Computer Vision Local Invariant Features Mehdi Faraji [email protected] SLIDES have been prepared by: Dr. Ghassabi

Computer VisionLocal Invariant Features

Mehdi [email protected]

SLIDES have been prepared by:Dr. Ghassabi

2

Outline• Why do we care about matching features?• Problem Statement

– Properties of features– Types of invariance

• Introduction to feature matching– Matching using invariant descriptors

• Feature Detection– Corner Detection

» Moravec, harris» Harris properties (rotation, intensity, scale invariance)

– Low’s key point

• Feature description– SIFT (Scale Invariant Feature Transform)– SIFT Extensions: PCA-SIFT, GLoH ,SPIN image, RIFT,

• Feature matching

• Applications (examples)• Future Works • Conclusion

Outline

Motivation

Problems statement

How we solve it

Future Work

Reference

3

Motivation

• Why do we care about matching features?– image stitching, – object recognition, – Indexing and database retrieval,– Motion tracking– … Others

Outline

Motivation

Problems statement

How we solve it

Future Work

Reference

Example: How do we build panorama?

We need to match (align) images

Matching with FeaturesDetect feature points in both images

6

Matching with FeaturesDetect feature points in both imagesFind corresponding pairs

7

Matching with FeaturesDetect feature points in both imagesFind corresponding pairsUse these pairs to align images

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• Types of variance– Illumination– Scale– Rotation– Affine– Full Perspective

• Problems statements• Properties of good features

Outline

Motivation

Problems statement

How we solve it

Future Work

Reference

9

Types of variance

• IlluminationOutline

Motivation

Problems statement

• Types of variance

• Problem1• Problem2• Properties

of good features

How we solve it

Future Work

Reference

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Types of variance

• Illumination• Scale

Outline

Motivation

Problems statement



of good features

How we solve it

Future Work

Reference

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Types of variance

• Illumination• Scale• Rotation

Outline

Motivation

Problems statement



of good features

How we solve it

Future Work

Reference

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Types of variance

• Illumination• Scale• Rotation• Affine

Outline

Motivation

Problems statement



of good features

How we solve it

Future Work

Reference

13

Types of variance

• Illumination• Scale• Rotation• Affine• Full Perspective

Outline

Motivation

Problems statement



of good features

How we solve it

Future Work

Reference

14

Problems statement

Problem 1:– Detect the same point independently in

both images

no chance to match!

We need a repeatable detectorHow to find landmarks to match across two images?How achieve landmarks invariance to scale, rotation, illumination distortions?

Outline

Motivation

Problems statement



of good features

How we solve it

Future Work

Reference

15

Problems statement

Problem 2:– For each point correctly recognize the

corresponding one

?

We need a reliable and distinctive descriptor

How to distinguish one landmark from another?

Outline

Motivation

Problems statement



of good features

How we solve it

Future Work

Reference

16

Properties of features

• Distinctiveness• Invariance

– Invariance to illumination, scale, Rotation, Affine, full perspective

Good features should be robust to all sorts of distortions that can occur

between images.

Outline

Motivation

Problems statement



of good features

How we solve it

Future Work

Reference

17

Methods using invariant descriptors

• Methods using invariant descriptors Invariance to: transformation change in illumination image noise Distinctiveness

• Local features– Feature Detector– Feature descriptor– Feature-matching

Outline

Motivation

Problems statement

How we solve it

• Methods of Feature matching

• Invariant descriptors

Future Work

Reference

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• Local features– Feature Detector

• Point detector– Corner detectors

» Moravec, harris, SUSAN, Trajkovic operators– Low’s key point

• Region detector– Harris-Laplase, Harris affine, Hessian affine, edge-

based, Intensity-based, salient region detectors

– Feature descriptor– Feature-matching

Outline

Motivation

Problems statement

How we solve it



Future Work

Reference

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• Local features– Feature Detector– Feature descriptor

• Filter-based– Steerable filters– Gabor filters– Complex filters

• Distribution-based– Local

» SIFT, PCA-SIFT, GLOH, Spin image, RIFT,, SURF– global

» Shape context

• Textons• Derivative-based• Others

– Moment-based, Phase-based, Color-based

– Feature-matching

Outline

Motivation

Problems statement

How we solve it



Future Work

Reference

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Corner detectors

21

Moravec corner detector (1980):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

22

Moravec corner detector:Idea

flatno change in all directions

23


flat

24


flat edgeno change along the edge direction

25


flat edgecornerisolated point

significant change in all directions

26

Feature detection: the mathConsider shifting the window W by (u,v)

• how do the pixels in W change?• compare each pixel before and after by

summing up the squared differences (SSD)• this defines an SSD “error” of E(u,v):


W

2

,

( , ) ( , ) ( , ) ( , )x y

E u v w x y I x u y v I x y

27

Moravec corner detector:IdeaChange of intensity for the shift [u,v]:

2

,

( , ) ( , ) ( , ) ( , )x y

E u v w x y I x u y v I x y

IntensityShifted intensity

Window function

Four shifts: (u,v) = (1,0), (1,1), (0,1), (-1, 1)Look for local maxima in min{E}

E

u v

28

29

Problems of Moravec detector

• Noisy response due to a binary window function

• 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.

