ISO/TR 16610-32

Geometrical product specifications (GPS) — Filtration — Part 32: Robust profile filters: Spline filters

Spécification géométrique des produits (GPS) — Filtrage — Partie 32: Filtres de profil robustes: Filtres splines

General Information

• Status: Not Published
• ICS: 17.040.20 - Properties of surfaces
• ICS: 17.040.40 - Geometrical Product Specification (GPS)
• Technical Committee: ISO/TC 213 - Dimensional and geometrical product specifications and verification
• Drafting Committee: ISO/TC 213 - Dimensional and geometrical product specifications and verification
• Current Stage: 6000 - International Standard under publication
• Completion Date: 21-Aug-2023

Relations

• Revises: ISO/TS 16610-32:2009 - Geometrical product specifications (GPS) — Filtration — Part 32: Robust profile filters: Spline filters
• Effective Date: 06-Jun-2022

Standards Content (Sample)

© ISO 2023 – All rights reserved
ISO/PRF TR 16610-32:2023(E)
ISO/TC 213/WG 15
Secretariat: BSI
Date: 2023-07-24
Geometrical product specifications (GPS) — Filtration —
Part 32:
Robust profile filters: Spline filters

Technical Report

Warning for WDs and CDs
This document is not an ISO International Standard. It is distributed for review and comment. It is subject to
change without notice and may not be referred to as an International Standard.
Recipients of this draft are invited to submit, with their comments, notification of any relevant patent rights of
which they are aware and to provide supporting documentation.

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ISO #####-#:####(X)
Spécification géométrique des produits (GPS) — Filtrage —
Partie 32: Filtres de profil robustes: Filtres splines
FDIS stage
2 © ISO #### – All rights reserved

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ISO/PRF TR 16610-32:(E)
© ISO 2023, Published in Switzerland
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,
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Published in Switzerland
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© ISO 2023 – All rights reserved iii

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ISO/PRF TR 16610-32:(E)
Contents
Foreword . v
Introduction . vii
Part 32: Robust profile filters: Spline filters. 1
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Spline filter for uniform and non-uniform sampling . 2
4.1 General . 2
4.2 Filter equation for cubic spline filter . 3
4.2.1 General . 3
4.2.2 Regularization parameter . 4
4.2.3 Tension parameter . 4
4.2.4 Matrix V for linear cubic spline filter . 5
4.2.5 Matrix V for robust cubic spline filter . 5
4.2.6 Termination of the iteration of robust estimation . 6
4.2.7 Matrices of differentiation P and Q . 6
4.3 Transmission characteristics . 9
4.4 Alternative robust spline filter . 9
4.4.1 General . 9
4.4.2 Objective function with L2-norm without tension energy for the linear filter equation . 10
4.4.3 Objective function with L1-norm without tension energy for robust filtration . 10
5 Filter designation . 11
Annex A (informative) Example of spline filter applied to plateau structured profile . 12
Annex B (informative) Relationship to the filtration matrix model . 17
B.1 General . 17
B.2 Position in the filtration matrix model . 17
Annex C (informative) Relationship to the GPS matrix model . 18
C.1 General . 18
C.2 Information about this document and its use . 18
C.3 Position in the GPS matrix model . 18
C.4 Related International Standards . 19
Bibliography . 20

iv © ISO 2023 – All rights reserved

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ISO/PRF TR 16610-32:(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
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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 documentsdocument 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).
Field Code Changed
Attention is drawnISO draws attention to the possibility that some of the elementsimplementation of this
document may beinvolve the subjectuse 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,
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received (see ).
Any trade name used in this document is information given for the convenience of users and does not
constitute an endorsement.
For an explanation onof 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 the following URL:
www.iso.org/iso/foreword.html.
Field Code Changed
This document was prepared by Technical Committee ISO/TC 213, Dimensional and geometrical product
specifications and verification.
This Technical Reportdocument cancels and replaces the previous edition (ISO/TS 16610-32:2009),,
which has been technically revised and converted to a Technical Report to retain the information.
The main changes compared to the previous edition are as follows:
— — conversion to a Technical Report;
— inclusion of spline filtration for non-uniform sampling points;
— — addition of a generalized filter equation with a revision of the equation of the robust spline filter
harmonizing the statistical estimator with that of ISO 16610-31;
— — inclusion of a termination criterion of the iterations for the robust, therefore non-
linearnonlinear, filter;
— — addition of specifications of the tension parameter.
A list of all parts in the ISO 16610 series can be found on the ISO website.
© ISO 2023 – All rights reserved v

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ISO/PRF TR 16610-32:(E)
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.
vi © ISO 2023 – All rights reserved

