Visualizing Quadratics16-385 Computer Vision (Kris Kitani)

Carnegie Mellon University

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?

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?

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

�

-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’

f(x, y) =⇥x y

⇤ 2 00 1

� x

y

�

What happens if you increase coefficient on x?

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

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!

-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’

f(x, y) =⇥x y

⇤ 1 00 2

� x

y

�

What happens if you increase coefficient on y?

and slice at 1

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

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?

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’

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?

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

T

!"

#$%

&!"

#$%

&!"

#$%

&=!

"

#$%

&=

1001

1001

1001

1001

A

EigenvaluesEigenvectors

Eigenvectors

Eigenvector

Eige

nvec

tor

x yx

y

*not the size of the axis

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:

T

!"

#$%

&!"

#$%

&!"

#$%

&=!

"

#$%

&=

1001

1004

1001

1004

AEigenvalues

Eigenvectors Eigenvectors

Eigenvector

Eige

nvec

tor

x yx

y

*not the size of the axis (inverse relation)

T

!"

#$%

&

−−

−!"

#$%

&!"

#$%

&

−−

−=!

"

#$%

&=

50.087.087.050.0

4001

50.087.087.050.0

75.130.130.125.3

A

Eigenvalues

Eigenvectors Eigenvectors

Eige

nvec

tor

Eigenvector

T

!"

#$%

&

−−

−!"

#$%

&!"

#$%

&

−−

−=!

"

#$%

&=

50.087.087.050.0

10001

50.087.087.050.0

25.390.390.375.7

A

Eigenvalues

Eigenvectors Eigenvectors

Eige

nvec

tor

Eigenvector

Error function (for Harris corners)

The surface E(u,v) is locally approximated by a quadratic form

We will need this to understand…

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’