CS6476-Project 2 Camera Projection Matrix and Fundamental Matrix Estimation with RANSAC Solved

Part 1: Camera Projection Matrix Estimation             


Learning Objective: (1) Understanding the the camera projection matrix and (2) estimating it using fiducial objects for camera projection matrix estimation and pose estimation.

Introduction                                                                              
In this first part you will perform pose estimation in an image taken by an uncalibrated camera. As we saw in class, pose estimation is incredibly useful; it is used in VR, AR, controller tracking, autonomous driving, and even satellite docking. Recall that for a pinhole camera model, the camera matrix  is a projective mapping from world (3D) to pixel (2D) coordinates defined up to a scale.



Above  is an arbitrary scale factor. The camera matrix can also be decomposed into intrinsic parameters  and extrinsic parameters .

Let's look more carefully into what each of the individual parts of the decomposed matrix mean. The homogenous vector coordinates          of           indicate the position of a point in 3D space in the world coordinate system. The matrix               represents a translation and the matrix  represents a rotation. When combined they convert points from the world to the camera coordinate system. An intuitive way to understand this is to think about how aligning the axes of the world coordinate system to the ones of the camera coordinate system can be done with a rotation and a translation.



The distinction between camera coordinate and world coordinate systems is a rotation and a translation.

In this part of the project you will learn how to estimate the projection matrix using objective function minimization, how you can decompose the camera matrix, and what knowing these lets you do.

Part 1.1: Implement Camera Projection                                   
In projection_matrix.py you will implement camera projection in the projection(P,

points_3d) from homogenous world coordinates             to non-homogenous image coordinates        .

Given the projection matrix

Part 1.2: Implement Objective Function                                   
A camera projection matrix maps points from 3D into 2D. How can we use this to estimate its parameters? Assume that we have             known 2D-3D correspondences for a set of points, that is, for points with index          we have both access to the respective 3D coordinates  and 2D coordinates         . Let        be an estimation for the camera projection matrix. We can determine how accurate the estimation is by measuring the reprojection error

between the 3D points projected into 2D  and the known 2D points , both in non- homogeneous coordinates. Therefore we can estimate the projection matrix itself by minimizing the reprojection error with respect to the projection matrix



In this part, in projection_matrix.py you will implement the objective function objective_function() that will be passed to scipy.optimize.least_squares for minimization

with the Levenberg-Marquardt algorithm.

Part 1.3: Estimating the Projection Matrix Given Point Correspondences
Optimizing the reprojection loss using Levenberg-Marquardt requires a good initial estimate for . This can be done by having good initial estimates for            and        and  which you can multiply to then generate your estimated              . In this part, to make sure that you have the least squares optimization working properly we will provide you with an initial estimate. In the function you will have to implement in this part, estimate_projection_matrix() , you will have to pass the initial guess to scipy.optimize.least_squares and get the appropriate output.

Note: because       has only 11 degrees of freedom, we fix             .

Part 1.4: Decomposing the Projection Matrix                           
Recall that

.

Rewriting this gives us:

Where    is the first 3 columns of                . An operation known as RQ decomposition which will decompose  into an upper triangular matrix  and an orthonormal matrix  such that , where the upper triangular matrix will correspond to      and the orthonormal matrix to . In this part you will implement decompose_camera_matrix(P) where you will need to get the appropriate matrix elements of  to perform the RQ decomposition, and make the appropriate function call to

scipy.linalg.rq() .

Part 1.5: Calculating the Camera Center                                   
In this part in projection_matrix.py you will implement calculate_camera_center(P, K, R) that takes as input the projection , intrinsic  and rotation  matrix and outputs the camera position in world coordinates.

Part 1.6: Taking Your Own Images and Estimating the Projection Matrix + Camera Pose
In part 1.3 you were given a set of known points in world coordinates. In this part of the assignment you will learn how to use a fiducial---an object of known size that can be used as a reference. Any object for which you have measured the size given some unit (in this project you should use centimeters).



Example of how one cuboid of known dimensions can be used as a fiducial to create multiple world coordinate systems

The figure above illustrates how a cuboid of known dimension can be used to create a world coordinate system and a set of points with known 3D location. Choose an object that you will use as a fiducial (we recommend using a thick textbook) and measure it. Using a camera capture two images of the object (you will estimate the camera parameters for both images) keeping in mind the considerations discussed in class for part 2 and fundamental matrix estimation if you want to reuse these images. When taking the images, try to estimate the pose of the camera lens of your phone in the world coordinate system.

Now that you have the dimension of your object (3D points), you can use the Jupyter notebook to find the image coordinates of the 3D points create your own 2D-3D correspondences for each image. For each of your 2 images, make initial estimates for  and if your estimate is good, using your code from the previous part you should be able to estimate both the the projection matrix and the camera pose. Use the code available in the Jupyter notebook to visualize your findings.