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ISO/PRF TR 16610-32:(E)
Introduction
This document is a Technical Report which develops the terminology and concepts for spline filters.
Spline filters have the advantage of being implementable for non-uniform sampling positions and for
closed profiles. An example of application of spline filters is given in Annex AAnnex A.
Robust filters are tolerant against outliers. Spline filters offer one method for form removal.
For more detailed information of the relation of this document to the filtration matrix and the ISO GPS
standards, see Annex BAnnex B and Annex CAnnex C.
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ISO/PRF TR 16610-32:(E)
Geometrical product specifications (GPS) — Filtration —
Part 32:
Robust profile filters: Spline filters
1 Scope
This document provides information on a generalized version of the linear spline filter for uniform and
non-uniform sampling and the robust spline filters for surface profiles. It supplements ISO 16610-22, ISO
16610-30 and ISO 16610-31.
This document provides information on how to apply the robust estimation to the spline filter as specified
in ISO 16610-22, as well as its generalized form for non-uniform sampling. The weight function chosen
for the M-estimator is the Tukey biweight influence function as specified in ISO 16610-31.
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.
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions apply.
ISO and IEC maintain terminologicalterminology databases for use in standardization at the following
addresses:
— — ISO Online browsing platform: available at https://www.iso.org/obphttp://www.iso.org/obp
— — IEC Electropedia: available at https://www.electropedia.org/http://www.electropedia.org/
3.1
robust filter
filter that is insensitive against specific phenomena in the input data
Note 1 to entry: A robust filter is a filter that delivers output data with robustness.
Note 2 to entry: Robust filters are non-linearnonlinear filters.
[SOURCE: ISO/TS 16610-31:2016, 3.1, modified – Note 1— Definition revised and notes to entry and Note
2 to entry have been added.]
3.2
spline
linear combination of piecewise polynomials, with a smooth fit between the pieces
[SOURCE: ISO 16610-22:2015, 3.1], modified — Note 1 to entry removed.]
3.3
spline filter
linear filter based on splines (3.2(3.2))
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ISO/PRF TR 16610-32:(E)
Note 1 to entry: An example of spline filter application is given in Annex AAnnex A. .
3.4
robust spline filter
robust filter (3.1(3.1)) based on splines (3.2(3.2))
3.5
uniform sampling
sampling of data points at equidistant positions, i.e. with the width of spacing intervals between
neighboringneighbouring probing points being constant
3.6
non-uniform sampling
sampling of data points with non-equidistant spacing points
3.7
robust statistical estimator
rule that indicates how to calculate an estimate based on sample data from a population that is insensitive
against specific phenomena in the input data
Note 1 to entry: An example forof specific phenomena is significant deviation of the distribution of the input data
(amplitude distribution in the case of surface profiles) from a Gaussian distribution mostly in the form of long tails.
3.8
M-estimator
robust statistical estimator (3.7(3.7),) which uses an influence function, i.e. a function which is asymmetric
and scale invariant, to weight points according to their signed distance from the reference line
[SOURCE: ISO 16610-30:2015, 3.5], modified — Definition revised.]
3.9
Tukey’s biweight influence function
influence function which supresses specific phenomena in the input data 𝑥 anddata𝑥and is defined by:
2
2
𝑥
𝑥 1−  for |𝑥|≤𝑐
( ( ) )
𝜓 𝑥 =
( ) {
𝑐
   0            for |𝑥|>𝑐
𝑥
2 2
𝑥 (1−( ) ) for |𝑥|≤𝑐
𝜓(𝑥)={ 𝑐
0             for |𝑥|>𝑐
where c is a scale parameter
4 Spline filter for uniform and non-uniform sampling
4.1 General
The following low-pass filter equation for spline profile filters is based on cubic splines with a
regularization parameter depending on the nesting index, which complies with the cut-off wavelength in
the case of linear filters, for the smoothness of the resultant waviness profile (low-passed signal) and a
tension parameter influencing the slope of the transfer function.
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ISO/PRF TR 16610-32:(E)
4.2 Filter equation for cubic spline filter
4.2.1 General
The filter equation is given in Formula (1)by Formula (1)::
2 4 −1
𝑤 =  ( 𝑉 +  𝛽 𝛼  𝑃 +  ( 1−  𝛽) 𝛼  𝑄 )  𝑉 𝑧 (1)
where
() z is the n-dimensional column vector of input data, e.g. the primary profile of n sampling
Inserted Cells
𝒘 = points;
 ( 𝑽  +
2
 𝛽 𝛼 𝑷  +
  ( 1−
4 −1
𝛽) 𝛼 𝑸 ) 𝑽 𝒛
  w is the column vector of output data, e.g. the waviness profile or smoothed profile;
  V is the unity matrix in the case of the linear filter and the weighting matrix in the case of
the robust filter;
  P and Q are the matrices for the discretized differentiation;
   𝛽 is the tension parameter (see also 4.2.3);
   𝛼 is the parameter (see 4.2.2) depending on the smoothness, the nesting index (cut-off
wavelength in the case of linear filters) of the spline.
Formula (1)where
z is the n-dimensional column vector of input data, e.g. the primary profile of n sampling points;
w is the column vector of output data, e.g. the waviness profile or smoothed profile;
V is the unity matrix in case of the linear filter and the weighting matrix in case of the robust filter;
P and Q are the matrices for the discretized differentiation;is the tension parameter (see also
4.2.3);
𝛼 is the parameter (see 4.2.2) depending on the smoothness, the nesting index (cut-off
wavelength in case of linear filters) of the spline.

