GB/T 26958.29-2024 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
Note: GB/Z 26958.1-2011 Geometrical Product Specifications (GPS) - Filtration - Part 1 : Overview and basic concepts (ISO /TS16610-1:2006, IDT)
ISO 16610-20 Geometrical product specifications (GPS) - Filtration - Part 20 : Linear profile filters : Basic concepts
Note: GB/Z 26958.20-2011 Geometrical Product Specifications (GPS) - Filtration - Part 20 : Linear profile filters : Basic concepts (ISO /TS16610-20:2006, IDT)
ISO 16610-22 Geometrical product specifications (GPS) - Filtration - Part 22 : Linear profile filters : Spline filters
Note: GB/Z 26958.22-2011 Geometrical Product Specifications (GPS) - Filtration - Part 22 : Linear profile filters : Spline filters (ISO/TS 16610-22:2006, IDT)
ISO/IEC Guidance 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.
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: 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: The merit of the biorthogonal wavelet is the possibility to construct symmetric wavelet functions, which allows a linear phase filter.
3.2
wavelet family
gα,b
family of functions generated from the mother wavelet (3.1) by dilation (3.2.1) and translation (3.2.2)
3.2.1
dilation
transformation which scales the spatial variable x by a factor α
Note 1: This transformation takes the function g(x) to α−0.5 g(x/α) for an arbitrary positive real number α.
Note 2: The factor α−0.5 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: 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: The dilation parameters are usually powers of 2.
3.5
multiresolution analysis
decomposition of a profile by a filter bank into portions of different scales
Note 1: The portions at different scales are also referred to as resolutions (see ISO 16610-20).
Note 2: Multiresolution is also called multiscale.
Note 3: See Figure 2.
Note 4: 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: 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: 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: In order to avoid loss of information on the multiresolution ladder structure (3.5.3), the wavelet and scaling function are matched.
Note 2: 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: 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 : Typically, k is equal to 2.
GB/T 26958.29-2024 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
Note: GB/Z 26958.1-2011 Geometrical Product Specifications (GPS) - Filtration - Part 1 : Overview and basic concepts (ISO /TS16610-1:2006, IDT)
ISO 16610-20 Geometrical product specifications (GPS) - Filtration - Part 20 : Linear profile filters : Basic concepts
Note: GB/Z 26958.20-2011 Geometrical Product Specifications (GPS) - Filtration - Part 20 : Linear profile filters : Basic concepts (ISO /TS16610-20:2006, IDT)
ISO 16610-22 Geometrical product specifications (GPS) - Filtration - Part 22 : Linear profile filters : Spline filters
Note: GB/Z 26958.22-2011 Geometrical Product Specifications (GPS) - Filtration - Part 22 : Linear profile filters : Spline filters (ISO/TS 16610-22:2006, IDT)
ISO/IEC Guidance 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.
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: 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: The merit of the biorthogonal wavelet is the possibility to construct symmetric wavelet functions, which allows a linear phase filter.
3.2
wavelet family
gα,b
family of functions generated from the mother wavelet (3.1) by dilation (3.2.1) and translation (3.2.2)
3.2.1
dilation
transformation which scales the spatial variable x by a factor α
Note 1: This transformation takes the function g(x) to α−0.5 g(x/α) for an arbitrary positive real number α.
Note 2: The factor α−0.5 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: 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: The dilation parameters are usually powers of 2.
3.5
multiresolution analysis
decomposition of a profile by a filter bank into portions of different scales
Note 1: The portions at different scales are also referred to as resolutions (see ISO 16610-20).
Note 2: Multiresolution is also called multiscale.
Note 3: See Figure 2.
Note 4: 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: 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: 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: In order to avoid loss of information on the multiresolution ladder structure (3.5.3), the wavelet and scaling function are matched.
Note 2: 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: 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 : Typically, k is equal to 2.
