A method for computation of surface roughness of digital elevation model terrains via multiscale analysis

Computers & Geosciences 37 (2011) 177–192

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A method for computation of surface roughness of digital elevation model terrains via multiscale analysis Ahmad Fadzil Mohamad Hani a, Dinesh Sathyamoorthy b,n, Vijanth Sagayan Asirvadam a a b

Department of Electrical and Electronics Engineering, Universiti Teknologi Petronas, Malaysia Science and Technology Research Institute for Defence (STRIDE), Ministry of Defence, Malaysia

a r t i c l e in f o

abstract

Article history: Received 15 July 2009 Received in revised form 29 April 2010 Accepted 1 May 2010 Available online 3 November 2010

In this paper, an algorithm to compute surface roughness of digital elevation model (DEM) terrains via multiscale analysis is proposed. The algorithm employs the lifting scheme to generate multiscale DEMs. At each scale, the areas of pixels that are modified are computed. Granulometric analysis is employed to compute the average area of curvature regions in the terrain, and the average roughness of the terrain due the distribution of curvature regions. The selected case studies of the algorithm implementation demonstrated that the proposed algorithm provides a surface roughness parameter that is realistic with respect to the amplitudes and frequencies of the terrain, invariant with respect to rotation and translation, and has intuitive meaning. The algorithm allows for a good quantification of a region’s convexity/concavity over varying scales, distinguishing between shallow and deep incisions of valleys and ridges of the terrain, and hence, provides an accurate surface roughness parameter. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Lifting scheme Multiscale digital elevation models (DEMs) Curvature regions Granulometric analysis

1. Introduction Roughness is a measure of the texture of a surface which is quantified by the vertical deviations of a real surface from its ideal form. If these deviations are large, the surface is rough; if they are small, the surface is smooth. Surface roughness is a useful tool for terrain analysis as it reflects numerous geophysical parameters, such as landform characteristics, distribution of crenulations, and degree of erosivity. Hence, in the past few decades, quantitative computation of surface roughness of terrains for the purpose of numerical surface study has received increasing attention. A summary of algorithms developed to compute surface roughness of terrains can be found in Shepard et al. (2001) and Li et al. (2005). The most referred requirements for surface roughness measurement, as reported by Hoffman and Krotkov (1990), are as follows: i. Must discriminate between surfaces of different amplitudes, frequencies, and correlation. ii. Be an intrinsic property of the surface, invariant with respect to rotation or translation. iii. Be a local, not a global measure of the surface. iv. Have intuitive or physical meaning. The most commonly used roughness parameter, and the easiest to obtain, is the root-mean-square (RMS) deviation, or n

Corresponding author. Tel.: + 603 87324431; fax: + 603 87348695. E-mail address: [email protected] (D. Sathyamoorthy).

0098-3004/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.cageo.2010.05.021

Fig. 1. Lifting stage: split, predict, update; ke and ko normalize energy of underlying scaling and wavelet functions.

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Fig. 2. GTOPO30 DEM of Great Basin. Elevation values (minimum 1005 m and maximum 3651 m) are rescaled to interval of 0–255 (brightest pixel has highest elevation), with scale being approximately 1:3 900 000.

the standard deviation of heights above the mean (Bennett, 1992; Yokota et al., 2008). The profile is first detrended by subtracting a best fit linear function from the data, leaving a series of heights with a mean value of zero. This approach is not a good frequency discriminator. An alternative approach is to fit a plane to a surface, and use the error as an estimate of the surface roughness (Wilcox and Gennery, 1987). In the case of two sinusoidal surfaces of differing frequencies, plane fitting suffers from a fundamental shortcoming by producing the same roughness estimation. For accurate surface roughness computation, the algorithm needs to be able to discriminate surfaces of varying amplitude and frequency. Stone and Dugundji (1965) proposed a method of computing surface roughness using Fourier analysis. This method measures roughness along specific directions of a surface, and includes amplitude, frequency and autocorrelation terms. Although this approach provides surface roughness parameters that have consistent representation in the frequency domain (Cuthbert and Huynh, 1992; Zhou and Zhang, 2002; El-Nicklawy et al., 2007), it is affected by any rotation or translation of the surface.

Fig. 3. Multiscale DEMs generated using scales of (a) 1, (b) 3, (c) 5, (d) 10, (e) 15, and (f) 20.

