Computer Vision invariance
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This document discusses feature invariance in computer vision. It covers several key points: 1) Local feature detection aims to identify interest points, extract feature descriptors around each point, and match descriptors between views. 2) Features should be invariant to photometric transformations like intensity changes and equivariant to geometric transformations like rotation and scaling. 3) The Harris corner detector is partially invariant to affine intensity changes and equivariant to translation and rotation but not scaling. 4) Scale-space feature detection uses the Laplacian of Gaussian (LoG) or Difference of Gaussians (DoG) to find maxima/minima in position and scale for scale invariance.
![Local features: main components
1) Detection: Identify the
interest points
2) Description: Extract vector
feature descriptor surrounding
each interest point.
3) Matching: Determine
correspondence between
descriptors in two views
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11 dxx x
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12 dxx x
Kristen Grauman](https://image.slidesharecdn.com/lec05invariance-200719135053/85/Computer-Vision-invariance-3-320.jpg)
Computer Vision invariance
- 1. Lecture 5: Feature invariance CSCI 455: Computer Vision
- 3. Local features: main components 1) Detection: Identify the interest points 2) Description: Extract vector feature descriptor surrounding each interest point. 3) Matching: Determine correspondence between descriptors in two views ],,[ )1()1( 11 dxx x ],,[ )2()2( 12 dxx x Kristen Grauman
- 4. Harris features (in red)
- 5. Image transformations • Geometric – Rotation – Scale • Photometric – Intensity change
- 6. Image transformations • Geometric Rotation Scale • Photometric Intensity change
- 7. Invariance and equivariance • We want corner locations to be invariant to photometric transformations and equivariant to geometric transformations – Invariance: image is transformed and corner locations do not change – Equivariance: if we have two transformed versions of the same image, features should be detected in corresponding locations – (Sometimes “invariant” and “equivariant” are both referred to as “invariant”) – (Sometimes “equivariant” is called “covariant”)
- 8. Harris detector: Invariance properties -- Image translation • Derivatives and window function are equivariant Corner location is equivariant w.r.t. translation
- 9. Harris detector: Invariance properties -- Image rotation Second moment ellipse rotates but its shape (i.e. eigenvalues) remains the same Corner location is equivariant w.r.t. image rotation
- 10. Harris detector: Invariance properties – Affine intensity change • Only derivatives are used => invariance to intensity shift I I + b • Intensity scaling: I a I R x (image coordinate) threshold R x (image coordinate) Partially invariant to affine intensity change I a I + b
- 11. Harris Detector: Invariance Properties • Scaling All points will be classified as edges Corner Neither invariant nor equivariant to scaling
- 12. Harris Detector: Invariance Properties • Rotation Ellipse rotates but its shape (i.e. eigenvalues) remains the same Corner response is invariant to image rotation
- 13. Harris Detector: Invariance Properties • Affine intensity change: I aI + b Only derivatives are used => invariance to intensity shift I I + b Intensity scale: I a I R x (image coordinate) threshold R x (image coordinate) Partially invariant to affine intensity change
- 14. Harris Detector: Invariance Properties • Scaling All points will be classified as edges Corner Not invariant to scaling
- 15. Scale invariant detection Suppose you’re looking for corners Key idea: find scale that gives local maximum of f – in both position and scale – One definition of f: the Harris operator
- 16. Slide from Tinne Tuytelaars Lindeberg et al, 1996 Slide from Tinne Tuytelaars Lindeberg et al., 1996
- 24. Implementation • Instead of computing f for larger and larger windows, we can implement using a fixed window size with a Gaussian pyramid (sometimes need to create in- between levels, e.g. a ¾-size image)
- 25. Feature extraction: Corners and blobs
- 26. Another common definition of f • The Laplacian of Gaussian (LoG) 2 2 2 2 2 y g x g g (very similar to a Difference of Gaussians (DoG) – i.e. a Gaussian minus a slightly smaller Gaussian)
- 27. Laplacian of Gaussian • “Blob” detector • Find maxima and minima of LoG operator in space and scale * = maximum minima
- 28. Scale selection • At what scale does the Laplacian achieve a maximum response for a binary circle of radius r? r image Laplacian
- 29. Characteristic scale • We define the characteristic scale as the scale that produces peak of Laplacian response characteristic scale T. Lindeberg (1998). "Feature detection with automatic scale selection." International Journal of Computer Vision 30 (2): pp 77--116.
- 30. Find local maxima in 3D position-scale space K. Grauman, B. Leibe )()( yyxx LL 2 3 4 5 List of (x, y, s)
- 31. Scale-space blob detector: Example
- 32. Scale-space blob detector: Example
- 33. Scale-space blob detector: Example
- 34. covariantNote: The LoG and DoG operators are both rotation equivariant
- 35. Questions?
- 36. Feature descriptors We know how to detect good points Next question: How to match them? Answer: Come up with a descriptor for each point, find similar descriptors between the two images ?