6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and...
Transcript of 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and...
![Page 1: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/1.jpg)
Harris Corners16-385 Computer Vision (Kris Kitani)
Carnegie Mellon University
![Page 2: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/2.jpg)
How do you find a corner?
Easily recognized by looking through a small window
Shifting the window should give large change in intensity
[Moravec 1980]
![Page 3: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/3.jpg)
“edge”:no change along the edge
direction
“corner”:significant change in all
directions
“flat” region:no change in all
directions
Easily recognized by looking through a small window
Shifting the window should give large change in intensity
[Moravec 1980]
![Page 4: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/4.jpg)
Design a program to detect corners (hint: use image gradients)
![Page 5: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/5.jpg)
Finding corners (a.k.a. PCA)
1.Compute image gradients over small region
2.Subtract mean from each image gradient
3.Compute the covariance matrix
4.Compute eigenvectors and eigenvalues
5.Use threshold on eigenvalues to detect corners
I
x
=@I
@x
Iy =@I
@y
2
4
Pp2P
Ix
Ix
Pp2P
Ix
Iy
Pp2P
Iy
Ix
Pp2P
Iy
Iy
3
5
uv
�= �
2
4
Pp2P
Ix
It
Pp2P
Iy
It
3
5
![Page 6: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/6.jpg)
1. Compute image gradients over a small region (not just a single pixel)
![Page 7: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/7.jpg)
1. Compute image gradients over a small region (not just a single pixel)
I
x
=@I
@x
Iy =@I
@y
array of x gradients
array of y gradients
![Page 8: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/8.jpg)
visualization of gradients
image
X derivative
Y derivative
![Page 9: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/9.jpg)
I
x
=@I
@x
I
x
=@I
@x
I
x
=@I
@x
Iy =@I
@yIy =
@I
@yIy =
@I
@y
What does the distribution tell you about the region?
![Page 10: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/10.jpg)
I
x
=@I
@x
I
x
=@I
@x
I
x
=@I
@x
Iy =@I
@yIy =
@I
@yIy =
@I
@y
distribution reveals edge orientation and magnitude
![Page 11: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/11.jpg)
I
x
=@I
@x
I
x
=@I
@x
I
x
=@I
@x
Iy =@I
@yIy =
@I
@yIy =
@I
@y
How do you quantify orientation and magnitude?
![Page 12: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/12.jpg)
2. Subtract the mean from each image gradient
![Page 13: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/13.jpg)
2. Subtract the mean from each image gradient
plot intensities
constant intensity gradient
intensities along the line
![Page 14: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/14.jpg)
2. Subtract the mean from each image gradient
plot intensities
constant intensity gradient
I
x
=@I
@x
Iy =@I
@y
intensities along the line
plot of image gradients
subtract mean
![Page 15: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/15.jpg)
2. Subtract the mean from each image gradient
plot intensities
constant intensity gradient
I
x
=@I
@x
Iy =@I
@y
intensities along the line
plot of image gradients
I
x
=@I
@x
Iy =@I
@y
data is centered (‘DC’ offset is removed)
subtract mean
![Page 16: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/16.jpg)
3. Compute the covariance matrix
![Page 17: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/17.jpg)
3. Compute the covariance matrix
Where does this covariance matrix come from?
2
4
Pp2P
Ix
Ix
Pp2P
Ix
Iy
Pp2P
Iy
Ix
Pp2P
Iy
Iy
3
5
uv
�= �
2
4
Pp2P
Ix
It
Pp2P
Iy
It
3
5
I
x
=@I
@x
Iy =@I
@y
array of x gradients array of y gradients
*.=sum( )2
4
Pp2P
Ix
Ix
Pp2P
Ix
Iy
Pp2P
Iy
Ix
Pp2P
Iy
Iy
3
5
uv
�= �
2
4
Pp2P
Ix
It
Pp2P
Iy
It
3
5
![Page 18: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/18.jpg)
Error functionChange of intensity for the shift [u,v]:
IntensityShifted intensity
Window function
orWindow function w(x,y) =
Gaussian1 in window, 0 outside
Error function
Some mathematical background…
![Page 19: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/19.jpg)
Error function approximationChange of intensity for the shift [u,v]:
Second-order Taylor expansion of E(u,v) about (0,0) (bilinear approximation for small shifts):
first derivative second derivative
![Page 20: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/20.jpg)
Bilinear approximationFor small shifts [u,v] we have a ‘bilinear approximation’:
where M is a 2×2 matrix computed from image derivatives:
Change in appearance for a
shift [u,v]
M
‘second moment’ matrix ‘structure tensor’
![Page 21: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/21.jpg)
2
4
Pp2P
Ix
Ix
Pp2P
Ix
Iy
Pp2P
Iy
Ix
Pp2P
Iy
Iy
3
5
uv
�= �
2
4
Pp2P
Ix
It
Pp2P
Iy
It
3
5
By computing the gradient covariance matrix…
we are fitting a quadratic to the gradients over a small image region
![Page 22: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/22.jpg)
Visualization of a quadraticThe surface E(u,v) is locally approximated by a quadratic form
![Page 23: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/23.jpg)
Which error surface indicates a good image feature?
What kind of image patch do these surfaces represent?
