CS 558 COMPUTER VISIONLecture VI: Corner and Blob Detection
Slides adapted from S. Lazebnik
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
FEATURE EXTRACTION: CORNERS9300 Harris Corners Pkwy, Charlotte, NC
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
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 features
WHY EXTRACT FEATURES?• Motivation: panorama stitching
We have two images – how do we combine them?
Step 1: extract featuresStep 2: match featuresStep 3: align images
CHARACTERISTICS OF GOOD FEATURES
• Repeatability The same feature can be found in several images despite
geometric and photometric transformations • Saliency
Each feature is distinctive• Compactness and efficiency
Many fewer features than image pixels• Locality
A feature occupies a relatively small area of the image; robust to clutter and occlusion
APPLICATIONS Feature points are used for:
Image alignment 3D reconstructionMotion trackingRobot navigation Indexing and database retrievalObject recognition
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
FINDING CORNERS
• Key property: in the region around a corner, image gradient has two or more dominant directions
• Corners are repeatable and distinctiveC.Harris and M.Stephens. "A Combined Corner and Edge Detector.“ Proceedings of the 4th Alvey Vision Conference: pages 147—151, 1988.
CORNER DETECTION: BASIC IDEA• We should easily recognize the point by
looking through a small window• Shifting a window in any direction should
give a large change in intensity
“edge”:no change along the edge direction
“corner”:significant change in all directions
“flat” region:no change in all directionsSource: A. Efros
CORNER DETECTION: MATHEMATICS
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y
Change in appearance of window w(x,y) for the shift [u,v]:
I(x, y)E(u, v)
E(3,2)
w(x, y)
CORNER DETECTION: MATHEMATICS
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y
I(x, y)E(u, v)
E(0,0)
w(x, y)
Change in appearance of window w(x,y) for the shift [u,v]:
CORNER DETECTION: MATHEMATICS
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y
IntensityShifted intensity
Window function
orWindow function w(x,y) =
Gaussian1 in window, 0 outside
Source: R. Szeliski
Change in appearance of window w(x,y) for the shift [u,v]:
CORNER DETECTION: MATHEMATICS
2
,
( , ) ( , ) ( , ) ( , )x y
E 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
Change in appearance of window w(x,y) for the shift [u,v]:
E(u, v)
CORNER DETECTION: MATHEMATICS
vu
EEEE
vuEE
vuEvuEvvuv
uvuu
v
u
)0,0()0,0()0,0()0,0(
][21
)0,0()0,0(
][)0,0(),(
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y
Local quadratic approximation of E(u,v) in the neighborhood of (0,0) is given by the second-order Taylor expansion:
We want to find out how this function behaves for small shifts
Change in appearance of window w(x,y) for the shift [u,v]:
CORNER DETECTION: MATHEMATICS
vu
EEEE
vuEE
vuEvuEvvuv
uvuu
v
u
)0,0()0,0()0,0()0,0(
][21
)0,0()0,0(
][)0,0(),(
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y Second-order Taylor expansion of E(u,v) about (0,0):
),(),(),(),(2
),(),(),(2),(
),(),(),(),(2
),(),(),(2),(
),(),(),(),(2),(
,
,
,
,
,
vyuxIyxIvyuxIyxw
vyuxIvyuxIyxwvuE
vyuxIyxIvyuxIyxw
vyuxIvyuxIyxwvuE
vyuxIyxIvyuxIyxwvuE
xyyx
xyyx
uv
xxyx
xxyx
uu
xyx
u
CORNER DETECTION: MATHEMATICS 2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y Second-order Taylor expansion of E(u,v) about (0,0):
),(),(),(2)0,0(
),(),(),(2)0,0(
),(),(),(2)0,0(
0)0,0(0)0,0(0)0,0(
,
,
,
yxIyxIyxwE
yxIyxIyxwE
yxIyxIyxwE
EEE
yxyx
uv
yyyx
vv
xxyx
uu
v
u
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
vu
yxIyxwyxIyxIyxw
yxIyxIyxwyxIyxwvuvuE
yxy
yxyx
yxyx
yxx
,
2
,
,,
2
),(),(),(),(),(
),(),(),(),(),(][),(
2
,
( , ) ( , ) ( , ) ( , )x y
E u v w x y I x u y v I x y Second-order Taylor expansion of E(u,v) about (0,0):
),(),(),(2)0,0(
),(),(),(2)0,0(
),(),(),(2)0,0(
0)0,0(0)0,0(0)0,0(
,
,
,
yxIyxIyxwE
yxIyxIyxwE
yxIyxIyxwE
EEE
yxyx
uv
yyyx
vv
xxyx
uu
v
u
CORNER DETECTION: MATHEMATICS
vu
MvuvuE ][),(
The quadratic approximation simplifies to
2
2,
( , ) x x y
x y x y y
I I IM w x y
I I I
where M is a second moment matrix computed from image derivatives:
M
The surface E(u,v) is locally approximated by a quadratic form. Let’s try to understand its shape.