Formula (1) is obtained by minimization of the objective (cost) function 𝐽 asfunction𝐽as indicated in
Formula (2)Formula :
min𝐽 (2):)
𝑤
min𝐽
(2)
𝒘
with the objective function defined in Formula (3)Formula (3):
𝑇 2 T 4 T
𝐽= (𝒛−𝒘) 𝑽 (𝒛−𝒘)+ 𝛽 𝛼 𝒘 𝑷 𝒘+ (1−𝛽) 𝛼 𝒘 𝑸 𝑤 (3)
𝑇 2 T 4 T
𝐽=  (𝑧−𝑤)  𝑉 (𝑧−𝑤)+  𝛽 𝛼  𝑤  𝑃 𝑤+  (1−𝛽) 𝛼  𝑤 𝑄 𝑤 (3)
T
where𝑄=𝑃 𝑃.
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ISO/PRF TR 16610-32:(E)
where
T
𝑸= 𝑷 𝑷 .
A sufficient condition of a minimum is 𝛁 𝐽= 0 leading𝛻  𝐽=  0leading to the filter equation in
𝒘 𝑤
Formula (1)Formula (1).
NOTE 1 After extending the matrices of Formula (1)Formula (1) to tensors, the filter is as wellalso applicable to
[11 ]
areal data [7]. .
NOTE 2 Usually the objective function of smoothing splines is defined with a regularization parameter 𝜇
alsoparameter𝜇also fitted during the optimization process with an additional condition for the smoothness
measured according to the deviations 𝑧 −𝑠 𝑥 .(𝑥). Objective functions of the more common type of smoothing
( )
𝑖 𝑖 𝑖
𝑛
2
2 𝑥 𝜕
𝑛
𝑛 2
splines do not include non-zero tension 𝐽= ∑ (𝑧 −𝑠(𝑥)) + 𝜇 ( 𝑠(𝑥)) d𝑥 =  ∑ (𝑧 −𝑠(𝑥)) +
𝑖=1 𝑖 𝑖 ∫ 𝑖 𝑖
𝑥 𝜕𝑥
1
𝑖=1
𝑥
𝑛
𝜕
2
 𝜇  ( 𝑠(𝑥)) d𝑥with 𝑠 𝑥 (𝑥)being the spline polynomials and the regularization parameter 𝜇
∫ ( )
𝜕𝑥
𝑥
1
determining𝜇determining the degree of smoothing and hence following the data points vs. approximating them.
4.2.2 Regularization parameter
The parameter 𝜇 specifiesparameter𝜇specifies the regularization, i.e. the degree of smoothing. In the case
4
4
of minimum tension, it holds 𝜇=  𝛼 and is therefore related to the nesting index 𝑛 , which is in the case
i
of linear filtration equal to the cut-off wavelength  )λ as given in Formula (4)Formula (4)::
c c
1
𝛼 =
𝜋 Δ (4)
2sin
( )
𝑛
i
1
𝛼 =   (4)
𝜋 𝛥
2sin( )
𝑛
i
where
Δ is the sampling interval for uniformly sampled data and the average sampling interval as given in
Formula (5)Formula (5)::
𝑛−1
1
Δ=  (𝑥 − 𝑥 ) (5)
∑
𝑖+1 𝑖
𝑛−1
𝑖=1
1 𝑛−1
𝛥=     (𝑥 −  𝑥  ) (5)
∑
𝑖+1 𝑖
𝑛−1 𝑖=1
for data sampled non-uniformly at positions x with i = = 1, …, n-−1.
i
NOTE 1 Formula (4) Equation (4) has been is derived in Reference [12 [ 8].].
4
NOTE 2 For sampling intervals Δ≪ 𝑛 intervals𝛥≪  𝑛 the regularization parameter tends to infinity 𝛼 →
i i
∞ !. →  ∞.
NOTE 3 For non-minimal tension the factor 𝜇 offactor𝜇of the second order derivative term is also dependent on
4 4
the tension parameter 𝛽: 𝜇= 1−𝛽 𝛼 =(1−𝛽)𝛼 .
( )
4.2.3 Tension parameter
2
The product 𝛽product𝛽𝛼 is the tension factor with parameter β lying between 0 and 1. The parameter
β controls the degree of subsequent topography curvatures, where curvature means a local property of
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ISO/PRF TR 16610-32:(E)
a curve or a surface, which is defined at every point quantifying second-order deviations of a curve from
a straight line or a surface from a plane.
Following curvatures closely means optimal shape retainment of the low-pass result, the output data w.
For β = 0 the characteristics of the transfer function complies withconform to Formula (1) in ISO 16610-
22, a minimum tension which is equivalent to the steepest slope of the transfer function and therefore a
better shape retainment than for β > 0.
[14
For β = 0,625 242 the characteristics of the transfer function is similar to that of the Gaussian filter
]
[10] as specified in ISO 16610-21 and ISO 16610-61.
NOTE The shape retainment by the spline filter for β = 0 is global, while the shape retainment by the Gaussian
regression with a parabolic regression (p = 2) is local.
4.2.4 Matrix V for linear cubic spline filter
Matrix V for linear filters is the n × × n-dimensional unity matrix as given in Formula (6)Formula (6)::
1 … 0
𝑽 = (6)
(⋮ ⋱ ⋮)
0 … 1
1 … 0
𝑉=  (⋮ ⋱ ⋮) (6)
0 … 1
4.2.5 Matrix V for robust cubic spline filter
Matrix V contains the weights suppressing specific phenomena in the input data. They are derived from
Tukey’s biweight influence function as given in Formula (7)by Formula (7)::
(𝑚)
𝑧− 𝑤
𝛿 … 0 (𝑚)
𝑖
1 𝑖 2 2 (𝑚)
(1−  ( ) ) for |𝑧 −𝑤 |≤𝑐
𝑖
(𝑚) (𝑚) 𝑖
𝑐
𝑉 =  (⋮ ⋱ ⋮) with 𝛿 ={ (7)
𝑖
(𝑚)
(𝑚)
0 … 𝛿
0 for |𝑧 −𝑤 |>𝑐
𝑛
𝑖
𝑖
where
(7) i = 1, …, n;
(𝑚)
𝑽
=
𝛿 … 0
1
⋮ ⋱ ⋮ with 𝛿 =
( )
𝑖
0 … 𝛿
𝑛
2
2
(𝑚)
𝑧− 𝑤
𝑖 (𝑚)
𝑖 (𝑚)
1−  for 𝑧 −𝑤 ≤𝑐
( ( ) ) | |
𝑖
(𝑚) 𝑖
𝑐

