Download Solved Problems on Image Understanding - Assignment 4 | ENEE 731 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!
1
Homework 4
I. PROBLEM 1
Consider a camera moving along its optical axis toward a planar surface at right angles to the optical axis.
- (a) Show that the optical flow is given by
u =
W
Z
x and v =
W
Z
y (1)
where W is the velocity and Z the distance to the plane. (Note the lack of dependence on the focal length of the lens.)
- (b) Is the optical flow stationary (that is, independent of time)?
- (c) Is the Laplacian of the optical flow zero?
- (d) How could you predict the the time to impact? (Note that this can be done despite the fact that we cannot recover the absolute value of either height or velocity.)
II. PROBLEM 2 A rigid body rotates about an axis through the origin. The axis of rotation is parallel to the vector ω, while the angular velocity is given by the magnitude of this vector. The velocity of a point r on the body is the cross-product ω × r. Define r = (x, y, z)T^ and ω = (α, β, γ)T^. Show that the smoothness of the optical flow is related to the smoothness of the rotating body. What happens on the silhouette? Assume orthographic projection. Hint: Show that ∇^2 u and ∇^2 v are related to ∇^2 z.
III. PROBLEM 3
Here we explore the relationship between distance and disparity. Let the length of the baseline, the line connecting two camera stations, be b. Suppose that an object can be seen from both station points and that the lines from the left and right cameras to the object make angles θl and θr , respectively, with the baseline (see figure below).
Fig. 1. Figure for Problem 3
- (a) Show that the perpendicular distance to the object from the baseline (or its extension, if needed) is
h = b sinθr sinθl sin(θr − θl)
Note the inverse dependence on the disparity θr − θl.
- (b) What is the variance in the distance h if the variance in the measurements θr and θl is σ^2? Hint: Differentiate h with respect to θr and θl.
- (c) How does the accuracy of this stereo method degrade with distance?
IV. PROBLEM 4
This exercise exposes the redundancy encountered when the coordinates of many points are measured in two different coordinate systems. Let the transformation between two camera stations be given by
rr = Rrl + ro, where RT^ R = I. (3)
Show that the distance between two particular points is the same in the left and right coordinate systems; that is,
2
|rl, 2 − rl, 1 |^2 = |rr, 2 − rr, 1 |^2 , (4)
where we use the standard notation |x|^2 = x · x.
V. PROBLEM 5
- (a) Show that the Laplacian of the optical flow is zero when a camera viewing a planar surface is translating parallel to the image-plane. The plane being viewed need not be parallel to the image plane.
- (b) Show that the Laplacian of the optical flow is zero when a camera is translating relative to a plane that is orthogonal to the optical axis. The translation need not be parallel to the image plane.
- (c) Show that the Laplacian of the optical flow is not zero when the camera is rotating about an axis other than the optical axis.
VI. PROBLEM 6
Let R = (X, Y, Z)T^ be the vector to a point on an object, and r = (x, y, f )T^ the vector to the corresponding image point. In addition, let Er = (Ex, Ey , 0)T^ be the spatial brightness gradient, and Et the time derivative of brightness.
- (a) Show that in the case of rigid body motion, the constraint equation relating spatial and time derivatives of brightness can be written in the form
Et − (s × r) · ω +
s · t R · ˆz
where, s = (Er × ˆz) × r. Hint: Use the projection equation, r = R/(R · ˆz), to show that the time derivatives of r and R are related by
rt =
R · ˆz
(ˆz × (Rt × r)). (6)
Then use the rigid body motion equation, Rt = −ω×R−t, and the assumption of constant brightness, that is Er ·rt +Et =
- (b) Find the form of the constraint equation in the special case of a planar surface, R · n + 1 = 0.
- (c) In the case of a general surface, solve the constraint equation for the depth Z(x, y) = R · ˆz. Given any set of motion parameters t and ω, a surface Z(x, y) can be calculated that satisfies the constraint equation. The values of depth found may not be positive, however, if the parameters do not correspond to those of the motion generating the time-varying imagery. The problem is therefore to find parameters that make the depth calculated from the constraint equation positive everywhere in the image. In practice, because of noise in the brightness measurements, we expect there to be places where the computed depth is negative even with the correct motion parameters. We therefore turn this into an optimization problem.
- (d) Find the set of motion parameters t and ω that maximize ∫ ∫ sgn(s · t) sgn(Et − (s × r) · ω) dx dy (7)
where
sgn(Z) =
+1, for Z > 0 0 , for Z = 0 − 1 , for Z < 0
Warning: This is a research project, we do not know the answer (yet).
