Patterns in discretized parabolas and length estimation (extended version)

(1)

HAL Id: hal-00374372

https://hal.archives-ouvertes.fr/hal-00374372v5

Submitted on 6 May 2010

HAL

is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire

HAL, est

destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Patterns in discretized parabolas and length estimation (extended version)

Alain Daurat, Mohamed Tajine, Mahdi Zouaoui

To cite this version:

Alain Daurat, Mohamed Tajine, Mahdi Zouaoui. Patterns in discretized parabolas and length estima-

tion (extended version). DGCI 2009, Sep 2009, Montréal, Canada. pp.737-384, �10.1007/978-3-642-

04397-0�. �hal-00374372v5�

(2)

Patterns in discretized parabolas and length estimation

(extended version)

Alain Daurat, Mohamed Tajine, and Mahdi Zouaoui LSIIT CNRS UMR 7005, Universit´e de Strasbourg,

Pˆole API, Boulevard S´ebastien Brant, 67400 Illkirch-Graffenstaden, France {daurat,tajine,mzouaoui}@unistra.fr

Abstract. We estimate the frequency of the patterns in the discretization of parabolas, when the resolution tends to zero. We deduce that local estimators of length almost never converge to the length for the parabolas.

Keywords:Digital Curve, Pattern, Multigrid Convergence.

1 Introduction

Length estimation is an important domain of Image Analysis (see [1] for a re- view). In this paper, we will consider the problem of estimating the length of a curve from its discretizations at different resolutions. In particular we are interested in the comportment of estimators when the resolution tends to zero.

We also restrict our study to special estimators called “local estimators” which consist in consideringpatterns which are pieces of fixed length of the discretized curve. Local estimators simply consist to fix a weight to each pattern and summing these weights to obtain the estimation of length (See Fig. 4 for illustration).

So, if we want to study the estimated length by local estimators when the resolution tends to zero, we have at first to study the number of occurrences of a pattern of the discretization of digital curves. In fact an asymptotic result about the occurrence number of patterns for discretized general curves looks to be a quite hard problem, because the discretization process is not a continuous process (the integer part function is not continuous), so the estimation of the occurrence number of patterns cannot be deduced from Mathematical Analysis arguments, but by Number Theory arguments. The two first authors of this paper have already made this study for segments in [2]. In this paper we continue this work by considering another class of curves, the parabolas.

The paper is organized as follows: Section 2 describes the notations used in this paper, Section 3 will be devoted to the study of the frequency of patterns in parabolas, and finally Section 4 will apply the results of this study to the local estimators of length of parabolas. Appendix A contains the detailed proofs which are not in the main part of this paper.

(3)

2 Notations

In this section we precise the notations that will be used in all the paper.

– Forx∈Rwe denote by⌊x⌋(resp.⌈x⌉) the integerksuch thatk≤x < k+ 1 (resp.k−1< x≤k).

– The fractional part ofxis denotedhxiand is defined byx=⌊x⌋+hxi. – ForA, B∈Z, the discrete interval{A, A+1, . . . , B−1, B}is denotedJA, BK.

– Letmbe a positive integer. A pattern of sizemis a functionωfrom J0, mK toZ such thatω(0) = 0 andω(k+ 1)∈ {ω(k), ω(k) + 1} (see Fig. 1). The set of patterns of sizemis denoted P^m.

– IfX andY are two real numbers such thatY >0 thenXmodY is the real number such that 0≤XmodY < Y and ^X⁻^X_Y^mod^Y ∈Z.

– Forr∈RandE⊂R²,rE={(rx, ry)|(x, y)∈E}.

00

11 010011 01 01 00 11

0000 1111

0 1 010011

0 10011

00 11

0 10011

00 11

0 10011

0000 1111

0

1 010011 01

00 11

0000 1111

0 1 0011

00 11

00 11 010011

00 11

00 11 01

00 11

0 1

0000 1111

00 11

0 10011

00 11

0 1

00 11

0 00 1

0011 11 00

11 0000 1111

0 1

00 11

0 1 00

11 00 11

00 11

00 11 01

00 11 010011

00 11

00 11 0 1 0000

1111 0000 1111

00 11 00 11

Fig. 1. The 16 patterns of size m = 4. Only the encircled one are not digital segments.

3 Frequency of patterns in Discrete Parabolas

Leta, b∈Rsuch thata < band a derivable functiong: [a, b]→Rwhich satisfies 0≤g^′(x)≤1 for allx∈[a, b].

In all the following, for anyr >0 we use the notations:

• Ar=⌈^ar⌉,Br=⌊r^b⌋, Nr=Br−Ar+ 1,

• Cr^g=r{(X, Y)∈Z²|Ar≤X≤Br and Y =⌊^g(rX)r ⌋}.

The setCr^gis the “naive” discretization of the graph ofgat resolutionr, andNr

is its number of points.

(4)

Letm be a positive integer. The pattern at position X ∈ JAr, Br−mK of sizem ofCr^g, denoted ω_X,r,m^g is defined by:

ω_X,r,m^g (k) =⌊g(r(X+k))

r ⌋ − ⌊g(rX) r ⌋. The frequency of a patternω of sizem inCr^gis defined by:

F_r^g(ω) =card{X ∈JAr, Br−mK|ω_X,r,m^g =ω}

Nr−m .

