corner detection - carleton...
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Corner Detection
COMP 4900D
Winter 2006
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Motivation: Features for Recognition
Image search: find the book in an image.
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Motivation: Build a Panorama
M. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003
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How do we build panorama?
We need to match (align) images
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Matching with Features
•Detect feature points in both images
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Matching with Features
•Detect feature points in both images
•Find corresponding pairs
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Matching with Features
•Detect feature points in both images
•Find corresponding pairs
•Use these pairs to align images
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More motivation…
Feature points are used also for:• Image alignment (homography, fundamental matrix)
• 3D reconstruction
• Motion tracking
• Object recognition
• Indexing and database retrieval
• Robot navigation
• … other
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Corner Feature
Corners are image locations that have large intensity changes
in more than one directions.
Shifting a window in any direction should give a large change in intensity
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Examples of Corner Features
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Harris Detector: Basic Idea
“flat” region:
no change in
all directions
“edge”:
no change along
the edge direction
“corner”:
significant change
in all directions
C.Harris, M.Stephens. “A Combined Corner and Edge Detector”. 1988
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Change of Intensity
The intensity change along some direction can be quantified
by sum-of-squared-difference (SSD).
( )∑ −++=ji
jiIvjuiIvuD,
2),(),(),(
v
u
),( jiI
),( vjuiI ++
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Change Approximation
If u and v are small, by Taylor theorem:
vIuIjiIvjuiI yx ++≈++ ),(),(
wherey
IIand
x
II yx
∂
∂=
∂
∂=
therefore
( ) ( )
( )
[ ]
=
++=
+=
−++=−++
v
u
III
IIIvu
vIuvIIuI
vIuI
jiIvIuIjiIjiIvjuiI
yyx
yxx
yyxx
yx
yx
2
2
2222
2
22
2
),(),(),(),(
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Gradient Variation Matrix
[ ]
=
∑∑∑∑
v
u
III
IIIvuvuD
yyx
yxx
2
2
),(
This is a function of ellipse.
=
∑∑∑∑
2
2
yyx
yxx
III
IIIC
Matrix C characterizes how intensity changes
in a certain direction.
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Eigenvalue Analysis – simple case
=
=
∑∑∑∑
2
1
2
2
0
0
λ
λ
yyx
yxx
III
IIIC
First, consider case where:
This means dominant gradient directions align with x or y axis
If either λ is close to 0, then this is not a corner, so look for
locations where both are large.
Slide credit: David Jacobs
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General Case
It can be shown that since C is symmetric:
QQCT
=
2
1
0
0
λ
λ
So every case is like a rotated version of the one on last
slide.
(λmax)-1/2
(λmin)-1/2
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Gradient Orientation
Closeup
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Corner Detection Summary
• if the area is a region of constant intensity, both eigenvalues will be very
small.
• if it contains an edge, there will be one large and one small eigenvalue (the
eigenvector associated with the large eigenvalue will be parallel to the
image gradient).
• if it contains edges at two or more orientations (i.e., a corner), there will be
two large eigenvalues (the eigenvectors will be parallel to the image
gradients).
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Corner Detection Algorithm