Harris corner detector scale invariant feature transform sift
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Harris Corner Detector & Scale Invariant Feature Transform (SIFT). Harris Corner Detector. Harris Detector: Intuition. “flat” region: no change in all directions. “edge” : no change along the edge direction. “corner” : significant change in all directions. Moravec Corner Detector.

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Harris Corner Detector & Scale Invariant Feature Transform (SIFT)

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Harris corner detector scale invariant feature transform sift

Harris Corner Detector&Scale Invariant Feature Transform (SIFT)


Harris corner detector

Harris Corner Detector


Harris detector intuition

Harris Detector: Intuition

“flat” region:no change in all directions

“edge”:no change along the edge direction

“corner”:significant change in all directions


Moravec corner detector

Moravec Corner Detector

  • Shift in any direction would result in a significant change at a corner.

  • Algorithm:

  • Shift in horizontal, vertical, and diagonal directions by one pixel.

  • Calculate the absolute value of the MSE for each shift.

  • Take the minimum as the cornerness response.


Harris detector mathematics

Window function

Shifted intensity

Intensity

Window function w(x,y) =

or

1 in window, 0 outside

Gaussian

Harris Detector: Mathematics

Change of intensity for the shift [u,v]:


Harris detector mathematics1

Harris Detector: Mathematics

Apply Taylor series expansion:


Harris detector mathematics2

Harris Detector: Mathematics

For small shifts [u,v] we have the following approximation:

where M is a 22 matrix computed from image derivatives:


Harris detector mathematics3

Harris Detector: Mathematics

Intensity change in shifting window: eigenvalue analysis

1, 2 – eigenvalues of M

direction of the fastest change

Ellipse E(u,v) = const

direction of the slowest change

(max)-1/2

(min)-1/2


Harris detector mathematics4

Harris Detector: Mathematics

2

Classification of image points using eigenvalues of M:

“Edge” 2 >> 1

“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

“Flat” region

1


Harris corner detector1

Harris corner detector

Measure of corner response:

(k – empirical constant, k = 0.04-0.06)

No need to compute eigenvalues explicitly!


Harris corner detector scale invariant feature transform sift

Eliminate small responses.


Harris corner detector scale invariant feature transform sift

Find local maxima of the remaining.


Harris detector scale

Rmin= 0

Rmin= 1500

Harris Detector: Scale


Summary of the harris detector

Summary of the Harris detector


Harris detector some properties

Harris Detector: Some Properties

  • Rotation invariance

Ellipse rotates but its shape (i.e. eigenvalues) remains the same

Corner response R is invariant to image rotation


Harris detector some properties1

  • Intensity scale: I  aI

R

R

threshold

x(image coordinate)

x(image coordinate)

Harris Detector: Some Properties

  • Partial invariance to affine intensity change

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


Harris detector some properties2

Harris Detector: Some Properties

  • But: non-invariant to image scale!

All points will be classified as edges

Corner !


Harris detector some properties3

Harris Detector: Some Properties

  • Quality of Harris detector for different scale changes

Repeatability rate:

# correspondences# possible correspondences

C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000


Scale invariant detection

Scale Invariant Detection

  • Consider regions (e.g. circles) of different sizes around a point

  • Regions of corresponding sizes will look the same in both images


Scale invariant detection1

Scale Invariant Detection

  • The problem: how do we choose corresponding circles independently in each image?


Scale invariant detection2

scale = 1/2

Image 1

f

f

Image 2

region size

region size

Scale Invariant Detection

  • Solution:

    • Design a function on the region (circle), which is “scale invariant” (the same for corresponding regions, even if they are at different scales)

Example: average intensity. For corresponding regions (even of different sizes) it will be the same.

  • For a point in one image, we can consider it as a function of region size (circle radius)


Scale invariant detection3

scale = 1/2

Image 1

f

f

Image 2

s1

s2

region size

region size

Scale Invariant Detection

  • Common approach:

Take a local maximum of this function

Observation: region size, for which the maximum is achieved, should be invariant to image scale.


