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Estimation of the derivatives of a digital function with a convergent bounded error
Laurent Provot , Yan Gerard *
1
DGCI, April, 6th 2011
* speaker
Outline
2
State of the Art
Principle
Error bound
Experimental results
Problem Statement
Conclusion
Outline
3
State of the Art
Principle
Error bound
Experimental results
Problem Statement
Conclusion
Problem statement
A digital function
Input
Its ``derivative´´
Output
4
Which definition of digital derivative ?
Which criterion to assess a definition ?
Problem statement
5
Which criterion to assess a definition ?
and prove Lim fh’(x) = f ’(x)
Problem statement
continuous function f(x)
f ’(x)
derivative
digitization
digital function fh(x)
fh’(x)
digital derivative
~
6
grid resolution h
Aim: bound the error |f’(x) – fh’(x)| according to h
h 0
f(x) fh(x)
Which criterion to assess a definition ?
and prove Lim fh’(x) = f ’(x)
Problem statement
continuous function f(x)
f ’(x)
derivative
digitization
digital function fh(x)
fh’(x)
digital derivative
~
7
grid resolution h
Aim: bound the error |f’(x) – fh’(x)| according to h
h 0
f(x) fh(x)
Which criterion to assess a definition ?
Lim fh’(x) = f ’(x) h 0
and prove Lim fh’(x) = f ’(x)
Problem statement
continuous function f(x)
f ’(x)
derivative
digitization
digital function fh(x)
fh*’(x)
digital derivative
~
grid resolution h
Aim: bound the error |f’(x) – fh’(x)| according to h
h 0
f(x) fh(x)
Which criterion to assess a definition ?
Lim fh’(x) = f ’(x) h 0
This criterion is called Multigrid Convergence
8
Outline
9
State of the Art
Principle
Error bound
Experimental results
Problem Statement
Conclusion
10
State of the Art
Finite differences
Which definition for a digital derivative ?
f ’(x) = f(x+1)-f(x)
Drawback: only integer values (with a lot of small variations)
No Multigrid convergence
Idea: smooth with a convolution…
Digital function f(x)
11
State of the Art
Finite differences
Which definition for a digital derivative ?
f ’(x) = f(x+1)-f(x)
Idea: smooth with a convolution…
Digital function f(x)
12
State of the Art
Finite differences with a convolution
Which definition for a digital derivative ?
f ’(x) = f(x+1)-f(x) Digital function f(x)
Convolution with a binomial kernel (R. Malgouyres, S. Fourey, H.-A Esbelin, F. Brunet…), since 2008…
For the derivative of order k| f
(k) (x) - fh(k) (x) | < O( h (2/3) )
Property : Multigrid convergence | f ’(x) - fh’(x) | < O( h 2/3 )
k
13
State of the Art
Digital Segments 0≤ y - (ax+b) < 1 as Tangents
Digital function f(x)
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 )
Which definition for a digital derivative ?
14
State of the Art
Digital Segments 0≤ y - (ax+b) < 1 as Tangents
Digital function f(x)
Which definition for a digital derivative ?
