SIST EN ISO 16610-31:2025

SLOVENSKI STANDARD
01-april-2025
Specifikacije geometrijskih veličin izdelka (GPS) - Filtriranje - 31. del: Robustni
profilni filtri: Gaussovi regresijski filtri (ISO 16610-31:2025)
Geometrical product specifications (GPS) - Filtration - Part 31: Robust profile filters:
Gaussian regression filters (ISO 16610-31:2025)
Geometrische Produktspezifikation (GPS) - Filterung - Teil 31: Robuste Profilfilter:
Gaußsche Regressionsfilter (ISO 16610-31:2025)
Spécification géométrique des produits (GPS) - Filtrage - Partie 31: Filtres de profil
robustes: Filtres de régression gaussiens (ISO 16610-31:2025)
Ta slovenski standard je istoveten z: EN ISO 16610-31:2025
ICS:
17.040.20 Lastnosti površin Properties of surfaces
17.040.40 Specifikacija geometrijskih Geometrical Product
veličin izdelka (GPS) Specification (GPS)
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

EN ISO 16610-31
EUROPEAN STANDARD
NORME EUROPÉENNE
February 2025
EUROPÄISCHE NORM
ICS 17.040.20; 17.040.40 Supersedes EN ISO 16610-31:2016
English Version
Geometrical product specifications (GPS) - Filtration - Part
31: Robust profile filters: Gaussian regression filters (ISO
16610-31:2025)
Spécification géométrique des produits (GPS) - Filtrage Geometrische Produktspezifikation (GPS) - Filterung -
- Partie 31: Filtres de profil robustes: Filtres de Teil 31: Robuste Profilfilter: Gaußsche
régression gaussiens (ISO 16610-31:2025) Regressionsfilter (ISO 16610-31:2025)
This European Standard was approved by CEN on 15 February 2025.

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.

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Poland, Portugal, Republic of North Macedonia, Romania, Serbia, Slovakia, Slovenia, Spain, Sweden, Switzerland, Türkiye 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
© 2025 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 16610-31:2025 E
worldwide for CEN national Members.

Contents Page
European foreword . 3

European foreword
This document (EN ISO 16610-31:2025) 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 August 2025, and conflicting national standards shall
be withdrawn at the latest by August 2025.
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 16610-31:2016.
Any feedback and questions on this document should be directed to the users’ national standards
body/national committee. A complete listing of these bodies can be found on the CEN website.
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, Türkiye and the
United Kingdom.
Endorsement notice
The text of ISO 16610-31:2025 has been approved by CEN as EN ISO 16610-31:2025 without any
modification.
International
Standard
ISO 16610-31
Second edition
Geometrical product specifications
2025-02
(GPS) — Filtration —
Part 31:
Robust profile filters: Gaussian
regression filters
Spécification géométrique des produits (GPS) — Filtrage —
Partie 31: Filtres de profil robustes: Filtres de régression
gaussiens
Reference number
ISO 16610-31:2025(en) © ISO 2025

ISO 16610-31:2025(en)
© ISO 2025
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
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CH-1214 Vernier, Geneva
Phone: +41 22 749 01 11
Email: copyright@iso.org
Website: www.iso.org
Published in Switzerland
ii
ISO 16610-31:2025(en)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Characteristics of the robust Gaussian regression filter for open profiles . 4
4.1 General .4
4.2 Filter equations .4
4.2.1 Determination of the large-scale lateral component .4
4.2.2 Determination of the small-scale lateral component .5
4.3 Transmission characteristics .5
5 Characteristics of the robust Gaussian regression filter for closed profiles . 6
5.1 General .6
5.2 Filter equations .6
5.2.1 Determination of the large-scale lateral component .6
5.2.2 Determination of the small-scale lateral component .7
5.3 Transmission characteristics .7
6 Series of nesting index values . 8
7 Regression degree, p . 8
8 Iterative solution . 8
9 Filter designation . 9
Annex A (informative) Linear Gaussian regression filter for unbounded open profiles .10
Annex B (informative) Examples for the application of the robust Gaussian regression filter .13
Annex C (informative) Relationship to the filtration matrix model .20
Annex D (informative) Relationship to the GPS matrix model .21
Bibliography .22

