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Invariant Features

Invariant Features. We need better features, better representations, …. Categories. Instances. Find these two objects. Find a bottle:. Can’t do unless you do not care about few errors…. Can nail it. Building a Panorama. M. Brown and D. G. Low e. Recognising Panorama s. ICCV 2003.

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Invariant Features

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  1. Invariant Features

  2. We need better features, better representations, …

  3. Categories Instances Find these two objects Find a bottle: Can’t do unless you do not care about few errors… Can nail it

  4. Building a Panorama M. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003

  5. How do we build a panorama? • We need to match (align) images • Global methods sensitive to occlusion, lighting, parallax effects. So look for local features that match well. • How would you do it by eye?

  6. Matching with Features • Detect feature points in both images

  7. Matching with Features • Detect feature points in both images • Find corresponding pairs

  8. Matching with Features • Detect feature points in both images • Find corresponding pairs • Use these pairs to align images

  9. Matching with Features • Problem 1: • Detect the same point independently in both images no chance to match! We need a repeatable detector

  10. Matching with Features • Problem 2: • For each point correctly recognize the corresponding one ? We need a reliable and distinctive descriptor

  11. More motivation… • Feature points are used also for: • Image alignment (homography, fundamental matrix) • 3D reconstruction • Motion tracking • Object recognition • Indexing and database retrieval • Robot navigation • … other

  12. Selecting Good Features • What’s a “good feature”? • Satisfies brightness constancy—looks the same in both images • Has sufficient texture variation • Does not have too much texture variation • Corresponds to a “real” surface patch—see below: • Does not deform too much over time Bad feature Right eye view Left eye view Good feature

  13. Contents • Harris Corner Detector • Overview • Analysis • Detectors • Rotation invariant • Scale invariant • Affine invariant • Descriptors • Rotation invariant • Scale invariant • Affine invariant

  14. An introductory example: Harris corner detector C.Harris, M.Stephens. “A Combined Corner and Edge Detector”. 1988

  15. The Basic Idea • We should easily localize the point by looking through a small window • Shifting a window in anydirection should give a large change in intensity

  16. Harris Detector: Basic Idea “flat” region:no change as shift window in all directions “edge”:no change as shift window along the edge direction “corner”:significant change as shift window in all directions

  17. Window function Shifted intensity Intensity Window function w(x,y) = or 1 in window, 0 outside Gaussian Harris Detector: Mathematics Window-averaged change of intensity induced by shifting the image data by [u,v]:

  18. Taylor series approx to shifted image

  19. Harris Detector: Mathematics Expanding I(x,y) in a Taylor series expansion, we have, for small shifts [u,v], a bilinear approximation: where M is a 22 matrix computed from image derivatives: M is also called “structure tensor”

  20. 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 Iso-intensity contour of E(u,v) (min)-1/2

  21. Selecting Good Features l1 and l2 are large

  22. Selecting Good Features large l1, small l2

  23. Selecting Good Features small l1, small l2

  24. 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

  25. Harris Detector: Mathematics Measure of corner response: This expression does not requires computing the eigenvalues. (k – empirical constant, k = 0.04-0.06)

  26. Harris Detector: Mathematics 2 “Edge” “Corner” • R depends only on eigenvalues of M • R is large for a corner • R is negative with large magnitude for an edge • |R| is small for a flat region R < 0 R > 0 “Flat” “Edge” |R| small R < 0 1

  27. Harris Detector • The Algorithm: • Find points with large corner response function R (R > threshold) • Take the points of local maxima of R

  28. Harris Detector: Workflow

  29. Harris Detector: Workflow Compute corner response R

  30. Harris Detector: Workflow Find points with large corner response: R>threshold

  31. Harris Detector: Workflow Take only the points of local maxima of R

  32. Harris Detector: Workflow

  33. Harris Detector: Summary • Average intensity change in direction [u,v] can be expressed as a bilinear form: • Describe a point in terms of eigenvalues of M:measure of corner response • A good (corner) point should have a large intensity change in all directions, i.e. R should be large positive

  34. Ideal feature detector • Would always find the same point on an object, regardless of changes to the image. • i.e, insensitive to changes in: • Scale • Lighting • Perspective imaging • Partial occlusion

  35. Harris Detector: Some Properties • Rotation invariance?

  36. 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

  37. Harris Detector: Some Properties • Invariance to image intensity change?

  38. R R threshold x(image coordinate) x(image coordinate) Harris Detector: Some Properties • Only derivatives are used => invariance to intensity shift I I+b • Partial invariance to additive and multiplicative intensity changes • Intensity scale: I  aI Because of fixed intensity threshold on local maxima, only partial invariance to multiplicative intensity changes.

  39. Harris Detector: Some Properties • Invariant to image scale?

  40. Harris Detector: Some Properties • Not invariant to image scale! All points will be classified as edges Corner !

  41. 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

  42. Evaluation plots are from this paper

  43. Contents • Harris Corner Detector • Overview • Analysis • Detectors • Rotation invariant • Scale invariant • Affine invariant • Descriptors • Rotation invariant • Scale invariant • Affine invariant

  44. We want to: detect the same interest points regardless of image changes

  45. Models of Image Change • Geometry • Rotation • Similarity (rotation + uniform scale) • Affine (scale dependent on direction)valid for: orthographic camera, locally planar object • Photometry • Affine intensity change (I  aI + b)

  46. Contents • Harris Corner Detector • Overview • Analysis • Detectors • Rotation invariant • Scale invariant • Affine invariant • Descriptors • Rotation invariant • Scale invariant • Affine invariant

  47. Rotation Invariant Detection • Harris Corner Detector C.Schmid et.al. “Evaluation of Interest Point Detectors”. IJCV 2000

  48. Contents • Harris Corner Detector • Overview • Analysis • Detectors • Rotation invariant • Scale invariant • Affine invariant • Descriptors • Rotation invariant • Scale invariant • Affine invariant

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