Description
The goal of this project is to implement forward (3D point to 2D point) and inverse (2D point to
3D ray) camera projection, and to perform triangulation from two cameras to do 3D
reconstruction from pairs of matching 2D image points. This project will involve understanding
relationships between 2D image coordinates and 3D world coordinates and the chain of
transformations that make up the pinhole camera model that was discussed in class. Your
specific tasks will be to project 3D coordinates (sets of 3D joint locations on a human body,
measured by motion capture equipment) into image pixel coordinates that you can overlay on
top of an image, to then convert those 2D points back into 3D viewing rays, and then
triangulate the viewing rays of two camera views to recover the original 3D coordinates you
started with (or values close to those coordinates).
Your group will be provided:
• 3D point data for each of 12 body joints for a set of motion capture frames recorded of a
subject performing a Taiji exercise. The 12 joints represent the shoulders, elbows,
wrists, hips, knees, and ankles. Each joint will be provided for a time series that is
~30,000 frames long, representing a 5 minute performance recorded at 100 frames per
second in a 3D motion capture lab.
• Camera calibration parameters (Intrinsic and extrinsic) for two video cameras that were
also recording the performance. Each set of camera parameters contains all information
needed to project 3D joint data into pixel coordinates in one of the two camera views.
• An mp4 movie file containing the video frames recorded by each of the two video
cameras. The video was recorded at 50 frames per second.
While this project appears to be a simple task at first, you will discover that practical
applications have hurdles to overcome. Specifically, in each frame of data there are 12 joints
with ~30,000 frames of data to be projected into 2 separate camera coordinate systems. That is
over ~700,000 joint projections into camera views and ~350,000 reconstructions back into
world coordinates! Furthermore, you will need to have a very clear understanding of the
pinhole camera model that we covered in class, to be able to write functions to correctly
project from 3D to 2D and back again.
The specific project outcomes include:
• Experience in Matlab programming
• Understanding intrinsic and extrinsic camera parameters
2
• Projection of 3D data into 2D images coordinates
• Reconstruction of 3D locations by triangulation from two camera views
• Measurement of 3D reconstruction error.
2 The Basic Operations
The following steps will be essential to the successful completion of the project:
1. Input and parsing of mocap dataset. Read in and properly interpret the 3D joint data.
2. Input and parsing of camera parameters. Read in each set of camera parameters and
interpret with respect to our mathematical camera projection model.
3. Use the camera parameters to project 3D joints into each of the two image coordinate
systems.
4. Reconstruct the 3D location of each joint in the world coordinate system from the
projected 2D joints you produced in Step3, using two-camera triangulation.
5. Compute Euclidean (L²) distance between all joint pairs. This is a per joint, per frame L²
distance between the original 3D joints and the reconstructed 3D joints providing a
quantitative analysis of the distance between the joint pairs.
2.1 Reading the 3D joint data
The motion capture data is in file Subject4-Session3-Take4_mocapJoints.mat . Once you load it
in, you have a 21614x12x4 array of numbers. The first dimension is frame number, the second
is joint number, and the last is joint coordinates + confidence score for each joint. Specifically,
the following snippet of code will extract x,y,z locations for the joints in a specific mocap frame.
mocapFnum = 1000; %mocap frame number 1000
x = mocapJoints(mocapFnum,:,1); %array of 12 X coordinates
y = mocapJoints(mocapFnum,:,2); % Y coordinates
z = mocapJoints(mocapFnum,:,3); % Z coordinates
conf = mocapJoints(mocapFnum,:,4) %confidence values
Each joint has a binary “confidence” associated with it. Joints that are not defined in a frame
have a confidence of 0. Feel free to Ignore any frames don’t have all confidences = 1.
There are 12 joints, in this order:
1 Right shoulder
2 Right elbow
3 Right wrist
4 Left shoulder
5 Left elbow
6 Left wrist
7 Right hip
8 Right knee
3
9 Right ankle
10 Left hip
11 Left knee
12 Left ankle
2.2 Reading camera parameters
There are two cameras, called “vue2” and “vue4”, and two files specifying their calibration
parameters: vue2CalibInfo.mat and vue4Calibinfo.mat . Each of these contains a structure with
intrinsic, extrinsic, and nonlinear distortion parameters for each camera. Here are the values of
the fields after reading in one of the structures
vue2 =
struct with fields:
foclen: 1557.8
orientation: [-0.27777 0.7085 -0.61454 -0.20789]
position: [-4450.1 5557.9 1949.1]
prinpoint: [976.04 562.82]
radial: [1.4936e-07 4.3841e-14]
aspectratio: 1
skew: 0
Pmat: [3×4 double]
Rmat: [3×3 double]
Kmat: [3×3 double]
Part of your job will be figuring out what those fields mean in regards to the pinhole camera
model parameters we discussed in two class lectures. In particularly, which are the internal
parameters? Which are the external parameters? Which internal parameters combine to form
the matrix Kmat? Which external parameters combine to form the matrix Pmat? Hint: the field
“orientation” is a unit quaternion vector describing the camera orientation, which is also
represented by the 3×3 matrix Rmat. What is the location of the camera? Verify that location
of the camera and the rotation Rmat of the camera combine in the expected way (expected as
per one of the slides in our class lectures on camera parameters) to yield the appropriate
entries in Pmat.