Characteristic scale

Characteristic Scale

Ratio of scales corresponds to a scale factor between two images


Scale invariant detection4

f

f

Good !

bad

region size

f

region size

bad

region size

Scale Invariant Detection

  • A “good” function for scale detection: has one stable sharp peak

  • For usual images: a good function would be a one which responds to contrast (sharp local intensity change)


Scale invariant detection5

Scale Invariant Detection

  • Functions for determining scale

Kernels:

(Laplacian)

(Difference of Gaussians)

where Gaussian

L or DoG kernel is a matching filter. It finds blob-like structure. It turns out to be also successful in getting characteristic scale of other structures, such as corner regions.


Difference of gaussians

Difference-of-Gaussians


Scale space extrema

Scale-Space Extrema

  • Choose all extrema within 3x3x3 neighborhood.

X is selected if it is larger or smaller than all 26 neighbors


Scale invariant detectors

scale

 Laplacian 

y

x

 Harris 

scale

  • SIFT (Lowe)2Find local maximum of:

    • Difference of Gaussians in space and scale

 DoG 

y

x

 DoG 

Scale Invariant Detectors

  • Harris-Laplacian1Find local maximum of:

    • Harris corner detector in space (image coordinates)

    • Laplacian in scale

1 K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 20012 D.Lowe. “Distinctive Image Features from Scale-Invariant Keypoints”. Accepted to IJCV 2004


Harris laplace detector

Harris-Laplace Detector


Scale invariant detectors1

Scale Invariant Detectors

  • Experimental evaluation of detectors w.r.t. scale change

Repeatability rate:

# correspondences# possible correspondences

K.Mikolajczyk, C.Schmid. “Indexing Based on Scale Invariant Interest Points”. ICCV 2001


Affine invariant detection

Affine Invariant Detection

  • Above we considered:Similarity transform (rotation + uniform scale)

  • Now we go on to:Affine transform (rotation + non-uniform scale)


Affine invariant detection1

f

points along the ray

Affine Invariant Detection

  • Take a local intensity extremum as initial point

  • Go along every ray starting from this point and stop when extremum of function f is reached

T.Tuytelaars, L.V.Gool. “Wide Baseline Stereo Matching Based on Local, Affinely Invariant Regions”. BMVC2000.


Affine invariant detection2

Affine Invariant Detection

Such extrema occur at positions where intensity suddenly changes compared to the intensity changes up to that point.

In theory, leaving out the denominator would still give invariant positions. In practice, the local extrema would be shallow, and might result in inaccurate positions.

T.Tuytelaars, L.V.Gool. “Wide Baseline Stereo Matching Based on Local, Affinely Invariant Regions”. BMVC2000.


Affine invariant detection3

Affine Invariant Detection

  • The regions found may not exactly correspond, so we approximate them with ellipses

  • Find the ellipse that best fits the region


Affine invariant detection4

( p = [x, y]T is relative to the center of mass)

Affine Invariant Detection

  • Covariance matrix of region points defines an ellipse:

Ellipses, computed for corresponding regions, also correspond!


Affine invariant detection5

Affine Invariant Detection

  • Algorithm summary (detection of affine invariant region):

    • Start from a local intensity extremum point

    • Go in every direction until the point of extremum of some function f

    • Curve connecting the points is the region boundary

    • Compute the covariance matrix

    • Replace the region with ellipse

T.Tuytelaars, L.V.Gool. “Wide Baseline Stereo Matching Based on Local, Affinely Invariant Regions”. BMVC2000.


Affine invariant detection6

Affine Invariant Detection

  • Maximally Stable Extremal Regions

    • Threshold image intensities: I > I0

    • Extract connected components(“Extremal Regions”)

    • Find “Maximally Stable” regions

    • Approximate a region with an ellipse

J.Matas et.al. “Distinguished Regions for Wide-baseline Stereo”. Research Report of CMP, 2001.


Affine invariant detection summary

Affine Invariant Detection : Summary

  • Under affine transformation, we do not know in advance shapes of the corresponding regions

  • Ellipse given by geometric covariance matrix of a region robustly approximates this region

  • For corresponding regions ellipses also correspond

  • Methods:

  • Search for extremum along rays [Tuytelaars, Van Gool]:

  • Maximally Stable Extremal Regions [Matas et.al.]


Point descriptors

Point Descriptors

  • We know how to detect points

  • Next question:

    How to match them?

?

  • Point descriptor should be:

    • Invariant

    • Distinctive


Descriptors invariant to rotation

Descriptors Invariant to Rotation

  • Convert from Cartesian to Polar coordinates

  • Rotation becomes translation in polar coordinates

  • Take Fourier Transform

    • Magnitude of the Fourier transform is invariant to translation.


