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Feature extraction: Corners
9300 Harris Corners Pkwy, Charlotte, NC
Why extract features?• Motivation: panorama stitching
• We have two images – how do we combine them?
Why extract features?• Motivation: panorama stitching
• We have two images – how do we combine them?
Step 1: extract featuresStep 2: match featuresStep 2: match features
Why extract features?• Motivation: panorama stitching
• We have two images – how do we combine them?
Step 1: extract featuresStep 2: match featuresStep 2: match featuresStep 3: align images
Characteristics of good features
• Repeatability• The same feature can be found in several images despite geometric g p g
and photometric transformations
• Saliency• Each feature is distinctive
• Compactness and efficiency• Many fewer features than image pixels
• LocalityLocality• A feature occupies a relatively small area of the image; robust to
clutter and occlusion
Applications Feature points are used for:
• Image alignment • 3D reconstruction• Motion tracking
R b t i ti• Robot navigation• Indexing and database retrieval• Object recognition• Object recognition
Finding Corners
• Key property: in the region around a corner, i di h d iimage gradient has two or more dominant directionsC t bl d di ti ti• Corners are repeatable and distinctive
C H i d M St h "A C bi d C d Ed D t t “C.Harris and M.Stephens. "A Combined Corner and Edge Detector.“Proceedings of the 4th Alvey Vision Conference: pages 147--151.
Corner Detection: Basic Idea
• We should easily recognize the point by looking through a small windowShifti i d i di ti h ld• Shifting a window in any direction should give a large change in intensity
“edge”:no change
“corner”:significant
“flat” region:no change in no change
along the edge direction
significant change in all directions
no change in all directions
Source: A. Efros
Corner Detection: MathematicsChange in appearance of window w(x,y)
for the shift [u,v]:
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
I(x, y)E(u v)E(u, v)
E(3,2)
w(x, y)
Corner Detection: MathematicsChange in appearance of window w(x,y)
for the shift [u,v]:
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
I(x, y)E(u v)E(u, v)
E(0,0)
w(x, y)
Corner Detection: MathematicsChange in appearance of window w(x,y)
for the shift [u,v]:
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
IntensityShifted intensity
Window function intensityfunction
orWindow function w(x,y) =
G i1 i i d 0 id Gaussian1 in window, 0 outside
Source: R. Szeliski
Corner Detection: MathematicsChange in appearance of window w(x,y)
for the shift [u,v]:
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
We want to find out how this function behaves for small shiftssmall shifts
E(u, v)
Corner Detection: MathematicsChange in appearance of window w(x,y)
for the shift [u,v]:
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
We want to find out how this function behaves for small shifts
Local quadratic approximation of E(u,v) in the neighborhood of (0 0) is given by the second order
small shifts
neighborhood of (0,0) is given by the second-order Taylor expansion:
⎤⎡⎤⎡⎤⎡ uEEE )00()00(1)00(⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡+⎥
⎦
⎤⎢⎣
⎡+≈
vu
EEEE
vuEE
vuEvuEvvuv
uvuu
v
u
)0,0()0,0()0,0()0,0(
][21
)0,0()0,0(
][)0,0(),(
Corner Detection: Mathematics
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
Second-order Taylor expansion of E(u,v) about (0,0):
⎥⎤
⎢⎡⎥⎤
⎢⎡
+⎥⎤
⎢⎡
+≈uEE
vuE
vuEvuE uvuuu )0,0()0,0(][1)0,0(
][)0,0(),( ⎥⎦
⎢⎣⎥⎦
⎢⎣
+⎥⎦
⎢⎣
+≈vEE
vuE
vuEvuEvvuvv )0,0()0,0(
][2)0,0(
][)0,0(),(
[ ] ),(),(),(),(2),( vyuxIyxIvyuxIyxwvuE xu ++−++=∑
[ ]
),(),(),(2),(,
,
vyuxIvyuxIyxwvuE xxyx
uu
yx
++++=
∑
∑[ ]
),(),(),(2),(
),(),(),(),(2,
vyuxIvyuxIyxwvuE
vyuxIyxIvyuxIyxw
xyuv
xxyx
++++=
++−+++
∑
∑
[ ] ),(),(),(),(2,
,
vyuxIyxIvyuxIyxw xyyx
xyyx
uv
++−+++∑
∑
Corner Detection: Mathematics
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
Second-order Taylor expansion of E(u,v) about (0,0):
⎥⎤
⎢⎡⎥⎤
⎢⎡
+⎥⎤
⎢⎡
+≈uEE
vuE
vuEvuE uvuuu )0,0()0,0(][1)0,0(
][)0,0(),(
0)0,0(E =
⎥⎦
⎢⎣⎥⎦
⎢⎣
+⎥⎦
⎢⎣
+≈vEE
vuE
vuEvuEvvuvv )0,0()0,0(
][2)0,0(
][)0,0(),(
0)0,0(0)0,0(
EE
v
u
∑==
),(),(),(2)0,0(
),(),(),(2)0,0(,
yxIyxIyxwE
yxIyxIyxwE
yyvv
xxyx
uu
∑
∑=
=
),(),(),(2)0,0(,
,
yxIyxIyxwE yxyx
uv
yyyx
∑
∑=
Corner Detection: Mathematics
[ ]2
,( , ) ( , ) ( , ) ( , )
x yE u v w x y I x u y v I x y= + + −∑
