Interest Point Detectors (see CS485/685 notes for more details)

Interest Point Detectors(see CS485/685 notes for more details)

CS491Y/691Y Topics in Computer Vision

Dr. George Bebis

What is an Interest Point?

• A point in an image which has a well-defined position and can be robustly detected.

• Typically associated with a significant change of one or more image properties simultaneously (e.g., intensity, color, texture).

What is an Interest Point? (cont’d)

• Corners is a special case of interest points.

• However, interest points could be more generic than corners.

Why are interest points useful?

• Could be used to find corresponding points between images which is very useful for numerous applications!

stereo matching

panorama stitching

left camera right camera

How to find corresponding points?

• Need to define local patches surrounding the interest points and extract feature descriptors from every patch.

• Match feature descriptors to find corresponding points.

( ) ( )=?

featuredescriptor

featuredescriptor

?

Properties of good features

• Local:Local: robust to occlusion and clutter.• Accurate:Accurate: precise localization.

• CovariantCovariant• Robust:Robust: noise, blur, compression, etc.

• Efficient:Efficient: close to real-time performance.

RepeatableRepeatable

Interest point detectors should be covariant

• Features should be detected in corresponding locations despite geometric or photometric changes.

( ) ( )=?

featuredescriptor

featuredescriptor

?

Interest point descriptors should be invariant

Should be similar despite geometricor photometric transformations

Interest point candidates

Use features with gradients in at least two, significantly different orientations (e.g., corners, junctions etc)

Harris Corner Detector

• AW(x,y) is a 2 x 2 matrix called auto-correlation matrix.

• fx, fy are the horizontal and vertical derivatives.

• w(x,y) is a smoothing function (e.g., Gaussian)

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

• Assuming a W x W window, it computes the matrix:

2

2,

( , ) x x yW

x W y W x y y

f f fA w x y

f f f

Why is the auto-correlation matrix useful?

Describes the gradient distribution (i.e., local structure) inside thewindow!

2

2,

( , ) x x yW

x W y W x y y

f f fA w x y

f f f

Properties of the auto-correlation matrix

Aw is symmetric and can be decomposed :

11

2

0

0WA R R

(max)1/2(min)1/2

• We can visualize AW as an ellipse with axis lengths and directions determined by its eigenvalues and eigenvectors.

Harris Corner Detector (cont’d)

• The eigenvectors of AW encode direction of intensity change.

• The eigenvalues of AW encode strength of intensity change.

v1

v2

(max)1/2(min)1/2 direction of

the fastest change

direction of the slowest change


1

2

“Corner”1 and 2 are large,

1 ~ 2;

intensity changes in all directions

1 and 2 are small;

intensity is almost constant in all directions

“Edge” 1 >> 2

“Edge” 2 >> 1

“Flat” region

Classification of pixels using the eigenvalues of AW :


• To avoid computing the eigenvalues explicitly, the Harris detector uses the following function:

R(AW) = det(AW) – α trace2(AW)

which is equal to:

R(AW) = λ1 λ2- α (λ1+ λ2)2

α: is a const


“Corner”R > 0

“Edge” R < 0

“Edge” R < 0

“Flat” region

|R| small α: is usually between

0.04 and 0.06


Classification of image

points using R(AW):


• Other functions:

2 1

1 2

1 2

det

a

A

trA

Harris Corner Detector - Steps

( , , )* ( , )x D i if G x y f x yx

( , , )* ( , )y D i if G x y f x y

y

σD is called the “differentiation” scale

1. Compute the horizontal and vertical (Gaussian) derivatives

2. Compute the three images involved in AW :

2

2,

( , ) x x yW

x W y W x y y

f f fA w x y

f f f

Harris Detector - Steps


3. Convolve each of the three images with a larger Gaussian

w(x,y) :

Gaussian

σI is called the “integration” scale

4. Determine cornerness:

5. Find local maxima

Harris Corenr Detector - Example

Harris Detector - Example

Compute corner response R

Find points with large corner response: R>threshold

Take only the points of local maxima of R

Map corners on the original image (for visualization)


• Rotation invariant

• Sensitive to:• Scale change• Significant viewpoint change• Significant contrast change

Multi-scale Harris Detector

scale

x

y

Harris

• Detect interest points at varying scales.

R(AW) = det(AW(x,y,σI,σD)) – α trace2(AW(x,y,σI,σD))

σσnnσD= σn

σI=γσD

σn=knσ

Multi-scale Harris Detector (cont’d)

• The same interest point will be detected at multiple consecutive scales.

