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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

“flat” region:no change in all directions

“edge”:no change along the edge direction

“corner”:significant change in all directions

- 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.

Window function

Shifted intensity

Intensity

Window function w(x,y) =

or

1 in window, 0 outside

Gaussian

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

Apply Taylor series expansion:

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

where M is a 22 matrix computed from image derivatives:

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

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

Measure of corner response:

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

No need to compute eigenvalues explicitly!

Eliminate small responses.

Find local maxima of the remaining.

Rmin= 0

Rmin= 1500

- Rotation invariance

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

Corner response R is invariant to image rotation

- Intensity scale: I aI

R

R

threshold

x(image coordinate)

x(image coordinate)

- Partial invariance to affine intensity change

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

- But: non-invariant to image scale!

All points will be classified as edges

Corner !

- Quality of Harris detector for different scale changes

Repeatability rate:

# correspondences# possible correspondences

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

- Consider regions (e.g. circles) of different sizes around a point
- Regions of corresponding sizes will look the same in both images

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

scale = 1/2

Image 1

f

f

Image 2

region size

region size

- 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 = 1/2

Image 1

f

f

Image 2

s1

s2

region size

region size

- Common approach:

Take a local maximum of this function

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

Ratio of scales corresponds to a scale factor between two images

f

f

Good !

bad

region size

f

region size

bad

region size

- 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)

- 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.

- Choose all extrema within 3x3x3 neighborhood.

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

scale

Laplacian

y

x

Harris

scale

- SIFT (Lowe)2Find local maximum of:
- Difference of Gaussians in space and scale

DoG

y

x

DoG

- 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

- 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

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

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

f

points along the ray

- 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.

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.

- The regions found may not exactly correspond, so we approximate them with ellipses
- Find the ellipse that best fits the region

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

- Covariance matrix of region points defines an ellipse:

Ellipses, computed for corresponding regions, also correspond!

- 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.

- 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.

- 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.]

- We know how to detect points
- Next question:
How to match them?

?

- Point descriptor should be:
- Invariant
- Distinctive

- Convert from Cartesian to Polar coordinates
- Rotation becomes translation in polar coordinates
- Take Fourier Transform
- Magnitude of the Fourier transform is invariant to translation.

- Find local orientation

Dominant direction of gradient

- Compute image regions relative to this orientation

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

A1

A2

rotation

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

SIFT (Scale Invariant Feature Transform)

- 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

- 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

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

- Choose all extrema within 3x3x3 neighborhood.

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

- 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

- 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

- 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:

- 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

- 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

- Could sample intensities around point, but…

Edelman et al. 1997

- 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

- Very robust
- 80% Repeatability at:
- 10% image noise
- 45° viewing angle
- 1k-100k keypoints in database

- 80% Repeatability at:
- Best descriptor in [Mikolajczyk & Schmid 2005]’s extensive survey