Features
Digital Visual Effectsg Yung-Yu Chuang
with slides by Trevor Darrell Cordelia Schmid David Lowe Darya Frolova Denis Simakov with slides by Trevor Darrell Cordelia Schmid, David Lowe, Darya Frolova, Denis Simakov, Robert Collins and Jiwon Kim
Outline
• Features
H i d
• Harris corner detector
• SIFT
• Extensions
• Applications
• Applications
Features Features
Features
• Also known as interesting points, salient points or keypoints Points that you can easily point or keypoints. Points that you can easily point out their correspondences in multiple images using only local information
using only local information.
?
Desired properties for features
• Distinctive: a single feature can be correctly matched with high probability
matched with high probability.
• Invariant: invariant to scale, rotation, affine, ill i ti d i f b t t hi illumination and noise for robust matching across a substantial range of affine distortion,
i i t h d Th t i it i viewpoint change and so on. That is, it is repeatable.
Applications
• Object or scene recognition
S f i
• Structure from motion
• Stereo
• Motion tracking
•• …
Components
• Feature detection locates where they are
F d i i d ib h h
• Feature description describes what they are
• Feature matching decides whether two are the same one
Harris corner detector
Harris corner detector
Moravec corner detector (1980)
• We should easily recognize the point by looking through a small window
through a small window
• Shifting a window in any direction should give a
l h i i t it
large change in intensity
Moravec corner detector
flat
Moravec corner detector
flat
Moravec corner detector
flat edgeg
Moravec corner detector
flat edge corner
i l t d i t g isolated point
Moravec corner detector
Change of intensity for the shift [u,v]:
y x
y x I v y u x I y x w v
u E
,
) 2
, ( ) ,
( ) , ( )
, (
y x,
window shifted intensity
function intensity
intensity
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 O l f hif 45 d i
• Only a set of shifts at every 45 degree is considered
• Only minimum of E is taken into account
Harris corner detector (1988) solves these problems
problems.
Harris corner detector
Noisy response due to a binary window function
U G i f i
Use a Gaussian function
Harris corner detector
Only a set of shifts at every 45 degree is considered
C id ll ll hif b T l ’ i
Consider all small shifts by Taylor’s expansion
Harris corner detector
Only a set of shifts at every 45 degree is considered
C id ll ll hif b T l ’ i
Consider all small shifts by Taylor’s expansion
w x y I x u y v I x y
v u
E( , )
( , )
( , ) ( , )
2y x
y y
y
,
) , ( ) ,
( ) , ( )
, (
w(x,y) I u I v O(u2,v2) 2
Au Cuv Bv v
u
E( , ) 2 2 2
y x
y
xu I v O u v
I y x w
,
) , ( )
, (
y x
x x y
I y x w A
,
2( , ) )
, ( ) , (
y x
y x y
I y x w B
,
2( , ) )
, (
y x
y
x x y I x y
I y x w C
,
) , ( ) , ( ) , (
Harris corner detector
Equivalently, for small shifts [u,v] we have a bilinear i ti
approximation:
v v u
u v u
E( , ) M
, where M is a 22 matrix computed from image derivatives:
( ) Ix2 IxIyM
y
x x y y
y
I I y I x w
,
) 2
, M (
Harris corner detector (matrix form)
( ) | ( ) ( ) |
2)
( u w x
0I x
0u I x
0E
x u
x0 I 0
I( ) ( )|2
|
(p) x0 Wx u
T
I I
I
2 0
0 x
IT
2
x u I
x u u x
T
T I I
Mu u
x x
T
Harris corner detector
Only minimum of E is taken into account
A b i i i h
A new corner measurement by investigating the shape of the error function
represents a quadratic function; Thus, we can analyze E’s shape by looking at the property
Mu u
Tcan analyze E s shape by looking at the property of M
Harris corner detector
High-level idea: what shape of the error function will we prefer for features?
will we prefer for features?
