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

Laurent Provot , Yan Gerard *

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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

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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

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Outline

9

State of the Art

Principle

Error bound

Experimental results

Problem Statement

Conclusion

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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)

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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)

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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

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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 ?

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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

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State of the Art

Principle

Error bound

Experimental results

Problem Statement

Conclusion

Principle

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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

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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

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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

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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

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Definition : The roughness of f(x) of degree k roughnessk(f)= inf ( ||P(x)- f(x)|| )

P in R [X] Xk

Principle

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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

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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

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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

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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

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State of the Art

Principle

Error bound

Experimental results

Problem Statement

Conclusion

Error bound

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- 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

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Theorem : | f (k) (x) - f h (k) (x) | = O( h ) 1

k+1

Error bound

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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

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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

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1k+1

Outline

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State of the Art

Principle

Error bound

Experimental results

Problem Statement

Conclusion

Experimental Results

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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

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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

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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

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Second derivative at several resolutions

Experimental Results

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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

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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

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perspectives

- Experimental comparisons with other approaches

- Improve the computation of the derivatives at consecutive points

What remains to do …

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Thank you for your attention