## Description

2 Camera Calibration

Orginal image Undistorted image k > 0 k < 0

Figure 1: Lens distortion recification.

Using your cellphone camera, you will estimate intrinsic camera parameters (focal

length, principal points, and lens distortion parameter).

Write-up:

(1) Derive K matrix using your camera specification.

(2) Calibrate lens distortion using the radial distortion model (single parameter, k1).

Show the original image, undistorted image, and two image examples of wrong k1 as

shown in Figure 1.

2

CSCI 5980: Assignment #2

Image Transformation

3 Projective Line

(a) Image (b) UMN logo

Figure 2: (a) You will use your cellphone camera image to compute camera poses with

respect to the ground plane and measure the height of your friend given the height of

an object. (b) You will paste the UMN logo to the ground plane.

Take a picture of your friend with many 3D objects such as street lamps and chair where

two orthogonal directions on the ground plane are visible as shown in Figure 2(a). Apply

lens undistortion to make the straight lines straight.

Write-up:

(1) Derive and compute two vanishing points and a vanishing line, and visualize them

on your image similar to Figure 2(a).

(2) Compute camera rotation matrix, R, and visualize 3D camera axes with respect to

the ground plane axes using MATLAB plot3 function. Give a geometric interpretation

of the computed rotation matrix.

(3) Measure the heights of at least 3D objects given your friend’s height using the cross

ratio. Verify the height measurements.

(4) Project UMN logo onto the ground plane or any planar surface. You are also free

to choose different logo or image.

3

CSCI 5980: Assignment #2

Image Transformation

4 Panoramic Image

(a) Input images

h

,

x

h y

z

p

p

p

φ

=

p

Camera center

Cylindrical surface

φ

(b) Geometry

h

φ

(c) Cylindrical coordinate

(d) Panoramic image

Figure 3: Given a collection of input images, you will create a panoramic images by

projecting onto a cylindrical surface.

4

CSCI 5980: Assignment #2

Image Transformation

You will create a panoramic image from multiple images (at least 8 images) taken

by your cellphone camera using a cylindrical projection as shown in Figure 3(a). The

panoramic image will be created in (φ, h) where φ and h are angle and height coordinate

of the cylindrical surface, respectively, as shown in Figure 3(c). Note that the radius and

height of the cylinder are set to the focal length and height of the image, respectively.

Write-up:

(1) Express the direction vector pφ,h =

px py pz

T

using φ and h as shown in

Figure 3(c) and 3(b).

(2) Given the first and second images, compute homography, 2H1 using 4 correspondences and relative rotation from first to second, 2R1 where the first image rotation is

the identity matrix I3.

λu2 = Hu1

µu2 = K2R1pφ,h (1)

where u1 ↔ u2 is the corresponding points in the first and second image.

(3) For all images, compute the rotation matrix, iR1, and visualize the camera Z axis

in 3D using MATLAB plot3 function. Hint:

iR1 =i Ri−1

i−1Ri−2 · · ·

2 R1.

(4) Create a panoramic image by copying RGB value of original image coordinate (u, v)

to the cylindrical coordinate (φ, h) as shown in Figure 3(d). You may blend overlapping

pixels by taking average.

Note:

1. Rotate your camera about fixed rotation center (no translation). Translation of

your camera produces mis-alignment.

2. Choose 4 correspondences very carefully.

3. Lens distortion may introduce mis-alignment.

4. Objects at far distance often work better.

5