80 100
80 100
80 100
20 40 60
20 40 60
20 40 60
10 12 10
0 20
10 12 10
0 20
10 12 10
0 20
0 2 4 6 8
0 5
0 2 4 6 8 10
0 5
0 2 4 6 8 10
0 5
fl d
flat edge corner
Quadratic forms
• Quadratic form (homogeneous polynomial of degree two) of n variables x
degree two) of n variables xi
• Examples
=
Symmetric matrices
• Quadratic forms can be represented by a real symmetric matrix A where
symmetric matrix A where
Eigenvalues of symmetric matrices
Brad Osgood
Eigenvectors of symmetric matrices
Eigenvectors of symmetric matrices
T
x Q Λ Q x
Ax x
T T
T
1 z z
T Q Q x Λ Λ Q Q x x
x
T T
T
1 1q
2 2q
z z 1
Λy y
x Q Λ x Q
T Λ
21y
TΛ
12y
T x
Tx 1
z z
T
Harris corner detector
Intensity change in shifting window: eigenvalue analysis
1, 2 – eigenvalues of M
u
v u v u
E( , ) , M
direction of the
Elli E( )
v
direction of the slowest change direction of the
fastest change
Ellipse E(u,v) = const
g
(max)-1/2
(min)-1/2
Visualize quadratic functions
T
1 0 1 0 1 0 1 0
A
0 1 0 1 0 1 0 1
A
Visualize quadratic functions
T
4 0 1 0 4 0 1 0
A
0 1 0 1 0 1 0 1
A
Visualize quadratic functions
T
3.25 1.30 0.50 0.87 1 0 0.50 0.87
A
50 . 0 87 . 0 4 0 50 . 0 87 . 0 75
. 1 30 . A 1
Visualize quadratic functions
T
7.75 3.90 0.50 0.87 1 0 0.50 0.87
A
50 . 0 87 . 0 10 0 50 . 0 87 . 0 25
. 3 90 . A 3
Harris corner detector
2 edge Classification of 2
Corner
and are large
edge
2>> 1 image points
using eigenvalues
of M: 1and 2are large,
1 ~ 2;
Eincreases in all
of M:
directions
1and 2are small;
Eis almost constant edge
flat
1
s a ost co sta t
in all directions flat 1>> 2
1
Harris corner detector
4 )
( 00 11 2 10 01
11
00 a a a a a
a
Only for reference,
you do not need
Measure of corner response:
2
you do not need
them to compute R
Measure of corner response:
trace
2detM k M
R detM k
traceM
R
d M
2 1
2 1
trace det
M M
(k – empirical constant k = 0 04-0 06)
2 1
(k empirical constant, k 0.04 0.06)
Harris corner detector Another view
Another view Another view
Summary of Harris detector
1. Compute x and y derivatives of image I
G
Ix x Iy Gy I
2. Compute products of derivatives at every pixel
3 C t th f th d t f
x
x Ix I
I 2 Iy2 IyIy Ixy IxIy
3. Compute the sums of the products of derivatives at each pixel
2
2 ' x
x G I
S Sy2 G'Iy2 Sxy G'Ixy
Summary of Harris detector
4. Define the matrix at each pixel
) , ( )
, (
) , ( )
, ) (
, (
2 2
y x S y x S
y x S y x y S
x
M x xy
5 Compute the response of the detector at each
Sxy(x,y) Sy2(x, y)
5. Compute the response of the detector at each pixel
trace
2detM k M
R
6. Threshold on value of R; compute nonmax i
suppression.
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:
expressed as a bilinear form:
u
v u v u
E( , ) , M
• Describe a point in terms of eigenvalues of M:
v v u v u
E( , ) , M
p g
measure of corner response
1 2
22
1
k
R
• A good (corner) point should have a large intensity change in all directions i e R should be large
1 2
2
1
k R
change in all directions, i.e. R should be large positive
Now we know where features are
• But, how to match them?
Wh i h d i f f ? Th
• What is the descriptor for a feature? The
simplest solution is the intensities of its spatial i hb Thi i ht t b b t t
neighbors. This might not be robust to brightness change or small shift/rotation.
1 2 3
4 5 6
7 8 9
( )
7 8 9
(
1 2 3 4 5 6 7 8 9)
Harris detector: some properties
• Partial invariance to affine intensity change
Only derivatives are used =>
invariance to intensity shift I I + b
Intensity scale: I a I
R R
R
threshold
R
x(image coordinate) x(image coordinate)
Harris Detector: Some Properties
• Rotation invariance
Ellipse rotates but its shape (i.e. eigenvalues) remains the same
the same
Corner response Ris invariant to image rotation
Harris Detector is rotation invariant
Repeatability rate:
# correspondences
# correspondences
# possible correspondences
Harris Detector: Some Properties
• But: not invariant to image scale!
