1. Computer Vision Laboratory 1 B-Spline Channels & Channel Smoothing Michael Felsberg Computer Vision Laboratory Linköping University SWEDEN

2. Computer Vision Laboratory 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. Computer Vision Laboratory 3 Example for Single Value

4. Computer Vision Laboratory 4 Example for Multiple Values

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

6. Computer Vision Laboratory 6 Quadratic B-Splines

7. Computer Vision Laboratory 7 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 B-Splines Encoding We assume f to be shifted and rescaled such that

8. Computer Vision Laboratory 8 Example

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

10. Computer Vision Laboratory 10 Quantization Effect

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

12. Computer Vision Laboratory 12 Quadratic Decoding II –Step 2: detect zeros

13. Computer Vision Laboratory 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 shifted and rescaled to the original interval

14. Computer Vision Laboratory 14 Quantization Effect

15. Computer Vision Laboratory 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. Computer Vision Laboratory 16 Kernel Density Estimation II Requirements for kernel function: –Non-negative –Integrates to one Expectation of estimate:

17. Computer Vision Laboratory 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. Computer Vision Laboratory 18 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 Error norm Influence function

19. Computer Vision Laboratory 19 Robust Error Norm E f - f 0

20. Computer Vision Laboratory 20 Robust Influence Function E’ f - f 0

21. Computer Vision Laboratory 21 Influence Function of C.R. Obtained from linear decoding:

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

23. Computer Vision Laboratory 23 Channel Smoothing

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

25. Computer Vision Laboratory 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. Computer Vision Laboratory 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. Computer Vision Laboratory 27 Quantization Effect and C.S.

28. Computer Vision Laboratory 28 Outlier Rejection in C.S.

29. Computer Vision Laboratory 29 Applications Image denoising Infilling of information Orientation estimation Edge detection Corner detection Disparity estimation

30. Computer Vision Laboratory 30 Image Denoising

31. Computer Vision Laboratory 31 Infilling of Information

32. Computer Vision Laboratory 32 Orientation Estimation

33. Computer Vision Laboratory 33 Corner Detection

34. Computer Vision Laboratory 34 Corner Detection

35. Computer Vision Laboratory 35 Corner Detection

36. Computer Vision Laboratory 36 Disparity Estimation

37. Computer Vision Laboratory 37 Disparity Estimation

38. Computer Vision Laboratory 38 Further Reading B-Spline Channel Smoothing for Robust Estimation Felsberg, M., Forssén, P.-E., Scharr, H. LiTH-ISY-R-2579 January, 2004