Corner and Interest Point Detection

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Copyright 2012 Elsevier Inc. All rights reserved.

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Chapter 6

Corner and Interest Point

Detection


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6.2 Template Matching

The suitable templates for corner templateshave the general appearance of corners:

The complete set of eight templates would be

generated by successive 90 degree rotation of

the above operators.The masks are made to sum to zero , so that

corner detection is insensitive to absolute

changes in light intensity.Copyright 2012 Elsevier Inc. All rights reserved.

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4-4-4-

4-54-

555

444

554

554


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6.3 Second-Order Derivative

Schemes

The corner pixels arise where two relatively

straight-edged fragments intersect.

We consider local variations in image intensity

up to at least second order. Hence, the localintensity variation is expanded as follows:


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The symmetrical matrix of second derivatives is:

This gives information on the local curvature at

X0. In fact, a suitable rotation of coordinates

system transforms I(2)into diagonal form:

Where appropriate derivatives have been

reinterpreted as principal curvature at X0.Copyright 2012 Elsevier Inc. All rights reserved.

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We are particularly interested in rotationallyinvariant operators and obtain the Beaudet

(1978) operators:

The Laplacian operators give significant

responses along lines and edges.

DET operators gives significant signals in the

vicinity of corners.


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FIGURE 6.1


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KR corner detector (Kitchen and Rosenfeld,

1983).

DN corner detector (Dreschler and Nagel, 1984)

ZH corner detector (Zuniga and Haralick, 1983)


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6.4 A Median Filter-Based Corner

Detector

The technique involves applying a median filter to theinput image, and then forming another image that is the

difference between the input and the filtered images.

This difference image contains a set of signals that are

interpreted as local measures of corner strength.The median corner detector gives zero signal if the

horizontal curvature is locally zero.

Corner strength is closely related to corner contrast and

corner sharpness.To summarize, the signal from the median-based corner

detector is proportional to horizontal curvature and to

intensity gradient.


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FIGURE 6.2

FIGURE 6.2(a) Contours of constant

intensity within a small

neighborhood: ideally, these

are parallel, circular and of

approximately equalcurvature (the contour of

median intensity does not

pass through the center of

the neighborhood); (b) cross-

section of intensity variation,

indicating how the

displacement D of the

median contour leads to an

estimate of corner strength.

Source: Springer 1988


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FIGURE 6.3


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FIGURE 6.4 (1/2)

FIGURE 6.4

Comparison of the median and KR corner detectors: (a) original 128X128

grayscale image; (b) result of applying a median detector; (c) result of

including a suitable gradient threshold; (d) result of applying a KR detector.

The considerable amount of background noise is saturated out in (a) but is

evident from (b). To give a fair comparison between the median and KR

detectors, 5X5 neighborhoods are employed in each case, and

nonmaximum suppression operations are not applied: the same gradient

threshold is used in (c) and (d). Source: Springer 1988


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FIGURE 6.4 (2/2)

FIGURE 6.4

Comparison of the median and KR corner detectors: (a) original 128X128

grayscale image; (b) result of applying a median detector; (c) result of

including a suitable gradient threshold; (d) result of applying a KR detector.

The considerable amount of background noise is saturated out in (a) but is

evident from (b). To give a fair comparison between the median and KR

detectors, 5X5 neighborhoods are employed in each case, and

nonmaximum suppression operations are not applied: the same gradient

threshold is used in (c) and (d).

Source: Springer 1988


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6.5 The Harris Interest Point

Operator

The second-order derivative corner detector wasdesigned on the basis that corners are ideal,

smoothly varying differentiable intensity profiles.

The media filter-based detectors were found to

be suitable for processing curved step edges.The Harris operator only takes account of first-

order derivatives of the intensity function.

The Harris operator is defined simply, interms ofthe local components of intensity gradient Ix, Iyin

an image and requires a window region to be

defined and averages are taken over the

window.Copyright 2012 Elsevier Inc. All rights reserved..

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We start by computing the following matrix:

The determinant and trace are used to estimate

the corner signal:

We work the sums of quadratic products ofintensity gradients:


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Consider the operation of the detectorfor a

single edge (Fig. 6.5(a))

Next, consider the situation in a corner region(Fig. 6.5(b)):

where l1and l2are the lengths of the two edges

bounding the corner, and g is the edge contrast.


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FIGURE 6.5


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We now find:


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The above equation may be interpreted as theproduct of

(1)A strength factor , which depends on the edge

length within the window,

(2)A contrast factor ,

(3)A shape factor , which depends on the

edge sharpness

C is zero for , and is maximum for


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2g

2

sin

and02/


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Suppose we set L=l1+l2=constant, then l1=L-l2,

which is zero for l2=L/2, at which point l1=l2.

This means that the best way of obtainingmaximum corner signal is to place the corner

symmetrically within the window.


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FIGURE 6.6


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6.5.1 Corner Signals and Shifts for Various

Geometric Configurations

When , this leads to C=0.

When is small ( ), we can go on

increasing L by moving the corner symmetrically.

The optimum is reached exactly as the tip of thecorner reaches the far side of the window (Fig.

6.6(c) ).