All points will be
l ifi d d Corner !
classified as edges
Harris detector: some properties
• Quality of Harris detector for different scale changes
changes
Repeatability rate:
# d
# correspondences
# possible correspondences
Scale invariant detection
• Consider regions (e.g. circles) of different sizes around a point
around a point
• Regions of corresponding sizes will look the i b th i
same in both images
Scale invariant detection
• The problem: how do we choose corresponding i l i d d tl i h i ?
circles independently in each image?
• Aperture problem
SIFT
(Scale Invariant Feature Transform)
SIFT
• SIFT is an carefully designed procedure with empirically determined parameters for the empirically determined parameters for the invariant and distinctive features.
SIFT stages:
• Scale-space extrema detection
K i l li i detector
• Keypoint localization
• Orientation assignment
detector
d i t
• Keypoint descriptor descriptor
( ) ( )
local descriptor A 500 500 i i b 2000 f
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 across all possible scales using a continuous function of scale, scale space.
SIFT D G filt f l b it i
• SIFT uses DoG filter for scale space because it is efficient and as stable as scale-normalized L l i f G i
Laplacian of Gaussian.
DoG filtering
Convolution with a variable-scale Gaussian
Difference-of-Gaussian (DoG) filter
Convolution with the DoG filter
Scale space
doubles for the next octave the next octave
K=2(1/s)
Dividing into octave is for efficiency only.
Detection of scale-space extrema 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
tradeoff efficiency with completeness.
• Decide the best sampling frequency by
i ti 32 l i bj t t experimenting on 32 real image subject to synthetic transformations. (rotation, scaling,
ffi t t h b i ht d t t h
affine stretch, brightness and contrast change, adding noise…)
Decide scale sampling frequency
Decide scale sampling frequency
for detector, repeatability repeatability for descriptor, p , distinctiveness
s=3 is the best for larger s too many unstable features s=3 is the best, for larger s, too many unstable features
Pre-smoothing
=1 6 plus a double expansion
=1.6, plus a double expansion
Scale invariance 2. Accurate keypoint localization
• Reject points with low contrast (flat) and poorly localized along an edge (edge) poorly localized along an edge (edge)
• Fit a 3D quadratic function for sub-pixel i
maxima
6 6
5
1
1 00 +1
-1 +1
2. Accurate keypoint localization
• Reject points with low contrast (flat) and poorly localized along an edge (edge) poorly localized along an edge (edge)
• Fit a 3D quadratic function for sub-pixel i
maxima
6
2
2 ) 0 ( ) '' 0 ( ' ) 0 ( )
( f x
x f f x
f
3 61
6
5 2 6 2 3 2
2 2 6 6 )
(x x x x x
f
3 ˆ1 0 x
6 2 ) (
' x x f
1 1 1 2 1
1 0 +1
3 61 3 3 1 3 2 1 6 ˆ)
(
x f 0 1
-1 +1
3 1
2. Accurate keypoint localization
• Taylor series of several variables
• Two variables
f f 1 2f 2f 2f
2 2 2
2 ) 1
0 , 0 ( ) ,
( y
y y xy f y x x f
x x y f
y x f x f f
y x f
2f 2f
y x f f
y x
f x x
f y y x
x y f x f f
y
f x 2 2
2 1 0
0
y
y y
f y x
f y y
y 0 x 2
0
1 2
f f
f
f T
T x
x x x x
0
x 2
2
f f
f
f T
Accurate keypoint localization
• Taylor expansion in a matrix form, x is a vector, f maps x to a scalar
f maps x to a scalar
Hessian matrix
f 2f 2f 2f Hessian matrix (often symmetric)
f x f
1
2 2
2
1 2
1 2
1 n
f f
f
x x
f x
x f x
f
gradient
x
f
1
2 2
2 2 2
1 2
2
xn
x f x
f x
x f
f
2 2 2
2f f f
xn
xnx1 xnx2 xn2
2D illustration
2D example
17 1 1
-17 -1 -1 7 7
9
7 7 7
7 -9
9 7 7
-9
Derivation of matrix form
x g x) T (
h h
x g x) (
h
x
Derivation of matrix form
x g x) T (
h(x)g x h
h
x1
