Assignment 4 3D Reconstruction1

COMP5421 Computer Vision
Homework Assignment 4
3D Reconstruction1
Make sure to start early!
Part I: Theory
Before implementing our own 3D reconstruction, let’s take a look at some simple theory questions
that may arise. The answers to the below questions should be relatively short, consisting of a few
lines of math and text (maybe a diagram if it helps your understanding).
Q1.1 (5 points) Suppose two cameras fixate on a point P (see Figure 1) in space such that their
principal axes intersect at that point. Show that if the image coordinates are normalized so that
the coordinate origin (0, 0) coincides with the principal point, the F33 element of the fundamental
matrix is zero.
Figure 1: Figure for Q1.1. C1 and C2 are the optical centers. The principal axes intersect at point
P.
Q1.2 (5 points) Consider the case of two cameras viewing an object such that the second camera
differs from the first by a pure translation that is parallel to the x-axis. Show that the epipolar lines
in the two cameras are also parallel to the x-axis. Backup your argument with relevant equations.
Q1.3 (5 points) Suppose we have an inertial sensor which gives us the accurate positions (Ri and
ti
, where R is the rotation matrix and t is corresponding translation vector) of the robot at time i.
What will be the effective rotation (Rrel ) and translation (trel ) between two frames at different
time stamps? Suppose the camera intrinsics (K) are known, express the essential matrix (E) and
the fundamental matrix (F) in terms of K, Rrel and trel.
Q1.4 (10 points) Suppose that a camera views an object and its reflection in a plane mirror.
Show that this situation is equivalent to having two images of the object which are related
by a skew-symmetric fundamental matrix. You may assume that the object is flat, meaning that
all points on the object are of equal distance to the mirror (Hint: draw the relevant vectors to
understand the relationships between the camera, the object and its reflected image.)
1Credit to CMU Simon Lucey, Chengqian Che, Nate Chodosh, Allie Chang, Akshita Mittel, Gaurav Mittal,
Sowmya Munukutla
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Figure 2: Figure for Q1.3. C1 and C2 are the optical centers. The rotation and the translation is
obtained using inertial sensors. Rrel and trel are the relative rotation and translation between two
frames.
Part II: Practice
1 Overview
In this part you will begin by implementing the two different methods seen in class to estimate
the fundamental matrix from corresponding points in two images (Section 2). Next, given the fundamental matrix and calibrated intrinsics (which will be provided) you will compute the essential
matrix and use this to compute a 3D metric reconstruction from 2D correspondences using triangulation (Section 3). Then, you will implement a method to automatically match points taking
advantage of epipolar constraints and make a 3D visualization of the results (Section 4). Finally,
you will implement RANSAC and bundle adjustment to further improve your algorithm (Section
5).
2 Fundamental matrix estimation
In this section you will explore different methods of estimating the fundamental matrix given a
pair of images. In the data/ directory, you will find two images (see Figure 3) from the Middlebury multiview dataset2
, which is used to evaluate the performance of modern 3D reconstruction
algorithms.
The Eight Point Algorithm
The 8-point algorithm (discussed in class, and outlined in Section 10.1 of Forsyth & Ponce) is
arguably the simplest method for estimating the fundamental matrix. For this section, you can use
2
http://vision.middlebury.edu/mview/data/
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Figure 3: Temple images for this assignment
Figure 4: displayEpipolarF in helper.py creates a GUI for visualizing epipolar lines
provided correspondences you can find in data/some corresp.npz.
Q2.1 (10 points) Finish the function eightpoint in submission.py. Make sure you follow the
signature for this portion of the assignment:
F = eightpoint(pts1, pts2, M)
where pts1 and pts2 are N × 2 matrices corresponding to the (x, y) coordinates of the N points
in the first and second image repectively. M is a scale parameter.
• You should scale the data as was discussed in class, by dividing each coordinate by M (the
maximum of the image’s width and height). After computing F, you will have to “unscale”
the fundamental matrix.