2.3 Projecting 3D points into 2D pixel locations
Ignoring the nonlinear distortion parameters in the “radial” field for now, write a function from
scratch that takes either a single 3D point or an array of 3D points and projects it (or them) into
2D pixel coordinates. You will want to refer to our lecture notes for the transformation chain
that maps 3D world coordinates into 2D pixel coordinates.
For verification, it will be helpful to visualize your projected 2D joints by overlaying them as
points on the 2D video frame corresponding to the motion capture frame. Two video files are
given to you: Subject4-Session3-24form-Full-Take4-Vue2.mp4 is the video from camera vue2,
and Subject4-Session3-24form-Full-Take4-Vue4.mp4 is the video from camera vue4. To get a
4
video frame out of the mp4 file we can use VideoReader in matlab. It is nonintuitive to use, so
to help out, here is a snippet of code that can read the video frame from vue2 corresponding to
the motion capture frame number mocapFnum.
%initialization of VideoReader for the vue video.
%YOU ONLY NEED TO DO THIS ONCE AT THE BEGINNING
filenamevue2mp4 = ‘Subject4-Session3-24form-Full-Take4-Vue2.mp4’;
vue2video = VideoReader(filenamevue2mp4);
%now we can read in the video for any mocap frame mocapFnum.
%the (50/100) factor is here to account for the difference in frame
%rates between video (50 fps) and mocap (100 fps).
vue2video.CurrentTime = (mocapFnum-1)*(50/100)/vue2video.FrameRate;
vid2Frame = readFrame(vue2video);
The result is a 1088x1920x3 unsigned 8-bit integer color image that can be displayed by
image(vid2Frame).
If all went well with your projection of 3D to 2D, you should be able to plot the x and y
coordinates of your 2D points onto the image, and they should appear to be in roughly the
correct places. IMPORTANT NOTE: since we ignore nonlinear distortion for now, it might be the
case that your projected points look shifted off from the correct image locations. That is OK.
However, if the body points are grossly incorrect (body is much larger or smaller or forming a
really weird shape that doesn’t look like the arms and legs of the person in the image), then
something is likely wrong in your projection code.
2.4 Triangulation back into a set of 3D scene points
As a result of the above step, for a given mocap frame you now have two sets of corresponding
2D pixel locations, in the two camera views. Perform triangulation on each of the 12 pairs of 2D
points to estimate a recovered 3D point position. As per our class lecture on triangulation, this
will be done, for a corresponding pair of 2D points, by converting each into a viewing ray
represented by camera center and unit vector pointing along the ray passing through the 2D
point in the image and out into the 3D scene. You will then compute the 3D point location that
is closest to both sets of rays (because they might not exactly intersect). Go back and refer to
our slide on this to see how to do the computation.
5
2.5 Measure error between triangulated and original 3D points
Because you are projecting 3D points into 2D (forward projection) and then backprojecting
those into viewing rays (backward projection) to do triangulation back into 3D, you should get
reconstructed point locations that are very similar to the original locations. However, the
computed locations may not be perfectly the same as the original locations due to inaccuracies
in the projection equations or even just small but nonzero numerical inaccuracies of floating
point calculations. To quantitatively measure how close your reconstructed 3D points come to
the original set of 3D points, you can compute L2 (Euclidean) distance between them. If an
original point location is (X,Y,Z) and the recovered location is (X’,Y’,Z’), then the L2 distance
between them is sqrt( (X-X’)
2 + (Y-Y’)
2 + (Z-Z’)
2 ). Consider evaluating the average L2 distance or
sum of L2 distances to summarize the reconstruction accuracy for any given frame. More
rigorous error analysis is described in the next section.
3 Evaluation
As part of your report, and to convince us that your code is working correctly, we would like
you to show the following quantitative and qualitative evaluations of your work.
Quantitative
For this project, quantifying the accuracy of the algorithm’s implementation is difficult for a
human because of the large amount of data, but easy for a computer to calculate and analyze.
There is one fundamental method you are to implement and document for quantitative results,
the L² distance, described above in section 2.5. Once the L² distance is calculated for each joint
pair in each mocap frame (you only have to use the mocap frames where all 12 joints are valid,
i.e. all 12 have confidence values of 1), the following statistics should be calculated from the L²
distance data:
• For each of the twelve joint pairs (original,recovered), provide the mean, standard deviation,
minimum, median, and maximum of the L² distance computed over the entire time sequence.