x g
h
h 1 1
n n
x g
g1 g
x
h g
h
n
n gixi
n n
x g
i igi 1
Derivation of matrix form
Ax x x) T (
h
h
x h
Derivation of matrix form
a11 a1n x1Ax x x) T (
h
n
x a a
x
x
1 1
n n an1 annxn
i j
j i ijxx a
1 1
h n n
n
i
n
j
j j i
i x a x
x a h
h 1 1
1 1
1 A x Ax
x
T
n
nj
x a x
h a
h 1
A AT )
(
i j
j nj i
in n
x a x
x 1a 1 (AT A)x
Derivation of matrix form
f f
f f
f
f 12 2 T 2
f f x
f x f
f f
2 2
2 2
1
x x x
x x
x
Accurate keypoint localization
• x is a 3-vector
• Change sample point if offset is larger than 0 5
• Change sample point if offset is larger than 0.5
• Throw out low contrast (<0.03)
Accurate keypoint localization
• Throw out low contrast |D(xˆ)|0.03
1 2
T D
D
x
x x x x
x ˆ ˆ
2 ˆ 1 )
ˆ (
2 1 1 2
2 2 2
T T
T
T D
D D D
x x x x x x
x 2
ˆ 1
1 2 2 2 1 2
2 T 2
D D D D D
D D
x x x x x x
x 2
ˆ 1
1 2 2 2 2 2
2 T
T
T D D D D D
D D
x x x x
x 2
ˆ 1
1 2 2
T T
T
T D D D
D D
x x x x
1
ˆ) 2 (
ˆ 1
T
T T
D
D D D
x xˆ 2 1 DT
D
Eliminating edge responses
Hessian matrix at keypoint locationyp
L t Let
r=10 Keep the points with
Maxima in D
Remove low contrast and edges Keypoint detector
233x89 832 extrema
729 after con- t t filt i
536 after cur- trast filtering vature filtering
3. Orientation assignment
• By assigning a consistent orientation, the
keypoint descriptor can be orientation invariant keypoint descriptor can be orientation invariant.
• For a keypoint, L is the Gaussian-smoothed i ith th l t l
image with the closest scale,
(Lx Ly) (Lx, Ly) m
θ
orientation histogram (36 bins)
Orientation assignment
Orientation assignment Orientation assignment
Orientation assignment
σ=1.5*scale of the keypointy
Orientation assignment
Orientation assignment Orientation assignment
accurate peak position is determined by fittingy g
Orientation assignment
36-bin orientation histogram over 360°
36 bin orientation histogram over 360 , weighted by m and 1.5*scale falloff Peak is the orientation
Peak is the orientation
Local peak within 80% creates multiple orientations
orientations
About 15% has multiple orientations and they contribute a lot to stability
0 2
y y
0 2
SIFT descriptor
4. Local image descriptor
• Thresholded image gradients are sampled over 16x16 array of locations in scale space
• Create array of orientation histograms (w.r.t. key orientation)
8 orientations x 4x4 histogram array 128 dimensions
• 8 orientations x 4x4 histogram array = 128 dimensions
• Normalized, clip values larger than 0.2, renormalize
σ=0.5*width
Why 4x4x8?
Sensitivity to affine change Feature matching
• for a feature x, he found the closest feature x1 and the second closest feature x If the
and the second closest feature x2. If the
distance ratio of d(x, x1) and d(x, x1) is smaller than 0 8 then it is accepted as a match
than 0.8, then it is accepted as a match.
SIFT flow Maxima in D
Remove low contrast Remove edges
SIFT descriptor
Estimated rotation
• Computed affine transformation from rotated image to original image:
image to original image:
0.7060 -0.7052 128.4230 0 7057 0 7100 128 9491 0.7057 0.7100 -128.9491 0 0 1.0000
• Actual transformation from rotated image to i i l i
original image:
0.7071 -0.7071 128.6934 0.7071 0.7071 -128.6934 0 0 1.0000
SIFT extensions SIFT extensions
PCA PCA-SIFT
• Only change step 4
P i f l l di
• Pre-compute an eigen-space for local gradient patches of size 41x41
• 2x39x39=3042 elements
• Only keep 20 componentsy p p
• A more compact descriptor
GLOH (Gradient location-orientation histogram)
SIFT SIFT
17 location bins 17 location bins 16 orientation bins
Analyze the 17x16=272-d Analyze the 17x16 272 d
eigen-space, keep 128 components SIFT is still considered the best.