The signal will fall when the corner moves

further (Fig. 6.6(d) ).

When , the optimum will still occur when

the tip of the corner lies on the far side of

window (Fig.6.7(a)).Copyright 2012 Elsevier Inc. All rights reserved..

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0

2/

2/


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However, further increase of will result in adifferent optimum condition (Fig.6-7(b-d)).

We can see this in the symmetrical case l1=l2,

so reduction of L will reduce and C will fall.

This situation continues until , at which

point C again falls to zero.


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sym


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FIGURE 6.7


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The detector places the maximum output signalat the corner of the window in the cases where

the signal is stated to be optimum above.

The shift produced has s size equal to radius a

of a window for small corner angles, as then thetip of the corner is symmetrically placed on the

boundary of the window.

When , simple geometry (Fig.6.8(a))

shows that the shift is given by:


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FIGURE 6.8

6 5 2 P f ith C i


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6.5.2 Performance with Crossing

Points and Junctions

Crossing points (Fig. 6.9 a-b, Fig. 6.10 a):

Eq.(6.17) still applies. However, l1,l2must now

be taken as the sum of the edge lengths in each

of the two main directions.There is an important point to note that along the

two edge directions the signs of the contrast

values both reverse at the crossing point.

When the window is centered at the crossing

point, which is the tip of both constituent corners,

the values of l1and l2 are doubled.


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Another relevant factor is that the corner

configuration is now symmetric about the

crossing point.

The global maximum signal must occur whenthere is a maximum length of both edges within

the window, and they must therefore be closely

aligned along the window diameters.

These remarks apply both for and for

oblique crossovers (Fig. 6.9(a, b) and 6.10 a.


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2/


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FIGURE 6.9


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T-junction: a more general case in that they are

mediated by three regions with three different

intensities (Fig. 6.9 c-d, and Fig. 6.10b).

To calculate the corner signal, we first

generalize Eq. (6.17) to take into account thefact that one line will have higher contrast than

the other:

Where l1is taken as the straight edge with high

contrast g1, and l2is the straight edge with low

contrast g2.Copyright 2012 Elsevier Inc. All rights reserved..

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FIGURE 6.10


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FIGURE 6.11 (1/4)

FIGURE 6.11Application of the Harris interest point detector. (a) Original image. (b) Interest

point feature strength


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FIGURE 6.11 (2/4)

FIGURE 6.11

Application of the Harris interest point detector.

(c) Map of interest points showing only those giving greatest response over a

distance of 5 pixels: (d) their placement in the original image


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FIGURE 6.11 (3/4)

FIGURE 6.11


(e and f) Corresponding results for interest points giving greatest response over

a distance of 7 pixels.


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FIGURE 6.11 (4/4)

FIGURE 6.11


(g and h) Later frames in the sequence (also using maximum responses over a

distance of 7 pixels), showing a high consistency of feature identification,

which is important for tracking purposes. Note that interest points really do

indicate locations of interestXcorners, peoples feet, ends of white road

markings, and castle window and battlement features. Also, the greater the

significance as measured by the pixel suppression range, the greater

relevance the feature tends to have.


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6.6 Corner Orientation

Orientation information is valuable in

immediately eliminating a large number of

possible interpretations of an image, and hence

of quickly narrowing down the search problemand saving computation.

Once a corner has been located accurately, the

orientation can be estimated by finding the mean

intensity gradient over a small regionsurrounding the corner position, i.e. using the

components Ixand Iy.


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FIGURE 6.12

6 7 Local Invariant Feature


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6.7 Local Invariant Feature

Detectors and Descriptors

Feature detection must be consistent and

repeatable in spite of substantial change of view

point.

Features must embody descriptions of theirlocalities so that there is high probability that the

same physical feature will be positively identified

in each view.

Usually, the number of features of each image is

large (~1000), it is highly important to minimize

the feature matching task,


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Broadly speaking, obtaining consistent,repeatable feature detection involves allowing

for and normalizing the variations between

views.

The obvious candidates for normalizing arescale, affine distortion and perspective distortion.

Figure 6.13 shows how various transformations

affect a 2-D shape.


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FIGURE 6.13


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FORMULA


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Euclidean transformation(rotation + translation)

Similarity transformation

(scale)

Affine transformation(stretch + shear)


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6.7.1 Harris Scale and Affine-Invariant Detectors

and Descriptors

6.7.2 Hessian Scale and Affine-Invariant

Detectors and Descriptions6.7.3 The SIFT OperatorScale Invariant

Feature Transformation

Lowe (2004)

6.7.4 The SURF OperatorSpeed-up Robust

Features

Bay et al. (2006, 2008)


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FIGURE 6.15


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Performance Comparison


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Standard Repeatability Criterion:

Other Criteria:


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Table 6.1


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Table 6.2


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Conclusion

Scale invariant operators can normally be dealt

with adequately by a robustness capability for

viewpoint changes of less than 30 degree.

In different applications, different feature

properties may be important, and thus successdepends largely on appropriate selection of

features.

Repeatability may not always be the most

important feature performance characteristics.

There is a need for work focusing on

complementarity of features.

Corner and Interest Point Detection

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