6.1 Visualizing Quadratics
Transcript of 6.1 Visualizing Quadratics
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Visualizing Quadratics16-385 Computer Vision (Kris Kitani)
Carnegie Mellon University
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f(x, y) = x
2 + y
2
1 = x
2 + y
2
Equation of a circle
Equation of a ‘bowl’ (paraboloid)
If you slice the bowl atf(x, y) = 1
what do you get?
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f(x, y) = x
2 + y
2
1 = x
2 + y
2
Equation of a circle
Equation of a ‘bowl’ (paraboloid)
If you slice the bowl atf(x, y) = 1
what do you get?
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f(x, y) = x
2 + y
2
can be written in matrix form like this…
f(x, y) =⇥x y
⇤ 1 00 1
� x
y
�
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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
f(x, y) =⇥x y
⇤ 1 00 1
� x
y
�
‘sliced at 1’
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f(x, y) =⇥x y
⇤ 2 00 1
� x
y
�
What happens if you increase coefficient on x?
and slice at 1
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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
f(x, y) =⇥x y
⇤ 2 00 1
� x
y
�
What happens if you increase coefficient on x?
and slice at 1decrease width in x!
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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
f(x, y) =⇥x y
⇤ 2 00 1
� x
y
�
What happens if you increase coefficient on x?
and slice at 1decrease width in x!
What happens to the gradient in x?
increases gradient in x‘thins the bowl in x’
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f(x, y) =⇥x y
⇤ 1 00 2
� x
y
�
What happens if you increase coefficient on y?
and slice at 1
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f(x, y) =⇥x y
⇤ 1 00 2
� x
y
�
What happens if you increase coefficient on y?
and slice at 1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
decrease width in y
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f(x, y) =⇥x y
⇤ 1 00 2
� x
y
�
What happens if you increase coefficient on y?
and slice at 1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
decrease width in y
What happens to the gradient in y?
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f(x, y) =⇥x y
⇤ 1 00 2
� x
y
�
What happens if you increase coefficient on y?
and slice at 1
-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5
-2
-1.5
-1
-0.5
0.5
1
1.5
2
decrease width in y
What happens to the gradient in y?
increases gradient in y‘thins the bowl in y’
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f(x, y) = x
2 + y
2
can be written in matrix form like this…
f(x, y) =⇥x y
⇤ 1 00 1
� x
y
�
What’s the shape? What are the eigenvectors? What are the eigenvalues?
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f(x, y) = x
2 + y
2
can be written in matrix form like this…
f(x, y) =⇥x y
⇤ 1 00 1
� x
y
�
1 00 1
�=
1 00 1
� 1 00 1
� 1 00 1
�>eigenvalues
along diagonaleigenvectors
Result of Singular Value Decomposition (SVD)
axis of the ‘ellipse slice’
gradient of the quadratic along
the axis
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T
!"
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1001
1001
1001
1001
A
EigenvaluesEigenvectors
Eigenvectors
Eigenvector
Eige
nvec
tor
x yx
y
*not the size of the axis
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f(x, y) =⇥x y
⇤ 1 00 1
� x
y
�
you can smash this bowl in the y direction
f(x, y) =⇥x y
⇤ 1 00 4
� x
y
�
you can smash this bowl in the x direction
f(x, y) =⇥x y
⇤ 4 00 1
� x
y
�
Recall:
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T
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&=!
"
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1001
1004
1001
1004
AEigenvalues
Eigenvectors Eigenvectors
Eigenvector
Eige
nvec
tor
x yx
y
*not the size of the axis (inverse relation)
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50.087.087.050.0
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50.087.087.050.0
75.130.130.125.3
A
Eigenvalues
Eigenvectors Eigenvectors
Eige
nvec
tor
Eigenvector
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Eigenvalues
Eigenvectors Eigenvectors
Eige
nvec
tor
Eigenvector
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Error function (for Harris corners)
The surface E(u,v) is locally approximated by a quadratic form
We will need this to understand…
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Conic section of Error function
Since M is symmetric, we have
We can visualize M as an ellipse with axis lengths determined by the eigenvalues and orientation determined by R
direction of the slowest change (smaller gradient)
direction of the fastest change (larger gradient)
(λmax)-1/2(λmin)-1/2
Ellipse equation:⇥u v
⇤M
uv
�= 1
‘isocontour’
but smaller axis on ‘slice’
but larger axis on ‘slice’