Hint: If xnormalized = T x, then Funnormalized = T
T FT.
You must enforce the singularity condition of the F before unscaling.
• You may find it helpful to refine the solution by using local minimization. This probably won’t
fix a completely broken solution, but may make a good solution better by locally minimizing
a geometric cost function.
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For this we have provided a helper function refineF in helper.py taking in F and the two
sets of points, which you can call from eightpoint before unscaling F.
• Remember that the x-coordinate of a point in the image is its column entry, and y-coordinate
is the row entry. Also note that eight-point is just a figurative name, it just means that you
need at least 8 points; your algorithm should use an over-determined system (N > 8 points).
• To visualize the correctness of your estimated F, use the supplied function displayEpipolarF
in helper.py, which takes in F, and the two images. This GUI lets you select a point in one
of the images and visualize the corresponding epipolar line in the other image (Figure 4).
• Output: Save your matrix F, scale M to the file q2 1.npz.
In your write-up: Write your recovered F and include an image of some example output of
displayEpipolarF.
The Seven Point Algorithm
Q2.2 (15 points) Since the fundamental matrix only has seven degrees of freedom, it is possible to
calculate F using only seven point correspondences. This requires solving a polynomial equation.
In the section, you will implement the seven-point algorithm (described in class, and outlined in
Section 15.6 of Forsyth and Ponce). Manually select 7 points from provided point in data/some
corresp.npz, and use these points to recover a fundamental matrix F. The function should have
the signature:
Farray = sevenpoint(pts1, pts2, M)
where pts1 and pts2 are 7 × 2 matrices containing the correspondences and M is the normalizer
(use the maximum of the images’ height and width), and Farray is a list array of length either 1
or 3 containing Fundamental matrix/matrices. Use M to normalize the point values between [0, 1]
and remember to “unnormalize” your computed F afterwards.
• Use displayEpipolarF to visualize F and pick the correct one.
• Output: Save your matrix F, scale M, 2D points pts1 and pts2 to the file q2 2.npz.
In your write-up: Write your recovered F and print an output of displayEpipolarF.
Also, include an image of some example output of displayEpipolarF using the seven point
algorithm.
• Hints: You can use Numpy’s function roots(). The epipolar lines may not match exactly
due to imperfectly selected correspondences, and the algorithm is sensitive to small changes
in the point correspondences. You may want to try with different sets of matches.
3 Metric Reconstruction
You will compute the camera matrices and triangulate the 2D points to obtain the 3D scene structure. To obtain the Euclidean scene structure, first convert the fundamental matrix F to an essential
matrix E. Examine the lecture notes and the textbook to find out how to do this when the internal
camera calibration matrices K1 and K2 are known; these are provided in data/intrinsics.npz.
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Q3.1 (5 points) Write a function to compute the essential matrix E given F, K1 and K2 with the
signature
E = essentialMatrix(F, K1, K2)
In your write-up: Write your estimated E using F from the eight-point algorithm. Given
an essential matrix, it is possible to retrieve the projective camera matrices M1 and M2 from it.
Assuming M1 is fixed at [I, 0], M2 can be retrieved up to a scale and four-fold rotation ambiguity.
For details on recovering M2 , see section 7.2 in Szeliski. We have provided you with the function
camera2 in python/helper.py to recover the four possible M2 matrices given E.
Note: The M1 and M2 here are projection matrices of the form: M1 = [I|0] and M2 = [R|t].
Q3.2 (10 points) Using the above, write a function to triangulate a set of 2D coordinates in the
image to a set of 3D points with the signature:
[P, err] = triangulate(C1, pts1, C2, pts2)
where pts1 and pts2 are the N ×2 matrices with the 2D image coordinates and P is an N ×3 matrix
with the corresponding 3D points per row. C1 and C2 are the 3×4 camera matrices. Remember that
you will need to multiply the given intrinsics matrices with your solution for the canonical camera
matrices to obtain the final camera matrices. Various methods exist for triangulation probably the
most familiar for you is based on least squares (see Szeliski Chapter 7 if you want to learn about
other methods).
