# SIST EN ISO 16610-29:2020

(Main)## Geometrical product specifications (GPS) - Filtration - Part 29: Linear profile filters: Wavelets (ISO 16610-29:2020)

## Geometrical product specifications (GPS) - Filtration - Part 29: Linear profile filters: Wavelets (ISO 16610-29:2020)

This document specifies biorthogonal wavelets for profiles and contains the relevant concepts. It gives

the basic terminology for biorthogonal wavelets of compact support, together with their usage.

## Geometrische Produktspezifikation (GPS) - Filterung - Teil 29: Lineare Profilfilter: Wavelets (ISO 16610-29:2020)

Dieses Dokument legt biorthogonale Wavelets für Profile fest und enthält die entsprechenden Konzepte. Es gibt die grundlegende Terminologie für biorthogonale Wavelets mit kompaktem Träger, gemeinsam mit ihrer Verwendung, an.

## Spécification géométrique des produits (GPS) - Filtrage - Partie 29: Filtres de profil linéaires: Ondelettes (ISO 16610-29:2020)

Le présent document spécifie les caractéristiques des ondelettes biorthogonales utilisées pour les profils ainsi que les concepts pertinents. Elle définit la terminologie de base pour les ondelettes biorthogonales à support compact, ainsi que leur usage.

## Specifikacija geometrijskih veličin izdelka (GPS) - Filtriranje - 29. del: Linearni profilni filtri: valjčki (ISO 16610-29:2020)

### General Information

### Relations

### Standards Content (Sample)

SLOVENSKI STANDARD

SIST EN ISO 16610-29:2020

01-julij-2020

Nadomešča:

SIST EN ISO 16610-29:2015

Specifikacija geometrijskih veličin izdelka (GPS) - Filtriranje - 29. del: Linearni

profilni filtri: valjčki (ISO 16610-29:2020)

Geometrical product specifications (GPS) - Filtration - Part 29: Linear profile filters:

Wavelets (ISO 16610-29:2020)

Geometrische Produktspezifikation (GPS) - Filterung - Teil 29: Lineare Profilfilter:

Wavelets (ISO 16610-29:2020)

Spécification géométrique des produits (GPS) - Filtrage - Partie 29: Filtres de profil

linéaires: Ondelettes (ISO 16610-29:2020)

Ta slovenski standard je istoveten z: EN ISO 16610-29:2020

ICS:

17.040.20 Lastnosti površin Properties of surfaces

17.040.40 Specifikacija geometrijskih Geometrical Product

veličin izdelka (GPS) Specification (GPS)

SIST EN ISO 16610-29:2020 en,fr,de

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST EN ISO 16610-29:2020

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SIST EN ISO 16610-29:2020

EN ISO 16610-29

EUROPEAN STANDARD

NORME EUROPÉENNE

April 2020

EUROPÄISCHE NORM

ICS 17.040.20 Supersedes EN ISO 16610-29:2015

English Version

Geometrical product specifications (GPS) - Filtration - Part

29: Linear profile filters: Wavelets (ISO 16610-29:2020)

Spécification géométrique des produits (GPS) - Filtrage Geometrische Produktspezifikation (GPS) - Filterung -

- Partie 29: Filtres de profil linéaires: Ondelettes(ISO Teil 29: Lineare Profilfilter: Wavelets (ISO 16610-

16610-29:2020) 29:2020)

This European Standard was approved by CEN on 23 March 2020.

CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this

European Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references

concerning such national standards may be obtained on application to the CEN-CENELEC Management Centre or to any CEN

member.

This European Standard exists in three official versions (English, French, German). A version in any other language made by

translation under the responsibility of a CEN member into its own language and notified to the CEN-CENELEC Management

Centre has the same status as the official versions.

CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Croatia, Cyprus, Czech Republic, Denmark, Estonia,

Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway,

Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and

United Kingdom.

EUROPEAN COMMITTEE FOR STANDARDIZATION

COMITÉ EUROPÉEN DE NORMALISATION

EUROPÄISCHES KOMITEE FÜR NORMUNG

CEN-CENELEC Management Centre: Rue de la Science 23, B-1040 Brussels

© 2020 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 16610-29:2020 E

worldwide for CEN national Members.

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SIST EN ISO 16610-29:2020

EN ISO 16610-29:2020 (E)

Contents Page

European foreword . 3

2

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SIST EN ISO 16610-29:2020

EN ISO 16610-29:2020 (E)

European foreword

This document (EN ISO 16610-29:2020) has been prepared by Technical Committee ISO/TC 213

"Dimensional and geometrical product specifications and verification" in collaboration with Technical

Committee CEN/TC 290 “Dimensional and geometrical product specification and verification” the

secretariat of which is held by AFNOR.