INTERPRETING THE SECOND MOMENT MATRIX
vu
MvuvuE ][),(
yx yyx
yxx
IIIIII
yxwM,
2
2
),(
2
1
,2
2
00
),(
yx yyx
yxx
IIIIII
yxwM
First, consider the axis-aligned case (gradients are either horizontal or vertical)
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):
INTERPRETING THE SECOND MOMENT MATRIX
const][
vu
Mvu
This is the equation of an ellipse.
Consider a horizontal “slice” of E(u, v):
INTERPRETING THE SECOND MOMENT MATRIX
const][
vu
Mvu
This is the equation of an ellipse.
RRM
2
11
00
The axis lengths of the ellipse are determined by the eigenvalues and the orientation is determined by R
direction of the slowest change
direction of the fastest change
(max)-1/2
(min)-1/2
Diagonalization of M:
VISUALIZATION OF SECOND MOMENT MATRICES
VISUALIZATION OF SECOND MOMENT MATRICES
INTERPRETING THE EIGENVALUES
1
2
“Corner”1 and 2 are large,
1 ~ 2;
E increases in all directions
1 and 2 are small;
E is almost constant in all directions
“Edge” 1 >> 2
“Edge” 2 >> 1
“Flat” region
Classification of image points using eigenvalues of M:
CORNER RESPONSE FUNCTION
“Corner”R > 0
“Edge” R < 0
“Edge” R < 0
“Flat” region
|R| small
22121
2 )()(trace)det( MMR
α: constant (0.04 to 0.06)
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
HARRIS DETECTOR: STEPS
1. Compute Gaussian derivatives at each pixel2. Compute second moment matrix M in a Gaussian
window around each pixel 3. Compute corner response function R4. Threshold R5. Find local maxima of response function
(nonmaximum suppression)
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: STEPSCompute corner response R
HARRIS DETECTOR: STEPSFind points with large corner response: R>threshold
HARRIS DETECTOR: STEPSTake 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
change Covariance: if we have two transformed versions of the same
image, features should be detected in corresponding locations
AFFINE INTENSITY CHANGE
• Only derivatives are used => invariance to intensity shift I I + b
• Intensity scaling: I a I
R
x (image coordinate)
threshold
R
x (image coordinate)
Partially invariant to affine intensity change
I a I + b
IMAGE TRANSLATION
• Derivatives and window function are shift-invariant
Corner location is covariant w.r.t. translation
IMAGE ROTATION
Second moment ellipse rotates but its shape (i.e. eigenvalues) remains the same
Corner location is covariant w.r.t. rotation
SCALING
All points will be classified as edges
Corner
Corner location is not covariant to scaling!
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
BLOB DETECTION
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
• Goal: independently detect corresponding regions in scaled versions of the same image
• Need scale selection mechanism for finding characteristic region size that is covariant with the image transformation
ACHIEVING SCALE COVARIANCE
RECALL: EDGE DETECTION
gdxdf
f
gdxd
Source: S. Seitz
Edge
Derivativeof Gaussian
Edge = maximumof derivative
EDGE DETECTION, TAKE 2
gdxdf 2
2
f
gdxd
2
2
Edge
Second derivativeof Gaussian (Laplacian)
Edge = zero crossingof second derivative
Source: S. Seitz
FROM EDGES TO BLOBS• Edge = ripple• Blob = superposition of two ripples
Spatial selection: the magnitude of the Laplacianresponse will achieve a maximum at the center ofthe blob, provided the scale of the Laplacian is“matched” to the scale of the blob
maximum
• We want to find the characteristic scale of the blob by convolving it with Laplacians at several scales and looking for the maximum response
• However, Laplacian response decays as scale increases:
SCALE SELECTION
Why does this happen?
