime]_\times P^\prime P^+$ and $H_\pi = P^\prime P^+$. Wow! Weâ€™ve discovered quite a few important relations between Homography, Camera Matrices and the Fundamental Matrix.
Properties of the Fundamental Matrix
After all the details above, it is easy to arrive and prove at the following result
$x^{\prime T}Fx = 0$where $x \leftrightarrow x^\prime$ are pointtopoint correspondences in the two image planes. Solution to the above equation is a fairly popular research problem. For all practical purposes, the above problem generally remains overdetermined with an exact solution determined by 3 noncollinear point correspondences. Hence, techniques like the family of RANSAC are used to nondeterministically determine the best fit model via a relevant cost function like the reprojection error.
Demo Illustration
Setup
Each execution requires a stereo image pair (for this purpose I shot twoview images around my house) which represent the two image planes discussed above. To arrive at $x\leftrightarrow x^\prime$ point correspondences, I use SURF to detect keypoints in both images, match them via FLANN and apply Loweâ€™s ratio test to filter out matches that are too far away and might be false positives. If all above sounds alien, consider it as a black box as it is out of scope for this discussion.
Then we estimate the Fundamental Matrix via RANSAC. RANSAC was covered in a previous post. Once we get that we draw the epilines on both images via $l = F^T x^\prime$ for the left image plane and $l^\prime = Fx$ for the right image plane. These line equations are a direct consequence of the result $x^{\prime T}Fx = 0$.
Code
Full code is available upon request but for the purpose of this discussion, the one below should be easy to follow.
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# Load the stereo image pair
#
image_l = cv2.pyrDown(cv2.imread('images/view_left.jpg', 0))
image_r = cv2.pyrDown(cv2.imread('images/view_right.jpg', 0))
##
# Consider this matcher a black box for the purpose of this discussion
#
matcher = SURFKeyPointMatcher()
kp_l, des_l, kp_r, des_r = matcher.detect_and_compute(image_l, image_r)
pts_l, pts_r, good_matches = matcher.find_good_matches(kp_l, des_l, kp_r, des_r)
##
# Estimate the Fundamental Matrix via RANSAC
#
pts_l = np.int32(pts_l)
pts_r = np.int32(pts_r)
F, mask = cv2.findFundamentalMat(pts_l, pts_r, cv2.FM_RANSAC)
##
# Use only the points which RANSAC determined to be model inliers
#
pts_l = pts_l[mask.ravel() == 1]
pts_r = pts_r[mask.ravel() == 1]
##
# Compute epilines. Note that right image points are used for left view
# and left image points are used for right view
#
epilines_l = cv2.computeCorrespondEpilines(pts_r, 2, F).reshape(1, 3)
epilines_r = cv2.computeCorrespondEpilines(pts_l, 1, F).reshape(1, 3)
image_epilines_l = draw_epilines(image_l, epilines_l)
image_epilines_r = draw_epilines(image_r, epilines_r)
cv2.imwrite('epi_left.jpg', image_epilines_l)
cv2.imwrite('epi_left.jpg', image_epilines_r)
Results
Take a look at the following resulting images from the above code which show a general sense of direction of both the camera centers. The epipoles (the point of intersection of the epilines) lie outside the visible image planes.
It goes without saying that the epilines constructed are only as good as the $x \leftrightarrow x^\prime$ point correspondences. The robustness of the point correspondences is dependent on how well our keypoint detection algorithm works.
Footnotes

Harltey, A., & Zisserman, A. (2003). Multiple view geometry in computer vision (2. ed.). â†©