Estimation of the derivatives of a digital function with a convergent bounded error

1. Estimation of the derivatives of a digital function with a convergent bounded error Laurent Provot , Yan Gerard * DGCI, April, 6th 2011 * speaker

2. Outline Problem Statement State of the Art Principle Error bound Experimental results Conclusion 2

3. Outline Problem Statement State of the Art Principle Error bound Experimental results Conclusion 3

4. Problem statement Input A digital function Output Its ``derivative´´ Which definition of digital derivative ? Which criterion to assess a definition ? 4

5. Problem statement Which criterion to assess a definition ? 5

6. Problem statement Which criterion to assess a definition ? grid resolution h f(x) fh(x) digitization continuous function f(x) digital function fh(x) derivative digital derivative ~ f ’(x) fh’(x) Aim: bound the error|f’(x) – fh’(x)|according toh and proveLim fh’(x) = f’(x) h 0 6

7. Problem statement Which criterion to assess a definition ? grid resolution h f(x) fh(x) digitization continuous function f(x) digital function fh(x) derivative digital derivative ~ f ’(x) fh’(x) Aim: bound the error|f’(x) – fh’(x)|according toh and proveLim fh’(x) = f’(x) Lim fh’(x) = f’(x) h 0 h 0 7

8. Problem statement Which criterion to assess a definition ? grid resolution h f(x) fh(x) digitization continuous function f(x) digital function fh(x) derivative digital derivative This criterion is called Multigrid Convergence ~ f ’(x) fh*’(x) Aim: bound the error|f’(x) – fh’(x)|according toh and proveLim fh’(x) = f’(x) Lim fh’(x) = f’(x) h 0 h 0 8

9. Outline Problem Statement State of the Art Principle Error bound Experimental results Conclusion 9

10. State of the Art Digital function f(x) f ’(x) = f(x+1)-f(x) Which definition for a digital derivative ? Finite differences Drawback: only integer values (with a lot of small variations) No Multigrid convergence Idea: smooth with a convolution… 10

11. State of the Art Digital function f(x) f ’(x) = f(x+1)-f(x) Which definition for a digital derivative ? Finite differences Idea: smooth with a convolution… 11

12. State of the Art Digital function f(x) f ’(x) = f(x+1)-f(x) Which definition for a digital derivative ? Finite differences with a convolution Convolution with a binomial kernel (R. Malgouyres, S. Fourey, H.-A Esbelin, F. Brunet…), since 2008… Property : Multigrid convergence | f’(x) - fh’(x) | < O( h 2/3 ) For the derivative of order k | f(k)(x) - fh(k)(x) | < O( h (2/3) ) k 12

13. State of the Art Digital function f(x) Which definition for a digital derivative ? Digital Segments 0≤ y - (ax+b) < 1 as Tangents This approach is mostly introduced on digital curves (4 or 8-connected) (J.-O Lachaud, A. Vialard, F. De Vieilleville, F. Feschet, L. Tougne) since 2004… Property : Multigrid convergence | f’(x) - fh’(x) | < O( h 1/3 ) 13

14. State of the Art Digital function f(x) Which definition for a digital derivative ? Digital Segments 0≤ y - (ax+b) < 1 as Tangents Generalization Idea: use digital primitives of higher degree 0≤ y – P(x) < namely Digital Level Layers (see poster session) 14

15. Outline Problem Statement State of the Art Principle Error bound Experimental results Conclusion 15

16. Principle Input A discrete function f : X R R A maximal degree k In most cases, if card X > k+1 , there is no polynomial P of degree k such that for all x in X, P(x)=f(x) No interpolation we expand each value in intervals, so that we can fit there is a threshold Rk such that there exists a polynomial P of degree k with for all x in X, |P(x)-f(x)|≤Rk 16

17. Principle Input A discrete function f : X R R A maximal degree k there is a threshold Rk such that there exists a polynomial P of degree k with for all x in X, |P(x)-f(x)|≤Rk 17

18. Principle Input A discrete function f : X R R A maximal degree k there is a threshold Rk such that there exists a polynomial P of degree k with for all x in X, |P(x)-f(x)|≤Rk Remark : it is equivalent to expand the values of f in intervalls or to expand the polynomial P The expansion of y=P(x) in |y-P(x)|≤ leads to digital primitives called Digital Level Layers 18