For each point i, we want to solve for 3D coordinates Pi = [xi
, yi
, zi
]
T
, such that when they
are projected back to the two images, they are close to the original 2D points. To project the
3D coordinates back to 2D images, we first write Pi
in homogeneous coordinates, and compute
C1Pi and C2Pi to obtain the 2D homogeneous coordinates projected to camera 1 and camera 2,
respectively.
For each point i, we can write this problem in the following form:
AiP i = 0,
where Ai
is a 4×4 matrix, and Pi
is a 4×1 vector of the 3D coordinates in the homogeneous form.
Then, you can obtain the homogeneous least-squares solution (discussed in class) to solve for each
Pi
.
In your write-up: Write down the expression for the matrix Ai
.
Once you have implemented triangulation, check the performance by looking at the reprojection
error:
err =
X
i
kp1i
, pˆ1ik
2 + kp2i
, pˆ2ik
2
where ˆp1i = Proj(C1, Pi) and ˆp2i = Proj(C2, Pi).
Note: C1 and C2 here are projection matrices of the form: C1 = K1M1 = K1[I|0] and
C2 = K2M2 = K2[R|t].
Q3.3 (10 points) Write a script findM2.py to obtain the correct M2 from M2s by testing the four
solutions through triangulations. Use the correspondences from data/some corresp.npz.
Output: Save the correct M2, the corresponding C2, and 3D points P to q3 3.npz.
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4 3D Visualization
You will now create a 3D visualization of the temple images. By treating our two images as a
stereo-pair, we can triangulate corresponding points in each image, and render their 3D locations.
Q4.1 (15 points) Implement a function with the signature:
[x2, y2] = epipolarCorrespondence(im1, im2, F, x1, y1)
This function takes in the x and y coordinates of a pixel on im1 and your fundamental matrix
F, and returns the coordinates of the pixel on im2 which correspond to the input point. The match
is obtained by computing the similarity of a small window around the (x1, y1) coordinates in im1
to various windows around possible matches in the im2 and returning the closest.
Instead of searching for the matching point at every possible location in im2, we can use F and
simply search over the set of pixels that lie along the epipolar line (recall that the epipolar line
passes through a single point in im2 which corresponds to the point (x1, y1) in im1.
There are various possible ways to compute the window similarity. For this assignment, simple
methods such as the Euclidean or Manhattan distances between the intensity of the pixels should
suffice. See Szeliski Chapter 11, on stereo matching, for a brief overview of these and other methods.
Implementation hints:
• Experiment with various window sizes.
• It may help to use a Gaussian weighting of the window, so that the center has greater influence
than the periphery.
• Since the two images only differ by a small amount, it might be beneficial to consider matches
for which the distance from (x1, y1) to (x2, y2) is small.
To test your epipolarCorrespondence, we have included a helper function epipolarMatchGUI
in python/helper.py, which takes in two images the fundamental matrix. This GUI allows you to
click on a point in im1, and will use your function to display the corresponding point in im2. See
Figure 5.
It is not necessary for your matcher to get every possible point right, but it should get easy
points (such as those with distinctive, corner-like windows). It should also be good enough to
render an intelligible representation in the next question.
Output: Save the matrix F, points pts1 and pts2 which you used to generate the screenshot to
the file q4 1.npz.
In your write-up: Include a screenshot of epipolarMatchGUI with some detected correspondences.
Q4.2 (10 points) Included in this homework is a file data/templeCoords.npz which contains 288
hand-selected points from im1 saved in the variables x1 and y1.
Now, we can determine the 3D location of these point correspondences using the triangulate
function. These 3D point locations can then plotted using the Matplotlib or plotly package. Write
a script visualize.py, which loads the necessary files from ../data/ to generate the 3D reconstruction using scatter. An example is shown in Figure 6.
