Feature extraction: Corners

9300 Harris Corners Pkwy, Charlotte, NC

Why extract features?• Motivation: panorama stitching

• We have two images – how do we combine them?

Why extract features?• Motivation: panorama stitching

• We have two images – how do we combine them?

Step 1: extract featuresStep 2: match featuresStep 2: match features

Why extract features?• Motivation: panorama stitching

• We have two images – how do we combine them?

Step 1: extract featuresStep 2: match featuresStep 2: match featuresStep 3: align images

Characteristics of good features

• Repeatability• The same feature can be found in several images despite geometric g p g

and photometric transformations

• Saliency• Each feature is distinctive

• Compactness and efficiency• Many fewer features than image pixels

• LocalityLocality• A feature occupies a relatively small area of the image; robust to

clutter and occlusion

Applications Feature points are used for:

• Image alignment • 3D reconstruction• Motion tracking

R b t i ti• Robot navigation• Indexing and database retrieval• Object recognition• Object recognition

Finding Corners

• Key property: in the region around a corner, i di h d iimage gradient has two or more dominant directionsC t bl d di ti ti• Corners are repeatable and distinctive

C H i d M St h "A C bi d C d Ed D t t “C.Harris and M.Stephens. "A Combined Corner and Edge Detector.“Proceedings of the 4th Alvey Vision Conference: pages 147--151.

Corner Detection: Basic Idea

• We should easily recognize the point by looking through a small windowShifti i d i di ti h ld• Shifting a window in any direction should give a large change in intensity

“edge”:no change

“corner”:significant

“flat” region:no change in no change

along the edge direction

significant change in all directions

no change in all directions

Source: A. Efros

Corner Detection: MathematicsChange in appearance of window w(x,y)

for the shift [u,v]:

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

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

E(3,2)

w(x, y)

Corner Detection: MathematicsChange in appearance of window w(x,y)

for the shift [u,v]:

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

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

E(0,0)

w(x, y)

Corner Detection: MathematicsChange in appearance of window w(x,y)

for the shift [u,v]:

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

IntensityShifted intensity

Window function intensityfunction

orWindow function w(x,y) =

G i1 i i d 0 id Gaussian1 in window, 0 outside

Source: R. Szeliski

Corner Detection: MathematicsChange in appearance of window w(x,y)

for the shift [u,v]:

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

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

E(u, v)

Corner Detection: MathematicsChange in appearance of window w(x,y)

for the shift [u,v]:

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

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

Local quadratic approximation of E(u,v) in the neighborhood of (0 0) is given by the second order

small shifts

neighborhood of (0,0) is given by the second-order Taylor expansion:

⎤⎡⎤⎡⎤⎡ uEEE )00()00(1)00(⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡+⎥

⎦

⎤⎢⎣

⎡+≈

vu

EEEE

vuEE

vuEvuEvvuv

uvuu

v

u

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

][21

)0,0()0,0(

][)0,0(),(

Corner Detection: Mathematics

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

Second-order Taylor expansion of E(u,v) about (0,0):

⎥⎤

⎢⎡⎥⎤

⎢⎡

+⎥⎤

⎢⎡

+≈uEE

vuE

vuEvuE uvuuu )0,0()0,0(][1)0,0(

][)0,0(),( ⎥⎦

⎢⎣⎥⎦

⎢⎣

+⎥⎦

⎢⎣

+≈vEE

vuE

vuEvuEvvuvv )0,0()0,0(

][2)0,0(

][)0,0(),(

[ ] ),(),(),(),(2),( vyuxIyxIvyuxIyxwvuE xu ++−++=∑

[ ]

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

,

vyuxIvyuxIyxwvuE xxyx

uu

yx

++++=

∑

∑[ ]

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

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

vyuxIvyuxIyxwvuE

vyuxIyxIvyuxIyxw

xyuv

xxyx

++++=

++−+++

∑

∑

[ ] ),(),(),(),(2,

,

vyuxIyxIvyuxIyxw xyyx

xyyx

uv

++−+++∑

∑

Corner Detection: Mathematics

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

Second-order Taylor expansion of E(u,v) about (0,0):

⎥⎤

⎢⎡⎥⎤

⎢⎡

+⎥⎤

⎢⎡

+≈uEE

vuE

vuEvuE uvuuu )0,0()0,0(][1)0,0(

][)0,0(),(

0)0,0(E =

⎥⎦

⎢⎣⎥⎦

⎢⎣

+⎥⎦

⎢⎣

+≈vEE

vuE

vuEvuEvvuvv )0,0()0,0(

][2)0,0(

][)0,0(),(

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

EE

v

u

∑==

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

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

yxIyxIyxwE

yxIyxIyxwE

yyvv

xxyx

uu

∑

∑=

=

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

,

yxIyxIyxwE yxyx

uv

yyyx

∑

∑=

Corner Detection: Mathematics

[ ]2

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

x yE u v w x y I x u y v I x y= + + −∑

Second-order Taylor expansion of E(u,v) about (0,0):