• For all joint pairs (independent of their locations), provide the mean, standard deviation,
minimum, median, and maximum of the L² distance computed over the entire time sequence.
• For each mocap frame, compute the sum of L2 distances across the 12 joint pairs in that
frame, and plot that sequence of “total error” values to see whether there are any interesting
correlations between frame numbers and amount of reconstruction error.
Qualitative
The report should also provide visualizations of each step of the data processing starting with
the input 3D data, ending with reconstructed 3D, and including images of each of the 2D
projections in the two camera views. These example images should be of selected frames and
those specific frame indexes should be used throughout the report when showing examples.
6
The example frames should represent an accurate cross section of your results but must
include the frame indexes with the minimum and maximum total error for all joints.
For ease of viewing in both 2D and 3D, rather than visualizing joints as independent points
(using plot3 in 3D, and plot in 2D), we’d like you to also draw line segments between suitable
body joint pairs to form a skeleton. For example, draw a line between the right shoulder and
right elbow, between the right elbow and right wrist, etc. It should be obvious which joints to
connect together to draw each of the four limbs. However, please also draw lines connecting
the right shoulder to left shoulder, right hip to left hip, and a “spine” that connects the
midpoint of the two shoulder joints to the midpoint of the two hip joints.
Efficiency
Use the Matlab profile() function to analyze and quantify the efficiency of your implementation.
This tool allows for a full breakdown of the processor time used by each function call in a
Matlab script. It isn’t required that your code be super-efficient, and we won’t penalize slow
functions, but it’s good practice to check where the bulk of the running time is coming from, as
that may indicate places that you could speed up in the future. More information on the
Matlab profile function is here: https://www.mathworks.com/help/matlab/ref/profile.html
4 What Libraries Can I Use?
The intent is that you will implement the various computational modules described using
general processing functions (https://www.mathworks.com/help/matlab/functionlist.html) in
Matlab. The single notable exception is that you MAY use the image processing toolbox
(https://www.mathworks.com/help/images/index.html). Specifically, you MAY NOT use
anything from the computer vision toolbox, or any third-party libraries/packages. Don’t
plagiarize any code from anywhere or anyone else.
5 Project Reporting and Evaluation
Half of your grade will be based on submitting a fully operation program, and the other half will
be based on a written report discussing your program, design decisions, and experimental
observations. The following components will be submitted:
1) Submit a written report in which you discuss at least the following:
a) Summarize in your own words what you think the project was about. What were the
tasks you performed; what did you expect to achieve?
b) Present an outline of the procedural approach along with a flowchart showing the flow
of control and subroutine structure of your Matlab code. Explain any design decisions
you had to make. For example, how did you choose the camera parameters that were
needed and how did you use those parameters? Include answers to the questions
asked in Section 2.2. How did you choose the specific visualizations to convince us that
7
your code is working? What mathematical equations were used to calculate the
Euclidean distance statistics? Basically, explain at each step why you chose to do
whatever you did. Be sure to document any deviations you made from the above
project descriptions (and why), or any additional functionality you added to increase
robustness or generality or efficiency of the approach.
c) Experimental observations. What do you observe about the behavior of your program
when you run it? Does it seem to work the way you think it should? Run the different
portions of your code and show pictures of intermediate and final results that convince
us that the program does what you think it does.
d) Quantitative Results: The report should contain a table with 5 columns, one for each of
the statistical metrics, and 13 rows, one for each of the 12 joints plus one for the overall
joint results. The closer the entries in this table are to zero, the more accurate your
camera projection and 3D reconstruction algorithms. The Matlab code used for these
calculations should also be included in the report. Also include the plot of total error
across all frames of the sequence. Is the curve constant, or are there notable peaks and
valleys?
e) Qualitative results: At least one image showing the input 3D skeleton and output 3D
reconstructed skeleton on the same plot, with each skeleton being a different color.
Also include images of the two camera projections, preferable overlaid over the color
images extracted from the video files for cameras vue2 and vue4.
f) Algorithm Efficiency: Provide an image showing the Matlab profiler output. If you
decide to make performance improvements, show the profiler results both before and
after to convince us that the efficiency has been improved, and in what functions.
g) Document what each team member did to contribute to the project. It is OK if you
divide up the labor into different tasks (it is expected), and it is OK if not everyone
contributes precisely equal amounts of time/effort on each project. However, if two
people did all the work because the third team member could not be contacted until the
day before the project was due, this is where you get to tell us about it, so the grading
can reflect the inequity.