This European Standard shall be given the status of a national standard, either by publication of an

identical text or by endorsement, at the latest by October 2020, and conflicting national standards shall

be withdrawn at the latest by October 2020.

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. CEN shall not be held responsible for identifying any or all such patent rights.

This document supersedes EN ISO 16610-29:2015.

According to the CEN-CENELEC Internal Regulations, the national standards organizations of the

following countries are bound to implement this European Standard: Austria, Belgium, Bulgaria,

Croatia, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland,

Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Republic of

North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Turkey and the

United Kingdom.

Endorsement notice

The text of ISO 16610-29:2020 has been approved by CEN as EN ISO 16610-29:2020 without any

modification.

3

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SIST EN ISO 16610-29:2020

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SIST EN ISO 16610-29:2020

INTERNATIONAL ISO

STANDARD 16610-29

Second edition

2020-04

Geometrical product specifications

(GPS) — Filtration —

Part 29:

Linear profile filters: wavelets

Spécification géométrique des produits (GPS) — Filtrage —

Partie 29: Filtres de profil linéaires: ondelettes

Reference number

ISO 16610-29:2020(E)

©

ISO 2020

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

COPYRIGHT PROTECTED DOCUMENT

© ISO 2020

All rights reserved. Unless otherwise specified, or required in the context of its implementation, no part of this publication may

be reproduced or utilized otherwise in any form or by any means, electronic or mechanical, including photocopying, or posting

on the internet or an intranet, without prior written permission. Permission can be requested from either ISO at the address

below or ISO’s member body in the country of the requester.

ISO copyright office

CP 401 • Ch. de Blandonnet 8

CH-1214 Vernier, Geneva

Phone: +41 22 749 01 11

Fax: +41 22 749 09 47

Email: copyright@iso.org

Website: www.iso.org

Published in Switzerland

ii © ISO 2020 – All rights reserved

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

Contents Page

Foreword .iv

Introduction .v

1 Scope . 1

2 Normative references . 1

3 Terms and definitions . 1

4 General wavelet description . 4

4.1 General . 4

4.2 Basic usage of wavelets . 4

4.3 Wavelet transform. 4

4.4 Biorthogonal wavelets . 5

4.4.1 General. 5

4.4.2 Cubic prediction wavelets . 6

4.4.3 Cubic b-spline wavelets . . 6

5 Filter designation. 6

Annex A (normative) Cubic prediction wavelets . 7

Annex B (normative) Cubic b-spline wavelets .15

Annex C (informative) Relationship to the filtration matrix model .18

Annex D (informative) Relation to the GPS matrix model .19

Bibliography .20

© ISO 2020 – All rights reserved iii

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular, the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/ directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/ patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation of the voluntary nature of standards, the meaning of ISO specific terms and

expressions related to conformity assessment, as well as information about ISO's adherence to the

World Trade Organization (WTO) principles in the Technical Barriers to Trade (TBT) see www .iso .org/

iso/ foreword .html.

This document was prepared by Technical Committee ISO/TC 213, Dimensional and geometrical product

specifications and verification, in collaboration with the European Committee for Standardization (CEN)

Technical Committee CEN/TC 290, Dimensional and geometrical product specification and verification,

in accordance with the Agreement on technical cooperation between ISO and CEN (Vienna Agreement).

This second edition cancels and replaces the first edition (ISO 16610-29:2015), which has been

technically revised.

The main changes compared to the previous edition are as follows:

— The terminology and requirements around wavelets have been clarified and expanded to cover

biorthogonal wavelets more fully.

— The requirements for cubic prediction wavelets are set out in Annex A.

— The requirements for cubic b-spline wavelets are given in Annex B.

A list of all parts in the ISO 16610 series can be found on the ISO website.

Any feedback or questions on this document should be directed to the user’s national standards body. A

complete listing of these bodies can be found at www .iso .org/ members .html.

iv © ISO 2020 – All rights reserved

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

Introduction

This document is a geometrical product specification (GPS) standard and is to be regarded as a general

GPS standard (see ISO 14638). It influences chain links C and F of the chains of standards on profile and

areal surface texture.

The ISO GPS matrix model given in ISO 14638 gives an overview of the ISO GPS system of which this

document is a part. The fundamental rules of ISO GPS given in ISO 8015 apply to this document and

the default decision rules given in ISO 14253-1 apply to the specifications made in accordance with this

document, unless otherwise indicated.

