B-Spline Channels & Channel Smoothing

1. B-Spline Channels & Channel Smoothing Michael Felsberg Computer Vision LaboratoryLinköping UniversitySWEDEN

2. General Idea of Channels • Encode single value (linear or modular) in N-D coefficient vector (channel vector) • Locality of encoding • Similar values in same coefficients • Dissimilar values in different coefficients • Stability by smooth, monopolar basis functions • Small changes of value lead to small changes of coefficients • Non-negative coefficients

3. Example for Single Value

4. Example for Multiple Values

5. Overview • Encoding with quadratic B-splines • Decoding strategies • Relation to kernel-density estimation • Relation to robust M-estimation • Channel smoothing • Applications

6. Quadratic B-Splines

7. We assume f to be shiftedand rescaled such that B-Splines Encoding • The value of the nth channel at x is obtained by • Encoding in practice: • m=round(f) • c[m-1]=(f-m-0.5)2 /2 • c[m]=0.75-(f-m)2 • c[m+1]=(m-f-0.5)2 /2

8. Example

9. Linear Decoding • Normalized convolution of the channel vector • Choice of n by heuristics • Largest denominator (3-box filter) • Additional: local maximum 0

10. Quantization Effect

11. Quadratic Decoding I • Idea: detect local maximum of B-spline interpolated channel vector • Step 1: recursive filtering to obtain interpolation coefficients:

12. Quadratic Decoding II • Step 2: detect zeros

13. Quadratic Decoding III • Step 3: compute energy • Step 4: sort the decoded values according to their energy(the energy represents the confidence) The decoded values must be shiftedand rescaled to the original interval

14. Quantization Effect

15. Kernel Density Estimation I • Given: several realizations of a stochastic variable (samples of the pdf) • Goal: estimate pdf from samples • Method: convolve samples with a kernel function

16. Kernel Density Estimation II • Requirements for kernel function: • Non-negative • Integrates to one • Expectation of estimate:

17. Relation to C.R. • Adding channel representation of several realizations corresponds to a sampled kernel density estimation • Ideal interpolation with B-splines possible!

18. Error norm Influence function L2 vs. Robust Optimization • Outliers are critical for L2 optimization: • Idea of robust estimation: • error norm is saturated for outliers • Influence function becomes zero for outliers

19. Robust Error Norm E f - f0

20. Robust Influence Function E’ f - f0

21. Influence Function of C.R. Obtained from lineardecoding:

22. Error Norm of C.R. Obtained by integrating the influence function:

23. Channel Smoothing

24. Discontinuity is preserved Constant and linear regions are correctly estimated Channel Smoothing Example

25. Stochastic Signals • Stochastic signal: single realization of a stochastic process • Ergodicity assumption: • averaging over several realizations at a single point can be replaced with • averaging over a neighborhood of a single realization

26. Ergodicity & C.S. • Ergodicity often not fulfilled for signals / features, but trivial for channels • Ergodicity of channels implies that averaging of channels corresponds to (sampled) kernel density estimation

27. Quantization Effect and C.S.

28. Outlier Rejection in C.S.

29. Applications • Image denoising • Infilling of information • Orientation estimation • Edge detection • Corner detection • Disparity estimation

30. Image Denoising

31. Infilling of Information

32. Orientation Estimation

33. Corner Detection

34. Corner Detection

35. Corner Detection

36. Disparity Estimation

37. Disparity Estimation

38. Further Reading • B-Spline Channel Smoothing for Robust EstimationFelsberg, M., Forssén, P.-E., Scharr, H. LiTH-ISY-R-2579January, 2004