Idea: use digital primitives of higher degree 0≤ y – P(x) <namely Digital Level Layers (see poster session)
Generalization
Outline
15
State of the Art
Principle
Error bound
Experimental results
Problem Statement
Conclusion
Principle
16
A discrete function f : X R R A maximal degree k
Input
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
In most cases, if card X > k+1 , there is no polynomial P of degree ksuch that for all x in X, P(x)=f(x)
No interpolation we expand each value in intervals, so that we can fit
Principle
17
A discrete function f : X R R A maximal degree k
Input
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
Principle
18
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
The expansion of y=P(x) in |y-P(x)|≤ leads to digital primitives called Digital Level Layers
Remark : it is equivalent to expand the values of f in intervalls or to expand the polynomial P
A discrete function f : X R R A maximal degree k
Input
Principle
19
Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| )
P in R [X] Xk
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
A discrete function f : X R R A maximal degree k
Input
Principle
20
Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| )
P in R [X] Xk
Principle
21
Property : The roughness of f(x) is the half of the vertical thickness of { ((xi )1≤i≤k ; f(x) ) / x in X }
Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| )
P in R [X] Xk
Many strips contain S
vertical thickness (S)
vertical thickness
one has a minimal vertical thickness
Principle
22
Example
Roughness of degree 2 of function f(0)=2f(1)=3f(2)=2f(3)=1f(4)=0
=
Vertical thickness(0,02,2)(1,12,3)(2,22,2)(3,32,1)(4,42,0)
Property : The roughness of f(x) is the half of the vertical thickness of { ((xi )1≤i≤k ; f(x) ) / x in X }
Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| )
P in R [X] Xk
Principle
23
Example
Roughness of degree 3 of function f(0)=2f(1)=3f(2)=2f(3)=1f(4)=0
=
Vertical thickness(0,02,03,2)(1,12,13,3)(2,22,23,2)(3,32,33,1)(4,42,43,0)
Property : The roughness of f(x) is the half of the vertical thickness of { ((xi )1≤i≤k ; f(x) ) / x in X }
Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| )
P in R [X] Xk
Linear ProgrammingMin h,P h / V x in X: -h≤f(x)-P(x)≤h
Computational GeometryChord’s algorithm
Principle
24
A digital function
Input
Its « derivative » of order k at x
Output
Fix a parameter Rmax of maximal roughness of degree k } Rmax
Extend the neighborhood around x until the roughness of the restriction of f becomes greater than the maximal authorized roughness Rmax
degree k = 2
The best fitting polynomial provides the derivatives
x
Outline
25
State of the Art
Principle
Error bound
Experimental results
Problem Statement
Conclusion
Error bound
26
- Let f: R R be a C k+1 function with f (k+1) (y) < M in the neighborhood of x
f(hx) - f h(x): Z Z is the digitization of f(x) at resolution h : f h(x) = | |
grid resolution h
fh(x)f (x) digitization
- Maximal roughness Rmax(x) ≥ +2
1
k+1
M
Three conditions
h
Theorem : | f (k) (x) - f h (k) (x) | = O( h ) 1
k+1
Error bound
27
Theorem : | f (k) (x) - f h (k) (x) | = O( h ) 1
k+1
Error bound
28
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 }
Nm(P) ≤ 1 |ak|≤
We prove for the polynomial P(X)= ∑ ak X kn
k=1Uk,n
m k
Theorem : | f (k) (x) - f h (k) (x) | = O( h ) 1
k+1
Error bound
29
α Derivative of order k
k=1 k=2 k=3 k=4 k=5
Digital Straight Segments
(J-O Lachaud et al.)0.33
Convolutions(R. Malgouyres et al.) ( )k 0.66 0.44 0.296 0.2 0.13
Digital Level Layers(L. Provot, Y. G.) 0.5 0.33 0.25 0.2 0.16
The errors are bounded in hα : the greater the α value, the better the convergence
23
1k+1
Outline
30
State of the Art
Principle
Error bound
Experimental results
Problem Statement
Conclusion
Experimental Results
31
digitization of sin(x) at resolution h=0.05
First derivative
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
Experimental Results
32
digitization of sin(x) at resolution h=0.05
Second derivative
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
Experimental Results
33
digitization of sin(x) at resolution h=0.05
Third derivative
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
Experimental Results
34
Second derivative at several resolutions
Experimental Results
35
digitization of sin(x) at resolution h=0.05
Second derivative
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
-3 -2 -1 -0.6 -0.2 0 0.2 0.6 1 2 3
Experimental Results
36
For the time of computation (with GMP), we did no specific experiments :
- For the first order derivative of sin(x) , it took less than 0.2s on 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)
Outline
37
State of the Art
Principle
Error bound
Experimental results
Problem Statement
Conclusion
- Bound the error on multivariate functions and prove multigrid convergence
- The method allows to compute partial derivatives of multivariate digital function : there is nothing to add in practice!
Conclusion
38
perspectives
- Experimental comparisons with other approaches
- Improve the computation of the derivatives at consecutive points
What remains to do …
39
Thank you for your attention