{
(𝑚)
(𝑚)
0          for |𝑧 −𝑤 |>𝑐
𝑖
𝑖
T
   𝑧=  (𝑧 , …, 𝑧 ) ;
1 𝑛
(𝑚) (𝑚) (𝑚)
T

  𝑤 =  (𝑤 , …, 𝑤 ) ;
𝑛
𝑖 1
 superscript m denotes the iteration put into brackets, which is not to be confused with an exponent;
 superscript T denotes transposed.
where
i = 1, …, n
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ISO/PRF TR 16610-32:(E)
T
𝒛= 𝑧 ,…,𝑧
( )
1 𝑛
T
(𝑚) (𝑚) (𝑚)
𝒘 = 𝑤 ,…,𝑤 and
( )
𝑛
𝒊 1
the superscript m denotes the iteration put into brackets which is not to be confused with an
exponent,
the superscript T denotes transposed.Furthermore, the parameter c is specified as given in ISO/TS
16610-31:20102016, Formula (17 (8), and as shown in Formula (8)Formula (8)::
𝑐=𝑎 median𝑧−𝒘   with   𝑎 ≅ 4,447 8
| | (8)
𝑐=𝑎 median|𝑧−𝑤| with 𝑎 ≅  4,447 8 (8)
−1
NOTE The exact value of a is obtained by the inverse error functionerf : :
3
𝑎 =
−1
2 erf (0,5)
√
3
𝑎 =
−1
2
erf (0,5)
√
4.2.6 Termination of the iteration of robust estimation
The matrix V containing weights δ being dependent on output data w starts with w obtained by linear
i i i
(0)
(0)
non-robust filtration, i.e. 𝑽 𝑉 is the unity matrix. The iteration process terminates if the following
condition given in Formula (9) is reached, as given by Formula (9)::
(𝑚+1) (𝑚)
|𝑐 − 𝑐 |
−5
(9)
  ≤  10        or       𝑚 ≥ 12
(𝑚)
𝑐
(𝑚+1) (𝑚)
|𝑐 − 𝑐 |
−5
 ≤  10 or 𝑚 ≥  12 (9)
(𝑚)
𝑐
4.2.7 Matrices of differentiation P and Q
4.2.7.1 General
A profile can be sampled at lateral positions𝑥 that are not necessarily equidistant. The lateral positions
𝑖
are strictly monotonically increasing, i.e. 𝑥 < 𝑥  <  𝑥 .
𝑖 𝑖+1 𝑖+1
The samplings intervals are denoted Δ = 𝑥 − 𝑥 𝛥 =  𝑥 −  𝑥 and the quotient of the average
𝑖,𝑗 𝑖 𝑗 𝑖,𝑗 𝑖 𝑗
Δ
sampling interval Δ and𝛥and the distance between sampling positions 𝑥 and 𝑥 is denoted by: 𝐷 = =
𝑖 𝑗 𝑖,𝑗
Δ
𝑖,𝑗
𝛥
  .
𝛥
𝑖,𝑗
4.2.7.2 Differentiation matrix for first discretized derivative
For open profiles, matrix P is tri-diagonal as shown in Formula (10)follows as given by Formula (10)::
𝑃 𝑃 0
⋯
1,1 1,2
𝑃 𝑃 𝑃
0 ⋯
2,1 2,2 2,3
𝑷= ( ) (10)
0 𝑃 𝑃
𝑃 0 ⋯
3,2 3,3 3,4
⋱ ⋱ ⋱
⋱ ⋱
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ISO/PRF TR 16610-32:(E)
𝑃 𝑃 0
⋯
1,1 1,2
𝑃 𝑃 𝑃
0 ⋯
2,1 2,2 2,3
𝑃=  ( ) (10)
0 𝑃 𝑃
𝑃 0 ⋯
3,2 3,3
3,4
⋱ ⋱ ⋱ ⋱ ⋱
2 2
For rows 𝑖rows𝑖=2,…,𝑛−1 the1the main diagonal has the elements: 𝑃 = 𝐷 + 𝐷 .𝑃  =
𝑖,𝑖 𝑖,𝑖−1 𝑖+1,𝑖 𝑖,𝑖
2 2
 𝐷  +  𝐷 .
𝑖,𝑖−1 𝑖+1,𝑖
2 2 2 2
The two off-diagonals have the elements𝑃 = −𝐷 =  −𝐷 and 𝑃 = −𝐷 .=  −𝐷 .
𝑖,𝑖−1 𝑖,𝑖+1
𝑖,𝑖−1 𝑖,𝑖−1 𝑖+1,𝑖 𝑖+1,𝑖
The first two elements of the first row and the last two elements of the last row are as follows:
2 2 2 2 2 2
𝑃 = 𝐷  =  𝐷 and 𝑃 = −𝐷 and 𝑃 = −𝐷  =  −𝐷 and 𝑃 =  −𝐷 and
1,1 1,2 1,2 1,2 1,2 𝑛,𝑛−1 𝑛,𝑛−1 1,2 𝑛, 𝑛−1 𝑛,𝑛−1
2 2
𝑃 = 𝐷 𝑃  =  𝐷
𝑛,𝑛 𝑛,𝑛−1 𝑛, 𝑛 𝑛,𝑛−1
For closed profiles the first row and the last row of the matrix differ, having additional non-zero entries
at 𝑃 𝑃 and at 𝑃 𝑃 for the wrap around, i.e. the right neighbour of 𝑥 is 𝑥 and the left neighbour of
1 𝑛 1 𝑛 𝑛 1 𝑛 1 𝑛 1
𝑥 is 𝑥 .
1 𝑛
Then the first row has the following non-zeros entries:
2 2 2 2
𝑃 = −𝐷  and 𝑃 = 𝐷 + 𝐷  and  𝑃 = −𝐷 .
1,𝑛 1,𝑛 1,1 1,𝑛 2,1 1,2 2,1
2 2 2 2
𝑃  =  −𝐷  and 𝑃  =  𝐷  +  𝐷  and 𝑃  =  −𝐷
1,𝑛 1,𝑛 1,1 1,𝑛 2,1 1,2 2,1
and the last row has the following entries:
2 2 2 2
𝑃 = −𝐷  and 𝑃 = 𝐷 + 𝐷  and  𝑃 = −𝐷 .
𝑛,𝑛−1 𝑛,𝑛−1 𝑛,𝑛 𝑛,𝑛−1 1,𝑛 𝑛,1 1,𝑛+1
2 2 2 2
𝑃  =  −𝐷  and 𝑃  =  𝐷  +  𝐷  and 𝑃  =  −𝐷
𝑛,𝑛−1 𝑛,𝑛−1 𝑛,𝑛 𝑛,𝑛−1 1,𝑛 𝑛,1 1,𝑛+1
4.2.7.3 Differentiation matrix for second discretized derivative
For open profiles matrix Q is penta-diagonal as shown in Formula (11)follows as given by Formula (11)::
𝑄 𝑄 𝑄
0 ⋯
1,1 1,2 1,3
𝑄 𝑄 𝑄 𝑄 ⋱
2,1 2,2 2,3 2,4