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The slope and intercept of the logarithmic plot of the power spectrum of the terrain profile is reported as a roughness parameter by van Zyl et al. (1991). However, there is no simple correspondence between the intercept or the slope of the logarithmic plot of the power spectrum and commonly used roughness measures. Other reported roughness measures, developed to overcome these shortfalls, include effective slope (Miller and Parsons, 1990; Campbell and Garvin, 1993), autocorrelation length (Turcotte, 1997), radiosity models (Li et al., 1998), median and absolute slope (Kreslavsky and Head, 1999), granulometry (Tay et al., 2005), highorder statistics (Nikora, 2005), and extended Kalman filtering (Dabrowski and Banaszkiewicz, 2008).

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These algorithms operate at singular scales of measurement. At various scales, the complexity of the terrain changes, resulting in varying values of surface roughness. Spectral analysis (Pike and Wesley, 1975) has been used to compute surface roughness at various singular scales, and to provide a numerical average of the computed values. However, the methodology, derived from Fourier analysis, is direction dependent (Josso et al., 2002). Fractal analysis (Brown, 1987; Davies and Hall, 1999; Jahn and Truckenbrodt, 2004; Gan et al., 2007; Grzesik and Brol, 2009) has been employed to partition roughness characteristics of a surface into a scale-free component (fractal dimension) and properties that depend purely on scale. While fractal dimension is a scale-independent surface

Fig. 4. 3D models of corresponding multiscale DEMs in Fig. 3.

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roughness parameter that correlates well with surface amplitudes and frequencies, and is invariant to rotation and translation, Russ (1994) and Jahn and Truckenbrodt (2004) report that, in addition to the fractal dimension, fractal analysis also provides a second parameter, the coefficient of correlation. In many cases, there is no obvious association between the coefficient and any feature of roughness. Furthermore, filtering procedures applied on surfaces prior to fractal analysis results in fractal behaviour being limited to high spatial frequencies; in these cases, fractal analysis is not applicable to very coarse scales (Jahn and Truckenbrodt, 2004).

This paper proposes an algorithm to compute surface roughness of digital elevation model (DEM) terrains, via multiscale analysis, that correlates to both amplitudes and frequencies of the terrain, and is invariant to rotation and translation. The proposed algorithm employs the lifting scheme and granulometric analysis to compute the average size of curvature regions of a terrain, and the average roughness of the terrain due to the distribution of curvature regions in the terrain. Selected case studies of the algorithm implementation are employed to demonstrate the robustness of the proposed algorithm compared with existing commonly used surface roughness measures.

Fig. 5. Mask of pixels modified during multiscaling process for corresponding multiscale DEMs in Fig. 3. This indicates curvature regions removed during multiscaling process.

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2. Methodology 2.1. Generation of multiscale DEMs using the lifting scheme Scale variations can constrain the detail with which information can be observed, represented and analyzed. Changing the scale without first understanding the effects of such an action can result in the representation of patterns or processes that are different from those intended (Robinson et al., 1984; Lam and Quattrochi, 1992; Goodchild and Quattrochi, 1997; Lam et al., 2004; Summerfield, 2005; Wu et al., 2008). Hence, feature detection and characterization often need to be performed at different scales measurement. Wood (1996a, b)

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and Wu et al. (2008) demonstrated that analysis of a location at multiple scales allows for a greater amount of information to be extracted from a DEM about the spatial characteristics of a feature. The term scale refers to combination of both spatial extent, and spatial detail or resolution (Goodchild and Quattrochi, 1997; Tate and Wood, 2001; Li et al., 2005). In this paper, multiscaling is performed using the lifting scheme (Sweldens, 1996, 1997). The lifting scheme is a flexible technique that has been used in several different settings, for easy construction and implementation of traditional wavelets and of second generation wavelets, such as spherical wavelets. The lifting scheme has proven to be a powerful multiscale analysis tool in image and signal

Fig. 6. Surfaces with sine waves of amplitudes of: (a) 255, (b) 122, (c) 85, (d) 63, (e) 51, (f) 42, (g) 36, (h) 31, (i) 28, and (j) 25.