The aim of this section is the study ofF_r^g(ω) for some functions g. For this we will approximate the curve by its tangents which will also be discretized, so we need some notions about digital straight lines:

Foru∈[0,1] andv∈[0,1), let us denotes^u,v_m the pattern of sizemdefined by:

s^u,v_m (k) =⌊uk+v⌋.

A pattern of this form is called a digital segment of sizem.

For any patternω,

P I(ω) ={(u, v)∈[0,1]²| ⌊uk+v⌋=ω(k) for anyk∈J0, mK},

pinf_u(ω) = inf{v|(u, v)∈P I(ω)}, (1) psup_u(ω) = sup{v|(u, v)∈P I(ω)}, (2) F Lu(ω) = 0 if{v|(u, v)∈P I(ω)}=∅

= psup_u(ω)−pinf_u(ω) otherwise.

P I(ω) is called the preimage of ω, it is nonempty if and only if ω is a digital segment.F Lu(ω) is intuitively the frequency of the patternω in the discretized straight lines of slopeu. (See [2,3] for more details and [4] for the generalization to slopes of planes).

In all the paper, the considered curves are parabolas corresponding to the function g(x) = αx². We distinguish two cases: the case α irrational and the caseαrational. In Subsection 3.1, we will see that forαirrational, the frequency F_r^g(ω) converges, whenris rational and tends to zero, to a quantity which can be expressed by using the functionx7→F L_g′(x)(ω). In Subsection 3.2, we study the caseαrational, but we do not succeed to prove a similar result as in the case αirrational. Nevertheless we obtain a weaker result (The Tangent Lemma).

3.1 Parabolas of equation y= αx² with α irrational

In this subsection we consider curves Cr^g for g defined on [a, b] by g(x) = αx² with αirrational, and 0≤a < b ≤ 2α¹. This last hypothesis is needed to have 0≤g^′(x)≤1 forx∈[a, b].

The main result of this subsection is the following:

(5)

Theorem 1. If g(x) =αx² withα /∈Qthen for any patternω we have F_r^g(ω)−−−→_r

→0 r∈Q

1 b−a

Z b a

F L_g′(x)(ω)dx (3)

The rest of this subsection is devoted to the proof of this Theorem.

The first needed lemma shows that the discretization of a curve and the discretization of its tangent are similar near the origin of the tangent and when the resolution tends to zero:

Lemma 1 (Tangent Lemma). If g is defined on [a, b] by g(x) =αx² with α irrational, and0≤a < b≤ 2α¹ then

card{X ∈JAr, Br−mK|ω_X,r,m^g 6=s^g^′^(rX),^h

g(rX) r i

m }

Nr−m −−−→_r

→0 r∈Q

0.

Lemma 1 is illustrated by Fig. 2. In the last lemma we considerr∈Qbecause its proof needs the irrationality ofαr.

rX

Discretization of the curve Discretization of the tangent y=g(x)

tangent of the curve

r

Fig. 2. Comparison of the discretization of the curve and the discretization of its tangent : here we haveω^g_X,r,m=s^g^′^(rX),^h

g(rX) r i

m form= 9 but not form= 10.

Before starting the proof of Lemma 1 we need one more notation and two other lemmas. ForX ∈JAr, BrK, we define Pr,k(X) by:

Pr,k(X) =g(rX)

r +g^′(rX)k=αrX²+ 2αkrX =αr((X+k)²−k²).

This definition is motivated by the following lemma:

Lemma 2. If r < _αm¹2, then for anyX we have ω_X,r,m^g =s^g^′^(rX),^h

g(rX) r i

m if and

only if for allk∈J0, mK we havehPr,k(X)i<1−αrk².

(6)

Proof.

ω_X,r,m^g (k) =⌊g(r(X+k))

r ⌋ − ⌊g(rX) r ⌋

=⌊g(rX)

r +g^′(rX)k+αrk²⌋ − ⌊g(rX) r ⌋ We know that⌊u+v⌋=⌊u⌋+⌊v⌋iffhui+hvi<1.

Soω_X,r,m^g (k) =⌊^g(rX)r +g^′(rX)k⌋−⌊^g(rX)r ⌋iffh^g(rX)r +g^′(rX)ki+hαrk²i<1.

With the hypothesisr < _αm¹2 we havehαrk²i=αrk². But:

⌊g(rX)

r +g^′(rX)k⌋−⌊g(rX)

r ⌋=⌊hg(rX)

r i+g^′(rX)k⌋−⌊hg(rX)

r i⌋=s^g^′^(rX),^h

g(rX) r i

m (k)

Soω^g_X,r,m=s^g^′^(rX),h

g(rX) r i

m iff for allk∈J0, mKwe havehPr,k(X)i<1−αrk².

⊓

⊔ Lemma 3. Let I be an interval of[0,1]. We haveTr,k(I)−−−→

r→0 r∈Q

µ(I)where

Tr,k(I) = 1 Nr

card{X ∈JAr, BrK| hPr,k(X)i ∈I} andµ(I)is the usual length ofI.

The proof of this lemma uses Weyl’s argument, as in the proof of Theorem 1 of [2] or Appendix A.1 of [4], but extended to the quadratic case following the same ideas as in [5, p6-7]. It is given in Appendix A.1.