Harris corner detector : the math

Noisy response due to a binary window function Use a Gaussian function

31


32


( , ) ,u

E u v u v Mv

Equivalently, for small shifts [u,v] we have a bilinear approximation:

2

2,

( , ) x x y

x y x y y

I I IM w x y

I I I

, where M is a 22 matrix computed from image derivatives:

• You can move the center of the green window to anywhere on the blue unit circle

• Which directions will result in the largest and smallest E values?

• We can find these directions by looking at the eigenvectors of M

33

Harris corner detector

( , ) ,u

E u v u v Mv

Intensity change in shifting window: eigenvalue analysis

1, 2 – eigenvalues of M

direction of the slowest change

direction of the fastest change

(max)-1/2

(min)-1/2

Ellipse E(u,v) = const

34


1

2

Corner1 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

Classification of image points using eigenvalues of M:

35

Selecting Good Features

l1 and l2 are large

36


large l1, small l2

37


small l1, small l2

Harris corner detector: the mathResponds too strong for edges because only minimum of E is taken into accountA new corner measurement

39

Harris corner detector: the math

Measure of corner response:

2det traceR M k M

1 2

1 2

det

trace

M

M

(k – empirical constant, k = 0.04-0.06)

The Algorithm:Find points with large corner response function R (R > threshold)Take the points of local maxima of R

40

Harris Detector

1

2 “Corner”

“Edge”

“Edge”

“Flat”

• R depends only on eigenvalues of M

• R is large for a corner

• R is negative with large magnitude for an edge

• |R| is small for a flat region

R > 0

R < 0

R < 0|R| small

41

Another view

42

Another view

43

Another view

44

Harris corner detector (input)

45

Corner response R

46

Threshold on R

47

Local maximum of R

48


49

Summary of Harris detector

50

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 Mv

2

1 2 1 2R k

51

Harris Detector: Some Properties

• Invariance to image intensity change?

52


• Rotation invariance?

53


• Rotation invariance

Ellipse rotates but its shape (i.e. eigenvalues) remains the same

Corner response R is invariant to image rotation

54

Harris Detector is rotation invariant

Repeatability rate:

# correspondences# possible correspondences

55


• Invariant to image scale?

56


• But: non-invariant to image scale!

All points will be classified as edges

Corner !

57

Harris detector: some properties• Quality of Harris detector for different scale

changes

Repeatability rate:


C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000

Evaluation of Interest Point detectorsTwo Criterions

Repeatability rate:


Information content:

Measure of the distinctiveness of an interest point by using entropy.

C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000

Five Detectors

Harris (or impHarris) Cottier Horaud Heitger Forstner

59

Repeatability

Rotation change Scale change

60

Repeatability

Illumination change Viewing angle change

Information Content

62




» Moravec, harris, – Low’s key point

– Feature descriptor• SIFT• SIFT Extensions: GLoH, PCA-SIFT, RIFT, SPIN

Image,


Outline

Motivation

Problems statement

How we solve it



Applications

Future Work

Conclusion

63

We want to:

detect the same interest points regardless of image changes

64

Models of Image Change

• Geometry– Rotation– Similarity (rotation + uniform scale)

– Affine (rotation+scale+shearing)

• Photometry– Affine intensity change (I a I + b)

65

Scale Invariant Detection

• Consider regions (e.g. circles) of different sizes around a point

• Regions of corresponding sizes will look the same in both images

66

Scale invariant detection• The problem: how do we choose corresponding circles

independently in each image?• Aperture problem

67

Scale Invariant Detection• Solution:

– Design a function on the region (circle), which is “scale invariant” (the same for corresponding regions, even if they are at different scales)

Example: average intensity. For corresponding regions (even of different sizes) it will be the same.

– For a point in one image, we can consider f as a function of region size (circle radius)

f

region size

Image 1 f

region size

Image 2

scale = ?

Scale Invariant Detection

scale = 1/2

f

region size/scale

Image 1 f

region size/scale

Image 2

Take a local maximum of this function

Observation: region size (scale), for which the maximum is achieved, should be invariant to image scale.

s1 s2

Important: this scale invariant region size is found in each image independently!

