The document discusses the mean shift algorithm, a non-parametric technique for analyzing complex multimodal feature spaces and estimating the stationary points (modes) of the underlying probability density function without explicitly estimating it. It provides an intuitive description of mean shift using a distribution of billiard balls, and outlines how mean shift uses kernel density estimation to perform gradient ascent and converge at the densest regions, allowing it to be used for tasks like mode detection, clustering, and image segmentation.

1. Mean Shift A Robust Approach to Feature Space Analysis Kalyan Sunkavalli 04/29/2008 ES251R

2. An Example Feature Space

3. An Example Feature Space

4. An Example Feature Space Parametric Density Estimation?

5. Mean Shift A non-parametric technique for analyzing complex multimodal feature spaces and estimating the stationary points (modes) of the underlying probability density function without explicitly estimating it .

6. Outline Mean Shift An intuition Kernel Density Estimation Derivation Properties Applications of Mean Shift Discontinuity preserving Smoothing Image Segmentation

7. Outline Mean Shift An intuition Kernel Density Estimation Derivation Properties Applications of Mean Shift Discontinuity preserving Smoothing Image Segmentation

8. Intuitive Description Distribution of identical billiard balls Region of interest Center of mass Mean Shift vector Objective : Find the densest region Slide Credit: Yaron Ukrainitz & Bernard Sarel

9. Intuitive Description Distribution of identical billiard balls Region of interest Center of mass Mean Shift vector Objective : Find the densest region

10. Intuitive Description Distribution of identical billiard balls Region of interest Center of mass Mean Shift vector Objective : Find the densest region

11. Intuitive Description Distribution of identical billiard balls Region of interest Center of mass Mean Shift vector Objective : Find the densest region

12. Intuitive Description Distribution of identical billiard balls Region of interest Center of mass Mean Shift vector Objective : Find the densest region

13. Intuitive Description Distribution of identical billiard balls Region of interest Center of mass Mean Shift vector Objective : Find the densest region

14. Intuitive Description Distribution of identical billiard balls Region of interest Center of mass Objective : Find the densest region

15. Outline Mean Shift An intuition Kernel Density Estimation Derivation Properties Applications of Mean Shift Discontinuity preserving Smoothing Image Segmentation

16. Assumed Underlying PDF Estimate from data Data Samples Parametric Density Estimation The data points are sampled from an underlying PDF

17. Assumed Underlying PDF Data Samples Data point density Non-parametric Density Estimation PDF value

18. Assumed Underlying PDF Data Samples Non-parametric Density Estimation

19. Parzen Windows Kernel Properties Bounded Compact support Normalized Symmetric Exponential decay

20. Kernels and Bandwidths Kernel Types Bandwidth Parameter (product of univariate kernels) (radially symmetric kernel)

21. Various Kernels Epanechnikov Normal Uniform

22. Outline Mean Shift An intuition Kernel Density Estimation Derivation Properties Applications of Mean Shift Discontinuity preserving Smoothing Image Segmentation

23. Density Gradient Estimation Epanechnikov  Uniform Normal  Normal Modes of the probability density

24. Mean Shift KDE Mean Shift Mean Shift Algorithm compute mean shift vector translate kernel (window) by mean shift vector

25. Mean Shift Mean Shift is proportional to the normalized density gradient estimate obtained with kernel The normalization is by the density estimate computed with kernel

26. Outline Mean Shift An intuition Kernel Density Estimation Derivation Properties Applications of Mean Shift Discontinuity preserving Smoothing Image Segmentation

27. Properties of Mean Shift Guaranteed convergence Gradient Ascent algorithms are guaranteed to converge only for infinitesimal steps. The normalization of the mean shift vector ensures that it converges. Large magnitude in low-density regions, refined steps near local maxima  Adaptive Gradient Ascent. Mode Detection Let denote the sequence of kernel locations. At convergence Once gets sufficiently close to a mode of it will converge to the mode. The set of all locations that converge to the same mode define the basin of attraction of that mode.

28. Properties of Mean Shift Smooth Trajectory The angle between two consecutive mean shift vectors computed using the normal kernel is always less that 90° In practice the convergence of mean shift using the normal kernel is very slow and typically the uniform kernel is used.

29. Mode detection using Mean Shift Run Mean Shift to find the stationary points To detect multiple modes, run in parallel starting with initializations covering the entire feature space. Prune the stationary points by retaining local maxima Merge modes at a distance of less than the bandwidth. Clustering from the modes The basin of attraction of each mode delineates a cluster of arbitrary shape.

30. Mode Finding on Real Data initialization detected mode tracks

31. Mean Shift Clustering

32. Outline Mean Shift Density Estimation What is mean shift? Derivation Properties Applications of Mean Shift Discontinuity preserving Smoothing Image Segmentation

33. Joint Spatial-Range Feature Space Concatenate spatial and range (gray level or color) information

34. Discontinuity Preserving Smoothing

35. Discontinuity Preserving Smoothing

36. Discontinuity Preserving Smoothing

37. Discontinuity Preserving Smoothing

38. Outline Mean Shift Density Estimation What is mean shift? Derivation Properties Applications of Mean Shift Discontinuity preserving Smoothing Image Segmentation

39. Clustering on Real Data

40. Image Segmentation

41. Image Segmentation

42. Image Segmentation

43. Image Segmentation

44. Image Segmentation

45. Acknowledgements Mean shift: A robust approach toward feature space analysis. D Comaniciu, P Meer Pattern Analysis and Machine Intelligence, IEEE Transactions on , Vol. 24, No. 5. (2002), pp. 603-619. http://www.caip.rutgers.edu/riul/research/papers.html Slide credits: Yaron Ukrainitz & Bernard Sarel

46. Thank You