Second-order Taylor expansion of E(u,v) about (0,0):
⎥⎤
⎢⎡⎥⎥⎤
⎢⎢⎡
≈∑∑ u
yxIyxIyxwyxIyxwvuvuE yx
yxyx
x,,
2 ),(),(),(),(),(][)( ⎥
⎦⎢⎣⎥⎥⎦⎢
⎢⎣
≈∑∑ vyxIyxwyxIyxIyxw
vuvuE
yxy
yxyx
yy
,
2
,
,,
),(),(),(),(),(][),(
0)0,0(E =
0)0,0(0)0,0(
EE
v
u
∑==
),(),(),(2)0,0(
),(),(),(2)0,0(,
yxIyxIyxwE
yxIyxIyxwE
yyvv
xxyx
uu
∑
∑=
=
),(),(),(2)0,0(,
,
yxIyxIyxwE yxyx
uv
yyyx
∑
∑=
Corner Detection: MathematicsThe quadratic approximation simplifies to
⎤⎡u⎥⎦
⎤⎢⎣
⎡≈
vu
MvuvuE ][),(
where M is a second moment matrix computed from image derivatives:
2
2( , ) x x yI I IM w x y
⎡ ⎤= ⎢ ⎥
⎢ ⎥∑
de at es
2,
( , )x y x y y
yI I I⎢ ⎥⎢ ⎥⎣ ⎦
∑
M
Interpreting the second moment matrix
The surface E(u,v) is locally approximated by a quadratic form. Let’s try to understand its shape.y
⎤⎡⎥⎦
⎤⎢⎣
⎡≈
vu
MvuvuE ][),(
∑ ⎥⎤
⎢⎡
= yxx IIIyxwM
2
)(∑⎥⎥⎦⎢
⎢⎣
=yx yyx III
yxwM,
2),(
Interpreting the second moment matrix
First, consider the axis-aligned case (gradients are either horizontal or vertical)
⎤⎡⎤⎡ 2 0λIII
(gradients are either horizontal or vertical)
⎥⎦
⎤⎢⎣
⎡=
⎥⎥⎦
⎤
⎢⎢⎣
⎡=∑
2
1
,2 0
0),(
λλ
yx yyx
yxx
IIIIII
yxwM⎦⎣ yy
If either λ is close to 0, then this is not a corner, so look for locations where both are large.
Interpreting the second moment matrix⎤⎡
Consider a horizontal “slice” of E(u, v):
This is the equation of an ellipse
const][ =⎥⎦
⎤⎢⎣
⎡vu
Mvu
This is the equation of an ellipse.
Interpreting the second moment matrix⎤⎡
Consider a horizontal “slice” of E(u, v):
This is the equation of an ellipse
const][ =⎥⎦
⎤⎢⎣
⎡vu
Mvu
This is the equation of an ellipse.
RRM ⎥⎦
⎤⎢⎣
⎡= −
2
11
00λ
λDiagonalization of M:
⎦⎣ 20 λThe axis lengths of the ellipse are determined by the eigenvalues and the orientation is determined by R
direction of the
direction of the fastest change
slowest change
(λmax)-1/2
(λ i )-1/2(λmin)
Visualization of second moment matrices
Visualization of second moment matrices
Interpreting the eigenvalues
Classification of image points using eigenvalues of M:
λ2
“Corner”λ d λ l
“Edge” λ2 >> λ1
λ1 and λ2 are large,λ1 ~ λ2;E increases in all directions
λ1 and λ2 are small;E is almost constant in all directions
“Edge” λ1 >> λ2
“Flat” region
λ1
1 2region
Corner response function2
21212 )()(trace)det( λλαλλα +−=−= MMR
t t (0 04 t 0 06)
“Corner”R > 0
“Edge” R < 0
α: constant (0.04 to 0.06)
R > 0
|R| ll“Edge” R < 0
“Flat” region
|R| small
region
Harris detector: Steps
1. Compute Gaussian derivatives at each pixel2 Compute second moment matrix M in a2. Compute second moment matrix M in a
Gaussian window around each pixel 3 Compute corner response function R3. Compute corner response function R4. Threshold R5 Find local maxima of response function5. Find local maxima of response function
(nonmaximum suppression)
C H i d M St h “A C bi d C d Ed D t t ”C.Harris and M.Stephens. “A Combined Corner and Edge Detector.” Proceedings of the 4th Alvey Vision Conference: pages 147—151, 1988.
Harris Detector: Steps
Harris Detector: StepsC t RCompute corner response R
Harris Detector: StepsFi d i t ith l R>thresholdFind points with large corner response: R>threshold
Harris Detector: StepsT k l th i t f l l i f RTake only the points of local maxima of R
Harris Detector: Steps
Invariance and covariance• We want corner locations to be invariant to photometric
transformations and covariant to geometric transformations• Invariance: image is transformed and corner locations do not changeInvariance: image is transformed and corner locations do not change• Covariance: if we have two transformed versions of the same image,
features should be detected in corresponding locations
Affine intensity change
I → a I + b
• Only derivatives are used => invariance to intensity shift I → I + b
• Intensity scaling: I → a I
R RRthreshold
R
x (image coordinate) x (image coordinate)
Partially invariant to affine intensity change
Image translation
• Derivatives and window function are shift-invariant
Corner location is covariant w r t translationCorner location is covariant w.r.t. translation
Image rotation
Second moment ellipse rotates but its shapeSecond moment ellipse rotates but its shape (i.e. eigenvalues) remains the same
Corner location is covariant w r t rotationCorner location is covariant w.r.t. rotation
Scaling
Corner
All points will be classifiedbe classified as edges
Corner location is not covariant to scaling!
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