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project1.md

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+**ENGR4350 Computer Vision**
+
+**Project 1. Linear Approach to Camera Calibration**
+
+**Due Data: September 20, 2026**
+
+**Objective:** To use a linear approach to calibrate a camera and to use
+the camera parameters to predict the 2D image-domain location of a set
+of 3D points or a 3D moving object.
+
+Procedure:
+
+1.  Study the lecture note by showing the use of projection matrix that
+    maps objects from 3-D to 2-D, the definition of intrinsic and
+    extrinsic parameters of a camera, and the linear approach to
+    geometric camera calibration.
+2.  Given an image *test_image.bmp* captured by a camera which is posed
+    toward a 3D chess board, we manually selected 27 points whose 2-D
+    image coordinates are saved in the file *observe.dat*, and whose 3-D
+    coordinates in a world coordinate are stored in the file
+    *model.dat*. Please use the linear approach to calibrate the camera
+    system by computing the projection matrix (M) from which you are
+    required to compute the intrinsic and extrinsic parameters,
+    including , u0, v0, α, β, and the rotation matrix (R), and the
+    shift vector (t).
+3.  In order to verify the correctness of the camera calibration, please
+    use the projection matrix to map the three set of 3D points, whose
+    3-D coordinates are {x,y,0\|x,y=0,...,10}, {x,10,z\|x,z=0,...,10},
+    {10,y,z\|y,z=0,...,10}, respectively, into the 2-D image space.
+    Discuss your results.
+4.  Create a video file that shows a 3D object moving in the 3D scene
+    along a specific pre-defined path. For example, a 3D cube (1x1x1)
+    can be displayed by nine lines connecting seven vertexes (see the
+    example below). Each line can be drawn with around 100 samples. (You
+    can see a video example from Slide 13 of the Lecture 8 handout in
+    the slideshow mode.)
+
+**Report Requirements:**
+
+1.  Briefly discuss the basics of geometric camera modeling.
+2.  Briefly discuss the linear approach to camera calibration.
+3.  Show the simulation results by figures (Part 3).
+4.  Include some sample frames of the video (Part 4) in the report.
+5.  The Python source code should be included as the appendix of the
+    report with detailed comments. A separate Python file should also be
+    provided for testing.
+6.  Zip all files (DOC, AVI, M-file) into one package and upload it to
+    Blackboard by the due date.
+
+**Useful Python functions:**
+
+You need to import the required libraries:
+
+> import torch\
+> import numpy as np\
+> import matplotlib.pyplot as plt\
+> from PIL import Image\
+> import time
+
+> import cv2\
+> import os
+
+- Inner product: torch.dot(t1,t2), Cross product : torch.cross(t1,t2).
+
+> v0 = (s \*\* 2) \* torch.dot(a2, a3)\
+> \
+> cross_a1_a3 = torch.cross(a1, a3)
+
+> 
+
+- Singular value decomposition (SVD) :
+
+> \# \-\-- Solve for the projection matrix M \-\--\
+> \# The solution to Qm=0 is the last column of V from the SVD of Q\
+> \# In PyTorch, torch.linalg.svd returns U, S, Vh (V transpose)\
+> U, S, Vh = torch.linalg.svd(Q)\
+> \
+> \# The solution is the last row of Vh, which corresponds to the
+> smallest singular value\
+> m = Vh\[-1, :\]\
+> \
+> \# Reshape the 12x1 vector m into the 3x4 projection matrix M\
+> M = m.reshape(3, 4)
+
+- Norm of a vector: torch.linalg.norm(vector).
+
+> torch.linalg.norm(cross_a1_a3)
+
+> 
+
+- Inverse of a matrix: torch.linalg.inv(M)
+
+> torch.linalg.inv(M)
+
+> 
+
+- Reshape of a vector to a matrix:
+
+> \# Reshape the 12x1 vector m into the 3x4 projection matrix M\
+> M = m.reshape(3, 4)