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Fig. 6. (Continued)

processing (Claypoole and Baraniuk, 2000; Starck, 2002; Guo et al., 2008). Lifting consists of the following three basic operations (Fig. 1): Step 1:Split The original data set x[n] is divided into two disjoint subsets, even indexed points xe[n]¼ x[2n], and odd indexed points x0[n]¼x[2n+1]. Step 2: Predict The odd and even subsets are often highly correlated. This correlation structure is typically local and hence, it is possible to accurately predict the wavelet coefficients d[n] as the error in predicting x0[n] from xe[n] using the prediction operator P(xe[n]). The predict step uses a function that approximates the data set (Eq. (1)). The difference between the approximation and the actual data replaces the odd elements of the data set (Eq. (2)). The even elements are left unchanged and become the input for the next step in the transform. Pðxe ½nÞ ¼

1 ðxe ½n þ xe ½n þ 1Þ 2

d½n ¼ x0 ½nPðxe ½nÞ

ð1Þ ð2Þ

Step 3: Update The update step replaces the even elements with an average. This results in a smoother output c[n] that represents a coarse approximation to the signal x[n]. The update operator U is applied to the wavelet coefficients (Eq. (3)), and is added to xe[n] (Eq. (4)). Uðd½nÞ ¼

1 ðd½n1 þ d½n þ 1Þ 4

c½n ¼ xe ½n þ Uðd½nÞ

ð3Þ

ð4Þ

The above three steps form a lifting stage. A simple worked example of the algorithm is provided in the Appendix. The lifting scheme scans 2D images row-by-row. Using a DEM as the input, an iteration of the lifting stage generates the complete set of multiscale DEMs cs[n] and the elevation loss caused by the change of scale ds[n]. At each iteration, cs[n] only contains the even points of the input for the iteration, and hence, the resolution of the generated multiscale DEM is reduced by half. At each iteration, the pixels of the DEM that are modified are curvatures regions, while the unmodified pixels are planar regions. The iterations should be repeated until all curvatures in the DEM are removed, leaving only planar regions. For varying DEMs, the number of iterations required would be dependent on the resolution and terrain profile. The Global Digital Elevation Model (GTOPO30) of Great Basin, Nevada, USA, shown in Fig. 2, was downloaded from the USGS GTOPO30 website.1 GTOPO30 DEMs are available at a global scale, providing a digital representation of Earth’s surface at a 30 arc-seconds sampling interval. The land data used to derive GTOPO30 DEMs are obtained from digital terrain elevation data (DTED), the 11 DEM for USA and the digital chart of the world (DCW). The accuracy of GTOPO30 DEMs varies by location according to the source data. The DTED and the 11 dataset have a vertical accuracy of 730 m while the absolute accuracy of the DCW vector dataset is 72000 m horizontal error and 7650 m vertical error. The DEM was rectified and resampled to 925 m in both x- and y-directions (Miliaresis and Argialas, 2002). The terrain in the DEM is bounded by latitude 381 150 –421N and longitude 1181 300 –1151 300 W. 1

http://edcwww.cr.usgs.gov/landdaac/gtopo30/gtopo30.html.

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Tensional forces on terrain’s crust and thins by normal faulting cause the formation an array of tipped mountain blocks that are separated from broad plain basins, producing a basin-and-range physiography (Howell, 1995; Summerfield, 1996, 2000; Miliaresis and Argialas, 1999; Miliaresis, 2008). Multiscale DEMs of the Great Basin region are generated by implementing the lifting scheme on the DEM using scales s of 1–20. As shown in Figs. 3 and 4, as the scale increases, the merging of small regions into the surrounding grey level regions increases, causing removal of fine detail in the DEM. As a result, the generated multiscale DEMs possess lower resolutions at higher degrees of scaling. The fine detail in DEMs represent the curvature regions (consisting of convex and concave crenulations) of the terrain, which are used to extract hydrological features from DEMs. Convex crenulations are used to extract ridge networks while concave crenulations are used to extract drainage networks (Gilbert, 1909; Howard, 1994; Rodrı´guez-Iturbe and Rinaldo, 1997; Sagar et al., 2003). The distribution of curvature regions in a terrain determines the surface roughness of the terrain. The removal of curvature regions from the terrain during the multiscaling process results in the terrain becoming smoother. It is observed in Fig. 5 that as the scale increases, the area of individual curvature regions increase, while the number of individual curvature regions decrease. This observation indicates that the development of an accurate surface roughness parameter requires an accurate quantification of a region’s convexity/concavity over varying scales, distinguishing between shallow and deep incisions of valleys and ridges of the terrain.

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2.2. Computation of surface roughness The area of pixels modified A(s) at each scale s is computed by subtracting the DEM of that scale, with the DEM of the previous scale. The resulting image is thresholded into a binary image by converting all pixels with grey level more than 0 to binary 1, and all pixels with grey level 0 to binary 0. The normalised probability functions a(s) are computed as the ratio of the areas of pixels modified to the area of the original DEM A0. aðsÞ ¼ AðsÞ=A0

ð5Þ

A larger value of a(s) indicates a larger area of curvature regions removed at scale s. The computed values of a(s) are used to compute two important greyscale granulometric complexity measures (Maragos, 1989). Average area of curvature A: Indicates the average area of curvature regions in the terrain. A¼

20 X

saðsÞ

ð6Þ

s¼1

Average roughness R: Indicates average surface roughness of the terrain due to the distribution of curvature regions.

