## Description

In this assignment you will be implementing an algorithm to reconstruct a 3D point cloud from a pair of images taken at different angles. In Part I you will answer theory questions about 3D reconstruction. In Part II you will apply the 8-point algorithm and triangulation to find and visualize 3D locations of corresponding image points.

Instructions

1. You will submit both a pdf write-up and a zip of your code to Gradescope. Zip your code into a single file named <AndrewId>.zip. See the complete submission checklist at the end to ensure you have everything. Handwritten write-ups will not be accepted.

2. Each question (for points) is marked with a Q.

4. Attempt to verify your implementation as you proceed. If you don’t verify that your implementation is correct on toy examples, you will risk having a huge mess when you put everything together. The provided boilerplate code contains some simple tests which should help you verify.

6. Some questions will ask you to “Include your code in the write-up”. For those questions, you can either copy/paste the code into a verbatim environment, or include screenshots of your code.

Part I

Theory

Q1.1 (5 points) Suppose two cameras fixate on a point x (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 w (P in the figure).

Q1.3 (5 points) Suppose we have an inertial sensor which gives us the accurate positions (Ri and ti, the rotation matrix and 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.

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.

Hint: as shown in Figure 3, try to draw the relevant vectors to understand the relationships between the camera, the object and its reflected image.

Hint: Assume that the 3D points from object X and its reflection X′ are related by pure translation, such that

−1 0 0 0

X = 00 10 01 00X′ (1)

0 0 0 1

Part II Practice

1 Overview

In this part you will begin by implementing the 8-point algorithm 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

Figure 3: Figure for Q1.4

Figure 4: Temple images for this assignment

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 4) from the Middlebury multi-view dataset , which is used to evaluate the performance of modern 3D reconstruction algorithms.

Figure 5: displayEpipolarF in helper.py creates a GUI for visualizing epipolar lines

2.1 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 provided correspondences you can find in data/some corresp.npz.

Q2.1 (10 points) Finish the function eightpoint in q2 1 eightpoint.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 respectively. 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 = Tx, then Funnormalized = TTFT.

You must enforce the singularity condition of the F before unscaling.

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 ycoordinate 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 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 5).

• In addition to visualization, we also provide a test code snippet in q2 1 eightpoint.py which uses helper function calc epi error to evaluate the quality of the estimated fundamental matrix. This function calculates the distance between the estimated epipolar line and the corresponding points. For the eight point algorithm, the error should on average be < 1.

• 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. Please include the code snippet of eightpoint function in your write-up.

2.2 The Seven Point Algorithm

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 this section, you will implement the seven-point algorithm (described in class, and outlined in this post.

Q2.2 (15 points) Finish the function sevenpoint in q2 1 sevenpoint.py. Make sure you follow the signature for this portion of the assignment:

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.

Manually select 7 points from provided point in data/some corresp.npz, and use these points to recover a fundamental matrix F. Use calc epi error in helper.py to calculate the error to pick the best one, and use displayEpipolarF to visualize and verify the solution.

Output: In your write-up: Print your recovered F and include an image output of displayEpipolarF. Please include the code snippet of sevenpoint function in your write-up.

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.

Q3.1 (5 points) Complete the function essentialMatrix in q3 1 essential matrix.py to compute the essential matrix E given F, K1 and K2 with the signature:

E = essentialMatrix(F, K1, K2)

Output: Save your estimated E using F from the eight-point algorithm to q3 1.npz. Please include the code snippet of essentialMatrix function in your write-up.

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 11.3 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 matrices M1 and M2 here are of the form:

M and M .

Q3.2 (10 points) Using the above, complete the function triangulate in q3 2 triangulate.py to triangulate a set of 2D coordinates in the image to a set of 3D points with the signature:

[w, err] = triangulate(C1, pts1, C2, pts2)

where pts1 and pts2 are the N×2 matrices with the 2D image coordinates and w 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 w , 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 wi in homogeneous coordinates, and compute C1w˜i and C2w˜i 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:

Aiwi = 0,

where Ai is a 4×4 matrix, and w˜i 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 wi.

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∥x1i,xc1i∥2 + ∥x2i,xc2i∥2

i

where ) and ). You should see an error less than 500.

Ours is around 350.

Note: C1 and C2 here are projection matrices of the form: C and

C .

Q3.3 (10 points) Complete the function findM2 in q3 2 triangulate.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. Please include the code snippet of triangulate and findM2 function in your write-up.

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) In q4 1 epipolar correspondence.py finish the function epipolarCorrespondence 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.

• 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 help you test your epipolarCorrespondence, we have included a helper function epipolarMatchGUI in q4 1 epipolar correspondence.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 6.

epipolarCorrespondence

It’s 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. Please include the code snippet of epipolarCorrespondence function in your write-up.