Proof of Lemma 1. We recall that ω_X,r,m^g =s^g^′^(rX),h

g(rX) r i

m iff for allk ∈J0, mK we havehPr,k(X)i<1−αrk² so:

card{X ∈JAr, Br−mK|ω_X,r,m^g 6=s^g^′^(rX),^h

g(rX) r i

m }

Nr−m ≤ Nr

Nr−m maxm

k=0Tr,k(Ir,k) where Ir,k = [1−αrk²,1), so it is sufficient to show that Tr,k(Ir,k)−−−→_r

→0 r∈Q

0 for anyk∈J0, mK. Letε >0, there existsR1>0 such that for anyr < R1 we have αrk² < ₂^ε. We know by Lemma 3 that there exists R2 > 0 such that for any r < R2 we haveTr,k(IR0,k)≤µ(IR0,k) +^ε₂. Ifris such thatr <min(R1, R2), we have:

Sr,k(Ir,k)≤Sr,k(IR1,k)

≤µ(IR1,k) +ε 2

=αrk²+ε 2

≤ ε 2+ ε

2 =ε

This finishes the proof of Lemma 1. ⊓⊔

(7)

Sketch of Proof of Theorem 1. We present here the main ideas of the proof of Theorem 1. The detailed proof is in Appendix A.2.

By using Lemma 1 we know thatF_r^g(ω) has the same limit asrtends to zero as:

G^g_r(ω) =card{X ∈JAr, Br−mK|s^g^′^(rX),^h

g(rX) r i

m =ω}

Nr−m .

Buts^x,y_m =ω is equivalent to (x,hyi)∈P I(ω) so:

G^g_r(ω) =card{X ∈JAr, Br−mK|(g^′(rX),h^g(rX)_r i)∈P I(ω)} Nr−m

which has the same limit asHr(P I(ω)) where

Hr(E) =card{X ∈JAr, BrK|(g^′(rX),h^g(rX)r i)∈E}

Br−Ar+ 1 .

By applying Lemma 3 withk= 0 to the piece of the curve y =g(x) restricted to the domaing^′−¹(α1)≤x≤g^′−¹(α2), we can prove:

Hr([α1, α2)×I)−−−→_r

→0 r∈Q

g^′−¹(α2)−g^′−¹(α1) b−a µ(I) So by approximatingP I(ω) as the union of rectangles

n

[

i=1

[yi−1, yi)×[pinf_y_i(ω),psup_y_i(ω)) we approximateHr(P I(ω)) by:

n

X

i=1

g^′−¹(yi)−g^′−¹(yi−1)

b−a F Lyi(ω) which is a Riemann sum forRb

a F L_g′(x)(ω)dx. ⊓⊔

Corollary 1. Ifg(x) =αx² with α /∈Q, then for any patternω which is not a digital segment we have

F_r^g(ω)−−−→_r

→0 r∈Q

0.

Numerical Application: We illustrate Theorem 1 with an example. Consider the curveC defined y=g(x) = ^√¹₂x² forxbetweena= 0 andb= ^√¹₂, and the patternω of size m= 3 defined by (ω(0), ω(1), ω(2), ω(3)) = (0,1,2,2). We will compute the limit of the frequency ofω when the resolution tends to zero.

(8)

First we can compute easilyF Lα(ω) becauseα7→ F Lα(ω) is a continuous function which is affine between twom-Farey numbers (see [3]). So we deduce:

F Lα(ω) = 0 ifα∈[0,1 2]

= 2α−1 ifα∈[1 2,2

3]

= 1−α ifα∈[2 3,1]

Theorem 1 proves that:

F_r^g(ω)−−−→_r

→0 r∈Q

1 b−a

Z b a

F L_g′(x)(ω)dx

=√ 2

Z √¹ 2

0

F L^√_2x(ω)dx

=√ 2

Z ₃√² 2

1 2√ 2

(2√

2x−1)dx+ Z √¹

2

2 3√ 2

(1−√ 2x)dx

!

= 1 12

3.2 Parabolas of equation y= αx² with α rational

Now we are interested in the case whereαis rational. Theorem 1 can be general- ized toαrational and to the irrational resolutionsrbecause only the irrationality ofαr is used in the proof of this theorem.

On the contrary, in the case α rational and rational resolutions, the only result we will prove in this subsection is about the Tangent Lemma. Moreover, we must impose some restrictions about the resolutionrand the interval [a, b] of definition of the parabola:rα= ¹_p wherepis a prime number ;a= 0 andb=_2α¹ . In all this subsection we suppose that these conditions are satisfied. Actually, we do not succeed to prove the Tangent Lemma in the general case for αrational.

LetPr,k(X) =rα((X+k)²−k²). We will prove that for any intervalI⊂[0,1]

rlim→0

1

rα is prime

card{X ∈JAr, Br−kK| hPr,k(x)i ∈I}

Nr−k =µ(I)

Definition 1. Let pbe a prime number anda be an integer number, we define the Legendre symbol (^a_p)of a relatively to p by

(a p) =







0 ifpdividesa

1 ifathere exists t∈Z such thatamodp=t²modp

−1otherwise.