R¼

20 X

aðsÞlogðaðsÞÞ

s¼1

Fig. 7. Average roughness of sinusoidal surfaces with varying amplitudes computed using (a) RMS deviation (b) Fourier analysis (c) proposed algorithm.

ð7Þ

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Fig. 8. Surfaces with sine waves of frequencies of: (a) 144 (b) 72 (c) 36 (d) 18 (e) 12 (f) 6 (g) 3 (h) 0.

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At each scale, due to changing complexity of the terrain, the distribution of curvature regions, and hence, the roughness, of the terrain varies. For terrains with varying complexities, the value a(s) at each scale varies, with Eqs. (6) and (7) providing higher values of A and R to terrains of increasing complexity, and viceversa. Eq. (7) is derived from Shannon entropy, which is used to measure the uncertainty associated with a random variable. In this case, entropy is used to measure the amount of uncertainty of the terrain from a normalised plane (or rather a terrain with no curvature regions, and only planar regions), which is the surface roughness of the terrain. For the case of this manuscript, normalised plane refers to the terrain after all its curvature regions have been removed via the multiscaling process. The entropy computation is used to compute the uncertainty (which is the surface roughness) of the curvature regions removed at each scale from the normalised plane. The proposed algorithm is able distinguish various terrain profiles via the number of multiscaling iterations required to smoothen the respective terrains. For example, terrains with larger elevation differences between pixels would require more multiscaling iterations to smoothen the region. Terrains with few, large curvature regions would require more multiscaling iterations to smoothen as compared with terrains with many small curvature regions.

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3. Results and analysis In this section, the proposed algorithm is compared with two commonly used singular scale surface roughness parameters;

Fig. 10. A model DEM (800  800 pixels) with a singular mountain (area of 20,000 pixels).

Fig. 9. Average roughness of sinusoidal surfaces with varying frequencies computed using (a) RMS deviation (b) Fourier analysis (c) proposed algorithm.

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Fig. 11. Rotation of model DEM in Fig. 9 with angles: (a) 451, (b) 901, (c) 1351, (d) 1801, (e) 2251, (f) 2701, and (g) 3151.

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RMS deviation and Fourier analysis. In order to test the robustness of the algorithm to parameters of amplitude, frequency, and rotations and translations, model DEMs with user controlled parameters are employed. The algorithms are then compared using the real DEM. 3.1. Surfaces of varying amplitudes In order to determine the ability of the proposed algorithm to discriminate between surfaces of different amplitudes, it is tested on sinusoidal surfaces of varying amplitudes (Fig. 6). All three methods produce increasing average roughness for increasing amplitudes (Fig. 7). For RMS deviation (Fig. 7(a)) and Fourier analysis (Fig. 7(b)), the increase is linear. This occurs as both approaches assume that an increase in amplitude results in a proportional increase in average roughness, even if the frequency is constant. For the proposed algorithm (Fig. 7(c)), average roughness increases drastically with increase in amplitude, until amplitude 32. After this point, the increase in average roughness happens at a much slower rate. This is because at this stage, with the frequency being constant, the increase in amplitude only contributes small increase to the average roughness. Thereby, the proposed algorithm is able to better discriminate surfaces of varying amplitudes compared with RMS deviation and Fourier analysis.

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3.2. Surfaces of varying frequencies In order to determine the ability of the proposed algorithm to discriminate between surfaces of different frequencies, it is tested on sinusoidal surfaces of varying frequencies (Fig. 8). For RMS deviation (Fig. 9(a)), the average roughness increases from frequencies 0 to 36, but then decreases. This occurs as RMS deviation operates based on deviations of the waves from the mean amplitude. At high frequencies, the mean amplitude of the waves increase, and this deviation reduces. However, the average roughness of the surface is actually increasing. It is observed that Fourier analysis (Fig. 9(b)) and the proposed algorithm (Fig. 9(c)) produce increasing average roughness with increasing frequencies. For Fourier analysis, average roughness initially increases drastically with increasing frequency. As the frequency is further increased, the rate of increase in average roughness becomes much slower. This occurs as, at this stage, Fourier analysis assumes that the increase in average roughness is limited by the constant amplitude. The proposed algorithm (Fig. 9(c)) shows similar behaviour to Fourier analysis for lower constant amplitudes. However, for higher constant amplitudes, the average roughness continues proportionally increasing at higher frequencies. This occurs as at higher frequencies, the distribution of curvature regions increases, resulting in increasing average roughness. Hence, the proposed algorithm is able to better

Fig. 12. Various translations of model DEM in Fig. 9.

