cs 558 c omputer v ision lecture vii: corner and blob detection slides adapted from s. lazebnik

CS 558 COMPUTER VISIONLecture VII: 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

FEATURE EXTRACTION: CORNERS9300 Harris Corners Pkwy, Charlotte, NC

OUTLINE



WHY EXTRACT FEATURES?

• Motivation: panorama stitching We have two images – how do we combine

them?



them?

Step 1: extract featuresStep 2: match features



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



FINDING CORNERS

• Key property: in the region around a corner, image gradient has two or more dominant directions

• Corners are repeatable and distinctive

C.Harris and M.Stephens. "A Combined Corner and Edge Detector.“ Proceedings of the 4th Alvey Vision Conference: pages 147—151, 1988.

http://www.csse.uwa.edu.au/~pk/research/matlabfns/Spatial/Docs/Harris/A_Combined_Corner_and_Edge_Detector.pdf

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 directions

Source: 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)


2

,

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


I(x, y)E(u, v)

E(0,0)

w(x, y)



2

,

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


IntensityShifted intensity

Window function

orWindow function w(x,y) =

Gaussian1 in window, 0 outside

Source: R. Szeliski



2

,

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


We want to find out how this function behaves for small shifts


E(u, v)


v

u

EE

EEvu

E

EvuEvuE

vvuv

uvuu

v

u

)0,0()0,0(

)0,0()0,0(][

2

1

)0,0(

)0,0(][)0,0(),(

2

,

( , ) ( , ) ( , ) ( , )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



v

u

EE

EEvu

E

EvuEvuE

vvuv

uvuu

v

u

)0,0()0,0(

)0,0()0,0(][

2

1

)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


2

,

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


),(),(),(2)0,0(

),(),(),(2)0,0(

),(),(),(2)0,0(

0)0,0(

0)0,0(

0)0,0(

,

,

,

yxIyxIyxwE

yxIyxIyxwE

yxIyxIyxwE

E

E

E

yxyx

uv

yyyx

vv

xxyx

uu

v

u

v

u

EE

EEvu

E

EvuEvuE

vvuv

uvuu

v

u

)0,0()0,0(

)0,0()0,0(][

2

1

)0,0(

)0,0(][)0,0(),(


v

u

yxIyxwyxIyxIyxw

yxIyxIyxwyxIyxw

vuvuE

yxy

yxyx

yxyx

yxx

,

2

,

,,

2

),(),(),(),(),(

),(),(),(),(),(

][),(

2

,

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


),(),(),(2)0,0(

),(),(),(2)0,0(

),(),(),(2)0,0(

0)0,0(

0)0,0(

0)0,0(

,

,

,

yxIyxIyxwE

yxIyxIyxwE

yxIyxIyxwE

E

E

E

yxyx

uv

yyyx

vv

xxyx

uu

v

u


v

uMvuvuE ][),(

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

v

uMvuvuE ][),(

yx yyx

yxx

III

IIIyxwM

,2

2

),(

2

1

,2

2

0

0),(

yx yyx

yxx

III

IIIyxwM

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.


Consider a horizontal “slice” of E(u, v):


const][

v

uMvu

This is the equation of an ellipse.

Consider a horizontal “slice” of E(u, v):


const][

v

uMvu

This is the equation of an ellipse.

RRM

2

11

0

0

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

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



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.

http://www.bmva.org/bmvc/1988/avc-88-023.pdf



HARRIS DETECTOR: STEPSCompute corner response R


Find points with large corner response: R>threshold


Take only the points of local maxima of R

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



BLOB DETECTION

OUTLINE



• 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

gdx

df

f

gdx

d

Source: S. Seitz

Edge

Derivativeof Gaussian

Edge = maximumof derivative

EDGE DETECTION, TAKE 2

gdx

df

2

2

f

gdx

d2

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

2

1

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



BLOB DETECTION IN 2D

Laplacian of Gaussian: Circularly symmetric operator for blob detection in 2D

2

2

2

22

y

g

x

gg

BLOB DETECTION IN 2D

Laplacian of Gaussian: Circularly symmetric operator for blob detection in 2D

2

2

2

222

norm y

g

x

gg 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 scale

T. Lindeberg (1998). "Feature detection with automatic scale selection." International Journal of Computer Vision 30 (2): pp 77--116.

http://www.nada.kth.se/cvap/abstracts/cvap198.html

SCALE-SPACE BLOB DETECTOR

1. Convolve image with scale-normalized Laplacian at several scales

SCALE-SPACE BLOB DETECTOR: EXAMPLE

SCALE-SPACE BLOB DETECTOR

1. Convolve image with scale-normalized Laplacian at several scales

2. Find maxima of squared Laplacian response in scale-space

SCALE-SPACE BLOB DETECTOR: EXAMPLE

OUTLINE



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.

http://www.cs.ubc.ca/~lowe/papers/ijcv04.pdf

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



ACHIEVING AFFINE COVARIANCE

• Affine transformation approximates viewpoint changes for roughly planar objects and roughly orthographic cameras

ACHIEVING AFFINE COVARIANCE

RRIII

IIIyxwM

yyx

yxx

yx

2

112

2

, 0

0),(

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][

v

uMvu

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 p

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

cs 558 c omputer v ision lecture vii: corner and blob detection slides adapted from s. lazebnik

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