![Page 24: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/24.jpg)
flat edge ‘line’
corner ‘dot’
![Page 25: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/25.jpg)
4. Compute eigenvalues and eigenvectors
![Page 26: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/26.jpg)
eig(M)
![Page 27: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/27.jpg)
4. Compute eigenvalues and eigenvectors
eigenvector
eigenvalue
Me = �e (M � �I)e = 0
![Page 28: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/28.jpg)
1. Compute the determinant of (returns a polynomial)
eigenvector
eigenvalue
M � �I
Me = �e (M � �I)e = 0
4. Compute eigenvalues and eigenvectors
![Page 29: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/29.jpg)
1. Compute the determinant of (returns a polynomial)
eigenvector
eigenvalue
2. Find the roots of polynomial (returns eigenvalues)
det(M � �I) = 0
M � �I
Me = �e (M � �I)e = 0
4. Compute eigenvalues and eigenvectors
![Page 30: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/30.jpg)
1. Compute the determinant of (returns a polynomial)
eigenvector
eigenvalue
2. Find the roots of polynomial (returns eigenvalues)
det(M � �I) = 0
M � �I
Me = �e (M � �I)e = 0
3. For each eigenvalue, solve (returns eigenvectors)
(M � �I)e = 0
4. Compute eigenvalues and eigenvectors
![Page 31: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/31.jpg)
Visualization as an ellipseSince 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
direction of the fastest change
(λmax)-1/2
(λmin)-1/2
Ellipse equation:
![Page 32: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/32.jpg)
interpreting eigenvalues
λ1
λ2
λ2 >> λ1
λ1 >> λ2�1 ⇠ 0
�2 ⇠ 0
What kind of image patch does each region represent?
![Page 33: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/33.jpg)
interpreting eigenvalues‘horizontal’ edge
‘vertical’ edge
flat
corner
λ1
λ2
λ2 >> λ1
λ1 >> λ2
λ1 ~ λ2
![Page 34: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/34.jpg)
interpreting eigenvalues‘horizontal’ edge
‘vertical’ edge
flat
corner
λ1
λ2
λ2 >> λ1
λ1 >> λ2
λ1 ~ λ2
![Page 35: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/35.jpg)
5. Use threshold on eigenvalues to detect corners
![Page 36: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/36.jpg)
flat
λ1
λ2
5. Use threshold on eigenvalues to detect corners
Think of a function to score ‘cornerness’
![Page 37: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/37.jpg)
flat
λ1
λ2
5. Use threshold on eigenvalues to detect corners
Think of a function to score ‘cornerness’
strong corner
![Page 38: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/38.jpg)
flat
corner
λ1
λ2
5. Use threshold on eigenvalues to detect corners^
R = min(�1,�2)
Use the smallest eigenvalue as the response function
![Page 39: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/39.jpg)
flat
corner
λ1
λ2
5. Use threshold on eigenvalues to detect corners^
R = �1�2 � (�1 + �2)2
Eigenvalues need to be bigger than one.
Can compute this more efficiently…
![Page 40: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/40.jpg)
flat
corner
λ1
λ2
R < 0 R > 0
R < 0R ⌧ 0
R = det(M)� �trace2(M)
5. Use threshold on eigenvalues to detect corners^
![Page 41: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/41.jpg)
R = det(M)� �trace2(M)
R =det(M)
trace(M) + �
Harris & Stephens (1988)
Kanade & Tomasi (1994)
Nobel (1998)
R = min(�1,�2)
![Page 42: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/42.jpg)
1. Compute x and y derivatives of image
2. Compute products of derivatives at every pixel
3. Compute the sums of the products of derivatives at each pixel
Harris DetectorC.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988.
IGI xx ∗= σ IGI y
y ∗= σ
xxxIII ⋅=2 yyy III ⋅=2 yxxy III ⋅=
22 ' xxIGS ∗= σ 22 ' yy IGS ∗= σ xyxy IGS ∗= 'σ
![Page 43: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/43.jpg)
Harris DetectorC.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988.
4. Define the matrix at each pixel
5. Compute the response of the detector at each pixel
6. Threshold on value of R; compute non-max suppression.
!!"
#
$$%
&= ),(),(
),(),(),(
2
2
yxSyxSyxSyxS
yxMyxy
xyx
( )2tracedet MkMR −=
![Page 44: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/44.jpg)
![Page 45: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/45.jpg)
Corner response
![Page 46: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/46.jpg)
![Page 47: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/47.jpg)
Thresholded corner response
![Page 48: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/48.jpg)
Non-maximal suppression
![Page 49: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/49.jpg)
![Page 50: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/50.jpg)
Harris corner response is rotation invariant
Ellipse rotates but its shape (eigenvalues) remains the same
Corner response R is invariant to image rotation
![Page 51: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/51.jpg)
intensity changesPartial invariance to affine intensity changeü Only derivatives are used => invariance to intensity shift I → I + b
ü Intensity scale: I → a I
R
x (image coordinate)
threshold
R
x (image coordinate)
![Page 52: 6.2 Harris Corner Detector16385/s17/Slides/6.2_Harris_Corner_Detector.pdf · C.Harris and M.Stephens. “A Combined Corner and Edge Detector.”1988. 4. Define the matrix at each](https://reader034.fdocuments.in/reader034/viewer/2022043000/5f774d42aff1db611d7f0238/html5/thumbnails/52.jpg)
The Harris detector not invariant to changes in …