Report                                                                                       
Put these answers in your report!

   What would happen to the projected points if you increased/decreased the  coordinate, or the other coordinates of the camera center ? Write down a description of your expectations in the appropriate part of your writeup submission.

   Perform this shift for each of the camera coordinates and then recompose the projection matrix and visualize the result in your Jupyter notebook. Was the visualized result what you expected?

Part 2: Fundamental Matrix Estimation                      




Diagram of epipolar lines and epipoles. The fundamental matrix maps points from one image to an epipolar line on the other.

Learning Objective: (1) Understanding the fundamental matrix and (2) estimating it using selfcaptured images to estimate your own fundamental matrix.

In this part, given a set of corresponding 2D points, we will estimate the fundamental matrix. Now that we know how to project a point from a 3D coordinate to a 2D coordinate, next we’ll look at how to map corresponding 2D points from two images of the same scene. You can think of the fundamental matrix as something that maps points from one view into a line in the other view. We get a line because a point in one image is only a projection to 2D,  which means we can’t actually know the “depth” of that point. As such, from the viewpoint of the other camera, we can see the entire “line” that our first point could exist on.

The fundamental matrix constraint between two points  and        in the left and right views, respectively, is given by the following equation:

 are homogenous coordinates in the two views, and                  is the  fundamental

matrix. We can write out the equation above as:

Since       takes points in one image and maps them to a line in another image, the constraint says that we want the point           to be on the line represented by              . If the point is exactly on the line, then the constraint     and the line-to-point distance between them is also zero.

Hence to estimate  we can minimize the line-to-point distance between the point  and the line

, for all matching points . This makes estimating the fundamental matrix a least squares problem.

Part 2.1: Distance Formula                                                       
The line-to-point distance formula  is defined by the following equation:

where  is a line and     is the homogenous coordinate for the point. You will need to implement this in point_line_distance() in fundamental_matrix.py .

Part 2.2: Symmetric line-to-point error                                     
Given a set of matching points     we can set up the following symmetric line-to-point error function for every match

where  is the distance formula between a line and a point,            is one of the homogeneous points,    is the fundamental matrix, and  is the other homogeneous point. In the first term we use , which maps from the second to the first view, and in second term we use , which maps from the first to the second view. SciPy handles the squaring and summing for you so you just need to implement

point_line_distance() .

By applying this symmetric line-to-point error calculation to every pair of points, we get this equation for the objective function to minimize, as a function of

This is the equation you will be implementing in signed_point_line_errors() which is in the file fundamental_matrix.py .

Part 2.3: Least Squares Optimization                                       
Since this is a least squares optimization problem, you will also be making the call to SciPy's least squares optimizer. The documentation for this is available here.

You'll need to give as input: the objective function, your initial estimate of the fundamental matrix, optimization algorithm, method for computing the Jacobian, and the point pairs. There are more details on this in the code.

Part 2.4: Try it Yourself                                                              
Similar to Part 1, you'll have to take two images of the same scene and estimate the fundamental matrix between the two images. Recall that these two images must be from different positions, and you cannot simply just rotate the camera or zoom the image. You'll have to save the images in the same project folder and use the Jupyter notebook to run your fundamental matrix estimator on your images.

Report                                                                                       
Put these answers in your report!

Why is it that when you take your own images, you can't just rotate the camera or zoom the image for your two images of the same scene?

Why is it that points in one image are projected by the fundamental matrix onto epipolar lines in the other image?

What happens to the epipoles and epipolar lines when you take two images where the camera centers are within the images? Why?

What does it mean when your epipolar lines are all horizontal across the two images?

Why is the fundamental matrix defined up to a scale? Why is the fundamental matrix rank 2?

Part 3: RANSAC                                                             


Learning objective: (1) Understand the parameters of the RANSAC algorithm. (2) Apply RANSAC to find the fundamental matrix for a pair of images given imperfect point correspondences (ie outliers).

Now you have a function which can calculate the fundamental matrix  from matching pairs of points in two different images. However, having to manually extract the matching points is undesirable. Later in the course, you will learn to to automate the process of identifying matching points in two images. For now we will assume that someone has already run code to find the matches for you; however, the automated matching is not perfect, and we will use RANSAC to find a set of true matches (inliers) to calculate . Fortunately, to calculate the fundamental matrix , we need only 9 matching points, but we want to automate the process of finding them. Below is a highlevel description of how we will implement this section. See the Jupyter notebook and code documentation for implementation details.

