Midterm Review

Midterm Review

Jonathan Krause Vignesh Ramanathan

Zixuan Wang Kevin Wong Jinchao Ye

2012.10.26

Midterm Review

Outline

• Face Recognition

• Filters and Line Fitting

• Segmentation and Clustering

• Pinhole Geometry and Camera Models

• Epipolar Geometry and Stereo

• Q&A

2012.10.26

Midterm Review

AdaBoost for Face Detection

Boosting

Defines a classifier using an additive model

For each round of boosting:

Evaluate each rectangle filter on each example

Select best filter/threshold combination

Reweight examples

ℎ 𝑥 = 𝑎1ℎ1 𝑥 + 𝑎2ℎ2 𝑥 + 𝑎3ℎ3 𝑥 + …

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Midterm Review

AdaBoost

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Midterm Review

AdaBoost

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Midterm Review

AdaBoost

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Midterm Review

AdaBoost

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Midterm Review

AdaBoost: Algorithm

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Midterm Review

Eigenfaces and Fisherfaces • Representation of faces in the face space

– The eigenfaces v1, …, vK span the face space

Lect ure 2 - !

!

!

Fei-Fei Li!

Projec<ng'onto'the'Eigenfaces'

• The'eigenfaces'v1,'...,'vK'span'the'space'of'faces''

– A'face'is'converted'to'eigenface'coordinates'by'

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Midterm Review

Eigenfaces and Fisherfaces

• Faces should be aligned.

• Not robust on the illumination change.

• Not robust on the deformable object recognition.

• Difference between two: – Eigenfaces use Principle Component Analysis

(PCA)

– Fisherfaces use Linear Discriminant Analysis (LDA)

2012.10.26

Midterm Review

Outline

• Face Recognition

• Filters and Line Fitting

• Segmentation and Clustering

• Pinhole Geometry and Camera Models

• Epipolar Geometry and Stereo

• Q&A

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Midterm Review

RANSAC for Model Fitting

RANSAC loop:

1. Randomly select a minimum number of points for model fitting

2. Compute a model from these points

3. Find inliers to this transformation

4. If the number of inliers is sufficiently large, re-compute least-squares estimate of model on all of the inliers

• Keep the model with the largest number of inliers

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Midterm Review

RANSAC Pros/Cons

Pros:

• General method suited for a wide range of model fitting problems

• Easy to implement and easy to calculate its failure rate

Cons:

• 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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Midterm Review

Hough Transform

• A voting technique that can be used for model fitting problems

• Main idea:

1. Record all possible models on which all given points belong to.

2. Look for models that get many votes.

• Line fitting carried out in (d, θ) space

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Midterm Review

Hough Transform Pros/Cons Pros • 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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Midterm Review

Linear Filter

• Linear filtering: – Form a new image whose pixels are a weighted sum of

original pixel values

• 1D linear filter and 2D linear filters

• Convolution – Linear Shift Invariant system

– Gaussian filter

– DFT can be used to perform fast convolution

• Cross-correlation

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Midterm Review

Canny Edge detector

1. Filter image with derivative of Gaussian

2. Find magnitude and orientation of gradient

3. Non‐maximum suppression: – Thin multi-pixel wide “ridges” down to single

pixel width

4. Linking and thresholding (hysteresis): – Define two thresholds: low and high

– Use the high threshold to start edge curves and the low threshold to continue them

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Midterm Review

Outline

• Face Recognition

• Filters and Line Fitting

• Segmentation and Clustering

• Pinhole Geometry and Camera Models

• Epipolar Geometry and Stereo

• Q&A

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Midterm Review

Segmentation

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Techniques

• K-Means

• Mixture of Gaussians

• Mean-Shift

• Graph Cut (eigenvalues)

• Min Cut

• Min Normalized Cut

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Midterm Review

Mean-Shift

For each point in the image

1. Set window center to be point

2. Get mean of points in window

3. Set window center to mean

4. If not converged, goto 2.

Speed-ups exist

e.g. basin of attraction

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Midterm Review

Mean-Shift

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• “Model-Free”

– Don't need to specify number of clusters

– Free parameter: size of window

Midterm Review

Normalized Cut

• Problem with min cut:

• Solution:

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Normalized Cut

• Reduces to generalized eigenvalue problem

• Optimal solution is second smallest eigenvector of a particular matrix

• Need to discretize

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Example questions

• Which methods have explicit parameters for the number of clusters?