𝑸=
𝑄 𝑄 𝑄 𝑄 𝑄
3,1 3,2 3,3 3,4 3,5


0 𝑄 𝑄 𝑄 𝑄 ⋱
4,2 4,3
4,4 4,5
⋱
⋱ ⋱ ⋱ ⋱
( )

For rows 𝑖=3,…,𝑛−2 the elements of the main diagonal are as follows:

2 2 2 2
(11)
𝑄 = 4 ( 𝐷 (𝐷 + 𝐷 𝐷 ) + 𝐷 (𝐷 𝐷 + 𝐷 ) ).
𝑖,𝑖 𝑖+1,𝑖−1 𝑖+1,𝑖 𝑖+1,𝑖−1 𝑖,𝑖−1
𝑖,𝑖−1 𝑖,𝑖−2 𝑖+1,𝑖 𝑖+2,𝑖

The elements of the off-diagonal next to the main are as follows in the case of the upper:

2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 𝐷 +𝐷 )
( )
𝑖,𝑖+1 𝑖+1,𝑖 𝑖+1,𝑖 𝑖+1,𝑖−1 𝑖,𝑖−1 𝑖+2,𝑖 𝑖+2,𝑖+1 𝑖+1,𝑖

And in the case of the lower as follows:
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 𝐷 +𝐷 )
( )
𝑖,𝑖−1 𝑖,𝑖−1 𝑖+1,𝑖 𝑖+1,𝑖−1 𝑖,𝑖−1 𝑖,𝑖−2 𝑖−1,𝑖−2 𝑖,𝑖−1

The elements of the second off-diagonals are in the case of the upper as follows:

© ISO 2023 – All rights reserved 7

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ISO/PRF TR 16610-32:(E)
2 2
𝑄 =4 𝐷 𝐷 𝐷  and of the lower 𝑄 =4 𝐷 𝐷 𝐷 .
𝑖,𝑖+2 𝑖+1,𝑖 𝑖+2,𝑖+1 𝑖+2,𝑖 𝑖,𝑖−2 𝑖,𝑖−2 𝑖−1,𝑖−2 𝑖,𝑖−1
𝑄 𝑄 𝑄
1,1 1,2 1,3
0 ⋯
𝑄 𝑄 𝑄 𝑄
2,1 2,2 2,3 2,4 ⋱
𝑄 𝑄 𝑄
𝑄=  ( 𝑄 𝑄 ) (11)
3,1 3,2 3,3 3,4 3,5
𝑄 𝑄 𝑄 𝑄
0      4,2 4,3 4,4 4,5
⋱
⋱ ⋱ ⋱ ⋱ ⋱
For rows𝑖=3,…,𝑛−2the elements of the main diagonal are as follows:
2 2 2 2
𝑄  =  4 ( 𝐷 (𝐷  +  𝐷 𝐷 ) +  𝐷  (𝐷 𝐷  +  𝐷 ) )
𝑖, 𝑖 𝑖+1,𝑖−1 𝑖+1,𝑖 𝑖+1,𝑖−1 𝑖,𝑖−1
𝑖,𝑖−1 𝑖,𝑖−2 𝑖+1,𝑖 𝑖+2,𝑖
The elements of the off-diagonal next to the main are as follows in the case of the upper:
2
𝑄  =  −4 𝐷  (𝐷 𝐷 𝐷  +  𝐷 (𝐷  +  𝐷 ))
𝑖,𝑖+1 𝑖+1,𝑖 𝑖+1,𝑖 𝑖+1,𝑖−1 𝑖,𝑖−1 𝑖+2,𝑖+1 𝑖+1,𝑖
𝑖+2,𝑖
and in the case of the lower as follows:
2
𝑄  =  −  4 𝐷  (𝐷 𝐷 𝐷  +  𝐷 (𝐷  +  𝐷 ))
𝑖, 𝑖−1 𝑖,𝑖−1 𝑖+1,𝑖 𝑖+1,𝑖−1 𝑖,𝑖−1 𝑖−1,𝑖−2 𝑖,𝑖−1
𝑖,𝑖−2
The elements of the second off-diagonals are in the case of the upper as follows:
2
𝑄 =4 𝐷 𝐷 𝐷
𝑖, 𝑖+2 𝑖+1,𝑖 𝑖+2,𝑖+1
𝑖+2,𝑖
and in the case of the lower as follows:
2
𝑄 =4 𝐷 𝐷 𝐷
𝑖, 𝑖−2 𝑖−1,𝑖−2 𝑖,𝑖−1
𝑖,𝑖−2
For the first row and in the case of open profiles, the elements are as follows:
2 2 2 2 2 2
𝑄 =4 𝐷  𝐷 𝐷 and 𝑄 = −=  −4 𝐷  𝐷 𝐷 𝐷 and 𝑄 =4 𝐷  𝐷 𝐷 𝐷 .
1,1 2,1 2,1 3,1 1,2 2,1 2,1 3,2 3,1 1,3 3,2 3,2 3,1 2,1
For the second row and in the case of open profiles they are as follows:
2 2 2 2
𝑄 =𝑄 and 𝑄 =4 𝐷 (𝐷 + 𝐷 ) and 𝑄 = −4 𝐷 ( 𝐷 𝐷 + 𝐷 𝐷 )
2,1 1,2 2,2 3,2 2,1 4,2 2,3 3,2 2,1 3,1 4,2 4,3
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 𝐷 +𝐷 )
( )
2,3 3,2 3,2 3,1 2,1 4,2 4,3 3,2
2 2 2 2
𝑄 =𝑄  𝑎𝑛𝑑 𝑄 =4 𝐷  (𝐷  +  𝐷 ) 𝑎𝑛𝑑 𝑄 =  −4 𝐷  ( 𝐷 𝐷  +  𝐷 𝐷  )
2,1 1,2 2,2 3,2 2,1 4,2 2,3 3,2 2,1 3,1 4,2 4,3