(9)

Property 1. (P´olya-Vinogradov inequality [6], [7, Chap. 23]) Letp be a prime number and M andN two positive integers then

|

M+N

X

n=M

(n

p)|<√plog(p)

Corollary 2. LetJ =JM, M+NKbe an integer interval. Then

|card(J)

2 −card{y∈I|(y

p) = 1}|<√plog(p).

Lemma 4. Let α be a rational number and assume that a = 0 and b = _2α¹. Then, for any interval I⊂[0,1]

r→lim0

card{X ∈JAr, Br−kK| hPr,k(X)i ∈I}

Nr−k =µ(I)

where the limit is taken on the rsuch thatr >0andrα= ¹_p wherepis a prime number.

Proof. The functionX 7→X²modpfromJ1,^p⁻₂¹Kto{Y |(^Y_p) = 1}is a bijection.

PutHr=card{X ∈JAr, Br−kK| hPr,k(x)i ∈I} Nr

. Then

Hr= card{X∈Jk,⌊^p2⌋K| h^X²p^−k²i ∈I}

Nr−k = 2card{X∈Jk,^p⁻₂¹K|^X²^modpp ⁻^k² ∈I}

p+ 1−2k (4)

Thus

card{X ∈J1,p−1

2 K|X²modp∈J}= card{Y ∈J|(Y

p) = 1}whereJ =pI+k². So|card{X∈Jk,^p−₂¹K|X²modp∈J} −card{Y ∈J|(^Y_p) = 1}|< k.

By using P´olya-Vinogradov inequality we have:

|card(J)

2 −card{X ∈J1,p−1

2 K|X²modp∈J}|<√plog(p) So,|^card(J)2 −card{X ∈Jk,^p⁻₂¹K|X²modp∈J}|<√plog(p) +k.

By (4) we have

| 2 p+ 1−2k

card(J)

2 − 2

p+ 1−2kcard{X∈Jk,p−1

2 K|X²modp∈J}|< 2√plog(p) + 2k p+ 1−2k Thus, limr→0

card{X∈JAr, Br−kK| hPr,k(X)i ∈I}

Nr−k =µ(I)

⊓

⊔

(10)

Theorem 2 (Tangent Lemma for rational slopes and special rational resolutions).For any rationalα >0and anya, bsuch that0≤a < b≤2α¹ we have

card{X∈JAr, Br−mK|ω^g_X,r,m6=s^g^′^(rX),^h

g(rX) r i

m }

Nr−m −−−−−−−−→_r

→0

1

rα is prime

0

Proof. The case wherea= 0,b= _2α¹ can be proved exactly in the same way as for Lemma 1 by using Lemma 4 instead of Lemma 3. Consider the general case [a, b]⊂[0,_2α¹ ]. Then we know that:

card{X∈JAr, Br−mK|ω^g_X,r,m6=s^g^′^(rX),h

g(rX) r i

m }

Nr−m

≤ ⌊2αr¹ ⌋+ 1−m

Nr−m · card{X ∈J0,⌊^br⌋ −mK|ω^g_X,r,m6=s^g^′^(rX),^h

g(rX) r i

m }

⌊2αr¹ ⌋+ 1−m

−−−−−−−→_r

→0

1

rαis prime 1 2α

b−a·0 = 0 because of the casea= 0, b= 1 2α.

⊓

⊔ Unfortunately we do not successfully generalize Theorem 1 to rational α and some rational resolutions even if experimentally Theorem 1 seems to be true in all the cases.

4 Application to local estimators of length

A local estimator is given by a weight functionpfrom the setP^mof patterns of size mtoR. The estimated length of the curvey=g(x) where g: [a, b]→Rat resolutionris given by:

l(p, g, r) =r

⌊^Br−^mm^−Ar⌋

X

k=0

p(ω_A^g_r_+km,r,m).

Theorem 3. Let a, bsuch that 0≤a < b. The length estimated by a local esti- mator of a parabola y=αx², x∈[a, b], (α≤ 2b¹) converges when the resolution r is rational and tends to zero, to the length of the parabola only for a finite number of irrational numbers α.

Sketch of proof. For this proof we use the notation [f(x)]^b_a=f(b)−f(a).

It is easy to see that:

l(p, g, r)−(b−a) m

X

ω∈Pm

p(ω)F^′^g_r(ω)−−−→_r→₀ 0

(11)

00 11

0000 1111

00 11 00 11

00 11

0000 1111

00 11

0000 1111

00 11

C1 2

C1 4

r= 12 r= 14

C y=g(x)

ω2 ω4 ω2 ω2 ω1 ω1 ω2ω4ω3ω4ω2ω2ω2ω2ω1ω3ω1ω1ω1

a) b) p(ω4) = 2.8 p(ω2) = 2.2

p(ω3 ) = 2.2 p(ω1 ) = 2 ω1

ω2

ω3

ω4

Fig. 3. Estimation of length of a curve from its discretization for two different resolutions: a)l(p, g,¹₂) = ¹₂(2p(ω1) + 3p(ω2) + 1p(ω4)) = 6.7, b)l(p, g,¹₄) =

1

4(4p(ω1) + 5p(ω2) + 2p(ω3) + 2p(ω4)) = 7.1

where

F^′^g_r(ω) = card{X ∈JAr, Br−mK∩(Ar+mZ)|ω^g_X,r,m=ω}

⌊^B^r⁻^mm⁻^A^r⌋ .