• Interest point location will shift as scale increases (i.e., due to smoothing).

i.e., the size of each circle corresponds to the scale at which the interest point was detected.

How do we match them?

• Corresponding features might appear at different scales.

• How do we determine these scales?• We need a scale selection mechanism!

Exhaustive Search

• Simple approach for scale selection but not efficient!

Characteristic Scale

• Find the characteristic scale of each feature (i.e., the scale revealing the spatial extent of an interest point).

characteristic scale characteristic scale

Characteristic Scale (cont’d)

• Only a subset of interest points are selected using the characteristic scale of each feature.

• i.e., the size of the circles is related to the scale at which the interest points were selected.

Matching can be simplified!

Automatic Scale Selection – Main Idea

• Design a function F(x,F(x,σσnn)) which provides some local measure.

• Select points at which F(x,F(x,σσnn)) is maximal over σσnn.

T. Lindeberg, "Feature detection with automatic scale selection" International Journal of Computer Vision, vol. 30, no. 2, pp 77-116, 1998.

max of F(x,F(x,σσnn))corresponds to characteristic scale!

σσnn

F(x,F(x,σσnn))

Slide from Tinne Tuytelaars

Lindeberg et al, 1996

Automatic Scale Selection (cont’d)

• Using characteristic scale, the spatial extent of interest points becomes covariant to scale transformations.

• The ratio σ1/σ2 reveals the scale factor between the images.

σ1/σ2 = 2.5

σ1 σ2

How to choose F(x,F(x,σσnn)) ?

• Typically, F(x,F(x,σσnn)) is defined using derivatives, e.g.:

• LoG (Laplacian of Gaussian) yielded best results in a recent evaluation study.

• DoG (Difference of Gaussians) was second best.

2 2 2

2

1

2

: ( ( , ) ( , ))

: | ( ( , ) ( , )) |

:| ( )* ( ) ( )* ( ) |

: det( ) ( )

x y

xx yy

n n

W W

Square gradient L x L x

LoG L x L x

DoG I x G I x G

Harris function A trace A

C. Schmid, R. Mohr, and C. Bauckhage, "Evaluation of Interest Point Detectors", International Journal of Computer Vision, 37(2), pp. 151-172, 2000.

LoG and DoG

• LoG can be approximated by DoG:

2 2( , , ) ( , , ) ( 1)G x y k G x y k G

σσnn

Harris-Laplace Detector

scale

x

y

Harris

L

oG

• Multi-scale Harris with scale selection.• Uses LoG maxima to find characteristic scale.

Harris-Laplace Detector (cont’d)

(1) Find interest points at multiple scales using Harris detector.- Scales are chosen as follows: σn =knσ

- At each scale, choose local maxima assuming 3 x 3 window

where2( , ) det( ) ( )n W wF x A trace A

( , ) ( , )

( , )n W n W

n h

F x F x x W

F x t

(σD =σn, σI =γσD )

(i.e., non-maximal suppression)


(2) Select points at which the normalized LoG is maximal across scales and the maximum is above a threshold.

K. Mikolajczyk and C. Schmid, “Indexing based on scale invariant interest points" IEEE Int. Conference on Computer Vision, pp 525-531, 2001.

2( , ) | ( ( , ) ( , )) |n xx n yy nF x L x L x

1 1( , ) ( , ) ( , ) ( , )

( , )n n n n

n

F x F x F x F x

F x t

where:σn

σn+1

σn-1

Example

• Interest points detected at each scale using Harris-Laplace– Few correspondences between levels corresponding to same σ.

– More correspondences between levels having a ratio of σ = 2.

images differ by a scale factor of 1.92

σ=1.2 σ=2.4 σ=4.8 σ=9.6

σ=1.2 σ=2.4 σ=4.8 σ=9.6

Example (cont’d)

-More than 2000 points would have been detected without scale selection.-Using scale selection, 190 and 213 points were detected in the left and right images, respectively.

(same viewpoint – change in focal length and orientation)

Example (cont’d)

58 points are initially matched (some were not correct)

Example (cont’d)

• Reject outliers (i.e., inconsistent matches) using RANSAC• Left with 32 matches, all of which are correct.•Estimated scale factor is 4:9 and rotation angle is 19 degrees.