iii
ISO 16610-31:2025(en)
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 document 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).
ISO draws attention to the possibility that the implementation of this document may involve the use of (a)
patent(s). ISO takes no position concerning the evidence, validity or applicability of any claimed patent
rights in respect thereof. As of the date of publication of this document, ISO had not received notice of (a)
patent(s) which may be required to implement this document. However, implementers are cautioned that
this may not represent the latest information, which may be obtained from the patent database available at
www.iso.org/patents. ISO shall not be held responsible for identifying any or all such patent rights.
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 16610-31:2016), which has been technically
revised.
The main changes compared to the previous edition are as follows:
— providing continuous Gaussian regression filters for open and for closed profiles;
— providing a normative iterative solution for continuous Gaussian regression filters.
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 16610-31:2025(en)
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 E in the GPS matrix structure.
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 information on the relationship of this document to the filtration matrix model, see Annex C.
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 of robust Gaussian regression filters for surface
profiles. It separates the large- and small-scale lateral components of surface profiles in such a way that the
surface profiles can be reconstructed without altering. The robust Gaussian regression filter for surface
profiles reduces the influence of protruding dales and hills. Depending on the selected nesting index and
regression degree, robust Gaussian regression filters offer one possible method for the F-Operation.

v
International Standard ISO 16610-31:2025(en)
Geometrical product specifications (GPS) — Filtration —
Part 31:
Robust profile filters: Gaussian regression filters
1 Scope
This document specifies robust Gaussian regression filters for the filtration of surface profiles. It defines,
in particular, how to separate large- and small-scale lateral components of surface profiles with protruding
dales and hills.
The concept presented for closed profiles are applicable to the case of roundness filtering. Where appropriate,
these concept can be extended to generalized closed profiles, especially for surface profiles with re-entrant
features.
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 16610-21, Geometrical product specifications (GPS) — Filtration — Part 21: Linear profile filters: Gaussian filters
ISO 16610-22, Geometrical product specifications (GPS) — Filtration — Part 22: Linear profile filters: Spline filters
ISO 16610-30, Geometrical product specifications (GPS) — Filtration — Part 30: Robust profile filters: Basic
concepts
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 16610-21, ISO 16610-22, ISO 16610-30, ISO/IEC Guide 99 and the following apply.
ISO and IEC maintain terminology databases for use in standardization at the following addresses:
— ISO Online browsing platform: available at https:// www .iso .org/ obp
— IEC Electropedia: available at https:// www .electropedia .org/
3.1
surface profile
line resulting from the intersection between a surface portion and an ideal plane
Note 1 to entry: The orientation of the ideal plane is usually perpendicular to the tangent plane of the surface portion.
Note 2 to entry: See ISO 17450-1:2011, 3.3 and 3.3.1, for the definition of an ideal plane.

ISO 16610-31:2025(en)
[SOURCE: ISO 16610-1:2015, 3.1.2, modified — Note 2 to entry replaced.]
3.1.1
open profile
finite length surface profile (3.1) with two ends
Note 1 to entry: An open profile has a compact support, i.e. within a certain interval the height values of an open
profile can be equal to any real number. Outside the interval, the height values of an open profile are set to zero.
[SOURCE: ISO 16610-1:2015, 3.7, modified — Note 1 to entry replaced.]
3.1.2
unbounded open profile
infinite length surface profile (3.1) without ends
Note 1 to entry: In this document, the term “unbounded” refers to the X-axis.
Note 2 to entry: The concept of the unbounded open profile is ideal and do not apply to real surface profiles.
3.1.3
closed profile
connected finite length surface profile (3.1) without ends
Note 1 to entry: A closed profile is a closed curve which is periodic with the finite period length L.
Note 2 to entry: A typical example of a closed profile is one from a roundness measurement.
[SOURCE: ISO 16610-1:2015, 3.8, modified — Note 1 to entry replaced and Note 2 to entry added.]
3.2
linear profile filter
profile filter which separates surface profiles (3.1) into large- and small-scale lateral components and is also
a linear function
Note 1 to entry: If F is a function and X and Y are surface profiles, and if a and b are independent from X and Y, then F
being a linear function implies F (a X + b Y) = a F(X) + b F(Y).
[SOURCE: ISO 16610-20:2015, 3.1, modified — In definition “profiles” replaced by “surface profiles” and “long
wave” and “short wave” replaced by “large-scale lateral” and “small-scale lateral”; Note 1 to entry replaced.]
3.3
weighting function
function to calculate large-scale lateral components by convolution of the surface profile heights with this
function
Note 1 to entry: The convolution (see ISO 16610-20:2015, 4.1) performs a weighted moving average of the surface
profile heights. The weighting function, reflected at the X-axis, defines the weighting coefficients for the averaging
process.
3.4
transmission characteristic of a filter
characteristic that indicates the amount by which the amplitude of a sinusoidal surface profile is attenuated
as a function of its wavelength
Note 1 to entry: The transmission characteristic is the Fourier transformation of the weighting function (3.3).
Note 2 to entry: [SOURCE: ISO 16610-20:2015, 3.4]
3.5
cut-off wavelength
λ
c
wavelength of a sinusoidal surface profile of which 50 % of the amplitude is transmitted by the profile
Note 1 to entry: Linear profile filters are identified by the filter type and the cut-off wavelength value.