Output: Again, save the matrix F, matrices M1, M2, C1, C2 which you used to generate the screenshots to the file q4 2.npz.
In your write-up: Take a few screenshots of the 3D visualization so that the outline of the temple
is clearly visible, and include them with your homework submission.
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Figure 5: epipolarMatchGUI shows the corresponding point found by calling
epipolarCorrespondence
5 Bundle Adjustment
Q5.1 (15 points) In some real world applications, manually determining correspondences is infeasible and often there will be noisy coorespondances. Fortunately, the RANSAC method seen in
class can be applied to the problem of fundamental matrix estimation.
Implement the above algorithm with the signature:
[F, inliers] = ransacF(pts1, pts2, M)
where M is defined in the same way as in Section 2 and inliers is a boolean vector of size equivalent
to the number of points. Here inliers is set to true only for the points that satisfy the threshold
defined for the given fundamental matrix F.
We have provided some noisy coorespondances in some corresp noisy.npz in which around
75% of the points are inliers. Compare the result of RANSAC with the result of the eightpoint
when ran on the noisy coorespondances. Briefly explain the error metrics you used, how you decided
which points were inliers and any other optimizations you may have made.
Hints: Use the seven point to compute the fundamental matrix from the minimal set of points.
Then compute the inliers, and refine your estimate using all the inliers.
Q5.2 (15 points)
So far we have independently solved for camera matrix, Mj and 3D projections, Pi
. In bundle
adjustment, we will jointly optimize the reprojection error with respect to the points Pi and the
camera matrix Mj .
err =
X
ij
kpij − Proj(Cj , Pi)k
2
where Cj = KjMj , same as in Q3.2.
For this homework we are going to only look at optimizing the extrinsic matrix. To do this we
will be parametrizing the rotation matrix R using Rodrigues formula to produce vector r ∈ R
3
.
Write a function that converts a Rodrigues vector r to a rotation matrix R
R = rodrigues(r)
as well as the inverse function that converts a rotation matrix R to a Rodrigues vector r
r = invRodrigues(R)
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Figure 6: An example point cloud
Q5.3 (10 points)
Using this parameterization, write an optimization function
residuals = rodriguesResidual(K1, M1, p1, K2, p2, x)
where x is the flattened concatenation of P, r2 , and t2 . P are the 3D points; r2 and t2 are
the rotation (in the Rodrigues vector form) and translation vectors associated with the projection
matrix M2. The residuals are the difference between original image projections and estimated
projections (the square of 2-norm of this vector corresponds to the error we computed in Q3.2):
residuals = numpy.concatenate([(p1-p1 hat).reshape([-1]),
(p2-p2 hat).reshape([-1])])
Use this error function and Scipy’s nonlinear least square optimizer leastsq write a function
to optimize for the best extrinsic matrix and 3D points using the inlier correspondences from some
corresp noisy.npz and the RANSAC estimate of the extrinsics and 3D points as an initialization.
[M2, P] = bundleAdjustment(K1, M1, p1, K2, M2 init, p2, P init)
In your write-up: include an image of the original 3D points and the optimized points as well as
the reprojection error with your initial M2 and P, and with the optimized matrices.
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6 Deliverables
If your andrew id is <ustid-login>, your submission should be the writeup <ustid-login>.pdf
contained in a zip file <ustid-login>.zip. Please submit the zip file to CANVAS.
The zip file should include the following directory structure:
• submission.py: your implementation of algorithms.
• findM2.py: script to compute the correct camera matrix.
• visualize.py: script to visualize the 3d points.
• q2 1.npz: file with output of Q2.1.
• q2 2.npz: file with output of Q2.2.
• q3 3.npz: file with output of Q3.3.
• q4 1.npz: file with output of Q4.1.
• q4 2.npz: file with output of Q4.2.
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