CSE 559A: Computer Vision

Fall 2018: T-R: 11:30-1pm @ Lopata 101

Instructor: Ayan Chakrabarti (ayan@wustl.edu).
Course Staff: Zhihao Xia, Charlie Wu, Han Liu

Oct 9, 2018

# General

• Problem Set 2 Due 11:59pm tonight.
• My office hours switched to by appointment
• Make a private post on Piazza to setup a time
• Let me know a list of times that work for you over the next two days
• Don't wait till the day before a deadline
• Problem Set 3 out and ready to clone
• Due two weeks from Thursday.

Szeliski Section 2.1

Szeliski Chapter 6

• Other means of camera calibration
• Minimizing other error metrics
• Lens Distortions and Dealing with them

# Two View Geometry

$p_l^T F p_r = 0$

• $$p_l, p_r$$ are 2D homogenous co-ordinates of left and right points.
• Co-ordinates in "image space". $$F$$ is called the fundamental matrix.
• So given a specific point $$p_r$$, says $$p_l^T (F p_r) = 0$$.
• This is the equation of a line !
• Same for the other way round.
• Has rank 2. Why ?
• Vector $$p$$ such that $$Fp = [0,0,0]^T$$.
• Means that this vector $$p$$ will satisfy $$p_l^T F p$$ for every $$p_l$$.
• $$p$$ is the homogeneous co-ordinate for the epipole in the right image.

# Two View Geometry

• Fundamental matrix has seven free parameters.
• One free parameter from scale.

# Two View Geometry

• Fundamental matrix has seven free parameters.
• One free parameter from scale.
• Require that $$\text{det}(F)=0$$
• Estimate using correspondences.
• (see "eight point algorithm" in Szeliski 7.2 / Wikipedia)
• If both cameras are calibrated, then only five unknowns
• Three for rotation
• Only two for translation !
• Only direction of translation matters. Epipolar lines stay the same irrespective of magnitude (how far you move the second camera in the same direction).

# Global Optimization

• Going back, we want to express the fact that our disparity map is smooth.
• Cost volume filtering is an ad-hoc way of doing that.
• Still making independent decisions at each pixel.
• Averaging each disparity level promotes disparity maps where values are "equal" not close.
• If $$C[x,y,d]$$ is a good match, then $$C[x+1,y,d\pm 1]$$ gets no benefit from filtering.
• Not good for slanted surfaces.
• Could be fixed by smoothing

$\min_{\delta = \{-1,0,1\}} C[x,y,d+\delta]$

• But generally, would prefer expressing this as optimizing a well-defined cost.