For more detailed information on the relation of this document to other standards and the GPS matrix

model, see Annex D.

This document develops the terminology and concepts for wavelets.

© ISO 2020 – All rights reserved v

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SIST EN ISO 16610-29:2020

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SIST EN ISO 16610-29:2020

INTERNATIONAL STANDARD ISO 16610-29:2020(E)

Geometrical product specifications (GPS) — Filtration —

Part 29:

Linear profile filters: wavelets

1 Scope

This document specifies biorthogonal wavelets for profiles and contains the relevant concepts. It gives

the basic terminology for biorthogonal wavelets of compact support, together with their usage.

2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies. For

undated references, the latest edition of the referenced document (including any amendments) applies.

ISO 16610-1, Geometrical product specifications (GPS) — Filtration — Part 1: Overview and basic concepts

ISO 16610-20, Geometrical product specifications (GPS) — Filtration — Part 20: Linear profile filters: Basic

concepts

ISO 16610-22, Geometrical product specifications (GPS) — Filtration — Part 22: Linear profile filters:

Spline filters

ISO/IEC Guide 99, International vocabulary of metrology — Basic and general concepts and associated

terms (VIM)

3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO 16610-1, ISO 16610-20,

ISO 16610-22 and ISO/IEC Guide 99 and the following apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at https:// www .iso .org/ obp

— IEC Electropedia: available at http:// www .electropedia .org/

3.1

mother wavelet

function of one or more variables which forms the basic building block for wavelet analysis, i.e. an

expansion of a signal/profile as a linear combination of wavelets

Note 1 to entry: A mother wavelet, which usually integrates to zero, is localized in space and has a finite

bandwidth. Figure 1 provides an example of a real-valued mother wavelet.

3.1.1

biorthogonal wavelet

wavelet where the associated wavelet transform (3.3) is invertible but not necessarily orthogonal

Note 1 to entry: The merit of the biorthogonal wavelet is the possibility to construct symmetric wavelet functions,

which allows a linear phase filter.

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

Figure 1 — Example of a real-valued mother wavelet

3.2

wavelet family

g

α ,b

family of functions generated from the mother wavelet (3.1) by dilation (3.2.1) and translation (3.2.2)

Note 1 to entry: If g(x) is the mother wavelet (3.1), then the wavelet family gx() is generated as shown in

α ,b

Formula (1):

xb−

−05,

gx()=×α g (1)

α ,b

α

where

α is the dilation parameter for the wavelet of frequency band [1/α, 2/α];

b is the translation parameter.

3.2.1

dilation

transformation which scales the spatial variable x by a factor α

−0,5

Note 1 to entry: This transformation takes the function g(x) to α g(x/α) for an arbitrary positive real number α.

−0,5

Note 2 to entry: The factor α keeps the area under the function constant.

3.2.2

translation

transformation which shifts the spatial position of a function by a real number b

Note 1 to entry: This transformation takes the function g(x) to g(x − b) for an arbitrary real number b.

3.3

wavelet transform

unique decomposition of a profile into a linear combination of a wavelet family (3.2)

3.4

discrete wavelet transform

DWT

unique decomposition of a profile into a linear combination of a wavelet family (3.2) where the

translation (3.2.2) parameters are integers and the dilation (3.2.1) parameters are powers of a fixed

positive integer greater than 1

Note 1 to entry: The dilation parameters are usually powers of 2.

2 © ISO 2020 – All rights reserved

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

3.5

multiresolution analysis

decomposition of a profile by a filter bank into portions of different scales

Note 1 to entry: The portions at different scales are also referred to as resolutions (see ISO 16610-20).

Note 2 to entry: Multiresolution is also called multiscale.

Note 3 to entry: See Figure 2.

Note 4 to entry: Since by definition there is no loss of information, it is possible to reconstruct the original profile

from the multiresolution ladder structure (3.5.3).

3.5.1

low-pass component

smoothing component

component of the multiresolution analysis (3.5) obtained after convolution with a smoothing filter (low-

pass) and a decimation (3.5.6)

3.5.2

high-pass component

difference component

component of the multiresolution analysis (3.5) obtained after convolution with a difference filter (high-

pass) and a decimation (3.5.6)

Note 1 to entry: The weighting function of the difference filter is defined by the wavelet from a particular family

of wavelets, with a particular dilation (3.2.1) parameter and no translation (3.2.2).