Part 3.1: RANSAC Iterations                                                      
We will use a method called RANdom SAmple Consensus (RANSAC) to search through the points returned by SIFT and find true matches to use for calculating the fundamental matrix . Please review the lecture slides on RANSAC to refresh your memory on how this algorithm works. Additionally, you can find a simple explanation of RANSAC at https://www.mathworks.com/discovery/ransac.html. See section 6.1.4 in the textbook for a more thorough explanation of how RANSAC works.

You may wonder how many iterations of this algorithm we need to run in order to acheive success. It turns out RANSAC actually provides probabilistic guarentees of finding the correct model for our dataset given the number of iterations we run, the size of our sample, and the proportion of true matches in our dataset. You will right a function to determine how many iterations of RANSAC to run in order to find the fundamental matrix with a high probability of success. The jupyter notebook will guide you through this derivation.

Part 3.2: RANSAC Implementation                                            
Next we will implement the RANSAC algorithm. Remember the steps from the link above:

1.  Randomly selecting a subset (k=9) of the data set

2.  Fitting a model to the selected subset

3.  Determining the number of outliers

4.  Repeating steps 1-3 for a prescribed number of iterations

For the application of finding true point pair matches and using them to calculate the fundamental matrix, our subset of the data will be the minimum number of point pairs needed to calculate the fundamental matrix.

The model we are fitting is the fundamental matrix.

Outliers will be found by using the point_line_distance() error function from part 2 and thresholding with a certain margin of error.

Part 3.3: RANSAC Visualization                                                 
Finally, we will demonstrate running RANSAC on a real image pair, and plot the epipolar lines from the resulting fundamental matrix. You do not need to write any new code for this section, but use it as an opportunity in addition to the unit tests to check your results and better understand epipolar geometry and RANSAC.

Report                                                                                       
Put these answers in your report!

   What is the minimum number of RANSAC iterations we would we need to find the fundamental matrix with 99.9% certainty from a set of proposed matches that have a 90% point correspondence accuracy? Keep in mind that we need at least 9 point correspondences for our optimization to find the fundamental matrix in part 2

         One might imagine that if we had more than 9 point correspondences, it would be better to use more of them to solve for the fundamental matrix. Investigate this by finding the number of RANSAC iterations you would need to run for the above situation with 18 points.           If our dataset had a lower point correspondence accuracy, say 70%, what is the minimum number of iterations needed to find the fundamental matrix with 99.9% certainty?

Part 4: Recovering relative camera pose from epipolar geometry


Note: This section is compulsory for graduate students. Undergraduate students are welcome to attempt this section for extra credits.

Once we have the fundamental matrix, we will try and recover the relative  rotation and translation between two camera poses.

4.1 Recovering essential matrix from the fundamental matrix
Let's work out some of the math and introduce the essential matrix.

Recall the epipolar constraint . If we assume that both these camera share the same calibration matrix , we can write the projection equation for               and       . Let        and  be the extrinsic matrix for the two cameras. Without loss of generality, we can fix the world coordinate system such that         is           . We can then represent                 as           where  and  are the relative rotation and translation between the two cameras.

 

 are called normalized image coordinates (image coordinates with the effect of

camera intrinsics removed), and                 are called normalized camera matrices.

We define the essential matrix     as the fundamental matrix corresponding to the a pair of normalized cameras. We can write the epipolar constraint as :

You can now write the essential matrix  in terms of . Derive the equation and write the code in recover_E_from_F() function in recover_rot_translation.py .

Please note that this writeup is not thorough and we expect you to refer to the lecture slides. Section 9.6 from the book Multi-view geometry by Hartley or Zisserman (or any other material of your preference) is also a good reference

 

4.2 Recovering relative rotation and translation                      
We have now computed the essential matrix. Please follow the derivation in the lecture slides for the essential matrix equation in terms of  and               :              .

We will follow section 9.6.2 from the book multi-view geometry for this part. Note that the proof in this writeup is not comprehensive and we expect you to go through the detailed proof.

Let us create two constant matrices to help with the notation:



Note that W is an orthogonal matrix, and Z is a skew-symmetric matrix. We can use SVD to write

. where  is a skew symmetric matrix and  is a rotation

matrix.

As  and  have the same left null space, we can write . As rotation matrices are orthonormal, we can write        in terms of another rotation matrix         , using    and        which are orthonormal:     .

 

Plugging in the values of  and       in the SVD decomposition of , we get two possible values of  or .

We have to derive the value of  from . As we can only derive  upto scale, we will restrict the l2norm of  to be 1. As  is a proxy for        , we can use the relationship        to calculate  as the the third column of          .

Hence we have two possible values of rotation, and two possible values of normalized translation (positive and negative sign).

Complete the function recover_rot_translation_from_E in recover_rot_translation.py . There is no deliverable in report for this section.