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

http://www.cse.wustl.edu/~ayan/courses/cse559a/

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.

Further Reading (optional)

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

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

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).

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Two View Geometry

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Rectified Binocular Stereo

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

Cost Volumes

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.