CS 549 -Computer Vision-HW #12 Solved

Description

 

 

 

1. Stereo Motion (7 pts): In stereo imaging we can compute a point’s world coordinates from left and right images as

𝐵𝐵⃗2

𝑋𝑋⃗𝑊𝑊 = 𝑋𝑋⃗𝐴𝐴𝐴𝐴𝐴𝐴

𝐵𝐵⃗ ∙ ∆⃗

1. Show that if the world point is in motion, we can compute its velocity as

𝐵𝐵⃗2             𝐵𝐵⃗

𝑉𝑉⃗^𝑊𝑊 = 𝑉𝑉⃗_{𝐴𝐴𝐴𝐴𝐴𝐴}  − 𝑋𝑋⃗_𝑊𝑊 

𝐵𝐵⃗ ∙ ∆⃗       𝐵𝐵⃗ ∙ ∆⃗

 

1. Suppose that a moving world point is imaged as

𝑐𝑐               −𝑐𝑐

𝑋𝑋⃗𝐿𝐿𝐼𝐼 = 20𝑑𝑑 , 𝑋𝑋⃗𝑅𝑅𝐼𝐼 = 20𝑑𝑑

𝑓𝑓               𝑓𝑓

With the usual imaging geometry of 𝐵𝐵⃗ = [𝑏𝑏  0  0]^𝑇𝑇, 𝐹𝐹⃗ = [0  0 𝑓𝑓]^𝑇𝑇, what is 𝑉𝑉⃗^𝑊𝑊?

Express 𝑉𝑉⃗^𝑊𝑊 in the simplest terms.

 

 

2. Shading (6 pts): Consider the two surfaces

𝑧𝑧₁ = (𝑥𝑥² + 𝑦𝑦²) and 𝑧𝑧₂ = 2𝑥𝑥𝑦𝑦

1. Find 𝑝𝑝(𝑥𝑥, 𝑦𝑦) and 𝑞𝑞(𝑥𝑥, 𝑦𝑦) for both surfaces.
2. Show that 𝑧𝑧₁ and 𝑧𝑧₂ give rise to the same shading when a rotationally symmetric reflectance map applies, that is, when 𝑅𝑅(𝑝𝑝, 𝑞𝑞) = 𝑅𝑅(𝑝𝑝² + 𝑞𝑞²).

 

 

3. SLAM: Following Bailey & Durrant-Whyte Part II, at time k the vehicle pose 𝐱𝐱_{𝑣𝑣𝑘𝑘}and landmark locations m can be combined into a single state 𝐱𝐱_𝑘𝑘.

𝐱𝐱𝑣𝑣𝑘𝑘

𝐱𝐱𝑘𝑘 =  𝐦𝐦

The covariance matrix of 𝐱𝐱_𝑘𝑘, denoted by 𝐏𝐏_{𝑘𝑘|𝑘𝑘}, is partitioned into

𝐏𝐏𝑘𝑘|𝑘𝑘 = 𝐏𝐏𝐏𝐏_{𝑣𝑣𝑣𝑣}𝑇𝑇𝑣𝑣𝑣𝑣 𝐏𝐏𝐏𝐏_{𝑣𝑣𝑣𝑣}𝑣𝑣𝑣𝑣 _{𝑘𝑘|𝑘𝑘}

Control input 𝐮𝐮_𝑘𝑘 only affects the vehicle pose, not the landmark locations, via 𝐱𝐱_{𝑣𝑣𝑘𝑘} =

𝐟𝐟_𝒗𝒗(𝐱𝐱_{𝑣𝑣𝑘𝑘−1}, 𝐮𝐮_𝑘𝑘). Thus, the update equation for state 𝐱𝐱_𝑘𝑘 is

 

𝐱𝐱𝑘𝑘 = 𝐟𝐟(𝐱𝐱𝑘𝑘−1, 𝐮𝐮𝑘𝑘) = 𝐟𝐟𝒗𝒗(𝐱𝐱𝑣𝑣𝑘𝑘−1, 𝐮𝐮𝑘𝑘)

𝐦𝐦

When control 𝐮𝐮_𝑘𝑘 is applied, but before any landmark measurement updates are made, the covariance prediction 𝐏𝐏_{𝑘𝑘|𝑘𝑘−1} is given by

𝐏𝐏_𝑘𝑘|𝑘𝑘−1 = ∇𝐟𝐟_𝐱𝐱𝐏𝐏_𝑘𝑘−1|𝑘𝑘−1∇𝐟𝐟_𝐱𝐱^𝑇𝑇 + ∇𝐟𝐟_𝐮𝐮𝐔𝐔_𝑘𝑘∇𝐟𝐟_𝐮𝐮^𝑇𝑇

𝝏𝝏𝐟𝐟                 𝝏𝝏𝐟𝐟 where ∇𝐟𝐟_𝐱𝐱 = , ∇𝐟𝐟_𝐮𝐮 = , and 𝐔𝐔_𝑘𝑘 is the control convariance. Show that the covariance

 

𝝏𝝏𝐱𝐱𝑘𝑘−1            𝝏𝝏𝐮𝐮𝑘𝑘

∇𝐟𝐟𝒗𝒗𝐱𝐱𝐏𝐏𝑣𝑣𝑣𝑣,𝑘𝑘−1|𝑘𝑘−1∇𝐟𝐟𝒗𝒗𝑇𝑇𝐱𝐱 + ∇𝐟𝐟𝒗𝒗𝐮𝐮𝐔𝐔𝑘𝑘∇𝐟𝐟𝒗𝒗𝑇𝑇𝐮𝐮 𝐏𝐏 𝐏𝐏𝑘𝑘|𝑘𝑘−1 =         𝑣𝑣𝑣𝑣𝑇𝑇 ,𝑘𝑘−1|𝑘𝑘−1∇𝐟𝐟𝒗𝒗𝑇𝑇𝐱𝐱 𝝏𝝏𝐟𝐟 where ∇𝐟𝐟𝒗𝒗𝐱𝐱 = 𝝏𝝏𝐱𝐱𝑣𝑣𝒗𝒗 and ∇𝐟𝐟𝒗𝒗𝐮𝐮 = 𝝏𝝏𝐮𝐮𝝏𝝏𝐟𝐟𝒗𝒗𝑘𝑘.

∇𝐟𝐟𝒗𝒗𝐱𝐱𝐏𝐏𝑣𝑣𝑣𝑣,𝑘𝑘−1|𝑘𝑘−1   𝐏𝐏𝑣𝑣𝑣𝑣,𝑘𝑘−1|𝑘𝑘−1

prediction simplifies (!) to

𝑘𝑘−1

 

Hint: Expand  ∇𝐟𝐟_𝐱𝐱 and ∇𝐟𝐟_𝐮𝐮. This problem is not as hard as it looks once you understand what is being asked.