Descriptors invariant to rotation1

Descriptors Invariant to Rotation

  • Find local orientation

Dominant direction of gradient

  • Compute image regions relative to this orientation


Descriptors invariant to scale

Descriptors Invariant to Scale

  • Use the characteristic scale determined by detector to compute descriptor in a normalized frame


Affine invariant descriptors

A1

A2

rotation

Affine Invariant Descriptors

A

  • Find affine normalized frame

  • Compute rotational invariant descriptor in this normalized frame

J.Matas et.al. “Rotational Invariants for Wide-baseline Stereo”. Research Report of CMP, 2003


Affine covariant regions

Affine covariant regions


Sift scale invariant feature transform

SIFT (Scale Invariant Feature Transform)


Sift scale invariant feature transform 1

SIFT – Scale Invariant Feature Transform1

  • Empirically found2 to show very good performance, invariant to image rotation, scale, intensity change, and to moderate affine transformations

Scale = 2.5Rotation = 450

1 D.Lowe. “Distinctive Image Features from Scale-Invariant Keypoints”. Accepted to IJCV 20042 K.Mikolajczyk, C.Schmid. “A Performance Evaluation of Local Descriptors”. CVPR 2003


Sift scale invariant feature transform1

SIFT – Scale Invariant Feature Transform

  • Descriptor overview:

    • Determine scale (by maximizing DoG in scale and in space), local orientation as the dominant gradient direction.Use this scale and orientation to make all further computations invariant to scale and rotation.

    • Compute gradient orientation histograms of several small windows (to produce 128 values for each point)

D.Lowe. “Distinctive Image Features from Scale-Invariant Keypoints”. Accepted to IJCV 2004


Scale space extrema detection

Scale-space extrema detection

  • Need to find “characteristic scale” for each feature point.


Difference of gaussians1

Difference-of-Gaussians


Scale space extrema1

Scale-Space Extrema

  • Choose all extrema within 3x3x3 neighborhood.

X is selected if it is larger or smaller than all 26 neighbors


Keypoint localization filtering

Keypoint Localization & Filtering

  • Now we have much less points than pixels.

  • However, still lots of points (~1000s)…

    • With only pixel-accuracy at best

    • And this includes many bad points

Brown & Lowe 2002


Keypoint filtering low contrast

Keypoint Filtering - Low Contrast

  • Reject points with bad contrast:

    • DoG smaller than 0.03 (image values in [0,1])

  • Reject edges

    • Similar to the Harris detector; look at the autocorrelation matrix


Maxima in d

Maxima in D


Remove low contrast and edges

Remove low contrast and edges


Orientation assignment

Orientation assignment

  • By assigning a consistent orientation, the keypoint descriptor can be orientation invariant.

  • Let, for a keypoint, L is the image with the closest scale.

    • Compute gradient magnitude and orientation using finite differences:


Orientation assignment1

Orientation assignment


Orientation assignment2

Orientation assignment


Orientation assignment3

Orientation assignment


Orientation assignment4

Orientation assignment


Orientation assignment5

Orientation Assignment

  • Any peak within 80% of the highest peak is used to create a keypoint with that orientation

  • ~15% assigned multiplied orientations, but contribute significantly to the stability


Sift descriptor

SIFT descriptor


Sift descriptor1

SIFT Descriptor

  • Each point so far has x, y, σ, m, θ

  • Now we need a descriptor for the region

    • Could sample intensities around point, but…

      • Sensitive to lighting changes

      • Sensitive to slight errors in x, y, θ

Edelman et al. 1997


Sift descriptor2

SIFT Descriptor

  • 16x16 Gradient window is taken. Partitioned into 4x4 subwindows.

  • Histogram of 4x4 samples in 8 directions

  • Gaussian weighting around center( is 0.5 times that of the scale of a keypoint)

  • 4x4x8 = 128 dimensional feature vector

Image from: Jonas Hurrelmann


Performance

Performance

  • Very robust

    • 80% Repeatability at:

      • 10% image noise

      • 45° viewing angle

      • 1k-100k keypoints in database

  • Best descriptor in [Mikolajczyk & Schmid 2005]’s extensive survey


Recognition under occlusion

Recognition under occlusion


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