Max. is called characteristic scale

Automatic Scale Selection

69

)),(( )),((11

xIfxIfmm iiii

Same operator responses if the patch contains the same image up to scale factorHow to find corresponding patch sizes?


K. Grauman, B. Leibe

)),(( )),((11

xIfxIfmm iiii

How to find corresponding patch sizes?


• Function responses for increasing scale (scale signature)

K. Grauman, B. Leibe)),((

1xIf

mii )),((1

xIfmii




1xIf

mii )),((1

xIfmii




1xIf

mii )),((1

xIfmii




1xIf

mii )),((1

xIfmii




1xIf

mii )),((1

xIfmii




1xIf

mii )),((1

xIfmii

77

Scale Invariant Detection• A “good” function for scale detection:

has one stable sharp peak

f

region size

bad

f

region size

bad

f

region size

Good !

• For usual images: a good function would be a one which responds to contrast (sharp local intensity change)

78

Scale Invariant Detection• Laplacian-of-Gaussian (LoG)• Difference of Gaussian (DOG)

79

Scale Invariant Detectors• Harris-Laplacian1

Find local maximum of:– Harris corner detector in

space (image coordinates)– Laplacian in scale

1 K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 20012 D.Lowe. “Distinctive Image Features from Scale-Invariant Keypoints”. Accepted to IJCV 2004

scale

x

y

Harris

Lap

laci

an

• SIFT (Lowe)2

Find local maximum of:– Difference of Gaussians in space

and scale

scale

x

y

DoG

DoG

Difference of Gaussian (DOG)

• Difference of Gaussian approximates the Laplacian )()( GkGDOG

Compare to human vision: eye’s response

Scale Invariant Detection• Functions for determining scale

2 2

21 22

( , , )x y

G x y e

2 ( , , ) ( , , )xx yyL G x y G x y

( , , ) ( , , )DoG G x y k G x y

Kernel Imagef Kernels:

where Gaussian

Note: both kernels are invariant to scale and rotation

(Laplacian)

(Difference of Gaussians)

82


Laplacian-of-Gaussian (LoG)for Harris-Laplace

• Local maxima in scale space of Laplacian-of-Gaussian

83

)()( yyxx LL

s

s2

s3

s4

s5

List of (x, y, s)

Computing Harris function

Detecting

local maxima

84

Harris-Laplace

• Two Parts:– Multiscale-Harris detector– Characteristic scale identification

85

Harris-Laplacepart 1

• Multiscale-Harris detector

The sets of scales:

Harris-Laplace [Mikolajczyk ‘01]

1. Initialization: Multiscale Harris corner detection2. Scale selection based on Laplacian

(same procedure with Hessian Hessian-Laplace)

86

Harris points

Harris-Laplace points

87

Harris-Laplace-part 2

• Characteristic scale identification

Choose the scale that maximizes the Laplacian-of-Gaussians (LoG) over a predefined range of neighboring scales.

88


Difference-of-Gaussian (DoG)

• Difference of Gaussians as approximation of the Laplacian-of-Gaussian

89K. Grauman, B. Leibe

- =

Convolution with the DoG filter

90

DoG DetectorScale space theory

Down-sample doubles for the next octave

K=2(1/s) = 2(1/4)

IkG *

IG *

IkG *2

IGkGD * Images separated by a constant factor k

D

kD

2kD

Convolution with a variable-scale GaussianDifference-of-Gaussian (DoG) filter

Sampling withstep s4 =2

91

DoG Detector

Scale Space Peak Detection

Find local maximum of Difference of Gaussians

92

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.

93

1. Detection of scale-space extrema (cont’d)

• Extract local extrema (i.e., minima or maxima) in DoG pyramid.- Compare each point to its 8 neighbors at the same level, 9 neighbors– in the level above, and 9 neighbors in the level below (i.e., 26 total).

D

kD

2kD

94

Choosing SIFT parameters• Experimentally using a matching task:

- 32 real images (outdoor, faces, aerial etc.)

- Images subjected to a wide range of transformations (i.e., rotation, scaling, shear, change in brightness, noise).

- Keypoints are detected in each image.

- Parameters are chosen based on keypoint repeatability, localization, and matching accuracy.

95

1. Scale-space Extrema Detection (cont’d)

• How many scales sampled per octave?3 scales

• S=3, for larger s, too many unstable features

# of keypoints increases but they are not stable!

96

1. Scale-space Extrema Detection (cont’d)

• Smoothing is applied to the first level of each octave.• How to choose σ? (i.e., integration scale)

σ =1.6

97

2. Accurate keypoint localization

• There are still a lot of points, some of them are not good enough.

