Scale SIFT: Invariant Feature Transform
With slides from Sebastian Thrun
Stanford CS223B Computer Vision, Winter 2006
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Pattern Recognition
Want to find ... in here
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SIFT
Invariances:
Scaling Rotation Illumination Deformation Yes Yes Yes Maybe Yes
Provides
Good localization
Distinctive image features from scale-invariant keypoints. David G. Lowe, International Journal of Computer Vision, 60, 2 (2004), pp. 91-110.
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Invariant Local Features
Image content is transformed into local feature coordinates that are invariant to translation, rotation, scale, and other imaging parameters
SIFT Features
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Advantages of invariant local features
Locality: features are local, so robust to occlusion and clutter (no prior segmentation) Distinctiveness: individual features can be matched to a large database of objects Quantity: many features can be generated for even small objects Efficiency: close to real-time performance Extensibility: can easily be extended to wide range of differing feature types, with each adding robustness
SIFT On-A-Slide
1. 2. 3. 4. Enforce invariance to scale: Compute Gaussian difference max, for may different scales; non-maximum suppression, find local maxima: keypoint candidates Localizable corner: For each maximum fit quadratic function. Compute center with sub-pixel accuracy by setting first derivative to zero. Eliminate edges: Compute ratio of eigenvalues, drop keypoints for which this ratio is larger than a threshold. Enforce invariance to orientation: Compute orientation, to achieve scale invariance, by finding the strongest second derivative direction in the smoothed image (possibly multiple orientations). Rotate patch so that orientation points up. Compute feature signature: Compute a "gradient histogram" of the local image region in a 4x4 pixel region. Do this for 4x4 regions of that size. Orient so that largest gradient points up (possibly multiple solutions). Result: feature vector with 128 values (15 fields, 8 gradients). Enforce invariance to illumination change and camera saturation: Normalize to unit length to increase invariance to illumination. Then threshold all gradients, to become invariant to camera saturation. 7
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SIFT On-A-Slide
1. 2. 3. 4. Enforce invariance to scale: Compute Gaussian difference max, for may different scales; non-maximum suppression, find local maxima: keypoint candidates Localizable corner: For each maximum fit quadratic function. Compute center with sub-pixel accuracy by setting first derivative to zero. Eliminate edges: Compute ratio of eigenvalues, drop keypoints for which this ratio is larger than a threshold. Enforce invariance to orientation: Compute orientation, to achieve scale invariance, by finding the strongest second derivative direction in the smoothed image (possibly multiple orientations). Rotate patch so that orientation points up. Compute feature signature: Compute a "gradient histogram" of the local image region in a 4x4 pixel region. Do this for 4x4 regions of that size. Orient so that largest gradient points up (possibly multiple solutions). Result: feature vector with 128 values (15 fields, 8 gradients). Enforce invariance to illumination change and camera saturation: Normalize to unit length to increase invariance to illumination. Then threshold all gradients, to become invariant to camera saturation. 8
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Find Invariant Corners
1. Enforce invariance to scale: Compute Gaussian difference max, for may different scales; non-maximum suppression, find local maxima: keypoint candidates
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Finding "Keypoints" (Corners)
Idea: Find Corners, but scale invariance Approach: Run linear filter (diff of Gaussians) At different resolutions of image pyramid
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Difference of Gaussians
Minus
Equals
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Difference of Gaussians
surf(fspecial('gaussian',40,4)) surf(fspecial('gaussian',40,8)) surf(fspecial('gaussian',40,8) - fspecial('gaussian',40,4))
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Find Corners with DiffOfGauss
im =imread('bridge.jpg'); bw = double(im(:,:,1)) / 256; for i = 1 : 10 gaussD = fspecial('gaussian',40,2*i) - fspecial('gaussian',40,i); res = abs(conv2(bw, gaussD, 'same')); res = res / max(max(res)); imshow(res) ; title(['\bf i = ' num2str(i)]); drawnow end
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Gaussian Kernel Size i=1
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Gaussian Kernel Size i=2
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Gaussian Kernel Size i=3
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Gaussian Kernel Size i=4
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Gaussian Kernel Size i=5
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Gaussian Kernel Size i=6
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Gaussian Kernel Size i=7
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Gaussian Kernel Size i=8
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Gaussian Kernel Size i=9
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Gaussian Kernel Size i=10
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Key point localization
Detect maxima and minima of difference-of-Gaussian in scale space
s p e m Ra e l r u l B a r t b ut Sc
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Example of keypoint detection
(a) 233x189 image (b) 832 DOG extrema (c) 729 above threshold
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SIFT On-A-Slide
1. 2. 3. 4. Enforce invariance to scale: Compute Gaussian difference max, for may different scales; non-maximum suppression, find local maxima: keypoint candidates Localizable corner: each For maximum fit quadratic function. Compute center with sub-pixel accuracy by setting first derivative to zero. Eliminate edges: Compute ratio of eigenvalues, drop keypoints for which this ratio is larger than a threshold. Enforce invariance to orientation: Compute orientation, to achieve scale invariance, by finding the strongest second derivative direction in the smoothed image (possibly multiple orientations). Rotate patch so that orientation points up. Compute feature signature: Compute a "gradient histogram" of the local image region in a 4x4 pixel region. Do this for 4x4 regions of that size. Orient so that largest gradient points up (possibly multiple solutions). Result: feature vector with 128 values (15 fields, 8 gradients). Enforce invariance to illumination change and camera saturation: Normalize to unit length to increase invariance to illumination. Then threshold all gradients, to become invariant to camera saturation. 26
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Example of keypoint detection
Threshold on value at DOG peak and on ratio of principle curvatures (Harris approach)
(c) 729 left after peak value threshold (from 832) (d) 536 left after testing ratio of principle curvatures
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SIFT On-A-Slide
1. 2. 3. 4. Enforce invariance to scale: Compute Gaussian difference max, for may different scales; non-maximum suppression, find local maxima: keypoint candidates Localizable corner: For each maximum fit quadratic function. Compute center with sub-pixel accuracy by setting first derivative to zero. Eliminate edges: Compute ratio of eigenvalues, drop keypoints for which this ratio is larger than a threshold. Enforce invariance to orientation: Compute orientation, to achieve scale invariance, by finding the strongest second derivative direction in the smoothed image (possibly multiple orientations). Rotate patch so that orientation points up. Compute feature signature: Compute a "gradient histogram" of the local image region in a 4x4 pixel region. Do this for 4x4 regions of that size. Orient so that largest gradient points up (possibly multiple solutions). Result: feature vector with 128 values (15 fields, 8 gradients). Enforce invariance to illumination change and camera saturation: Normalize to unit length to increase invariance to illumination. Then threshold all gradients, to become invariant to camera saturation. 28
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Select canonical orientation
Create histogram of local gradient directions computed at selected scale Assign canonical orientation at peak of smoothed histogram Each key specifies stable 2D coordinates (x, y, scale, orientation)
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SIFT On-A-Slide
1. 2. 3. 4. Enforce invariance to scale: Compute Gaussian difference max, for may different scales; non-maximum suppression, find local maxima: keypoint candidates Localizable corner: For each maximum fit quadratic function. Compute center with sub-pixel accuracy by setting first derivative to zero. Eliminate edges: Compute ratio of eigenvalues, d...