DD2429 Computational Photography :00-19:00

Transcription

1 . Examination: DD2429 Computational Photography :00-9:00 Each problem gives max 5 points. In order to pass you need about 0-5 points. You are allowed to use the lecture notes and standard list of mathematical formulas. A perfect sphere is viewed by a normalised camera x y = λ a the -axis intersects the center of the sphere and the original image is therefore a perfect circle with center on the origin of the image (x, y) coordinate system If we change the camera parameters the projection is described by: x y = λ σ x γ x 0 0 σ y y r r 2 r 3 r 2 r 22 r 23 r 3 r 32 r Y The figure shows the image of the sphere before and after a change of the camera parameters Denote for each of the changes in camera parameters a-e below the associated images -5. One image for each parameter change. Different parameter alternatives could map to the same image a) Move the camera position 0 forwards along the -axis b) Change of the principal point c) Rotate the camera around the Y-axis d) Move the camera position 0 backwards along the -axis and change the focal length f so that the image plane remains in the same position e) Change of σ x Whatever looks like a perfect circle (images, 2, 4, 5) should be considered as a perfect circle. 2. A normalized camera is located at (t, t Y, t ) The camera can be expressed as: x y = λ 0 0 t 0 0 t Y 0 0 t

2 Figure : 2

3 a) Formulate a set of linear equations for finding the camera location (t, t Y, t ) assuming that you know the 3D and image coordinates of a set of points. b Show that this linear system of three unknowns can be separated into two systems of two unknowns (t, t ) and (t Y, t ) c) Solve one of this systems using a minimal set of points. I.e give an explicit solution of (t, t ) in terms of image and 3D coordinates 3 A general uncalibrated perspective camera is looking at a planar surface where there is a unit square. Any point on the plane can then be expressed in the coordinate system (x, y ) defined by the unit square where one of the corners is used as the origin. a) What is the general exact relation between the coordinates (x, y ) of a point on the plane and it s image coordinates (x, y)? (No derivations, you only have to state the result) b) Describe in words how you would go about computing the coordinates (x, y ) of a point on the plane from knowledge of it s image coordinates, using the image coordinates of the corners of the square Again you don t have to compute anything just formulate the equations, what operations have to be performed in order to solve them, what information you use and in what order 4 Figure 2: We have two cameras: a) A normalized camera located at the origin and b) the same camera rotated the angle θ around the axis and moved distance t along the axis xa y a = λ cos(θ) sin(θ) 0 y b = λ sin(θ) cos(θ) xb t 3

4 Suppose the cameras are looking at a unit circle 2 + Y 2 = located in a plane orthogonal to the axis at distance d from the origin. a) Compute the images of this unit circle in camera a and b b) Can the translation t and rotation θ of the second camera be determined from knowledge of the images of the unit circle in the two cameras? Give a motivation from the equations of the circles in the two images as well as a geometric motivation 5 A line in 3-D space passes through two points P = (, Y, ) T and P 2 = ( 2, Y 2, 2 ) T. All points on the line can then be expressed as with P = (, Y,, ) T where µ is a real number P = P + µ(p 2 P ) A camera with general 3 4 matrix M is looking at the line a) Assume that the line in space is infinitely long i.e. < µ < + and that only the part of the line in front of the camera will be projected The projection of the line in the image will then be of finite extent Sketch a geometric explanation for this. (Use the figure in the problem) b) If we assume that the line is in front of the camera for all µ > 0 then the point on the line corresponding to lim µ will get projected to a specific point in the image. Compute the image coordinates of this point in terms of the camera matrix M and the points P and P 2 c) If the line in space was moved parallel to it s original position What would be the respective solutions to a) and b)? Hint: The crucial problem is to understand how λ(µ) varies with µ You can assume a specific scale factor for M and use the form: p = λ(µ)m P Note that the real number λ(µ) is only there to ensure that the image coordinate p = (x, y, ) 4

5 Figure 3: 5