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.

Figure 7: An example point cloud

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. Complete the script q4 2 visualize.py and compute3D pts function, which loads the necessary files from ../data/ to generate the 3D reconstruction using scatter function matplotlib. An example is shown in Figure 7.

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. Please include the code snippet of compute3D pts function in your write-up.

5 Bundle Adjustment

Bundle Adjustment is commonly used as the last step of every feature-based 3D reconstruction algorithm. Given a set of images depicting a number of 3D points from different viewpoints, bundle adjustment is the process of simultaneously refining the 3D coordinates along with the camera parameters. It minimizes reprojection error, which is the squared sum of distances between image points and predicted points. In this section, you will implement bundle adjustment algorithm by yourself (make use of q5 bundle adjustment.py

file). Specifically,

• In Q5.1, you need to implement a RANSAC algorithm to estimate the fundamental

matrix F and all the inliers.

• In Q5.2, you will need to write code to parameterize Rotation matrix R using Rodrigues formula (please check this pdf for a detailed explanation), which will enable the joint optimization process for Bundle Adjustment.

• In Q5.3, you will need to first write down the objective function in rodriguesResidual, and do the bundleAdjustment.

Q5.1 RANSAC for Fundamental Matrix Recovery (15 points) In some real world applications, manually determining correspondences is infeasible and often there will be noisy correspondences. 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, nIters, tol)

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 correspondences in some corresp noisy.npz in which around 75% of the points are inliers.

• Hints: Use the Eight or Seven point algorithm 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 Rodrigues and Invsere Rodrigues (15 points)

So far we have independently solved for camera matrix, Mj and 3D points wi. In bundle adjustment, we will jointly optimize the reprojection error with respect to the points wi and the camera matrix Cj.

err = X∥xij − Proj(Cj,wi)∥2 ,

ij

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 parameterizing the rotation matrix R using Rodrigues formula to produce vector r ∈ R3. 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)

Reference: Rodrigues formula and this pdf.

Please include the code snippet of rodrigues and invRodrigues functions in your write-up.

Q5.3 Bundle Adjustment (10 points)

Using this parameterization, write an optimization function

residuals = rodriguesResidual(K1, M1, p1, K2, p2, x)

where x is the flattened concatenation of x, r2, and t2. w 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 L2-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 optimizer minimize 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, w, o1, o2] = bundleAdjustment(K1, M1, p1, K2, M2 init, p2, w init)

Try to extract the rotation and translation from M2 init, then use invRodrigues you implemented previously to transform the rotation, concatenate it with translation and the 3D points, then the concatenate vector are variables to be optimized. After obtaining optimized vector, decompose it back to rotation using Rodrigues you implemented previously, translation and 3D points coordinates.

In your write-up: include an image of the original 3D points and the optimized points (use the provided plot 3D dual function) as well as the reprojection error with your initial M2 and w, and with the optimized matrices. Please include the code snippets in your write-up.

Figure 8: Visualization of 3D points for noisy correspondences before and after using bundle adjustment

6 Multi View Keypoint Reconstruction (Extra Credit)

You will use multi-view capture of moving vehicles and reconstruct the motion of a car. The first part of the problem will be using a single time instance capture from Three views (Figure 9(Top)) and reconstruct vehicle keypoints and render from multiple views(Figure 9(Bottom)). Make use of q6 ec multiview reconstruction.py file and data/q6 folder contains the images.

Q6.1 (Extra Credit – 15 points) Write a function to compute the 3D keypoint locations P given the 2d part detections pts1, pts2 and pts3 and the camera projection matrices C1, C3, C3. The camera matrices are given in the numpy files.

[P, err] = MultiviewReconstruction(C1, pts1, C2, pts2, C3, pts3, Thres)

In your write-up: Describe the method you used to compute the 3D locations and include an image of the Reconstructed 3D points with the points connected using the helper function plot 3d keypoint(P) with the reprojection error. Please include the code snippets in your write-up.

The 2D part detections(pts) are computed using a neural network and correspond to different locations on a car like the wheels, headlights etc. The third column in pts is the confidence of localization of the keypoints. Higher confidence value represents more accurate localization of the keypoint in 2D. To visualize the 2D detections run visualize keypoints(image, pts, Thres) helper function. Thres is defined as the confidence threshold of the 2D detected keypoints. The camera matrices (C) are computed by running an SFM from multiple views and are given in the numpy files with the 2D locations. By varying confidence Threshold

Figure 9: An example detections on the top and the reconstructions from multiple views

Thres(.i.e. considering only the points above the threshold), We get different reconstruction and accuracy. Try varying the thresholds and analyze its effects on the accuracy of the reconstruction. Save the best reconstruction(the 3D locations of the parts) from these parameters into a q6 1.npz file.