2) Turn in a running version of your main program and all subroutines in a single zip/tar
archive file (e.g. all code/data in a single directory, then make a zip file of that directory).
a) Include a demo routine that can be invoked with no arguments and that loads the input
data from the local directory and displays intermediate and final results showing the
different portions of your program are working as intended.
b) We might be running the code ourselves on other input datasets, so include a routine
that takes as input a joint dataset and that outputs the Euclidean distance between
input and output with the statistics that are calculated for the report.
c) Include enough comments in your functions so that we have a clear understanding of
what each section of code does. The more thought and effort you put in to
demonstrating / illustrating in your written report that your code works correctly, the
less likely we feel the need to poke around in your code, but in case we do, make your
code and comments readable.
8
6 Grading Criteria
Project 2 grade = (100 − Deductions + Additions) × (1− Late Penalty)
Deductions – Implementation
• Implementation of camera projections [10pts]: visualizations for specific frame indexes of
skeletons projected into both camera viewpoints.
• Implementation of 3D reconstruction [10pts]: visualization of both the input and output 3D
skeletons for specific frame indexes (same indexes as the 2D visualizations)
• Implementation of Euclidean distance statistics [10pts]: Euclidean distance calculated for each
joint pair (input 3D to reconstructed 3D) with all 5 statistical measurements calculated for
each of the 12 joints individually plus the overall (all 12 joints combined) each over the full
time series.
• Program structure and readability [10pts]: The code should have enough comments to explain
the functions and procedures while being properly divided into small modules (function calls).
• Code should be runnable [10pts]: Each function created should operate properly when called
as part of your implementation and should not cause failures in any demonstration code.
Functions may be spot checked to verify proper coding style and commenting.
Deductions – Report
• Implementation Description [10pts]: outline of your code along with proper diagrams. Unless
the operation is predefined in Matlab, explanation of what operation is being applied is
required. This is a step by step of WHAT was done to achieve the desired outcome.
• Experiment Observations and Explanations [10pts]: This is a step by step explanation of WHY
an operation was used in the implementation.
• Experiment Visualizations [20pts]: This is a step by step visual representation of intermediate
and final results including the recommended qualitative and quantitative evaluation methods.
• Evaluation and Discuss [10pts]: This is an explanation of the results, the overall successes and
failures of the implementation as compared to the ideal result (small Euclidean distances).
Additions
• Good demo routine [5pts]: clearly demonstrating the functionality and correctness of your
code. It can be a big demo for the whole project or small demos for function parts. A good
demo is a script that can be opened in Matlab and run to demonstrate the code producing the
reported results.
• Adding Gaussian Noise [5pts]: demonstrating the impact of additive zero-mean, Gaussian
noise on both the quantitative and qualitative results. Specifically, show the correlation of
output reconstruction error vs increasing amount of noise added to the intermediate 2D joint
location data. Results of this experiment must be provided and explained in the report.
• Adding Nonlinear Lens Distortion to the Pinhole Camera Model [5pts]: You will notice that
projected 2D points overlaid on the video camera images will not always line up well with
where those joints should be located. The offset error will be especially bad when the
9
performer is near the left or right borders of the image, and less when they are near the
center. These offset errors are due to nonlinear lens distortion, which our linear-inhomogeneous-coordinates camera model does not take into account. However, the “radial”
field of each camera calibration parameter structure contains numbers that will allow us to
add nonlinear lens effects to our model. The files given to you include two functions,
performRadialDistortion.m and performRadialCorrection.m . The first takes perspectively
projected 2D film coordinates and applies nonlinear lens distortion prior to converting from
film coordinates to pixels. The second is the inverse function that take distorted film
coordinates and corrects for the nonlinear distortion, prior to backprojecting a viewing ray out
into the scene to use for triangulation. Unfortunately, applying these nonlinear functions
breaks the nice end-to-end linear transformation chain from 3D world to 2D pixel coordinates.
You job is to figure out how to revise the forward and backward projection functions you
wrote earlier to insert these nonlinear functions into the chain. You should find that the
locations of projected 2D joints should overlay much more accurately onto the 2D video
frames. Show that. Also, repeat your 3D reconstruction error analysis in two ways: 1) using
the nonlinear distortion/correction in the forward/backward projection functions; and 2) for
more interesting results, using nonlinear distortion in the forward projections, but DO NOT do
the correction in the backward projections. The error analysis should be very interesting for
this second case, which models a common scenario where joints are detected directly from
2D images but an approximate, linear camera model is used to do the triangulation to get
reconstructed 3D points.
• Make a movie comparing original and reconstructed skeletons [5pts]: Learn how to use
Matlab’s movie() function to record 3D plots of original vs reconstructed skeletons (make the
skeletons different colors so we know which is which). Do this for at least 2 minutes of the
performance, but preferably for all of it (or for the range of frames that have all 12 mocap
joints defined).