Note 2 to entry: The filter coefficients require the evaluation of an integral over a continuous space unless there

exists a complementary function to form the basis expanding the signal/profile.

3.5.3

multiresolution ladder structure

structure consisting of all the orders of the difference components and the highest order smooth

component

3.5.4

scaling function

function which defines the weighting function of the smoothing filter used to obtain the smooth

component

Note 1 to entry: In order to avoid loss of information on the multiresolution ladder structure (3.5.3), the wavelet

and scaling function are matched.

Note 2 to entry: The low-pass component (3.5.1) is obtained by convolving the input data with the scaling function.

3.5.5

wavelet function

function which defines the weighting function of the difference filter used to obtain the detail

component

Note 1 to entry: The high-pass component (3.5.2) is obtained by convolving the input data with the wavelet

function.

3.5.6

decimation

action which samples every k-th point in a sampled profile, where k is a positive integer

Note 1 to entry: Typically, k is equal to 2.

3.6

lifting scheme

fast wavelet transform (3.3) that uses splitting, prediction and updating stages (3.6.1), (3.6.2), (3.6.3)

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ISO 16610-29:2020(E)

3.6.1

splitting stage

partition of a profile into “even” and “odd” subsets, in which each sequence contains half as many

samples as the original profile

3.6.2

prediction stage

calculation which predicts the odd subset from the even subset and then removes the predicted value

from the odd subset value

3.6.3

updating stage

calculation which updates the even subset from the odd subset, in order to preserve as many profile

moments as possible

4 General wavelet description

4.1 General

A cubic prediction wavelet claiming to conform with this document shall satisfy the procedure given in

Annex A.

A cubic spline wavelet claiming to conform with this document shall satisfy the procedure given in

Annex B.

NOTE The relationship to the filtration matrix model is given in Annex C.

4.2 Basic usage of wavelets

Wavelet analysis consists of decomposing a profile into a linear combination of wavelets g (x), all

a,b

[4]

generated from a single mother wavelet . This is similar to Fourier analysis, which decomposes a

profile into a linear combination of sinewaves, but unlike Fourier analysis, wavelets are finite in both

spatial and frequency domain. Therefore, they can identify the location as well as the scale of a feature

in a profile. As a result, they can decompose profiles where the small-scale structure in one portion of

the profile is unrelated to the structure in a different portion, such as localized changes (i.e. scratches,

defects or other irregularities). Wavelets are also ideally suited for non-stationary profiles. Basically,

wavelets decompose a profile into building blocks of constant shape, but of different scales.

4.3 Wavelet transform

[5]

The discrete wavelet transform of a profile, s(x), given as height values, s(x ), at uniformly sampled

i

positions, x = (i−1) Δx (where Δx is the sampling interval, i = 1, ., n and n being the number of

i

sampling points), with the wavelet function g((x−b)/a), is given by the differences (or details), d (i),

k

and the smoothed data, s (i), and a subsequent decimation (down-sampling) for each level or rung, k,

k

of decomposition. The smoothed data and differences are obtained by convolving the signal with the

scaling function, h, and the wavelet, g, as shown in Formula (2a) and Formula (2b):

si =−hs ij (2a)

() ()

k ∑ jk−1

j

di()=−gs ()ij (2b)

∑

k jk−1

j

where j = −m, ., −2, −1, 0, 1, 2, ., m; (m is the number of coefficients of the filter on one side from the

centre).

The dilation parameter, a, is determined by the level of decomposition, k, and by down-sampling the

−k k

smoothed data commonly by a factor of two, i.e. a = 2 , respectively. a = 1/(2 Δx), such that for each

step of the decomposition ladder the number of smoothed data points reduces by a factor of two.

4 © ISO 2020 – All rights reserved

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

The decomposition starts with the original signal values, s(x ), denoted as s (i).

i 0

The mother wavelet of the discrete wavelet transform is defined as a set of discrete high-pass

filter coefficients, g , and the scaling function as a set of discrete low-pass filter coefficients, h . As

j j

the decimation is carried out by keeping every second value of the smooth and every second of the

difference signal, the total number of data points is conserved, such that n/2 of the s (i) are saved and

1

n/2 of the d (i) and the distance between the i-th and the (i+1)-th is then 2Δx.

1

For the second decomposition step, the set of n/2 differences, d (i), will be kept until termination but

1

the set of the s (i) is subdivided half and half, such that n/4 values s (i) and n/4 values d (i) are obtained.

1 2 2

k k

For the k-th step of decomposition and decimation n/2 of s (i) and n/2 values d (i) are evaluated and

k k

k

the distance between the i-th and the (i+1)-th is then 2 Δx.