2
𝑄 =  −4 𝐷  (𝐷 𝐷 𝐷 +𝐷 (𝐷 +𝐷 ))
2,3 3,2 3,2 3,1 2,1 4,2 4,3 3,2
For closed profiles the first row has the following non-zero entries:
2 2 2 2
𝑄 = 4 ( 𝐷 ( 𝐷 + 𝐷 𝐷 ) + 𝐷 (𝐷 𝐷 + 𝐷 ) )
1,1 1,𝑛 1,𝑛−1 2,𝑛 2,1 2,1 2,𝑛 1,𝑛 3,1
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 (𝐷 +𝐷 ))
1,2 2,1 2,1 2,𝑛 1,𝑛 3,1 3,2 2,1
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 (𝐷 +𝐷 ))
1,𝑛 1,𝑛 2,1 2,𝑛 1,𝑛 1,𝑛−1 𝑛,𝑛−1 1,𝑛
2 2
𝑄 =4 𝐷 𝐷 𝐷  and 𝑄 =4 𝐷 𝐷 𝐷 .
1,3 2,1 3,2 3,1 1,𝑛−1 1,𝑛−1 𝑛,𝑛−1 1,𝑛
2 2 2 2
𝑄  =  4 ( 𝐷  ( 𝐷  +  𝐷  𝐷 ) +  𝐷  (𝐷  𝐷  +  𝐷 ) )
1,1 1,𝑛 1,𝑛−1 2,𝑛 2,1 2,1 2,𝑛 1,𝑛 3,1
2
𝑄  =  −4 𝐷  (𝐷 𝐷 𝐷  +  𝐷 (𝐷  +  𝐷 ))
1,2 2,1 2,1 2,𝑛 1,𝑛 3,1 3,2 2,1
Furthermore, the
2
𝑄  =  −4 𝐷  (𝐷 𝐷 𝐷  +  𝐷 (𝐷  +  𝐷 ))
1,𝑛 1,𝑛 2,1 2,𝑛 1,𝑛 1,𝑛−1 𝑛,𝑛−1 1,𝑛
2 2
𝑄  =  4 𝐷 𝐷 𝐷  and 𝑄  =  4 𝐷 𝐷 𝐷
1,3 2,1 3,2 3,1 1, 𝑛−1 1,𝑛−1 𝑛,𝑛−1 1,𝑛
The last row, in the case of closed profiles, has the following non-zero entries:
2 2 2 2
𝑄 = 4 𝐷 𝐷 + 𝐷 𝐷 + 𝐷 𝐷 𝐷 + 𝐷
𝑛,𝑛 ( 𝑛,𝑛−1( 𝑛,𝑛−2 1,𝑛−1 1,𝑛−1) 1,𝑛( 1,𝑛−1 𝑛,𝑛−1 1,𝑛))
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 𝐷 +𝐷 )
𝑛 1 1,𝑛 1,𝑛 1,𝑛−1 𝑛,𝑛−1 2,𝑛( 2,1 1,𝑛)
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 𝐷 +𝐷 )
𝑛,𝑛−1 𝑛,𝑛−1 1,𝑛 1,𝑛−1 𝑛,𝑛−1 𝑛,𝑛−2( 𝑛−1,𝑛−2 𝑛,𝑛−1)
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ISO/PRF TR 16610-32:(E)
2 2
𝑄 =4 𝐷 𝐷 𝐷  and 𝑄 =4 𝐷 𝐷 𝐷 .
𝑛,2 𝑛+1,𝑛 2,1 2,𝑛 𝑛,𝑛−2 𝑛,𝑛−2 𝑛−1,𝑛−2 𝑛,𝑛−1
2 2 2 2
𝑄  =  4 ( 𝐷  (𝐷 +𝐷  𝐷 ) +  𝐷  (𝐷  𝐷 +𝐷 ) )
𝑛,𝑛 𝑛,𝑛−1 𝑛,𝑛−2 1,𝑛−1 1,𝑛−1 1,𝑛 1,𝑛−1 𝑛,𝑛−1 1,𝑛
2
𝑄  =  −4 𝐷  (𝐷 𝐷 𝐷 +𝐷 (𝐷 +𝐷 ))
𝑛 1 1,𝑛 1,𝑛 1,𝑛−1 𝑛,𝑛−1 2,𝑛 2,1 1,𝑛