Hint: You can modify the triangulation function to take three views as input. After you do the threshold lets say m points lie above the threshold and n points lie below the threshold. Now your task is to use these m good points to compute the reconstruction. For each 3D location use two view or three view triangulation for intialization based on visibility after thresholding.

Q6.2 (Extra Credit – 15 points)

From the previous question you have done a 3D reconstruction at a time instance. Now you are going to iteratively repeat the process over time and compute a spatio temporal reconstruction of the car. The images in the data/q6 folder shows the motion of the car at an intersection captured from multiple views. The images are given as (cam1 time0.jpg,

…, cam1 time9.jpg) for camera 1 and (cam2 time0.jpg, …, cam2 time9.jpg) for camera2 and (cam3 time0.jpg, …, cam3 time9.jpg) for camera3. The corresponding detections and camera matrices are given in (time0.npz, …, time9.npz). Use the above details and compute the spatio temporal reconstruction of the car for all 10 time instances and plot them by completing the plot 3d keypoint video function. A sample plot with the first and last time instance reconstuction of the car with the reprojections shown in the Figure 10. Please include the code snippets in your write-up.

Figure 10: Spatiotemporal reconstruction of the car (right) with the projections at two different time instances in a single view(left)

Deliverables

The assignment (code and write-up) should be submitted to Gradescope. The write-up should be named <AndrewId> hw4.pdf and the code should be a zip named <AndrewId> hw4.zip. The zip should have the following files in the structure defined below. (Note: Neglecting to follow the submission structure will incur a huge score penalty!). You can run the included checkA4Submission.py script to ensure that your zip folder structure is correct.

• <AndrewId> hw4.pdf: your write-up.

• q2 1eightpoint.py: script for Q2.1.

• q2 2sevenpoint.py: script for Q2.2.

• q3 1 essential matrix.py: script for Q3.1.

• q3 2triangulate.py: script for Q3.2.

• q4 1epipolar correspondence.py: script for Q4.1.

• q4 2visualize.py: script for Q4.2.

• q5 bundle adjustment.py: script for Q5.

• q6 ec multiview reconstruction.py: script for (extra-credit) Q6.

• helper.py: helper functions (optional to include).

• q2 1.npz: file with output of Q2.1.

• q2 2.npz: file with output of Q2.2.

• q3 1.npz: file with output of Q3.1.

• 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.

• q6 1.npz: (extra-credit) file with output of Q6.1.

*Do not include the data directory in your submission.

FAQs

Credits: Paul Nadan

Q2.1: Does it matter if we unscale F before or after calling refineF?

Q2.1: Why does the other image disappear (or become really small) when I select a point using the displayEpipolarF GUI?

This issue occurs when the corresponding epipolar line to the point you selected lies far away from the image. Something is likely wrong with your fundamental matrix.

Q2.1 Note: The GUI will provide the correct epipolar lines even if the program is using the wrong order of pts1 and pts2 in calculating the eightpoint algorithm. So one thing to check is that the optimizer should only take ¡10 iterations (shown in the output) to converge if the ordering is correct.

Q3.2: How can I get started formulating the triangulation equations?

One possible method: from the first camera, x1i = P1ω1 =⇒ x1i × P1ω1 = 0 =⇒ A1iωi = 0. This is a linear system of 3 equations, one of which is redundant (a linear combination of the other two), and 4 variables. We get a similar equation from the second camera, for a total of 4 (non-redundant) equations and 4 variables, i.e. Aiωi = 0.

Q3.2: What is the expected value of the reprojection error?

The reprojection error for the data in some corresp.npz should be around 352 (or 89 without using refineF). If you get a reprojection error of around 94 (or 1927 without using refineF) then you have somehow ended up with a transposed F matrix in your eightpoint function.

Q3.2: If you are getting high reprojection error but can’t find any errors in your triangulate function?

one useful trick is to temporarily comment out the call to refineF in your 8-point algorithm and make sure that the epipolar lines still match up. The refineF function can sometimes find a pretty good solution even starting from a totally incorrect matrix, which results in the F matrix passing the sanity checks even if there’s an error in the 8-point function. However, having a slightly incorrect F matrix can still cause the reprojection error to be really high later on even if your triangulate code is correct.

Q4.2 Note: Figure 7 in the assignment document is incorrect – if you look closely you’ll notice that the z coordinates are all negative. Don’t worry if your solution is different from the example as long as the 3D structure of the temple is evident.

Q5.1: How many inliers should I be getting from RANSAC?

The correct number of inliers should be around 106. This provides a good sanity check for whether the chosen tolerance value is appropriate.

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