Therefore, the dilation is done by down-sampling, i.e. managing the indices of the signal rather than

changing the wavelet and scaling functions. Thus, for discrete wavelet transformations only the two

sets of filter coefficients, the set {h , j = −m,.,0,.m} for the low-pass and {g , j = −m,.,0,.m} for the high-

j j

pass, define the analysis filter.

Figure 2 — Ladder structure of multiresolution separation using a discrete wavelet transform

Figure 2 illustrates the ladder structure of the consecutive steps with action of the low-pass

−k

(smoothing) filter with subsequent decimation, H = {2 , h , j = −m, ., 0, …, m}, and the high-pass filter,

k j

−k

G = {2 , g , j = −m, ., 0, ., m}, with decimation reducing the number of smooth profile points by half for

k j

each rung.

The reconstruction is performed by up-sampling and the subsequent application of the matching

synthesis filters. The original profile can be regained if all difference signals are included to the

recovery.

The multiresolution form of the wavelet transform consists of constructing a ladder of smooth

approximations to the profile (see Figure 2). The first rung, i.e. rung number 0, is the original profile.

Each rung in the ladder consists of a filter bank. To apply discrete wavelet transformations to the

multiresolution concept in the sense of ISO 16610-20, a decomposition is performed until a desired level

k, i.e. rung or resolution, is achieved. Then the signal is reconstructed without the details d (i) . d (i),

1 k

i.e. the up-sampling is done for s (i) and thereafter the convolution with the synthesis low-pass yielding

k

the desired smoothed signal.

4.4 Biorthogonal wavelets

4.4.1 General

The application addressed with this document is to recognize features of differing scales (resolutions)

by smoothing accordingly. The biorthogonal wavelets specified in this document are all symmetrical

wavelets and the decomposed signal can be reconstructed without loss.

© ISO 2020 – All rights reserved 5

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

4.4.2 Cubic prediction wavelets

A fast implementation of the wavelet decomposition and reconstruction has been employed using a

lifting scheme with three stages: splitting, prediction and updating, originally introduced by Sweldens,

in which the Neville polynomials are employed to implement the prediction stage by interpolating

[6,7]

between sampling positions . The cubic prediction wavelets in Annex A using Sweldens’ lifting

[6]

scheme has been validated as an efficient tool for fast and in-place wavelet transform for geometrical

[9]

products applications, for example surface metrology .

4.4.3 Cubic b-spline wavelets

Spline wavelets are based on the spline function. In this document a cubic b-spline function is used,

which has a compact support. The particular cubic spline wavelets used are the biorthogonal wavelets

CDF 9/7 with four vanishing moments, detailed in Annex B. This was original introduced by Cohen et

[8]

al. and has been used in geometrical products applications, for example multiscale analysis. The cubic

spline wavelet transform can be implemented using both the Fourier method and the lifting scheme

(however, it is a five-stage process) with relevant precision.

5 Filter designation

Lifting schemes using cubic interpolation for the wavelet transform in conformity with this document

are designated:

FPLWCP

CDF 9/7 Spline wavelets in conformity with this document are designated:

FPLWCS

See also ISO 16610-1:2015, Clause 5.

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

Annex A

(normative)

Cubic prediction wavelets

A.1 General

The lifting scheme is used to define a fast in-place wavelet transform (see References [7], [9]). Starting

with the original profile, each rung in the multiresolution ladder is calculated from the previous rung in

three stages. These stages are called:

— splitting;

— prediction;

— updating.

The lifting scheme using cubic polynomial interpolation for the prediction stage described in this annex

[7]

was introduced by Sweldens in 1996 for image processing purposes. Jiang et al. have applied the

[9]

method to surface metrology (see Figure A.1).

Figure A.1 — Forward transform using the lifting scheme for wavelet defined in this annex

A.2 Splitting

The lifting algorithm of the wavelet transform first of all divides the smoothed profile from the jth

rung, A , into “even” and “odd” subsets, in which each sequence contains half as many samples as A .

j,k j,k

The operator is given by Formula (A.1):

aA=

jk+12,,jk

(A.1)

dA=

jk++12,,jk 1

where A = Z , the original profile.

