Feature matching
Digital Visual Effects, Spring 2005 Yung-Yu Chuang
2005/3/16
with slides by Trevor Darrell Cordelia Schmid, David Lowe, Darya Frolova, Denis Simakov, Robert Collins and Jiwon Kim
Announcements
• Project #1 is online, you have to write a program, not just using available software.
• Send me the members of your team.
• Sign up for scribe at the forum.
Blender
http://www.blender3d.com/cms/Home.2.0.html Blender could be used for your project #3 matchmove.
In the forum
• Barycentric coordinate
• RBF
Outline
• Block matching
• Features
• Harris corner detector
• SIFT
• SIFT extensions
• Applications
Correspondence by block matching
• Points are individually ambiguous
• More unique matches are possible with small regions of images
Correspondence by block matching Sum of squared distance
Image blocks as a vector Image blocks as a vector
Matching metrics Features
• Properties of features
• Detector: locates feature
• Descriptor and matching metrics: describes and matches features
• In the example for block matching:
– Detector: none – Descriptor: block – Matching: distance
Desired properties for features
• Invariant: invariant to scale, rotation, affine, illumination and noise for robust matching across a substantial range of affine distortion, viewpoint change and so on.
• Distinctive: a single feature can be correctly
matched with high probability
Harris corner detector
Moravec corner detector (1980)
• 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
Moravec corner detector
flat
Moravec corner detector
flat
Moravec corner detector
flat edge
Moravec corner detector
flat edge corner
isolated point
Moravec corner detector
Change of intensity for the shift [u,v]:
[ ]
2,
( , ) ( , ) ( , ) ( , )
x y
E u v =
∑
w x y I x u y v+ + −I x yIntensity Shifted
intensity Window
function
Four shifts: (u,v) = (1,0), (1,1), (0,1), (-1, 1) Look for local maxima in min{E}
Problems of Moravec detector
• Noisy response due to a binary window function
• Only a set of shifts at every 45 degree is considered
• Responds too strong for edges because only minimum of E is taken into account
ÖHarris corner detector (1988) solves these problems.
Harris corner detector
Noisy response due to a binary window function
¾Use a Gaussian function
Harris corner detector
Only a set of shifts at every 45 degree is considered
¾Consider all small shifts by Taylor’s expansion
∑
∑
∑
=
=
=
+ +
=
y x
y x
y x
y y
x
x
y x I y x I y x w C
y x I y x w B
y x I y x w A
Bv Cuv Au
v u E
, ,
2 ,
2
2 2
) , ( ) , ( ) , (
) , ( ) , (
) , ( ) , (
2 )
, (
Harris corner detector
[ ]
( , ) , u
E u v u v M v
≅ ⎡ ⎤⎢ ⎥
⎣ ⎦
Equivalently, for small shifts [u,v] we have a bilinear approximation:
2
2 ,
( , ) x x y
x y x y y
I I I M w x y
I I I
⎡ ⎤
= ⎢ ⎥
⎢ ⎥
⎣ ⎦
∑
, where M is a 2×2 matrix computed from image derivatives:
Harris corner detector
Responds too strong for edges because only minimum of E is taken into account
¾A new corner measurement
Harris corner detector
[ ]
( , ) , u
E u v u v M v
≅ ⎡ ⎤⎢ ⎥
⎣ ⎦
Intensity change in shifting window: eigenvalue analysis
λ1, λ2 – eigenvalues of M
direction of the slowest change direction of the
fastest change
(λmax)-1/2
(λmin)-1/2 Ellipse E(u,v) = const
Harris corner detector
λ1 λ2
Corner
λ1and λ2are large,
λ1 ~ λ2;
Eincreases in all directions
λ1and λ2are small;
Eis almost constant in all directions
edge λ1>> λ2 edge
λ2>> λ1
flat Classification of
image points using eigenvalues of M:
Harris corner detector
Measure of corner response:
( )
2det trace R = M k− M
1 2
1 2
det trace
M M
λ λ λ λ
=
= +
(k – empirical constant, k = 0.04-0.06)
Another view Another view
Another view Summary of Harris detector
Harris corner detector (input) Corner response R
Threshold on R Local maximum of R
Harris corner detector Harris Detector: Summary
• Average intensity change in direction [u,v] can be expressed as a bilinear form:
• Describe a point in terms of eigenvalues of M:
measure of corner response
• A good (corner) point should have a large intensity change in all directions, i.e. R should be large positive
[ ]
( , ) , u
E u v u v M v
≅ ⎡ ⎤⎢ ⎥
⎣ ⎦
( )
21 2 1 2
R=λ λ −k λ λ+
Harris Detector: Some Properties
• Partial invariance to affine intensity change 9 Only derivatives are used =>
invariance to intensity shift I → I + b 9 Intensity scale: I → a I
R
x(image coordinate) threshold
R
x(image coordinate)
Harris Detector: Some Properties
• Rotation invariance
Ellipse rotates but its shape (i.e. eigenvalues) remains the same
Corner response Ris invariant to image rotation
Harris Detector is rotation invariant
Repeatability rate:
# correspondences
# possible correspondences
Harris Detector: Some Properties
• But: non-invariant to image scale!