2
𝑄 =  −4 𝐷  (𝐷 𝐷 𝐷 +𝐷 (𝐷 +𝐷 ))
𝑛, 𝑛−1 𝑛,𝑛−1 1,𝑛 1,𝑛−1 𝑛,𝑛−1 𝑛,𝑛−2 𝑛−1,𝑛−2 𝑛,𝑛−1
2 2
𝑄 =4 𝐷 𝐷 𝐷  and 𝑄 =4 𝐷 𝐷 𝐷
𝑛, 2 𝑛+1,𝑛 2,1 2,𝑛 𝑛, 𝑛−2 𝑛,𝑛−2 𝑛−1,𝑛−2 𝑛,𝑛−1
For closed profiles, the second row has the following non-zero entries:
2 2 2 2
𝑄 = 4 𝐷 𝐷 + 𝐷 𝐷 + 𝐷 𝐷 𝐷 + 𝐷
( ( ) ( ))
2,2 2,1 2,𝑛 3,1 3,2 3,2 3,1 2,1 4,2
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 𝐷 +𝐷 )
( )
2,3 3,2 3,2 3,1 2,1 4,2 4,3 3,2
2
𝑄 = −4 𝐷 (𝐷 𝐷 𝐷 +𝐷 𝐷 +𝐷 )
( )
2,1 2,1 3,2 3,1 2,1 2,𝑛 1,𝑛 2,1
2 2
𝑄 =4 𝐷 𝐷 𝐷  and 𝑄 =4 𝐷 𝐷 𝐷
2,4 3,2 4,3 4,2 2,𝑛 2,𝑛 1,𝑛 2,1
2 2 2 2
𝑄 =  4 ( 𝐷  ( 𝐷 +  𝐷  𝐷 ) +  𝐷
...

TECHNICAL ISO/TR
REPORT 16610-32
First edition
Geometrical product specifications
(GPS) — Filtration —
Part 32:
Robust profile filters: Spline filters
Spécification géométrique des produits (GPS) — Filtrage —
Partie 32: Filtres de profil robustes: Filtres splines
PROOF/ÉPREUVE
Reference number
ISO/TR 16610-32:2023(E)
© ISO 2023

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ISO/TR 16610-32:2023(E)
COPYRIGHT PROTECTED DOCUMENT
© ISO 2023
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
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ISO/TR 16610-32:2023(E)
Contents Page
Foreword .iv
Introduction .v
1 Scope . 1
2 Normative references . 1
3 Terms and definitions . 1
4 Spline filter for uniform and non-uniform sampling . 2
4.1 General . 2
4.2 Filter equation for cubic spline filter . 2
4.2.1 General . 2
4.2.2 Regularization parameter . 3
4.2.3 Tension parameter . 4
4.2.4 Matrix V for linear cubic spline filter . 4
4.2.5 Matrix V for robust cubic spline filter . 4
4.2.6 Termination of the iteration of robust estimation . 5
4.2.7 Matrices of differentiation P and Q . 5
4.3 Transmission characteristics . 8
4.4 Alternative robust spline filter . 8
4.4.1 General . 8
4.4.2 Objective function with L2-norm without tension energy for the linear
filter equation . 9
4.4.3 Objective function with L1-norm without tension energy for robust
filtration . 9
5 Filter designation . 9
Annex A (informative) Example of spline filter applied to plateau structured profile .11
Annex B (informative) Relationship to the filtration matrix model .13
Annex C (informative) Relationship to the GPS matrix model .14
Bibliography .15
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ISO/TR 16610-32:2023(E)
Foreword
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bodies (ISO member bodies). The work of preparing International Standards is normally carried out
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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
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www.iso.org/iso/foreword.html.
This document was prepared by Technical Committee ISO/TC 213, Dimensional and geometrical product
specifications and verification.
This document cancels and replaces ISO/TS 16610-32:2009, which has been technically revised.
The main changes are as follows:
— conversion to a Technical Report;
— inclusion of spline filtration for non-uniform sampling points;
— addition of a generalized filter equation with a revision of the equation of the robust spline filter
harmonizing the statistical estimator with that of ISO 16610-31;
— inclusion of a termination criterion of the iterations for the robust, therefore nonlinear, filter;
— addition of specifications of the tension parameter.
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.
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ISO/TR 16610-32:2023(E)
Introduction
This document develops the terminology and concepts for spline filters. Spline filters have the advantage
of being implementable for non-uniform sampling positions and for closed profiles. An example of
application of spline filters is given in Annex A.
Robust filters are tolerant against outliers. Spline filters offer one method for form removal.
For more detailed information of the relation of this document to the filtration matrix and the ISO GPS
standards, see Annex B and Annex C.
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TECHNICAL REPORT ISO/TR 16610-32:2023(E)
Geometrical product specifications (GPS) — Filtration —
Part 32:
Robust profile filters: Spline filters
1 Scope
This document provides information on a generalized version of the linear spline filter for uniform and
non-uniform sampling and the robust spline filters for surface profiles. It supplements ISO 16610-22,
ISO 16610-30 and ISO 16610-31.
This document provides information on how to apply the robust estimation to the spline filter as
specified in ISO 16610-22, as well as its generalized form for non-uniform sampling. The weight function
chosen for the M-estimator is the Tukey biweight influence function as specified in ISO 16610-31.
2 Normative references
There are no normative references in this document.
3 Terms and definitions
For the purposes of this document, the following terms and definitions 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
robust filter
filter that is insensitive against specific phenomena in the input data
Note 1 to entry: A robust filter is a filter that delivers output data with robustness.
Note 2 to entry: Robust filters are nonlinear filters.
[SOURCE: ISO 16610-31:2016, 3.1, modified — Definition revised and notes to entry added.]
3.2
spline
linear combination of piecewise polynomials, with a smooth fit between the pieces
[SOURCE: ISO 16610-22:2015, 3.1, modified — Note 1 to entry removed.]
3.3
spline filter
linear filter based on splines (3.2)
Note 1 to entry: An example of spline filter application is given in Annex A.
3.4
robust spline filter
robust filter (3.1) based on splines (3.2)
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ISO/TR 16610-32:2023(E)
3.5
uniform sampling
sampling of data points at equidistant positions, i.e. with the width of spacing intervals between
neighbouring probing points being constant
3.6
non-uniform sampling
sampling of data points with non-equidistant spacing points
3.7
robust statistical estimator
rule that indicates how to calculate an estimate based on sample data from a population that is
insensitive against specific phenomena in the input data
Note 1 to entry: An example of specific phenomena is significant deviation of the distribution of the input data
(amplitude distribution in the case of surface profiles) from a Gaussian distribution mostly in the form of long
tails.
3.8
M-estimator
robust statistical estimator (3.7) which uses an influence function, i.e. a function which is asymmetric
and scale invariant, to weight points according to their signed distance from the reference line
[SOURCE: ISO 16610-30:2015, 3.5, modified — Definition revised.]
3.9
Tukey’s biweight influence function
influence function which supresses specific phenomena in the input data x and is defined by:
2
 2
 