Consider again the curve defined by g(x) = αx² for α /∈ Q. The proof of Theorem 1 can be extended to prove that:

F^′^g_r(ω)−−−→_r

→0 r∈Q

1 b−a

Z b a

F L_g′(x)(ω)dx.

So,

l(p, g, r)−−−→

r→0 r∈Q

1 m

X

ω∈Pm

p(ω) Z b

a

F Lg^′(x)(ω)dx. (5)

(12)

We know thatx7→F Lx(ω) is piecesewisely affine ([3]). From this property we deduce that we can partition the interval [0,_2b¹] in a finite number of intervals (Ik)0≤k≤n such that Lest(α) = lim_r→0

r∈Ql(p, g, r) is of the form ^A_α +Bα+C on each intervalIk. (See Appendix A.3 for the details).

LetLreal(α) be the length of the parabola{(x, αx²)|x∈[a, b]}. We have:

Lreal(α) = Z b

a

p1 + (2αx)²dx=

"

xp

1 + (2αx)²

2 +arg sinh(2αx) 4α

#b

a

Suppose that Lreal(α) = Lest(α) for an infinite number of irrational numbers α, then there exists an interval Ik of the previous partition of [0,_2b¹] such that Lreal(α) =Lest(α) for an infinite number of irrational numbers α∈Ik. OnIk

we know thatLest(α) has the form ^A_α +Bα+C. The functions α7→αLreal(α) andα7→α(^A_α +Bα+C) are holomorphic in a open set of Ccontaining [0,_2b¹] and are equal for an infinite number of α∈Ik ⊂[0,_2b¹]. So by Theorem on the zeros of holomorphic functions [8, Cha. 10] they are equal on [0,_2b¹]. So:

αLreal(α) =A+Bα²+Cα for allα∈[0, 1 2b] We have:

∂(αLreal(α)

∂α =bp

1 + (2αb)²−ap

1 + (2αa)²

=b−a+ 2(b³−a³)α²+o(α²) when α→0

But ^∂(A+Bα_∂α²^+Cα) = 2Bα+C, so 2(b³−a³) = 0 which is impossible ifb > a. So the hypothesis that Lreal(α) =Lest(α) for an infinite number of irrationalαis

absurd. ⊓⊔

Corollary 3. Let a, b such that 0 ≤ a < b. The length estimated by a local estimator of a parabolay=αx²,x∈[a, b], does notconverge when the resolution is rational and tends to zero, to the length of the curve for almost allα∈[0,_2b¹].

Numerical application: Again, we take the curve y = g(x) = √¹

2x² for x betweena= 0 andb=√¹

2. Suppose that we consider the local estimator Chamfer 5-7-11 ([9]) with m= 2, p(000) = 2, p(001) =p(011) = ²²₁₀, p(012) = ²⁸₁₀. With Equation (13), we can prove that the estimation of the length of the parabola given by this local estimator converges toLest= ²³₄₀√

2≈0.813172, this limit is different from the length of the parabola which is¹₂+¹₄√

2 log(1+√

2)≈0.811612.

Moreover Figure 4 shows how the length given by the estimator converges to its limit when the resolution tends to zero. It seems on this example thatl(p, g, r)− Lest=O(r).

(13)

-16 -14 -12 -10 -8 -6 -4 -2 0

2 4 6 8 10 12 14

log(|estimated length-limit|)

log(1/r)

Fig. 4. Figure showing Convergence Speed of the Chamfer 5-7-11 to its limits for the parabolay=^√¹₂x², 0≤x≤^√¹₂.

5 Conclusion and Perspectives

In this paper, we have proved some local properties of discretizations of parabolas: First we show that locally discretization of parabola and discretization of its tangent often coincide (Tangent Lemma: Lemmas 1 and Theorem 2). In particular, asymptotically, the local patterns of a discretized parabola are digital segments. From this, we also give an explicit formula for the limit of frequency of a pattern of a parabola when the resolution tends to zero (Theorem 1). This has the important consequence that we can know to what tend local estimators of length for the parabolas, moreover it can be proved that this limit is often different from the length of the curve.

This work mainly brings two perspectives:

– The extension of Formula (3), which gives the limit of the frequency of pattern when the resolution tends to zero, to more general curves, in particular to the curvesy=P(x) whenP is a polynomial of degree greater than 2.

– The application of this work for recognition of curve by just looking at patterns. For example, if the frequencies of patterns of a curve does not satisfy Theorem 1 then it is not a parabola of equationy=αx².