19. Principle Input A discrete function f : X R R A maximal degree k there is a threshold Rk such that there exists a polynomial P of degree k with for all x in X, |P(x)-f(x)|≤Rk Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| ) k X P in R [X] 19

20. Principle Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| ) k X P in R [X] 20

21. Principle Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| ) k X P in R [X] Property : The roughness of f(x) is the half of the vertical thickness of { ((xi )1≤i≤k; f(x) ) / x in X } vertical thickness Many strips contain S one has a minimal vertical thickness vertical thickness (S) 21

22. Principle Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| ) k X P in R [X] Property : The roughness of f(x) is the half of the vertical thickness of { ((xi )1≤i≤k; f(x) ) / x in X } Example Roughness of degree 2 of function f(0)=2 f(1)=3 f(2)=2 f(3)=1 f(4)=0 Vertical thickness (0,02,2) (1,12,3) (2,22,2) (3,32,1) (4,42,0) = 22

23. Principle Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| ) k X P in R [X] Property : The roughness of f(x) is the half of the vertical thickness of { ((xi )1≤i≤k; f(x) ) / x in X } Example Roughness of degree 3 of function f(0)=2 f(1)=3 f(2)=2 f(3)=1 f(4)=0 Vertical thickness (0,02,03,2) (1,12,13,3) (2,22,23,2) (3,32,33,1) (4,42,43,0) = Linear Programming Min h,P h / V x in X: -h≤f(x)-P(x)≤h Computational Geometry Chord’s algorithm 23

24. Principle Input A digital function Output Its « derivative » of order katx x Fix a parameter Rmax of maximal roughness of degree k } Rmax degree k = 2 Extend the neighborhood around x until the roughness of the restriction of fbecomes greater than the maximal authorized roughness Rmax The best fitting polynomial provides the derivatives 24

25. Outline Problem Statement State of the Art Principle Error bound Experimental results Conclusion 25

26. Error bound 1 M 2 1 k+1 k+1 Three conditions -Let f: R Rbe aC k+1 function with f (k+1) (y) < M in the neighborhood of x - Maximal roughness Rmax(x) ≥ + grid resolution h digitization f(x) fh(x) f(hx) - f h(x): Z Z is the digitization of f(x) at resolution h : f h(x) = | | h Theorem : | f (k) (x) - f h(k) (x) | = O( h ) 26

27. Error bound 1 k+1 Theorem : | f (k) (x) - f h(k) (x) | = O( h ) 27

28. Error bound 1 k+1 Theorem : | f (k) (x) - f h(k) (x) | = O( h ) Proof : Taylor Lagrange inequality + A discrete norm on polynomials of degree n Nm(P)= max { |P(x)| / for all integers x from –m to m } We prove for the polynomial P(X)= ∑ ak X k n k=1 Uk,n |ak|≤ Nm(P) ≤ 1 m k 28

29. Error bound The errors are bounded in hα: the greater the α value, the better the convergence 2 3 1 k+1 29

30. Outline Problem Statement State of the Art Principle Error bound Experimental results Conclusion 30

31. Experimental Results digitization of sin(x) at resolution h=0.05 -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 First derivative -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 31

32. Experimental Results digitization of sin(x) at resolution h=0.05 -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 Second derivative -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 32

33. Experimental Results digitization of sin(x) at resolution h=0.05 -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 Third derivative -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 33

34. Experimental Results Second derivative at several resolutions 34

35. Experimental Results digitization of sin(x) at resolution h=0.05 -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 Second derivative -3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3 35

36. Experimental Results For the time of computation (with GMP), we did no specific experiments : - For the first order derivative of sin(x) , it took less than 0.2son a laptop. - For the derivatives of higher order, it goes to some seconds (<10s) Nevertheless, the code which is used is made in order to compute only one value without taking account of the neighbors Many improvements have to be done for the computation of the derivatives at consecutive points (as for the computation of the tangential cover) 36

37. Outline Problem Statement State of the Art Principle Error bound Experimental results Conclusion 37

38. Conclusion What remains to do … - Experimental comparisons with other approaches - Improve the computation of the derivatives at consecutive points perspectives - The method allows to compute partial derivatives of multivariate digital function : there is nothing to add in practice! - Bound the error on multivariate functions and prove multigrid convergence 38

39. Thank you for your attention 39