x
 

x1−  for xc≤

 
 
ψ x =
() c
  
 

0 for xc>


where c is a scale parameter
4 Spline filter for uniform and non-uniform sampling
4.1 General
The following low-pass filter equation for spline profile filters is based on cubic splines with a
regularization parameter depending on the nesting index, which complies with the cut-off wavelength
in the case of linear filters, for the smoothness of the resultant waviness profile (low-passed signal) and
a tension parameter influencing the slope of the transfer function.
4.2 Filter equation for cubic spline filter
4.2.1 General
The filter equation is given in Formula (1):
−1
24
wV=+βα PQ+−1 βα Vz (1)
()
()
2
PROOF/ÉPREUVE © ISO 2023 – All rights reserved

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ISO/TR 16610-32:2023(E)
where
 z is the n-dimensional column vector of input data, e.g. the primary profile of n sampling points;
 w is the column vector of output data, e.g. the waviness profile or smoothed profile;
 V is the unity matrix in the case of the linear filter and the weighting matrix in the case of
the robust filter;
 P and Q are the matrices for the discretized differentiation;
β
is the tension parameter (see also 4.2.3);
α
is the parameter (see 4.2.2) depending on the smoothness, the nesting index (cut-off wave-
length in the case of linear filters) of the spline.
Formula (1) is obtained by minimization of the objective (cost) function J as indicated in Formula (2):
min J (2)
w
with the objective function defined in Formula (3):
T 24TT
J =−zw Vz−ww++βα Pw 1−βα wQw (3)
() () ()
T
where QP= P .
A sufficient condition of a minimum is ∇∇ J= 0 leading to the filter equation in Formula (1).
w
[11]
NOTE 1 After extending the matrices of Formula (1) to tensors, the filter is also applicable to areal data .
NOTE 2 Usually the objective function of smoothing splines is defined with a regularization parameter μ also
fitted during the optimization process with an additional condition for the smoothness measured according to
the deviations zs− x . Objective functions of the more common type of smoothing splines do not include
()
ii
2
x
n ∂
n
2  
non-zero tension Jz=−sx +μ sx dx with sx being the spline polynomials and
()() () ()
 
∑ ii
∫
i=1
x
∂x 
1
the regularization parameter μ determining the degree of smoothing and hence following the data points vs
approximating them.
4.2.2 Regularization parameter
The parameter μ specifies the regularization, i.e. the degree of smoothing. In the case of minimum
4
tension, it holds μα= and is therefore related to the nesting index n , which is in the case of linear
i
filtration equal to the cut-off wavelength λ as given in Formula (4):
c
1
α = (4)
π Δ
2sin
 
n
 
i
where Δ is the sampling interval for uniformly sampled data and the average sampling interval as given
in Formula (5):
n−1
1
Δ= xx− (5)
()
ii+1
∑
n−1
i=1
for data sampled non-uniformly at positions x with i = 1, …, n−1.
i
NOTE 1 Formula (4) is derived in Reference [12].
3
© ISO 2023 – All rights reserved PROOF/ÉPREUVE

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ISO/TR 16610-32:2023(E)
4
NOTE 2 For sampling intervals Δ n the regularization parameter tends to infinity α →∞ .
i
NOTE 3 For non-minimal tension the factor μ of the second order derivative term is also dependent on the
4
tension parameter β : μβ=−()1 α .
4.2.3 Tension parameter
2
The product βα is the tension factor with parameter β lying between 0 and 1. The parameter β
controls the degree of subsequent topography curvatures, where curvature means a local property of a
curve or a surface, which is defined at every point quantifying second-order deviations of a curve from
a straight line or a surface from a plane.
Following curvatures closely means optimal shape retainment of the low-pass result, the output data w.
For β = 0 the characteristics of the transfer function conform to Formula (1) in ISO 16610-22, a minimum
tension which is equivalent to the steepest slope of the transfer function and therefore a better shape
retainment than for β > 0.
[14]
For β = 0,625 242 the characteristics of the transfer function is similar to that of the Gaussian filter
as specified in ISO 16610-21 and ISO 16610-61.
NOTE The shape retainment by the spline filter for β = 0 is global, while the shape retainment by the Gaussian
regression with a parabolic regression (p = 2) is local.
4.2.4 Matrix V for linear cubic spline filter
Matrix V for linear filters is the n × n-dimensional unity matrix as given in Formula (6):
10…
 
 
V =   (6)
 
 
01…
 
4.2.5 Matrix V for robust cubic spline f
...