References

1. Coeurjolly, D., Klette, R.: A comparative evaluation of length estimators. IEEE Trans. Pattern Anal. Mach. Intell.26(2) (2004) 252–258

2. Tajine, M., Daurat, A.: On local definitions of length of digital curves. In: Proc.

of DGCI 2003. Volume 2886 of Lecture Notes in Comp. Sci. (2003) 114–123 3. Tajine, M.: Digital segments and Hausdorff discretization. In: Proc. of IWCIA

2008. Volume 4958 of Lecture Notes in Comp. Sci. (2008) 75–86

4. Daurat, A., Tajine, M., Zouaoui, M.: About the frequencies of some patterns in digital planes. Application to area estimators. Comput. Graph.33(1) (2008) 11–20

(14)

5. Granville, A., Rudnick, Z.: Uniform distribution. In Granville, A., Rudnick, Z., eds.: Equidistribution in Number Theory, An Introduction. Springer (2007) 1–13 6. Vinogradov, I.M.: Elements of Number Theory, 5th rev. ed. Dover (1954) 7. Davenport, H.: Multiplicative Number Theory, 2nd ed. Springer-Verlag, New York

(1980)

8. Rudin, W.: Real and complex analysis. McGraw-Hill, New York (1966)

9. Borgefors, G.: Distance transformations in digital images. Computer Vision, Graphics, and Image Processing34(3) (1986) 344–371

10. McIlroy, M.D.: A note on discrete representation of lines. ATT Tech. J. 64(2) (1985) 481–490

11. Wikipedia: (2008) http://en.wikipedia.org/wiki/Pigeonhole principle ac- cessed 14-November-2008.

12. Munroe, M.E.: Measure and integration, 2nd ed. Addison-Wesley (1971)

A Proofs

Additional notations

– The distance betweenxandZis denotedkxk. Sokxk= min(hxi,1− hxi).

– Ifz is complex numberz denotes its conjugate, and Re(z) denotes its real part.

– gcd(p, q) denotes the greatest common divisor ofpandq.

A.1 Proof of Lemma 3 For anyf ∈L¹([0,1]) we define:

Sr,k(f) = 1 Nr

Br

X

X=Ar

f(hPr,k(X)i).

so we haveTr,k(I) =Sr,k(χI) whereχI is the indicator function ofI. We denote e(t) =e^2πit andec(t) =e(ct).

Sublemma 5 For any c∈Z\ {0}we have Sr,k(ec)−−−→_r

→0 r∈Q

0.

Proof.

|Sr,k(ec)|²=Sr,k(ec)Sr,k(ec)

= 1 N_r²

X

Ar≤X,Y≤Br

ec(Pr,k(X))ec(Pr,k(Y))

= 1 Nr

+ 2 N_r²Re





X

Ar≤X<Y≤Br

ec(Pr,k(Y)−Pr,k(X))





(15)

Let us poseY =X+h, we have

Pr,k(Y)−Pr,k(X) =rα(X+h)²+ 2αk(X+H)−αrX²−2αkrX

=Pr,k(h) + 2αrXh So:

|Sr,k(ec)|²= 1 Nr

+ 2 N_r²Re

Br−1

X

X=Ar

Br−X

X

h=1

ec(Pr,k(h))ec(2αrXh)

!

= 1 Nr

+ 2 N_r²Re

Nr−1

X

h=1 Br−h

X

X=Ar

ec(Pr,k(h))ec(2αrXh)

!

= 1 Nr

+ 2 N_r²Re

Nr−1

X

h=1

ec(Pr,k(h))

Br−h

X

X=Ar

ec(2αrXh)

!!

Subsublemma 6 For any β∈R\Z,u, v∈Z such thatu≤v we have:

v

X

k=u

e(βk)

≤min(v−u+ 1, 1 2kβk).

Proof.

v

X

k=u

e(βk) =e(βu)1−e(β(v−u+ 1)) 1−e(β)

so

v

X

k=u

e(βk)

≤ 2

|1−e(β)|

But|1−e(β)|=|e(^β₂)(e(−^β2)−e(^β₂))|= 2|sin(πβ)|. Moreover sin(πβ)≥2kβk (because sin(πx)≥2xforx∈[0,¹₂]). So:

v

X

k=u

e(βk)

≤ 1 2kβk With the clear inequality|Pv

k=ue(βk)| ≤v−u+ 1, it ends the proof of Subsub-

lemma 6. ⊓⊔

As Re(x)≤ |x|, Subsublemma 6 shows that:

|Sr,k(ec)|²≤ 1 Nr

+ 2 N_r²

Nr−1

X

h=1

min(Nr, 1

2k2αrchk) (6)

(16)

Subsublemma 7 Let C0 ≥0, u∈R\Q, N, p, q ∈Nwhich satisfy q ≤C0N,

|uq−p| ≤ _C₀¹_N, gcd(p, q) = 1. We have

N−1

X

h=1

min(N, 1

2kuhk)≤4(1 +C0) N²

q +Nlogq

.

Proof.

N−1

X

h=1

min(N, 1 2kuhk)≤

⌊^Nq⁻¹⌋

X

H=0 qH+q

X

h=qH+1

min(N, 1 2kuhk)

But: qH+q

X

h=qH+1

min(N, 1 2kuhk) =

q

X

h=1

min(N, 1

2ku(qH+h)k) ku(qH+h)k=k(p+ε

q )(qH+h)k whereε=uq−p

=kpH+p

qh+εH+εh q k

=kp

qh+εH+εh q k Let γ = (εH) mod ¹_q

, and k0 such that εH = γ + ^k_q⁰. We have: (^p_qh+ εH) mod 1 = γ+ⁱ_q^h where ih = (ph+k0q) modq. As gcd(p, q) = 1, {ih|h ∈ J1, qK}=J0, q−1K.