– The locations of keypoints may be not accurate.– Eliminating edge points.

• Thus– Reject points with low contrast and poorly

localized along an edge– Fit a 3D quadratic function for sub-pixel maxima

98






99


• Determine the location and scale of keypoints to sub-pixel and sub-scale accuracy by fitting a 3D quadratic function at each keypoint.

• Substantial improvement to matching and stability!

( , , )i i i i iX x y X X

2. Accurate keypoint localization (cont’d)

• Use Taylor expansion of D(x,y,σ) (i.e., DoG function) around the sample point

where is the offset from this point.

2

2

( ) ( )1( ) ( )

2

TTi i

i

D X D XD X D X

( , , )i i iX x x y y

( , , )i i i iX x y

Taylor expansion with sample point as the origin

where

2

2

2

1)(

DDDD T

T

Tyx ),,(

101

2. Accurate keypoint localization (cont’d)

• Change sample point if offset is larger than 0.5

• Throw out low contrast (<0.03)

102






103

2. Keypoint Localization (cont’d)

Eliminating edge responsesReject points lying on edges (or being close to edges)• Such a point has large principal curvature across the

edge but a small one in the perpendicular direction• The principal curvatures can be calculated from a

Hessian function

• The eigenvalues of H are proportional to the principal curvatures, so two eigenvalues shouldn’t diff too much


• Reject points lying on edges (or being close to edges)

• Harris uses the 2nd order moment matrix:

2

2,

( , ) x x yW

x W y W x y y

f f fA x y

f f f

R(AW) = det(AW) – α trace2(AW)

or R(AW) = λ1 λ2- α (λ1+ λ2)2

2. Keypoint Localization (cont’d)• SIFT uses the Hessian matrix for efficiency.

– i.e., encodes principal curvatures

α: largest eigenvalue (λmax)β: smallest eigenvalue (λmin)(proportional to principal curvatures)

(SIFT uses r = 10)

(r = α/β)


(a) 233x189 image

(b) 832 DoG extrema

(c) 729 left after low contrast threshold

(d) 536 left after testing ratio based on Hessian

107

Scale Invariant Detectors

K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 2001

• Experimental evaluation of detectors w.r.t. scale change

Repeatability rate:


108

Scale Invariant Detection: Summary

• Given: two images of the same scene with a large scale difference between them

• Goal: find the same interest points independently in each image

• Solution: search for maxima of suitable functions in scale and in space (over the image)

Methods:

1. Harris-Laplacian [Mikolajczyk, Schmid]: maximize Laplacian over scale, Harris’ measure of corner response over the image

2. SIFT [Lowe]: maximize Difference of Gaussians over scale and space

109




» Moravec, harris, – Low’s key point

– Feature descriptor• SIFT


Outline

Motivation

Problems statement

How we solve it



Applications

Future Work

Conclusion

110

Feature descriptorsWe know how to detect good pointsNext question: How to match them?

?

Point descriptor should be:1. Invariant2. Distinctive

111

Feature descriptorsWe know how to detect good pointsNext question: How to match them?

Lots of possibilities (this is a popular research area)– Simple option: match square windows around the point– State of the art approach: SIFT

• David Lowe, UBC http://www.cs.ubc.ca/~lowe/keypoints/

?

http://www.cs.ubc.ca/~lowe/keypoints/

Feature descriptors

112

N p

ixel

s

N pixels

Similarity measureAf

e.g. color

Bf

e.g. color

B1

B2

B3A1

A2 A3

Tffd BA ),(

1. Find a set of distinctive keypoints

3. Extract and normalize the region content

2. Define a region around each keypoint

4. Compute a local descriptor from the normalized region

5. Match local descriptors

113

Feature Descriptorsmatch square windows around the point

114

SIFT

• Scale Invariant Feature Transform

D. Lowe, “Distinctive Image Features from Scale-Invariant Keypoints”, International Journal of Computer Vision, 60(2):91-110, 2004.

Cited 9589 times (as of 3/7/2011)

115

SIFT stages:

• 1.Scale-space extrema detection• 2.Keypoint localization• 3.Orientation assignment• 4.Keypoint descriptor

( )local descriptor

detector

descriptor

A 500x500 image gives about 2000 features

116

3. Orientation assignment

• By assigning a consistent orientation, the keypoint descriptor can be orientation invariant.

