How does bilateral filter relates with other methods?

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1 A Gentle Introduction to Bilateral Filtering and its Applications How does bilateral filter relates with other methods? Fredo Durand (MIT CSAIL) Slides by Pierre Kornprobst (INRIA) 0:35

2 Many people worked on edge-preserving restoration Bilateral filter Partial differential equations Anisotropic diffusion Local mode filtering Robust statistics

3 Goal: Understand how does bilateral filter relates with other methods Bilateral filter Partial differential equations Local mode filtering Robust statistics

4 Local mode filtering principle Spatial window Smoothed local histogram You are going to see that BF has the same effect as local mode filtering

5 Let s prove it! Define global histogram Define a smoothed histogram Define a local smoothed histogram What does it mean to look for local modes? What is the link with bilateral filter?

6 Definition of a global histogram Formal definition of histogram H at intensity i: Where is the Dirac symbol (zero everywhere except at 0) A sum of Dirac, «a sum of ones»

7 # pixels intensity

8 # pixels Smoothing the histogram intensity

9 Smoothing the histogram

10 # pixels intensity

11 # pixels intensity

12 # pixels intensity

13 # pixels intensity

14 # pixels intensity

15 # pixels This is it! intensity

16 Definition of a local smoothed histogram We introduce a «smooth window» where Smoothing of intensities Spatial window And that s the formula to have in mind!

17 Definition of local modes A local mode i verifies

18 Local modes? Given We look for Result:

19 Summary A local mode i verifies: Hey! That looks like bilateral filter!!! One iteration of the bilateral filter amounts to converge to the local mode

20 DIscussion The bilateral filter goes to a LOCAL mode, not necessarily the global mode Often desirable: mode closest to input pixel Sometimes not: impulse noise case Recall the use of the median as pre-filter amounts to going to the global mode

21 Take home message # Bilateral filter is equivalent to mode filtering in local histograms [Van de Weijer, Van den Boomgaard, 200, etc]

22 Goal: Understand how does bilateral filter relates with other methods Bilateral filter Partial differential equations Local mode filtering Robust statistics

23 Robust statistics? Goals: Reduce the influence of outliers Minimizing a cost Error norm In standard robust statistics Iq are measured data, Ip is a robust average of the data [Huber 8, Hampel 86]

24 Robust statistics? In our case: the output at a pixel should be a robust smoothing of its neighbors Error norm Minimizing a cost Extended local formulation [Huber 8, Hampel 86]

25 How to choose the error norm? Least square pays a big penalty for big errors problem in the presence of outliers

26 How to choose the error norm? Strong differences must not be too penalizing, otherwise, everything will be smoothed!

27 How to minimize the cost function? Gradient descent and iterative scheme Doesn t cost too much No influence In gradient

28 Getting closer to bilateral filter Rewrite introducing a new function

29 Getting closer to bilateral filter Rewrite introducing a new function g has the same qualitative behavior than a Gaussian Now this operator reminds us about bilateral filter! No influence

30 Really the same? Iteratively reweighted least square M-estimator Weighted average of the data [Hampel etal, 986]: M-estimators and W_estimators are essentially equivalent and solve the same minimization problem W-estimator

31 Take home message #2 Bilateral filter can be interpreted in term of robust statistics since it is related to a cost function! [Durand, Dorsey, 2002, Black, Marimont, 998, etc]

32 Goal: Understand how does bilateral filter relates with other methods Bilateral filter Partial differential equations Local mode filtering Robust statistics

33 Disclaimer We will shrink the neighborhood This will lose some properties of the bilateral filter But although partial, this parallel is insightful

34 What do I mean by PDEs? Images live in a continuous domain Two kinds of formulations Variational approach Evolving a partial differential equation

35 Recall robust statistics

36 PDEs PDEs Images PDEs PDEs PDEs are PDEs PDEs continuous Robust statistics Images Robust statistics are Robust St Robust statistics discrete Robust statistics Robust St

37 Some technical results to establish Considering the Yaroslavsky Filter When (operation similar to M-estimators) At a very local scale, the asymptotic behavior of the integral operator corresponds to a diffusion operator [Buades, Coll, Morel, 2005]

38 The PDE world at a glance Tschumperle, Deriche Perez, Gangnet, Blake Tschumperle, Deriche Sussman Desbrun etal Breen, Whitaker

39 Discussion We shrunk the kernel

40 Take home message #3 Bilateral filter is a discretization of a particular kind of a PDEbased anisotropic diffusion. [Barash 200, Elad 2002, Durand 2002, Buades, Coll, Morel, 2005] Welcome to the PDE-world! [Kornprobst 2006]

41 Summary Bilateral filter is one technique for anisotropic diffusion and it makes the bridge between several frameworks. From there, you can explore news worlds! Bilateral filter Local mode filtering Anisotropic diffusion PDEs Robust statistics

42 Questions?

Examples. Bilateral filter. Input. Soft texture is removed

Examples. Bilateral filter. Input. Soft texture is removed A Gentle Introduction to Bilateral Filtering and its Applications Limitation? Pierre Kornprobst (INRIA) 0:20 Examples Soft texture is removed Input Bilateral filter Examples Constant regions appear Input