Ifih< ^q₂ then ku(qH+h)k=kγ+ih

q +εh q k

≥ kγ+ih

qk −|ε|h

q becausex7→ kxkis Lipschitz continuous with constant 1

=γ+ih

q −|ε|h q

≥ ih

q − |ε| ash≤q

≥ ih

q − 1

C0N. (7)

Similarly ifih≥ ^q2 then

ku(qH+h)k ≥ kγ+ih

qk − |ε|h q

= 1−(γ+ih

q)−|ε|h q

≥q−1−ih

q − 1

C0N. (8)

(17)

If we use the variable changeh7→ihfor ih< ^q₂,h7→q−1−ih forih≥ ^q2, the inequality min(N,₂_ku(qH+h)¹ _k)≤N forih= 0,1, q−2, q−1, and the inequalities (7),(8) for the otherih, we deduce:

q

X

h=1

min(N, 1

2ku(qH+h)k)≤4N+ 2

⌊^q−2¹⌋

X

i=2

1 2

i q −C0¹N

.

But ₁_−x¹ ≤1 + 2xforx∈(0,¹₂]. So fori≥2 we have:

1

i

q −_C₀¹_N =q i

1 1−iC^q0N

≤q i

1 + 2 q iC0N

because q

iC0N ∈(0,1

2] asi≥2 andq≤C0N

≤2q i So

q

X

h=1

min(N, 1

2ku(qH+h)k)≤4N+ 2q

⌊^q−12 ⌋

X

i=2

1 i. But:

⌊^q−12 ⌋

X

i=2

1 i =

⌊^q−12 ⌋

X

i=2

Z i i−1

1 idt

≤

⌊^q−2¹⌋

X

i=2

Z i i−1

1 tdt

=

Z _⌊^q−₂¹_⌋

1

1 tdt

= log(⌊q−1 2 ⌋) So:

N−1

X

h=1

min(N, 1 2kuhk)≤

⌊N−1

q ⌋+ 1 4N+ 2qlog(⌊q−1 2 ⌋)

≤ N

q + 1

(4N+ 2qlogq)

≤ N

q +C0N q

(4N+ 2qlogq) because q≤C0N

≤4(1 +C0)(N²+Nlogq)

This finishes the proof of Subsublemma 7. ⊓⊔

(18)

In the following we suppose thatris rational. Soαris irrational, so by Dirichlet’s principle ([11]) we know that there exist two coprime integersqr≤_αc(b−a)^2N^r and prsuch that|2αrcqr−pr| ≤ ^αc(b2N⁻r^a). By Subsublemma 7 withC0= _αc(b−a)² and Equation (6) we deduce:

|Sr,k(ec)|²≤ 1 Nr

+8(1 +C0) N_r²

N_r²

q +Nrlogqr

≤ 1 Nr

+8(1 +C0) qr

+8(1 +C0) logqr

Nr

AsNr−−−→_r

→0 +∞and ^log_N^q^r

r ≤^log(CN⁰r^N^r⁾ −−−→_r

→0 0, it remains to prove thatqr−−−→_r

→0

+∞.

We have:

pr

qr ≥2αrc−αc(b−a) 2Nr

We have Nr ≥ (_r^b −1)−(â_r + 1)−1 = ^b⁻â_r⁻^r. We suppose without loss of generality thatr≤^b⁻2â so Nr≥ ^b⁻2râ, so

pr

qr ≥2αrc−αrc >0, sopr≥1. We deduce:

1 qr ≤ pr

qr ≤2αrc+ 1

Nrqr ≤2αrc+ 1 Nr

Soqr≥ 2αrc+¹ ¹

Nr −−−→_r

→0 +∞. This ends the proof of Sublemma 5. ⊓⊔ To finish the proof of Lemma 3 we must prove thatSr,k(χI)−−−→_r

→0 r∈Q

µ(I) where χI is the indicator function of I. We have Sr,k(e0) = 1 and by Sublemma 5 Sr,k(ec) −−−→_r

→0 r∈Q

0 for c 6= 0, so by Weyl’s strategy (see for example [2] or [4,

Appendix A.1]) we can prove Lemma 3. ⊓⊔

A.2 Proof of Theorem 1 Now let:

G^g_r(ω) =card{X ∈JAr, Br−mK|s^g^′^(rX),^h

g(rX) r i

m =ω}

Nr−m .

By Lemma 1 we have

F_r^g(ω)−G^g_r(ω)−−−→

r→0 r∈Q

0. (9)

(19)

We have by definition ofP I(ω) (and because⌊h^g(rX)_r i⌋= 0):

G^g_r(ω) =card{X ∈JAr, Br−mK|(g^′(rX),h^g(rX)r i)∈P I(ω)} Nr−m

For any subsetE ofR², we define:

Hr(E) = card{X ∈JAr, BrK|(g^′(rX),h^g(rX)r i)∈E} Nr

We have

NrHr(P I(ω))−m

Nr−m ≤G^g_r(ω)≤ NrHr(P I(ω))

Nr−m . (10)

Suppose first thatE= [α1, α2)×I. The functiong^′is linear, so in particular is a bijection, we denote by g^′−¹ its reciprocal function. We suppose that α >

0, so that g^′ is an increasing function. (g^′(rX),h^g(rX)r i) ∈ E is equivalent to g^′(rX)∈[α1, α2) andh^g(rX)r i)∈I. So

Hr(E) =card{X ∈J⌈^g^′−¹r^(α¹⁾⌉,⌊^g^′−¹r^(α²⁾⌋K| h^g(rX)r i ∈I} Nr

.