• For a keypoint, create histogram of gradient directions, within a region around the keypoint, at selected scale (i.e., scale invariance):

( 1, ) ( 1, )

( , 1) ( , 1)

L x y L x yGradientVector

L x y L x y

117

3. Orientation assignment (cont’d)

118


119


120

3.Orientation assignment (cont’d)

σ=1.5*scale of the keypoint

121


122


123

4. Keypoint Descriptor

• Have achieved invariance to location, scale, and orientation.

• Next, tolerate illumination and viewpoint changes.

4. Keypoint Descriptor (cont’d)

16 histograms x 8 orientations = 128 features

Main idea:1. Take a 16 x16

window around detected interest point.

2. Divide into a 4x4 grid of cells.

3. Compute histogram in each cell.

(8 bins)

125


• Descriptor has 3 dimensions (x,y,θ)• Orientation histogram of gradient magnitudes• Position and orientation of each gradient

sample rotated relative to keypoint orientation


128 features

• Descriptor depends on two parameters:(1) number of orientations r(2) n x n array of orientation histograms

•

SIFT: r=8, n=4

rn2 features

Why 4x4x8?


• Invariance to affine (linear) illumination changes:– Normalization to unit length is sufficient.

• Non-linear illumination changes :– Threshold gradient magnitudes to be no larger

than 0.2 and renormalize to unit length

128


Sensitivity to affine change

Correctely matched

129

SIFT demo

Detection of scale-space extrema

130

Maxima in D Remove low contrast

Remove edgesSIFT descriptor

Actual SIFT stage output

Local Descriptors• The ideal descriptor should be

– Repeatable– Distinctive– Compact– Efficient

• Most available descriptors focus on edge/gradient information– Capture texture information– Color still relatively seldomly used

(more suitable for homogenous regions)

131

The most successful feature (probably the most successful paper in computer vision)

Applications of SIFT

• Object recognition• Object categorization• Location recognition• Robot localization• Image retrieval• Image panoramas

133

Future Work• Region Detectors

Harris-/Hessian-Laplace Harris-/Hessian-Affine

• Region Matching• Texture Descriptors

– Gabor Wavelet Feature, – Local Binary Pattern(LBP), – Local Gabor Binary Pattern, – Local Directional pattern(LDP),– Histogram of Gabor Phase Pattern (HGPP)

• Learning local image descriptors (Winder et al 2007): tuning parameters given their dataset.

• Multimodal Retinal Image Registration

Outline

Motivation

Problems statement

How we solve it

Future Work

reference

Many Existing Detectors AvailableHessian & Harris [Beaudet ‘78], [Harris ‘88]Laplacian, DoG [Lindeberg ‘98], [Lowe 1999]Harris-/Hessian-Laplace [Mikolajczyk & Schmid ‘01]Harris-/Hessian-Affine[Mikolajczyk & Schmid ‘04]EBR and IBR [Tuytelaars & Van Gool ‘04] MSER [Matas ‘02]Salient Regions [Kadir & Brady ‘01] Others…

Comparison of Keypoint Detectors

Tuytelaars Mikolajczyk 2008

136

Reference

Outline

Motivation

Problems statement

How we solve it

Future Work

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.• “Local Invariant Feature Detectors: A Survey”, Tinne Tuytelaars and

Krystian Mikolajczyk, Computer Graphics and Vision, Vol. 3, No. 3 (2007) 177–280

http://www.csie.ntu.edu.tw/~cyy/courses/vfx/papers/Harris1988ACC.pdf

http://www.csie.ntu.edu.tw/~cyy/courses/vfx/papers/Lowe2004DIF.pdf

http://www.csie.ntu.edu.tw/~cyy/courses/vfx/papers/Lowe2004DIF.pdf

http://www.csie.ntu.edu.tw/~cyy/courses/vfx/papers/Ke2004PSM.pdf

http://www.csie.ntu.edu.tw/~cyy/courses/vfx/papers/Ke2004PSM.pdf

http://www.csie.ntu.edu.tw/~cyy/courses/vfx/papers/Mikolajczyk2004APE.pdf




ftp://ftp.cs.utoronto.ca/pub/jepson/teaching/vision/2503/SIFTtutorial.tar.gz

ftp://ftp.cs.utoronto.ca/pub/jepson/teaching/vision/2503/SIFTtutorial.tar.gz

137

Reference

• http://en.wikipedia.org/wiki/Scale-invariant_feature_transform

Outline

Motivation

Problems statement

How we solve it

Future Work

Reference

http://en.wikipedia.org/wiki/Scale-invariant_feature_transform

http://en.wikipedia.org/wiki/Scale-invariant_feature_transform

Documents

Computer Vision Local Invariant Features Mehdi Faraji [email protected] SLIDES have been prepared by: Dr. Ghassabi