• How do we increase/decrease the number of clusters of each?

• Which are subject to local minima?

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Example questions

• What types of features could be used?

• Which scale well with feature dimension?

• Other strengths/weaknesses of each

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Midterm Review

Outline

• Face Recognition

• Filters and Line Fitting

• Segmentation and Clustering

• Pinhole Geometry and Camera Models

• Epipolar Geometry and Stereo

• Q&A

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Midterm Review

Pinhole Geometry

28 10/19/2012

• 𝑥, 𝑦, 𝑧 → (𝑓𝑥

𝑧, 𝑓

𝑦

𝑧) non-linear

Midterm Review

Camera Model

• 𝑥, 𝑦, 𝑧 → (𝑓𝑥

𝑧, 𝑓

𝑦

𝑧) non-linear

•𝑓𝑥𝑓𝑦𝑧

=𝑓 00 𝑓

0 00 0

0 0 1 0

𝑥𝑦𝑧1

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Camera Model

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• 𝑃′ = 𝐾 𝑅 𝑇 𝑃𝑤

• 𝐾 =

𝛼 𝑠 𝑐𝑥

0 𝛽 𝑐𝑦

0 0 1

Midterm Review

Camera Calibration

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Midterm Review

Camera Calibration

• 𝑃1 = 3, 4, 6, 1 𝑇; p1 = 9.5, 11.3333 𝑇;

• 𝑃2 = 5, 6, 7, 1 𝑇; p2 = 10.0769, 11.8462 𝑇;

• 𝑃3 = −7, 8, 10, 1 𝑇; p3 = 5.75, 11.75 𝑇;

• 𝑃4 = −3, −9 3, 1 𝑇; p4 = 7.00, 5.2222 𝑇;

• 𝑃5 = −56, −17, 8, 1 𝑇; p5 = −11.9286, 3.3571 𝑇;

• 𝑃6 = −34, 37, 3, 1 𝑇; p6 = −10.2222, 30.7788 𝑇;

• Solved by SVD in matlab:

– 𝑀′ =5.56 0 7.79 63.44

0 5.56 8.9 75.680 0 1.11 6.68

• Actual camera matrix:

– 𝑀 =5 0 7 570 5 8 680 0 1 6

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zeroBlock = [0 0 0 0]'; A = [ P1' zeroBlock' -p1(1)*P1'; zeroBlock' P1' -p1(2)*P1'; P2' zeroBlock' -p2(1)*P2'; zeroBlock' P2' -p2(2)*P2'; P3' zeroBlock' -p3(1)*P3'; zeroBlock' P3' -p3(2)*P3'; P4' zeroBlock' -p4(1)*P4'; zeroBlock' P4' -p4(2)*P4'; P5' zeroBlock' -p5(1)*P5'; zeroBlock' P5' -p5(2)*P5'; P6' zeroBlock' -p6(1)*P6'; zeroBlock' P6' -p6(2)*P6'; ]; [U, D, V] = svd(A, 0); m = V(:, end); m = [m(1:4)'; m(5:8)'; m(9:12)'];

Midterm Review

Outline

• Face Recognition

• Filters and Line Fitting

• Segmentation and Clustering

• Pinhole Geometry and Camera Models

• Epipolar Geometry and Stereo

• Q&A

2012.10.26

Midterm Review

Epipolar Geometry

• Epipolar lines, Epipolar plane, Epipoles, Baseline • Epipolar constraint 0)pR(TpT

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Midterm Review

Essential and Fundamental Matrix

Skew symmetric matrix:

0pFpT

01

pKRTKp TT 0 pRTpT

RTE

baba ][

0pEpT

1

KRTKF T

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Midterm Review

A Simple “Trick”

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Midterm Review

Essential Matrix Example

From :http://www.filmschoolrejects.com/features/cinematic-listology-six-incredibly-awesome-uses-of-camera-rigs-dbell.php

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Midterm Review

Essential Matrix Example Continued

How would you find the essential matrices between pairs of evenly spaced cameras around a semi circle?

X

Y

Need R and T for each pair in world coordinates.

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Midterm Review

Reconstruction

• Simplified Depth Estimation Example

2012.10.26

Midterm Review

Outline

• Face Recognition

• Filters and Line Fitting

• Segmentation and Clustering

• Pinhole Geometry and Camera Models

• Epipolar Geometry and Stereo

• Q&A

2012.10.26

Midterm Review

Q&A

2012.10.26