0,k k

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SIST EN ISO 16610-29:2020

ISO 16610-29:2020(E)

A.3 Prediction

The prediction of the wavelet algorithm consists of predicting the odd subset from the even subset and

then removing the predicted value from the odd subset value. The operator is given by Formula (A.2):

dd=−ρ a (A.2)

()

jk++11

**...**

SLOVENSKI STANDARD

oSIST prEN ISO 16610-29:2019

01-marec-2019

6SHFLILNDFLMDJHRPHWULMVNLKYHOLþLQL]GHOND*36)LOWULUDQMHGHO/LQHDUQL

SURILOQLILOWUL9DOMþNL,62',6

Geometrical product specifications (GPS) - Filtration - Part 29: Linear profile filters -

Wavelets (ISO/DIS 16610-29:2019)

Geometrische Produktspezifikation (GPS) - Filterung - Teil 29: Lineare Profilfilter -

Wavelets (ISO/DIS 16610-29:2019)

Spécification géométrique des produits (GPS) - Filtrage - Partie 29: Filtres de profil

linéaires - Ondelettes splines (ISO/DIS 16610-29:2019)

Ta slovenski standard je istoveten z: prEN ISO 16610-29

ICS:

17.040.20 Lastnosti površin Properties of surfaces

17.040.40 6SHFLILNDFLMDJHRPHWULMVNLK Geometrical Product

YHOLþLQL]GHOND*36 Specification (GPS)

oSIST prEN ISO 16610-29:2019 en,fr,de

2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

---------------------- Page: 1 ----------------------

oSIST prEN ISO 16610-29:2019

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oSIST prEN ISO 16610-29:2019

DRAFT INTERNATIONAL STANDARD

ISO/DIS 16610-29

ISO/TC 213 Secretariat: BSI

Voting begins on: Voting terminates on:

2019-01-08 2019-04-02

Geometrical product specifications (GPS) — Filtration —

Part 29:

Linear profile filters — Wavelets

Spécification géométrique des produits (GPS) — Filtrage —

Partie 29: Filtres de profil linéaires - Ondelettes splines

ICS: 17.040.20

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Contents Page

Foreword .iv

Introduction .v

1 Scope . 1

2 Normative references . 1

3 Terms and definitions . 1

4 General wavelet description . 4

4.1 General . 4

4.2 Basic usage of wavelets . 4

4.3 Wavelet transform. 4

4.4 Biorthogonal wavelets . 5

4.4.1 Cubic prediction wavelet . 5

4.4.2 Cubic b-spline wavelets . . 6

5 Filter designation. 6

Annex A (normative) Cubic prediction wavelets . 7

Annex B (normative) Cubic b-spline wavelets .15

Annex C (informative) Relationship to the filtration matrix model .18

Annex D (informative) Relation to the GPS matrix model .19

Bibliography .20

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Foreword

ISO (the International Organization for Standardization) is a worldwide federation of national standards

bodies (ISO member bodies). The work of preparing International Standards is normally carried out

through ISO technical committees. Each member body interested in a subject for which a technical

committee has been established has the right to be represented on that committee. International

organizations, governmental and non-governmental, in liaison with ISO, also take part in the work.

ISO collaborates closely with the International Electrotechnical Commission (IEC) on all matters of

electrotechnical standardization.

The procedures used to develop this document and those intended for its further maintenance are

described in the ISO/IEC Directives, Part 1. In particular the different approval criteria needed for the

different types of ISO documents should be noted. This document was drafted in accordance with the

editorial rules of the ISO/IEC Directives, Part 2 (see www .iso .org/directives).

Attention is drawn to the possibility that some of the elements of this document may be the subject of

patent rights. ISO shall not be held responsible for identifying any or all such patent rights. Details of

any patent rights identified during the development of the document will be in the Introduction and/or

on the ISO list of patent declarations received (see www .iso .org/patents).

Any trade name used in this document is information given for the convenience of users and does not

constitute an endorsement.

For an explanation on the meaning of ISO specific terms and expressions related to conformity assessment,

as well as information about ISO's adherence to the World Trade Organization (WTO) principles in the

Technical Barriers to Trade (TBT) see the following URL: www .iso .org/iso/foreword .html.

The committee responsible for this document is Technical Committee ISO/TC 213, Dimensional and

geometrical product specifications and verification.

This second edition cancels and replaces the first edition (ISO 16610-29:2015), which has been

technically revised.

A list of all parts in the ISO 16610- series can be found on the ISO website.

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Introduction

This document is a geometrical product specification (GPS) standard and is to be regarded as a general

GPS standard (see ISO 14638). It influences chain links C and F of the chains of standards on profile and

areal surface texture.