increasing σoriginal signal(radius=8)
SCALE NORMALIZATION• The response of a derivative of Gaussian
filter to a perfect step edge decreases as σ increases
21
SCALE NORMALIZATION• The response of a derivative of Gaussian
filter to a perfect step edge decreases as σ increases
• To keep response the same (scale-invariant), must multiply Gaussian derivative by σ
• Laplacian is the second Gaussian derivative, so it must be multiplied by σ2
EFFECT OF SCALE NORMALIZATION
Scale-normalized Laplacian response
Unnormalized Laplacian responseOriginal signal
maximum
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
BLOB DETECTION IN 2D Laplacian of Gaussian: Circularly symmetric
operator for blob detection in 2D
2
2
2
22
yg
xgg
BLOB DETECTION IN 2D Laplacian of Gaussian: Circularly symmetric
operator for blob detection in 2D
2
2
2
222
norm yg
xgg Scale-normalized:
• At what scale does the Laplacian achieve a maximum response to a binary circle of radius r?
SCALE SELECTION
r
image Laplacian
• At what scale does the Laplacian achieve a maximum response to a binary circle of radius r?
• To get maximum response, the zeros of the Laplacian have to be aligned with the circle
• The Laplacian is given by:
• Therefore, the maximum response occurs at
SCALE SELECTION
r
image
62/)(222 2/)2(222
yxeyx .2/r
circle
Laplacian
0
CHARACTERISTIC SCALE• We define the characteristic scale of a blob
as the scale that produces peak of Laplacian response in the blob center
characteristic scaleT. Lindeberg (1998). "Feature detection with automatic scale selection." International Journal of Computer Vision 30 (2): pp 77--116.
SCALE-SPACE BLOB DETECTOR1. Convolve image with scale-normalized
Laplacian at several scales
SCALE-SPACE BLOB DETECTOR: EXAMPLE
SCALE-SPACE BLOB DETECTOR: EXAMPLE
SCALE-SPACE BLOB DETECTOR1. Convolve image with scale-normalized
Laplacian at several scales2. Find maxima of squared Laplacian response
in scale-space
SCALE-SPACE BLOB DETECTOR: EXAMPLE
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
Approximating the Laplacian with a difference of Gaussians:
2 ( , , ) ( , , )xx yyL G x y G x y
( , , ) ( , , )DoG G x y k G x y
(Laplacian)
(Difference of Gaussians)
EFFICIENT IMPLEMENTATION
EFFICIENT IMPLEMENTATION
David G. Lowe. "Distinctive image features from scale-invariant keypoints.” IJCV 60 (2), pp. 91-110, 2004.
INVARIANCE AND COVARIANCE PROPERTIES• Laplacian (blob) response is invariant w.r.t.
rotation and scaling• Blob location and scale is covariant w.r.t.
rotation and scaling• What about intensity change?
OUTLINE Corner detection
Why detecting features? Finding corners: basic idea and mathematics Steps of Harris corner detector
Blob detection Scale selection Laplacian of Gaussian (LoG) detector Difference of Gaussian (DoG) detector Affine co-variant region
ACHIEVING AFFINE COVARIANCE• Affine transformation approximates viewpoint
changes for roughly planar objects and roughly orthographic cameras
ACHIEVING AFFINE COVARIANCE
RRIIIIII
yxwMyyx
yxx
yx
2
112
2
, 00
),(
direction of the slowest
change
direction of the fastest change
(max)-1/2
(min)-1/2
Consider the second moment matrix of the window containing the blob:
const][
vu
Mvu
Recall:
This ellipse visualizes the “characteristic shape” of the window
AFFINE ADAPTATION EXAMPLE
Scale-invariant regions (blobs)
AFFINE ADAPTATION EXAMPLE
Affine-adapted blobs
FROM COVARIANT DETECTION TO INVARIANT DESCRIPTION• Geometrically transformed versions of the same neighborhood
will give rise to regions that are related by the same transformation
• What to do if we want to compare the appearance of these image regions?
Normalization: transform these regions into same-size circles
• Problem: There is no unique transformation from an ellipse to a unit circle
We can rotate or flip a unit circle, and it still stays a unit circle
AFFINE NORMALIZATION
• To assign a unique orientation to circular image windows:
Create histogram of local gradient directions in the patch
Assign canonical orientation at peak of smoothed histogram
ELIMINATING ROTATION AMBIGUITY
0 2
FROM COVARIANT REGIONS TO INVARIANT FEATURES
Extract affine regions Normalize regionsEliminate rotational
ambiguityCompute appearance
descriptors
SIFT (Lowe ’04)
INVARIANCE VS. COVARIANCE Invariance:
features(transform(image)) = features(image)
Covariance: features(transform(image)) =
transform(features(image))
Covariant detection => invariant description