CS549-Homework 8 Solved

Smallest Optical Flow (4 pts): What velocity 𝑉⃗ 𝑀𝐼𝑁 that satisfies the Optical Flow
Constraint Equation 𝐼𝑥𝑢+𝐼𝑦𝑣+𝐼𝑡 = 0 has the smallest magnitude |𝑉⃗ |? Hint: This can be solved geometrically as was outlined in class by considering the OFCE in 𝑢,𝑣 space.

                                                                                                  𝐼𝑥𝑢 +𝐼𝑦𝑣+𝐼𝑡 = 0

 

Moving Gaussian Blob (6 pts): A Gaussian blob is observed over time to have brightness
 2 𝑥 𝑦 𝑥 𝑦 2 𝐼(𝑥,𝑦,𝑡) = 𝑒−(𝑡 −2(𝑘1+𝑘2)𝑡+(𝑘1+𝑘2) )

 

What are 𝐼𝑥, 𝐼𝑦, and 𝐼𝑡? Hint: You should find that these derivatives have a simple form.
The Optical Flow Constraint Equation is 𝐼𝑥𝑢 +𝐼𝑦𝑣+𝐼𝑡 = 0. Write this out using the results of Part a. and simplify it as much as possible. For example, you should be able to cancel terms that occur in each of 𝐼𝑥, 𝐼𝑦, and 𝐼𝑡.
 

Quadratic Optical Flow (8 pts): Suppose the image brightness is given by
𝐼(𝑥,𝑦,𝑡) = 𝐼0 +[(𝑥−𝑐1𝑡)2 +(𝑦−𝑐2𝑡)2]

What are Ix, Iy, and It? Hint: You should find that these derivatives have a simple form.
 

Express the Optical Flow Constraint Equation 𝐼𝑥𝑢 +𝐼𝑦𝑣+𝐼𝑡 = 0 in the simplest terms possible for this image sequence.
 

The equation from b. must hold for all x, y, and t. Find a constant solution for u and v that makes this true, that is, such that u and v do not depend on x, y, and t.
 

Iterative Optical Flow(8 pts): We saw in class an iterative method for computing optical flow, where at each iteration, the optical flow 𝑢 is updated according to
                              [             ]new = [𝜆𝐼𝑥2 +4                 𝜆𝐼𝑥𝐼𝑦 ]neighbors(𝑥,𝑦)𝑢old(𝑛)−𝜆𝐼𝑥𝐼𝑡   

𝑢(𝑥,𝑦)

                                 𝑣(𝑥,𝑦)                𝜆𝐼𝑥𝐼𝑦              𝜆𝐼                       𝑣old(𝑛)−𝜆𝐼𝑦𝐼𝑡

                                                                                                                [      𝑛neighbors(𝑥,𝑦)                                                   ]

Show that this is equivalent to
                 [𝑢 new                                         𝜆𝐼𝑥𝐼𝑦                         neighbors(𝑥,𝑦)𝑢old(𝑛)−𝜆𝐼𝑥𝐼𝑡   

]

                  𝑣                                                         𝜆𝐼𝑥𝐼𝑦            𝜆𝐼                          𝑣old(𝑛)−𝜆𝐼𝑦𝐼𝑡

                                                                                                                                      [      𝑛neighbors(𝑥,𝑦)                                                   ]

Show that this is equivalent to update equations
                                                          new(𝑥,𝑦) = 𝑢̅old −          𝐼𝑥               4(𝐼𝑥𝑢̅old +𝐼𝑦𝑣̅ old +𝐼𝑡)

𝑢

                                                                                                  𝐼𝑥2 +𝐼𝑦2 +𝜆                                   

𝐼

                                                      𝑣new(𝑥,𝑦) = 𝑣̅ old − 𝑦 4(𝐼𝑥𝑢̅old +𝐼𝑦𝑣̅ old +𝐼𝑡)

𝐼𝑥2 +𝐼𝑦2 +𝜆

where 𝑢̅old,𝑣̅ old are the averages of the 4 neighbors of 𝑢(𝑥,𝑦),𝑣(𝑥,𝑦). Hint: You only need to show this for 𝑢new because 𝑣new follows an identical derivation.

In the case that 𝜆 = 0, what do the update equations reduce to?