The ISO GPS matrix model given in ISO 14638 gives an overview of the ISO GPS system of which this

document is a part. The fundamental rules of ISO GPS given in ISO 8015 apply to this document. The

default decision rules given in ISO 14253-1 apply to the specifications made in accordance with this

document, unless otherwise indicated.

For more detailed information on the relation of this document to other standards and the GPS matrix

model, see Annex D.

This document develops the terminology and concepts for wavelets.

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DRAFT INTERNATIONAL STANDARD ISO/DIS 16610-29:2019(E)

Geometrical product specifications (GPS) — Filtration —

Part 29:

Linear profile filters — Wavelets

1 Scope

This document specifies biorthogonal wavelets for profiles and contains the relevant concepts. It gives

the basic terminology for biorthogonal wavelets of compact support, together with their usage.

2 Normative references

The following documents are referred to in the text in such a way that some or all of their content

constitutes requirements of this document. For dated references, only the edition cited applies. For

undated references, the latest edition of the referenced document (including any amendments) applies.

ISO 16610-1, Geometrical product specifications (GPS) — Filtration — Part 1: Overview and basic concepts

ISO 16610-20, Geometrical product specifications (GPS) — Filtration — Part 20: Linear profile filters: Basic

concepts

ISO 16610-22, Geometrical product specifications (GPS) — Filtration — Part 22: Linear profile filters:

Spline filters

ISO/IEC Guide 99:2007, International vocabulary of metrology — Basic and general concepts and

associated terms (VIM)

3 Terms and definitions

For the purposes of this document, the terms and definitions given in ISO/IEC Guide 99, ISO 16610-1,

ISO 16610-20, ISO 16610-22, and the following apply.

ISO and IEC maintain terminological databases for use in standardization at the following addresses:

— ISO Online browsing platform: available at http: //www .iso .org/obp

— IEC Electropedia: available at http: //www .electropedia .org

3.1

mother wavelet

function of one or more variables which forms the basic building block for wavelet analysis, i.e. an

expansion of a signal/profile as linear combination of wavelets

Note 1 to entry: A mother wavelet, which usually integrates to zero, is localized in space and has a finite

bandwidth. Figure 1 provides an example of a real-valued mother wavelet.

3.1.1

biorthogonal wavelet

wavelet where the associated wavelet transform is invertible but not necessarily orthogonal

Note 1 to entry: The merit of biorthogonal wavelet is the possibility to construct symmetric wavelet functions,

which allows a linear phase filter.

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Figure 1 — Example of a real-valued mother wavelet

3.2

wavelet family

g

ab,

family of functions generated from the mother wavelet (3.1) by dilation and translation

Note 1 to entry: If g(x) is the mother wavelet, then the wavelet family gx() is generated as follows:

ab,

xb−

−05,

gx()=×ag (1)

ab,

a

where

a is the dilation parameter for the wavelet of frequency band [1/a, 2/a];

b is the translation parameter.

3.2.1

dilation

〈wavelet〉 transformation which scales the spatial variable x by a factor a

−0,5

Note 1 to entry: This transformation takes the function g(x) to a g(x/a) for an arbitrary positive real number a.

−0,5

Note 2 to entry: The factor a keeps the area under the function constant

3.2.2

translation

transformation which shifts the spatial position of a function by a real number b

Note 1 to entry: This transformation takes the function g(x) to g(x − b) for an arbitrary real number b.

3.3

discrete wavelet transform

unique decomposition of a profile into a linear combination of a wavelet family (3.2) where the

translation (3.2.2) parameters are integers and the dilation (3.2.1) parameters are powers of a fixed

positive integer greater than 1

Note 1 to entry: The dilation parameters are usually powers of 2.

Note 2 to entry: Throughout the rest of this document, the discrete wavelet transform is referred to as the

wavelet transform.

3.4

multiresolution analysis

decomposition of a profile by a filter bank into portions of different scales

Note 1 to entry: The portions at different scales are also referred to as resolutions (see ISO 16610-20).

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Note 2 to entry: Multiresolution is also called multiscale.

Note 3 to entry: See Figure 2.

Note 4 to entry: Since by definition there is no loss of information, it is possible to reconstruct the original profile

from the multiresolution ladder structure.

3.4.1

low-pass component

component obtained after convolution with a smoothing filter (low pass) and a decimation

3.4.2

high-pass component

component obtained after convolution with a difference filter (high pass) and a decimation

Note 1 to entry: The weighting function of the difference filter is defined by the wavelet from a particular family

of wavelets, with a particular dilation parameter and no translation.