⎥⎤

⎢⎡⎥⎥⎤

⎢⎢⎡

≈∑∑ u

yxIyxIyxwyxIyxwvuvuE yx

yxyx

x,,

2 ),(),(),(),(),(][)( ⎥

⎦⎢⎣⎥⎥⎦⎢

⎢⎣

≈∑∑ vyxIyxwyxIyxIyxw

vuvuE

yxy

yxyx

yy

,

2

,

,,

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

0)0,0(E =

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

EE

v

u

∑==

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

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

yxIyxIyxwE

yxIyxIyxwE

yyvv

xxyx

uu

∑

∑=

=

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

,

yxIyxIyxwE yxyx

uv

yyyx

∑

∑=

Corner Detection: MathematicsThe quadratic approximation simplifies to

⎤⎡u⎥⎦

⎤⎢⎣

⎡≈

vu

MvuvuE ][),(

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

2

2( , ) x x yI I IM w x y

⎡ ⎤= ⎢ ⎥

⎢ ⎥∑

de at es

2,

( , )x y x y y

yI I I⎢ ⎥⎢ ⎥⎣ ⎦

∑

M

Interpreting the second moment matrix

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

⎤⎡⎥⎦

⎤⎢⎣

⎡≈

vu

MvuvuE ][),(

∑ ⎥⎤

⎢⎡

= yxx IIIyxwM

2

)(∑⎥⎥⎦⎢

⎢⎣

=yx yyx III

yxwM,

2),(

Interpreting the second moment matrix

First, consider the axis-aligned case (gradients are either horizontal or vertical)

⎤⎡⎤⎡ 2 0λIII

(gradients are either horizontal or vertical)

⎥⎦

⎤⎢⎣

⎡=

⎥⎥⎦

⎤

⎢⎢⎣

⎡=∑

2

1

,2 0

0),(

λλ

yx yyx

yxx

IIIIII

yxwM⎦⎣ yy

If either λ is close to 0, then this is not a corner, so look for locations where both are large.

Interpreting the second moment matrix⎤⎡

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

This is the equation of an ellipse

const][ =⎥⎦

⎤⎢⎣

⎡vu

Mvu

This is the equation of an ellipse.

Interpreting the second moment matrix⎤⎡

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

This is the equation of an ellipse

const][ =⎥⎦

⎤⎢⎣

⎡vu

Mvu

This is the equation of an ellipse.

RRM ⎥⎦

⎤⎢⎣

⎡= −

2

11

00λ

λDiagonalization of M:

⎦⎣ 20 λThe axis lengths of the ellipse are determined by the eigenvalues and the orientation is determined by R

direction of the

direction of the fastest change

slowest change

(λmax)-1/2

(λ i )-1/2(λmin)

Visualization of second moment matrices

Visualization of second moment matrices

Interpreting the eigenvalues

Classification of image points using eigenvalues of M:

λ2

“Corner”λ d λ l

“Edge” λ2 >> λ1

λ1 and λ2 are large,λ1 ~ λ2;E increases in all directions

λ1 and λ2 are small;E is almost constant in all directions

“Edge” λ1 >> λ2

“Flat” region

λ1

1 2region

Corner response function2

21212 )()(trace)det( λλαλλα +−=−= MMR

t t (0 04 t 0 06)

“Corner”R > 0

“Edge” R < 0

α: constant (0.04 to 0.06)

R > 0

|R| ll“Edge” R < 0

“Flat” region

|R| small

region

Harris detector: Steps

1. Compute Gaussian derivatives at each pixel2 Compute second moment matrix M in a2. Compute second moment matrix M in a

Gaussian window around each pixel 3 Compute corner response function R3. Compute corner response function R4. Threshold R5 Find local maxima of response function5. Find local maxima of response function

(nonmaximum suppression)

C H i d M St h “A C bi d C d Ed D t t ”C.Harris and M.Stephens. “A Combined Corner and Edge Detector.” Proceedings of the 4th Alvey Vision Conference: pages 147—151, 1988.

Harris Detector: Steps

Harris Detector: StepsC t RCompute corner response R

Harris Detector: StepsFi d i t ith l R>thresholdFind points with large corner response: R>threshold

Harris Detector: StepsT k l th i t f l l i f RTake only the points of local maxima of R

Harris Detector: Steps

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 changeInvariance: 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

I → a I + b

• Only derivatives are used => invariance to intensity shift I → I + b

• Intensity scaling: I → a I

R RRthreshold

R

x (image coordinate) x (image coordinate)

Partially invariant to affine intensity change

Image translation

• Derivatives and window function are shift-invariant

Corner location is covariant w r t translationCorner location is covariant w.r.t. translation

Image rotation

Second moment ellipse rotates but its shapeSecond moment ellipse rotates but its shape (i.e. eigenvalues) remains the same

Corner location is covariant w r t rotationCorner location is covariant w.r.t. rotation

Scaling

Corner

All points will be classifiedbe classified as edges

Corner location is not covariant to scaling!