Multi-Scale Oriented Patches
• Simpler than SIFT. Designed for image matching.
[Brown Szeliski Winder CVPR’2005]
[Brown, Szeliski, Winder, CVPR 2005]
• Feature detector
– Multi-scale Harris corners
– Orientation from blurred gradient – Geometrically invariant to rotation
• Feature descriptor
– Bias/gain normalized sampling of local patch (8x8) – Photometrically invariant to affine changes in y g
intensity
Multi-Scale Harris corner detector
2 s
• Image stitching is mostly concerned with g g y matching images that have the same scale, so sub-octave pyramid might not be necessary.py g y
Multi-Scale Harris corner detector
smoother version of gradients
Corner detection function:
Pick local maxima of 3x3 and larger than 10
Keypoint detection function
Experiments show roughly h f
the same performance.
Non-maximal suppression
• Restrict the maximal number of interest points, but also want them spatially well distributed but also want them spatially well distributed
• Only retain maximums in a neighborhood of di
radius r.
• Sort them by strength, decreasing r from infinity until the number of keypoints (500) is satisfied.
Non-maximal suppression Sub-pixel refinement
Orientation assignment
• Orientation = blurred gradient
Descriptor Vector
• Rotation Invariant Frame
S l iti ( ) i t ti () – Scale-space position (x, y, s) + orientation ()
MSOP descriptor vector
• 8x8 oriented patch sampled at 5 x scale. See TR for details
for details.
• Sampled from with i 5
spacing=5
8 pixels
MSOP descriptor vector
• 8x8 oriented patch sampled at 5 x scale. See TR for details
for details.
• Bias/gain normalisation: I’ = (I – )/
• Wavelet transform
8 pixels
Detections at multiple scales Summary
• Multi-scale Harris corner detector S b i l fi
• Sub-pixel refinement
• Orientation assignment by gradients
• Blurred intensity patch as descriptor
Feature matching
• Exhaustive search
f h f t i i l k t ll th th – for each feature in one image, look at all the other
features in the other image(s)
Hashing
• Hashing
– compute a short descriptor from each feature vector, or hash longer descriptors (randomly)
or hash longer descriptors (randomly)
• Nearest neighbor techniques
– k-trees and their variants (Best Bin First)
Wavelet-based hashing
• Compute a short (3-vector) descriptor from an 8x8 patch using a Haar “wavelet”
8x8 patch using a Haar wavelet
• Quantize each value into 10 (overlapping) bins
• Quantize each value into 10 (overlapping) bins (103 total entries)
[B S li ki Wi d CVPR’2005]
• [Brown, Szeliski, Winder, CVPR’2005]
Nearest neighbor techniques
• k-D tree and and
• Best Bin First (BBF)
Indexing Without Invariants in 3D Object Recognition, Beis and Lowe, PAMI’99
Applications Applications
Recognition
SIFT Features
3D object recognition
3D object recognition Office of the past
Video of desk Images from PDF Internal representation
Track &
recognize recognize
Desk Desk
T T+1
Scene Graph
Image retrieval
…
> 5000 5000 images
change in viewing angle
Image retrieval
22 correct matches
Image retrieval
…
> 5000 5000 images change in viewing angle
change in viewing angle + scale change
Robot location
Robotics: Sony Aibo
SIFT is used for
Recognizing
Recognizing charging station
Communicating g with visual cards
Teaching object recognition
soccer
Structure from Motion
• The SFM Problem
Reconstruct scene geometry and camera motion – Reconstruct scene geometry and camera motion
from two or more images
Track
2D Features EstimateEstimate
3D Optimize
(Bundle Adjust)
Fit Surfaces
SFM Pipelinep
Structure from Motion
Poor mesh Good mesh
Augmented reality
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
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
• SIFT Keypoint Detector, David Lowe.
• Matlab SIFT Tutorial, University of Toronto.