Note 2 to entry: The filter coefficients require the evaluation of an integral over a continuous space unless there

exists a complementary function to form the basis expanding the signal/profile.

3.4.3

multiresolution ladder structure

structure consisting of all the orders of the difference components and the highest order smooth

component

3.4.4

scaling function

function which defines the weighting function of the smoothing filter used to obtain the smooth

component

Note 1 to entry: In order to avoid loss of information on the multiresolution ladder structure, the wavelet and

scaling function are matched.

Note 2 to entry: Low-pass component is obtained by convolving the input data with the scaling function.

3.4.5

wavelet function

function which defines the weighting function of the difference filter used to obtain the detail

component

Note 1 to entry: High-pass component is obtained by convolving the input data with the wavelet function.

3.4.6

decimation

〈wavelet〉 action which samples every k-th point in a sampled profile, where k is a positive integer

Note 1 to entry: Typically, k is equal to 2.

3.5

multiresolution synthesis

reconstruction of a profile by the filter bank matching the analysis filter bank

3.6

lifting scheme

fast wavelet transform that uses splitting, prediction, and updating stages

3.6.1

splitting stage

partition of a profile into “even” and “odd” subsets, in which each sequence contains half as many

samples as the original profile

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3.6.2

prediction stage

calculation which predicts the odd subset from the even subset and then removing the predicted value

from the odd subset value

3.6.3

updating stage

calculation which updates the even subset from the odd subset, in order to preserve as many profile

moments as possible

4 General wavelet description

4.1 General

A cubic prediction wavelet claiming to comply with this document shall satisfy the procedure given in

Annex A.

A cubic spline wavelet claiming to comply with this document shall satisfy the procedure given in

Annex B.

NOTE : A concept diagram for the concepts for wavelets is given in Annex C, and the relationship to the

filtration matrix model is given in Annex D.

4.2 Basic usage of wavelets

Wavelet analysis consists of decomposing a profile into a linear combination of wavelets g (x), all

a,b

[2]

generated from a single mother wavelet. This is similar to Fourier analysis, which decomposes a

profile into a linear combination of sinewaves, but unlike Fourier analysis, wavelets are finite in both

spatial and frequency domain. Therefore, they can identify the location, as well as the scale of a feature

in a profile. As a result, they can decompose profiles where the small-scale structure in one portion of

the profile is unrelated to the structure in a different portion, such as localized changes (i.e. scratches,

defects or other irregularities). Wavelets are also ideally suited for non-stationary profiles. Basically,

wavelets decompose a profile into building blocks of constant shape, but of different scales.

4.3 Wavelet transform

[3]

The discrete wavelet transform of a profile s(x) given as height values s(x ) at uniformly sampled

i

positions x = (i-1) Δx, (where Δx is the sampling interval, i = 1, ., n and n being the number of sampling

i

points) with the wavelet function g((x-b)/a) is given by the differences (or details) d (i) and the smoothed

k

data s (i) and a subsequent decimation (down sampling) for each level or rung k of decomposition. The

k

smoothed data and differences are obtained by convolving the signal with the scaling function h and

the wavelet g:

si =−hs ij and di =−gs ij (2)

() () () ()

k ∑ jk−1 k ∑ jk−1

j j

with j = -m, ., -2, -1, 0, 1, 2, ., m

The dilation constant a is determined by the level of decomposition k and by down sampling the

-k k

smoothed data commonly by a factor of two, i.e. a = 2 resp. a = 1/(2 Δx), such that for each step of the

decomposition ladder the number of smoothed data points reduces by a factor of two.

The decomposition starts with the original signal values s(x ) denoted as s (i).

i 0

The mother wavelet of the discrete wavelet transform is defined as set of discrete high pass filter

coefficients g and the scaling function as set of discrete low pass filter coefficients h . As the decimation

j j

is carried out by keeping every second value of the smooth and every second of the difference signal,

the total number of data points is conserved, such that n/2 of the s (i) are saved and n/2 of the d (i) and

1 1

the distance between the i-th and the (i+1)-st then is 2Δx.

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For the second decomposition step, the set of n/2 differences d (i) will be kept until termination but the

1

set of the s (i) is subdivided half and half such that n/4 values s (i) and n/4 values d (i) are obtained.

1 2 2

k k

For the k-th step of decomposition and decimation n/2 of s (i) and n/2 values d (i) are evaluated and

k k

k

the distance between the i-th and the (i+1)-st then is 2 Δx.

Therefore, the dilation is done by down sampling, i.e. managing the indices of the signal rather than

changing the wavelet and scaling functions. Th

**...**

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