• Invariant to:– Scale

– Rotation

– Translation

• Robust to: – Illumination changes

– Limited viewpoint changes

Repeatability

Harris-Laplace using DoG

Look for local maxima in DoG pyramid

DoG pyramid

David G. Lowe, "Distinctive image features from scale-invariant keypoints.” Int. Journal of Computer Vision, 60 (2), pp. 91-110, 2004.

Handling Affine Changes

• Similarity transformations cannot account for perspective distortions; affine could be used for planar surfaces.

• Similarity transform

• Affine transform

Harris Affine Detector

• Use an iterative approach:– Extract approximate locations and scales using the Harris-

Laplace detector.

– For each point, modify the scale and shape of its neighborhood in an iterative fashion.

– Converges to stable points that are covariant to affine transformations.

.

Steps of Iterative Affine Adaptation

1. Detect initial locations and neighborhood using Harris-Laplace.

2. Estimate affine shape of neighborhood using 2nd order moment matrix μ(x, σI, σD).

Steps of Iterative Affine Adaptation (cont’d)

3. Normalize (i.e., de-skew) the affine region by mapping it to a circular one (i.e., “remove” perspective distortions).

4. Re-detect the new location and scale in the normalized image.

5. Go to step 2 if the eigenvalues of μ(x, σI, σD) for the new point are not equal (i.e., not yet adapted to the characteristic shape).

Iterative affine adaptation - Examples

Initial points

Example 1 Example 2

Iterative affine adaptation – Examples (cont’d)

Iteration #1

Example 1 Example 2


Iteration #2

Example 1 Example 2


Iteration #3, #4, …

K. Mikolajczyk and C. Schmid, “Scale and Affine invariant interest point detectors”, International Journal of Computer Vision, 60(1), pp. 63-86, 2004.

http://www.robots.ox.ac.uk/~vgg/research/affine/

Example 1 Example 2


De-skewing

• Consider a point xL with 2nd order matrix ML

• The de-skewing transformation is defined as follows:

xL

De-skewing corresponding regions

• Consider two points xL and xR which are related through an affine transformation:

xL xR

De-skewing corresponding regions (cont’d)

Normalized regionsare related by purerotation R.

Resolving orientation ambiguity

• Create histogram of local gradient directions in the patch.

• Smooth histogram and assign canonical orientation at peak of smoothed histogram.

Dominant gradientdirection!

0 2

(36 bins)

Resolving orientation ambiguity (cont’d)

• Resolve orientation ambiguity.

Compute R

Other Affine Invariant Blob/Region Detectors

• There are many other techniques for detecting affine invariant blobs or regions, for example:

– Intensity Extrema-Based Region (IER)

– Maximally Stable Extremal Regions (MSERs)

• No need to detect interest points.

Intensity Extrema-Based Region Detector(1) Take a local intensity extremum as

initial point.

(2) Examine the intensity along “rays” from the local extremum.

(3) The point on each ray for which fI(t) reaches an extremum is selected (i.e., invariant).

(4) Linking these points together yields an affinely invariant region, to which an ellipse is fitted.

Tuytelaars, T. and Van Gool, L. “Matching Widely Separated Views based on Affine Invariant Regions”, International Journal on Computer Vision, 59(1):61–85, 2004.

d>0: small const

Intensity Extrema-Based Region Detector (cont’d)

• The regions found may not exactly correspond, so we approximate them with ellipses using 2nd order moments.


• Double the size of the ellipse to make regions more distinctive.

• Final ellipse is not necessarily centered at original anchor point.



• Region extraction is fairly robust to inaccurate localization of intensity extremum.

Maximally Stable Extremal Regions (MSERs)

• Consider all possible thresholdings of a gray-scale image:

If I(x,y) > It then I(x,y)=255; else I(x,y)=0

Matas, J., Chum, O., Urban, M. and Pajdla, T., “Robust wide-baseline stereo from maximally stable extremal regions”, British Machine Vision Conf, pp. 384–393, 2002.

Maximally Stable Extremal Regions (MSERs)

• Extermal Regions– All regions formed using different thresholds.– Can be extracted using connected components.

• Maximally Stable Extremal Regions– Regions that remain stable over a large range of thresholds.

• Approximate MSER by an ellipse

Example

• Extermal regions can be extracted in O(nlog(logn)) time where n is the number of pixels.

• Sensitive to image blur.

Interest Point Detectors (see CS485/685 notes for more details)

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