Top-points in Image Matching

1. Top-points in Image Matching Bram Platel Evgenya Balmashnova Luc Florack Bart ter Haar Romeny

2. Introduction • Question: can top-points be used for object-retrieval tasks?

3. Introduction to Scale Space and Deep Structure

4. Importance of Scalein Image Analysis • Objects in images exist at different ranges of scale. • Usually it is not known a priory at what scale to look. Painting by Dali

5. BLUR • At the original scale of a dithered image we cannot calculate a derivative. • We need to observe the image at a certain scale.

6. Scale y x Solution? • Look at all scales simultaneously Scale Space

7. Scale Space in Human Vision • The human visual system is a multi-scale sampling device • The retina contains receptive fields; groups of receptors assembled in such a way that they form a set of apertures of widely varying size.

8. Practical Implementation • Convolve the image with a Gaussian Kernel

9. Original Image We can Calculate Derivatives and Combinations of them at all Scales Gradient Magnitude Laplacian

10. Main Topic • In this presentation we will show how we can exploit the deep structure of images to define invariant interest points and features which can be used for matching problems in computer vision. • We consider only grey-value images.

11. Interest Points • The locations of particularly characteristic points are called the interest points or key points. • These interest points have to be as invariant as possible, but at the same time they have to carry a lot of distinctive information.

12. Why Interest Points in Scale Space? • Information in interest points is defined by their neighborhood. But how big should we choose this neighborhood? • Let’s take the corners of the mouth as interest points. • The red circles are the areas in which the information is gathered. • If we make the picture bigger, the size of the neighborhood is too small. • The neighborhood should scale with the image

13.   Why Interest Points in Scale Space? • When the interest points are detected in scale space they do not only have spatial coordinates x and y, but also a scale . • This scale tells us how big the neighborhood should be.

14. Which Interest Points to Use? • Our interest points have to be detected in scale space. • They also have to… • …contain a lot of information • …be reproducible • …be stable • …be well understood

15. We Suggest Top-Points • The points we introduce have these desired properties.

16. Maxima Saddles Critical Points L=0 Minimum Critical Points, Paths and Top-Points

17. Top-Points Det(H)=0 Maxima Saddles Critical Points L=0 Minimum Critical Points, Paths and Top-Points

18. Possible to calculate them for every Function of the Image L(x,y,) Original Gradient Magnitude Laplacian Det(H)

19. Detecting Critical Paths • Since for a critical path L=0 • Intersection of Level Surfaces Lx=0 with Ly=0 • Will give the critical paths.

20. Detecting Top-Points • Since for a top-point both L=0 and Det[H]=Lxx Lyy-Lxy2=0 • We can find them by intersecting the paths with the level surface Det[H]=0

22. Allowed Trans. Invariance of Top Points • Top-points are invariant to certain transformations. • By invariant we mean that they move according to the transformation.

23. Reconstruction • It is possible to make a reconstruction of the original image from its top-points. • We can generate reconstructed images which give the same (plus more) top-points as the original image. • This reconstruction resembles the original image.

24. Original Image Reconstruction Top-Points and Features

25. Metameric Class Original By adjusting boundary and smoothness constraints we can improve the visual performance. For this 300x300 picture 1000 top-points with 6 features were used.

26. This tells us that the top-points indeed contain a lot of information about the image.

27.  y x Localization of Top-Points • For points close to top-points it is possible to calculate a vector pointing towards the position of the top-point. Approximated Top-Points Displacement Vectors Real Locations

28. Localization of Top-Points • For points close to top-points it is possible to calculate a vector pointing towards the position of the top-point. • This enables us to use fast top-point detection algorithms which do not have to be very accurate.

29. Stability of Top-Points • The locations of top-points change when noise is added to the image.

31. Stability of Top-Points • We can calculate the variance of the displacement of top- points under noise. • We need 4th order derivatives in the top-points for that.

32. Selecting Stable Paths • Large instabilities are found in flat areas in the image. Areas with a lot of structure result in more stable top points. • In the article we used an adapted version of the TV-norm (Total Variation) to measure the flatness around a top point. • We can give top points a weight in the EMD algorithm based on the stability of the point.

33. Integration Area x xc Taylor Expansion (2nd order)  Limiting Procedure Calculating the Stability Norm Koenderink’s deviation from flatness

34. Stable Paths Unstable Paths Thresholding on Stability

35. 1 2 3 4 5 8 7 9 6 Database Query Image Database Image Retrieval

36. Experiments • A simple image retrieval task. • Using a small version of the Olivetti Faces Database. • Consisting of 200 images of 20 different people (10 p.p.)

37. Scale y x The Deep Structure of an Image Look at all scales simultaneously

38. Top Points Det(H)=0 Maxima Saddles Critical Points u=0 Minimum Critical Points, Paths and Top Points

40. Comparing Top Points of Images Compare EMD

41. A B wi fij cij uj Earth Movers Distance (EMD) Piles Holes [*]Rubner, Tomasi, Guibas, 1998, IEEE Conf. on Computer Vision

42. Calculating the EMD • To calculate the earth movers distance we need: • Weights for our top points (e.g. more important points could contain more weight). • A cost function for transporting the weights between point sets (we incorporated a distance function between top points). MMA

43. Scale y x Measuring Distance in Scale Space • Scale Space is not a Euclidean Space. • The simple Euclidean Metric no longer holds.

44. For an infinitely small step in scale space the following distance function holds: 0 relates the importance of scale- to spatial measurements Given this distance function we can find a unique geodesic path connecting two points in scale space. The distance between two points is measured along the this geodesic path connecting these points. Eberly’s Distance Measure in Scale Space

45. Distance Between Points in Scale Space • By parameterizing along the geodesic path the distance function can be derived by solving an ordinary differential equation. • This yields the following equation describing the distance between two points in scale space:

46. We now have a distance measure which we can use in our EMD algorithm. • We have introduced a tunable parameter  (>0). • We now need to give a weight to our points.

47. Results • Using Euclidean Distance • Using Eberly Distance • As b. including stability norm • As c. including 2nd order derivatives.

48. Distinctive Features • To distinguish top-points from each other a set of distinctive features are needed in every top-point. • These local features describe the neighborhood of the top-point.

49. Differential Invariants • We use the complete set of irreducible 3rd order differential invariants. • These features are rotation and scaling invariant.

50. Matching • With the top-points as interest points and the differential invariants as the descriptors we can now start matching them across images.