CIS 580, Machine Perception, Spring 2015 Homework 1 Due: :59AM

Transcription

1 CIS 580, Machine Perception, Spring 2015 Homework 1 Due: :59AM Instructions. Submit your answers in PDF form to Canvas. This is an individual assignment. 1 Camera Model, Focal Length and Measurement Given the physical specification of a camera and the distance between the camera and a structure, we can measure the height of the structure and do fun things. Assume a pinhole camera model. 1.1 Measure the height of Levine building 1. Camera sensor size. Pick a digital camera you have, for example your cell phone. What is the width (mm) and height (mm) of CCD of you camera? Include the web link of the reference. 2. Pixel size. What is the sensor pixel resolution? Compute the the size of a pixel in mm. 3. Focal length. What is the focal length of your phone camera? You can find this answer by looking up the specification of your camera. You can also compute it by finding the field of view (FOV) of your camera, and size of the CCD. Are they similar? Autofocus will change your focal length slightly, we can ignore this change for this problem. 4. Measurement. Capture an image of Levine building and measure the distance between the capturing location and the building. Hint: you can approximately measure distance by using Google map or counting the number of your steps. Calculate the height of the Levine building with detailed procedure. 1.2 Dolly Zoom: depth compression due to focal length change An example of Dolly zoom is shown in Figure 1. The goal is to keep one object the same size, while making the rest of the scene smaller (or larger) by adjusting focal length and moving the camera. Assume we have two objects, a and b. The heights of each are H a = 60mm and H b = 120mm, and the distance between the origin of the world coordinate and each objects are D a = 155mm and D b = 315mm. The camera has a focal length f = 4.15mm. Denote the heights of the projected objects of a and b as h a and h b. To simplify computation, assume the optical center, o c, is located at origin of the world coordinate: o c = 0. The projection of the object a can be computed as: h a = f H a D a o c. (1) In general, we do not know the precise location of the optical center, since it could be in front of the lens depending on the lens design. As a result, in the lecture slides, we discussed the case where the distance is measure to the image plane rather than the optical center. 1. Compute h a and h b. 1

2 Figure 1: Dolly Zoom. Two frames of the dolly zoom are shown (bottom left and right), by adjusting focal length and moving the camera. 2

3 2. If we want to amplify the image of the object b (h b ) by 1.5 times while keep the height of the image of the object a (h a) same, what should be the focal length f and the center of camera o C? Note that if you want to test Dolly zoom by using your cell phone, you cannot change the focal length physically. In that case, you can still achieved the goal using digital zoom. 3

4 Figure 2: Single vanishing point. Locations of points are specified as in the figure (best view in color). 2 Recovering camera orientation and position By exploiting geometrical properties of camera projection of vanishing points or a planar surface, we can estimate a camera orientation and position. We have two pictures of Levine building. The world coordinate system is right handed with the origin at the door of the building; the vector from the door to S 34th St defines z-axis and the vector from the door to the sky defines y-axis. The camera intrinsic parameter K is given by all numbers are in pixels. 2.1 Single vanishing point K = , Compute the z-vanishing point, using the points given in Figure Compute the rotation angles, Pan α and Tilt β (defined in the lecture note) by using the z-vanishing point. 2.2 Two vanishing points 1. Compute the x/y-vanishing points in Figure Compute the rotation angles, Pan α, Tilt β, and Yaw γ (defined in the lecture note) by using x/y-vanishing points. 4

5 Figure 3: x-y vanishing points. Locations of points are specified as in the figure (best view in color) Figure 4: Using Homography matrix to estimate Rotation and Translation of the camera. Points used to compute homography H 1 (left), H 2 (right) are plotted. 5

6 2.3 Homography matrix 1. Compute R 1 and t 1 using a planar reference object (Grasp nameplate) shown in the left image of Figure 4. The homography from the 3D planar surface to the projected image is H 1 = Compute R 2 and t 2 using the same planar reference object (Grasp nameplate) shown in the right image of Figure 4. The homography from the 3D planar surface to the projected image is H 2 = The left picture in Figure 4 was taken in the eye-level and the right one was taken in the ground-level. What is the approximate height of the observer in terms of the distance between the ground and the eyes. Note that the camera coordinate is a right handed system with x-right and y-down and units are in mm. 6