If we apply Lemma 3 withk= 0 to the piece of the curvey=g(x) restricted to the domaing^′−¹(α1)≤x≤g^′−¹(α2). we find:

NrHr(E)

⌊^g^′−¹r^(α²⁾⌋ − ⌈^g^′−¹r^(α¹⁾⌉+ 1 −−−→_r

→0 r∈Q

µ(I)

We deduce that:

Hr([α1, α2)×I)−−−→_r

→0 r∈Q

g^′−¹(α2)−g^′−¹(α1)

b−a µ(I) (11)

In [10] it is shown that

P I(ω) ={(α, β)|α∈[αb, αe] and pinf_α(ω)≤β <psup_α(ω)

whereα7→pinf_α(ω) andα7→psup_α(ω) are two piecewise affine functions which slope is between−m and 0.^{1 2}

Letn∈N\ {0}andyi=αb+i^α^e⁻_n^α^b fori∈J0, nK.

P I(ω) is approximated by an union ofnrectangles:

n

[

i=1

[yi−1, yi)×[pinf_y_i₋₁(ω),psup_y_i(ω))⊂P I(ω)⊂

n

[

i=1

[yi−1, yi]×[pinf_y_i(ω),psup_y_i₋₁(ω))

1 In fact it is proved thatP I(ω) is a triangle or a quadrangle.

2 Formulas (1),(2) do not define pinf and psup forα=α^b, α^e, we define these values just by continuity.

(20)

Asyi−yi−1≤ ¹_n andα7→pinf_α(ω) andα7→psup_α(ω) are two piecewise affine functions which slope is between−mand 0, we have pinf_y_i₋₁(ω)≤pinf_y_i(ω)+^m_n and psup_y_i₋₁(ω)≤psup_y_i(ω) +^m_n. So:

n

[

i=1

[y_i−1, yi)×[pinf_y_i(ω) +m

n,psup_y_i(ω))⊂P I(ω)⊂

n

[

i=1

[yi−1, yi]×[pinf_y_i(ω),psup_y_i(ω) +m n) (12) Let

Fn,r=Hr n

[

i=1

[yi−1, yi)×[pinf_y_i(ω) +m

n,psup_y_i(ω))

! ,

F_n,r^′ =Hr n

[

i=1

[yi−1, yi]×[pinf_y_i(ω),psup_y_i(ω) +m n)

! .

So by Equation (12):

Fn,r≤Hr(P I(ω))≤F_n,r^′ . By summing equations of the form (11) we obtain:

Fn,r−−−→_r

→0 r∈Q

n

X

i=1

g^′−¹(yi)−g^′−¹(y_i−1) b−a

F Lyi(ω)−m n

F_n,r^′ −−−→

r→0 r∈Q

n

X

i=1

g^′−¹(yi)−g^′−¹(y_i−1) b−a

F Lyi(ω) +m n

Let

F_n^′′=

n

X

i=1

g^′−¹(yi)−g^′−¹(y_i−1)

b−a F Lyi(ω).

(limr→0 r∈Q

Fn,r)−F_n^′′=−

n

X

i=1

g^′−¹(yi)−g^′−¹(yi−1) b−a

m n

=−

n

X

i=1 1

2α(yi−yi−1) b−a

m n

=−m(αe−αb) 2α(b−a)

1 n Similarly

(limr→0 r∈Q

F_n,r^′ )−F_n^′′=m(αe−αb) 2α(b−a)

1 n.

(21)

Letzi=g^′−¹(yi), we have F_n^′′=

n

X

i=1

(zi−z_i−1)F L_g′(zi)(ω) b−a

which is a Riemann sum ofx7→ ^{F L}^g′(x)b−a^(ω), butt7→F Lt(ω) is continuous ([3]), so we have by [12, Chap. 5]:

F_n^′′−−−−→_n

→∞

1 b−a

Z b a

F L_g′(x)(ω)dx Letε >0, there existsN1 such that for alln > N1 we have

F_n^′′− 1 b−a

Z b a

F L_g′(x)(ω)dx

≤ ε 3. There existsN2such that for anyn > N2we have

m(αe−αb) 2α(b−a)

1 n <ε

3

LetN = max(N1, N2) + 1. There exists R1 >0 such that for any rational r < R1 we have:

|FN,r−(F_N^′′ −m(αe−αb) 2α(b−a)

1 N)| ≤ ε

3. There existsR2>0 such that for any rationalr < R2 we have:

|F_N,r^′ −(F_N^′′ +m(αe−αb) 2α(b−a)

1 N)| ≤ ε

3. Suppose thatr <min(R1, R2).

We have:

Hr(P I(ω))≥Fn,r

≥F_N^′′ −m(αe−αb) 2α(b−a)

1 N −ε

3

≥F_N^′′ −ε 3−ε

3

≥ 1 b−a

Z b a

F L_g′(x)(ω)dx−ε 3 −ε

3 −ε 3

= 1

b−a Z b

a

F Lg^′(x)(ω)dx−ε Similarly

Hr(P I(ω))≤F_n,r^′

≤ 1 b−a

Z b a

F Lg^′(x)(ω)dx+ε,