CS 558 Computer Vision

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

FEATURE EXTRACTION: CORNERS9300 Harris Corners Pkwy, Charlotte, NC

WHY EXTRACT FEATURES?• Motivation: panorama stitching

We have two images – how do we combine them?

Step 1: extract featuresStep 2: match features

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

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

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

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

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

E(0,0)

w(x, y)

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

IntensityShifted intensity

Window function

orWindow function w(x,y) =

Gaussian1 in window, 0 outside

Source: R. Szeliski

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

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

E(u, v)

vuEvuEvvuv

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

)0,0()0,0(

][)0,0(),(

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

vuEvuEvvuv

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

)0,0()0,0(

][)0,0(),(

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

CORNER DETECTION: MATHEMATICS 2

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

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

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

yxIyxIyxwE

vuEvuEvvuv

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

)0,0()0,0(

][)0,0(),(

yxIyxwyxIyxIyxw

yxIyxIyxwyxIyxwvuvuE

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

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

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

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

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

yxIyxIyxwE

MvuvuE ][),(

The quadratic approximation simplifies to

( , ) x x y

x y x y y

I I IM w x y

where M is a second moment matrix computed from image derivatives:

The surface E(u,v) is locally approximated by a quadratic form. Let’s try to understand its shape.

INTERPRETING THE SECOND MOMENT MATRIX

MvuvuE ][),(

yx yyx

IIIIII

yx yyx

IIIIII

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][

This is the equation of an ellipse.

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

const][

This is the equation of an ellipse.

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

“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

“Flat” region

|R| small

2 )()(trace)det( MMR

α: constant (0.04 to 0.06)

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: 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

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

x (image coordinate)

threshold

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!

BLOB DETECTION

• 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

Source: S. Seitz

Derivativeof Gaussian

Edge = maximumof derivative

EDGE DETECTION, TAKE 2

gdxdf 2

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

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

BLOB DETECTION IN 2D Laplacian of Gaussian: Circularly symmetric

operator for blob detection in 2D

BLOB DETECTION IN 2D Laplacian of Gaussian: Circularly symmetric

operator for blob detection in 2D

norm yg

xgg Scale-normalized:

• At what scale does the Laplacian achieve a maximum response to a binary circle of radius r?

SCALE SELECTION

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

62/)(222 2/)2(222

yxeyx .2/r

circle

Laplacian

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 DETECTOR1. Convolve image with scale-normalized

Laplacian at several scales2. Find maxima of squared Laplacian response

in scale-space

SCALE-SPACE BLOB DETECTOR: EXAMPLE

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

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?

ACHIEVING AFFINE COVARIANCE• Affine transformation approximates viewpoint

changes for roughly planar objects and roughly orthographic cameras

ACHIEVING AFFINE COVARIANCE

RRIIIIII

yxwMyyx

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][

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

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 Computer Vision

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