Computer Science - The City College of New York
CSC I6716 - Fall 2007 3D Computer Vision and Video Computing
Assignment 3 Camera Models and Camera Clibration ( Deadline:
Oct 23 before class)
(Those marked with * are optional for extra credits)
Note: All the writings must be hard copies in print. Do write
your **names** and IDs (last four
digits) in your submissions.
1 (Camera Models- 30 points) Prove that the vector
from the viewpoint of a pinhole camera to the vanishing point (in the
image plane) of a set of 3D parallel lines is parallel to the direction
of
the parallel
lines.
Hint: You can either use geometric reasoning or algebraic
calculation.
If you choose to use geometric reasoning, you can use the fact that the
projection of a 3D line in space is the intersection of its
“interpretation plane”
with the image plane. Here the interpretation plane (IP) is a
plane
passing through the 3D line and the center of projection (viewpoint) of
the camera. Also, the interpretation planes of two parallel lines
intersect in a line passing through the viewpoint, and the intersection
line is parallel to the parallel lines.
If you select to use algebraic calculation, you may use the parametric
representation of a 3D line: P = P0 +tV, where P= (X,Y,Z)T
is any point on the line (here T denote for
transpose), P0 =
(X0,Y0,Z0)T is a given fixed point on the line, vector V =
(a,b,c)T represents the direction of the line, and t is the
scalar parameter that controls the distance (with sign) between P and
P0.
2. (Camera Models- 30 points) Show that relation between any image
point (xim, yim)T of a plane (in the form of (x1,x2,x3)T
in projective space ) and its corresponding point (Xw, Yw, Zw)T
on the plane in 3D space can be represented by a 3x3 matrix. You should
start from the general form of the camera model (x1,x2,x3)T
=
MintMext (Xw, Yw, Zw, 1)T, where the
image
center (ox, oy), the focal length f, the scaling factors( sx and sy),
the
rotation matrix R and the translation vector T are all unknown. Note
that
in the course slides, I used a simplified model by assuming ox and oy
are
known and sx = sy =1 so you cannot directly copy those equations.
You can either use a special plane Zw = 0 or a general plane nx
Xw + ny
Yw + Zw = d.
3. (Calibration- 40 points +20 extra points) Prove the
Orthocenter
Theorem
by geometric arguments: Let T be the triangle on the image plane
defined
by the three vanishing points of three mutually orthogonal sets of
parallel
lines in space. Then the image center is the orthocenter of the
triangle
T (i.e., the common intersection of the three altitudes.
(1) Basic proof: use the
result
of Question 1, assuming the aspect ratio of the camera is 1. (40 points)
(2) *If you do not know the aspect ratio and the
focal
length of the camera, can you still find the image center using the
Orthocenter
Theorem?
Show why or why not. (10 points)
(3) *Can you estimate aspect ratio and the image
center
all together using the Orthocenter Theorem? Please give your solution. Could you further estimate the focal
length of the camera based on the fact that the three sets of
parallel lines are orthogonal to each other? (10 points)