ISO 16610-31:2025(en)
Note 2 to entry: The cut-off wavelength is the nesting index for linear profile filters.
[SOURCE: ISO 16610-20:2015, 3.5, modified — In Note 2 to entry “recommended” deleted.]
3.6
undulations per revolution
UPR
integer number of sinusoidal undulations contained in a closed profile (3.1.3)
Note 1 to entry: In this document, UPR is a frequency and is denoted by f.
3.7
cut-off frequency in undulations per revolution
f
c
frequency in UPR of a sinusoidal closed profile (3.1.3) of which 50 % of the amplitude is transmitted by the
profile filter
3.8
robust profile filter
profile filter which separates surface profiles (3.1) into large- and small-scale lateral components and is
insensitive against specific phenomena in the input data
Note 1 to entry: A robust profile filter is a nonlinear filter.
Note 2 to entry: See also ISO 16610-1:2015, 3.9.
Note 3 to entry: Outliers, scratches and steps are examples of specific phenomena. Further details can be found in
ISO 16610-30:2015.
Note 4 to entry: In particular, the robust Gaussian regression filter (3.11) in accordance with this document reduces the
influence of specific phenomena such as protruding dales and hills. Profile examples are given in Annex B.
3.9
biweight function of Beaton and Tukey
function used in M-estimation and defined by Formula (1)
 2
 
Δzx
()
 

1− for Δzx() ≤c
 

 
δ ()Δzx(),c = (1)
c
  
 
 

0 for Δzx() >c

where
x is the given x-coordinate;
Δzx()
are heights depending on x;
c is a scale value.
Note 1 to entry: The biweight function δ ()Δzx() ,c of Beaton and Tukey is almost constant and equals nearly 1 for
heights Δzx()c . For increasing heights Δzx() , the biweight function of Beaton and Tukey approaches zero.
Note 2 to entry: The biweight function of Beaton and Tukey is related to the influence function ψ ()Δzx() (see
ISO 16610-30:2015) used in M-estimation as follows: ψ ()ΔΔzx() = zx() Δδ ()zx() ,c .
Note 3 to entry: See also ISO 16610-30:2015.
3.10
regression filter
profile filter which based on a local polynomial modelling of the large-scale lateral component of a surface
profile (3.1)
ISO 16610-31:2025(en)
3.11
robust Gaussian regression filter
regression filter (3.10) based on the Gaussian weighting function and the biweight function of Beaton and
Tukey (3.9)
4 Characteristics of the robust Gaussian regression filter for open profiles
4.1 General
In this clause, the ideal filtration of open profiles is considered. Since the robust Gaussian regression filter is
nonlinear and no transmission characteristic by means of the Fourier transformation can be given, the
generic term nesting index N is used as the filter parameter instead of cut-off wavelength λ . But in many
i c
cases of application, values for the cut-off wavelength λ used for linear filtration are also suitable as a
c
nesting index N for robust filtration.
i
4.2 Filter equations
4.2.1 Determination of the large-scale lateral component
To determine the large-scale lateral component of an open profile, the robust Gaussian regression filter with
degree p is defined by Formula (2):
−1
 
T TT
 
wx() = ()10  0 vv()xu,, ()xu sx(),uu d zu()v ()xu,,sx()uu d (2)
pp p
∫∫
 
 ΩΩ
where
Ω
is the finite interval, expressed as a set, in which the open profile can be any real number;
x is the given x-coordinate with x∈Ω ;

u is the integration variable along the X-axis with u∈Ω ;

zu()
is the open profile depending on u;
th
v xu,
() is the vector space of polynomials up to p degree depending on x and u;
p
T
is the transpose of v ()xu, ;
p
v ()xu,
p
sx(),u
is t
...