## 这是一篇关于 (3D point to 2D) Point and Inverse (2D point to 3D ray) Camera Projection （3D指向2D）点和逆(2D指向3D射线）相机投影的计算机视觉基础代写，以下是作业相关内容：

1 Motivation

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).

You 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
- Projection of 3D data into 2D images coordinates
- Reconstruction of 3D locations by triangulation from two camera views
- Measurement of 3D reconstruction error
- Practical understanding of epipolar geometry.

2 The Basic Operations

The following steps will be essential to the successful completion of the project:

- Input and parsing of mocap dataset. Read in and properly interpret the 3D joint data.
- Input and parsing of camera parameters. Read in each set of camera parameters and interpret with respect to our mathematical camera projection model.

- Use the camera parameters to project 3D joints into pixel locations in each of the two image coordinate systems.

- 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.

- 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

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 class lectures. 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 fromscratch 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 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.