All points will be classified as edges
Corner !
Harris Detector: Some Properties
• Quality of Harris detector for different scale changes
Repeatability rate:
# correspondences
# possible correspondences
SIFT
(Scale Invariant Feature Transform)
SIFT
• SIFT is an carefully designed procedure with empirically determined parameters for the invariant and distinctive features.
SIFT stages:
• Scale-space extrema detection
• Keypoint localization
• Orientation assignment
• Keypoint descriptor
( )
local descriptor
detector descriptor
A 500x500 image gives about 2000 features
1. Detection of scale-space extrema
• For scale invariance, search for stable features across all possible scales using a continuous function of scale, scale space.
• SIFT uses DoG filter for scale space because it is efficient and as stable as scale-normalized Laplacian of Gaussian.
DoG filtering
Convolution with a variable-scale Gaussian
Difference-of-Gaussian (DoG) filter
Convolution with the DoG filter
Scale space
σ doubles for the next octave
K=2(1/s), s+3 images for each octave
Keypoint localization
X is selected if it is larger or smaller than all 26 neighbors
Decide scale sampling frequency
• It is impossible to sample the whole space, tradeoff efficiency with completeness.
• Decide the best sampling frequency by experimenting on 32 real image subject to synthetic transformations.
Decide scale sampling frequency
S=3, for larger s, too many unstable features
Decide scale sampling frequency Pre-smoothing
σ =1.6, plus a double expansion
Scale invariance 2. Accurate keypoint localization
• Reject points with low contrast and poorly localized along an edge
• Fit a 3D quadratic function for sub-pixel maxima
Accurate keypoint localization
If has offset larger than 0.5, sample point is changed.
If is less than 0.03 (low contrast), it is discarded.
Eliminating edge responses
r=10 Let
Keep the points with
Keypoint detector
(a) 233x189 image (b) 832 DOG extrema (c) 729 left after peak
value threshold (d) 536 left after testing
ratio of principle curvatures
3. Orientation assignment
• By assigning a consistent orientation, the
keypoint descriptor can be orientation invariant.
• For a keypoint, L is the image with the closest scale,
orientation histogram
Orientation assignment Orientation assignment
Orientation assignment Orientation assignment
Orientation assignment Orientation assignment
Orientation assignment Orientation assignment
0 2π
36-bin orientation histogram over 360°, weighted by m and 1.5*scale falloff Peak is the orientation
Local peak within 80% creates multiple orientations
About 15% has multiple orientations
Orientation invariance 4. Local image descriptor
• Thresholded image gradients are sampled over 16x16 array of locations in scale space
• Create array of orientation histograms
• 8 orientations x 4x4 histogram array = 128 dimensions
• Normalized, clip the components larger than 0.2
Why 4x4x8? Sensitivity to affine change
SIFT demo Maxima in D
Remove low contrast Remove edges
SIFT descriptor
Estimated rotation
• Computed affine transformation from rotated image to original image:
0.7060 -0.7052 128.4230 0.7057 0.7100 -128.9491 0 0 1.0000
• Actual transformation from rotated image to original image:
0.7071 -0.7071 128.6934 0.7071 0.7071 -128.6934 0 0 1.0000
SIFT extensions
PCA PCA-SIFT
• Only change step 4
• Pre-compute an eigen-space for local gradient patches of size 41x41
• 2x39x39=3042 elements
• Only keep 20 components
• A more compact descriptor
GLOH (Gradient location-orientation histogram)
17 location bins 16 orientation bins
Analyze the 17x16=272-d
eigen-space, keep 128 components
Applications
Recognition
SIFT Features
3D object recognition
3D object recognition Office of the past
Video of desk Images from PDF
Track &
recognize
T T+1
Internal representation
Scene Graph
Desk Desk
…
> 5000 images
change in viewing angle
Image retrieval
22 correct matches
Image retrieval
…
> 5000 images change in viewing angle
+ scale change
Image retrieval Robot location
Robotics: Sony Aibo
SIFT is used for
¾Recognizing charging station
¾Communicating with visual cards
¾Teaching object recognition
¾soccer
Structure from Motion
• The SFM Problem
– Reconstruct scene geometry and camera motion from two or more images
Track
2D Features Estimate
3D Optimize
(Bundle Adjust)
Fit Surfaces
SFM Pipeline
Structure from Motion
Poor mesh Good mesh
Augmented reality
Automatic image stitching Automatic image stitching
Automatic image stitching Automatic image stitching
Automatic image stitching Automatic image stitching
Reference
• Chris Harris, Mike Stephens, A Combined Corner and Edge Detector, 4th Alvey Vision Conference, 1988, pp147-151.
• David G. Lowe, Distinctive Image Features from Scale-Invariant Keypoints, International Journal of Computer Vision, 60(2), 2004, pp91-110.
• Yan Ke, Rahul Sukthankar, PCA-SIFT: A More Distinctive Representation for Local Image Descriptors, CVPR 2004.
• Krystian Mikolajczyk, Cordelia Schmid, A performance evaluation of local descriptors, Submitted to PAMI, 2004.
• SIFT Keypoint Detector, David Lowe.
• Matlab SIFT Tutorial, University of Toronto.
