cs231a midterm reviewmidterm logistics we will post the midterm on the course website at ~12:15pm...
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CS231A Midterm Review
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Overview
● General Logistics● Hough Transform and RANSAC examples and
overviews● Single View Metrology: Vanishing lines, points, etc.● Camera Matrix and geometry● Fundamental matrix and Affine SFM
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Midterm Logistics
● We will post the midterm on the course website at ~12:15pm Tuesday 2/25.
● Midterms must be submitted by Thursday 11am, either in class or in the Gates dropbox.
● SCPD can submit through the SCPD office.● Late submissions will get a 0%● Piazza will be frozen(no new posts),
clarifications can be requested via the staff mailing list.
● Open book, open notes, no collaboration.
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Midterm Structure
● 1-2 Implementation questions● 3-4 Theory questions● Coverage is up to and including PS3
material.● 6 hour estimated time.
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RANSAC + Hough Transform
RANSAC: A randomized iterative method to fit a parametric model to data based on a random sampling of dataHough Transform: A voting scheme to fit a parametric model to data by selecting the model with the most votes.
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Example Problem
Finding a circle using RANSAC:1. Decide how many points that we need2. Fit a model to the data3. Compute inliers4. Refine model based on all points
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Finding the circle from 3 points
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Distance to circle?
Outliers?
Figures: Brehar Raluca
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RANSAC Pros/ConsPros:•General method suited for a wide range of model fitting problems•Easy to implement and easy to calculate its failure rateCons:•Only handles a moderate percentage of outliers without cost blowing up•Many real problems have high rate of outliers (but sometimes selective choice of random subsets can help)
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Hough Transform to find a circle
We can start with a simplified scenario where the radius is known
Figures: Dr. Harvey Rhody, RIT
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Hough Transform: Circles
Each point maps to a circle in the parameter space spanned by candidate circle centers. What if there are multiple circles?
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Hough Transform: Circles
We can then vote on the most likely circle(s)What if we don't know the radius?● We have a 3d parameter
space, each point maps to a hollow cone.
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Hough Transform Pros/ConsPros•All points are processed independently, so can cope with occlusion•Some robustness to noise: noise points unlikely to contribute consistently to any single bin•Can detect multiple instances of a model in a single pass
Cons•Complexity of search time increases exponentially with the number of model parameters•Non-target shapes can produce spurious peaks in parameter space•Quantization: hard to pick a good grid size
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Single View Metrology
● Vanishing points● Vanishing lines
○ construction of lines from points○ directions and normals of vanishing points/planes
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Vanishing Points
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Vanishing Lines
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Vanishing Lines
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Projection
From pinhole camera
This is not linear
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Homogeneous Coordinates
Cartesian → Homogeneous
Homogeneous → Cartesian
camera scene
camera scene
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Projective Camera
intrinsic matrix extrinsic matrix
camera matrix
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Epipolar geometry
P: objectO: center of camerap: image pointe: epipole
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Fundamental Matrix F
F is rank 2, with 7 degree of freedomEpipolar lines: l1 = F p2
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Computation of F
corresponding points(u', v', 1), (u, v, 1):
8-point algorithm
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Normalized 8-point Algorithm
Normalize: qi = Tpi, qi’ = T’pi’8-point algorithm to solve Fq’ from
Force Fq’ to have rank 2De-normalize Fq to get F
→ SVD
→ SVD
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Structure From Motion
solve forknown
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Factorization
solve forknown
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solve forknown
(1)(2) SVD
(3) columns are scene points