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3.3. Rotation and translation of surfaces

rotations and translaions, both algorithms are able to provide consistent average roughness; 16.957 for RMS deviation, and 0.312 for the proposed algorithm. This demonstrates that both RMS deviation and the proposed algorithm are invariant to rotation and translation of surfaces.

In order to test the proposed methodology’s invariance to rotation and translation, a model DEM with a singular mountains object (Fig. 10) is used. The model DEM has a background of 800  800 pixels, while the mountain has an area of 20,000 pixels. For the non-mountain region of each model, being completely flat, the pixels would have roughness values of 0. The average roughness of surfaces of various rotations (Fig. 11), translations (Fig. 12), and combinations of rotations and translations (Fig. 13) of the model are computed using RMS deviation, Fourier analysis, and the proposed algorithm. A surface roughness measure that is invariant to rotation and translation of surfaces should be able to provide the same average roughness for all the cases of rotations and translations. Fourier analysis provides consistent average roughness for translations, 529.197. However, as it depends on direction of measurement, it provides inconsistent average roughness for rotations, as shown in Fig. 14. RMS deviation and the proposed algorithm do not depend on direction of measurement to compute surface roughness. Hence, for all cases of rotations, translations, and combinations of

Fig. 14. Average roughness of rotations of model DEM in Fig. 9, computed using Fourier analysis.

discriminate surfaces of varying frequencies compared with RMS deviation and Fourier analysis.

Fig. 13. Various combinations of rotations and translations of model DEM in Fig. 9.

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Fig. 15. Surface roughness of terrain over varying scales of measurement via (a) RMS deviation and (b) Fourier analysis.

3.4. Advantages of surface roughness computation via multiscale analysis For GTOPO30 DEM of Great Basin, A and R are 42.038 and 2.341, respectively. In order to demonstrate the importance of surface roughness computation via multiscale analysis, the average roughness of the terrain over varying scales of measurement is computed using RMS deviation and Fourier analysis. It is observed in Fig. 15 that as the scale is increased, the surface roughness of the terrain decreases. This is due to the reduction in fine detail that occurs with the loss of resolution with increasing scale. Singular scale roughness parameters only consider curvature regions at fixed levels of incisions of valleys and ridges of the terrain, and hence, vary significantly according to scale. The proposed algorithm allows for an accurate quantification of a region’s convexity/ concavity over varying scales, distinguishing between shallow and deep incisions, and hence, provides an accurate surface roughness parameter. The adaptability of the algorithm is further

tested by implementing on different types of DEMs with varying terrain profiles (Fig. 16).

4. Conclusion In this paper, an algorithm to compute surface roughness via the generation of multiscale DEMs was proposed. The lifting scheme was used to perform the generation of multiscale DEMs. The area of pixels modified at each scale was computed. The computed areas were divided with the area of the DEM to obtain the normalised probability functions, which were used to compute to the average size of curvature regions in the DEM, and the average roughness of the terrain of the DEM due to the distribution of curvature regions in the terrain. The proposed algorithm provides a surface roughness parameter that is realistic with respect to the amplitudes and frequencies of the terrain, invariant to with respect to rotation and translation, and has intuitive meaning. The algorithm allows for a

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Fig. 16. Implementation of proposed algorithm on different types of DEMs with varying terrain profiles: (a) USGS distributed Defence Mapping Agency (DMA) 11 DEM of Salt Lake City, with resolution of 90 m (downloaded from /http://edc2.usgs.gov/geodata/index.phpS). (b) National Elevation Dataset (NED) DEM of Alaska, with resolution of 60 m (downloaded from /http://ned.usgs.govS). (c) USGS Western Geographic Science Center generated DEM of Lake Tahoe, with resolution of 10 m (downloaded from /http://tahoe.usgs.govS).

good quantification of a region’s convexity/concavity over varying scales, distinguishing between shallow and deep incisions, and hence, provides an accurate surface roughness parameter. Acknowledgements

Iteration 1 Seperation of the signal into even and odd xo ½n ¼ xe ½n ¼

10

30

50

360

10

40

90

90

40

20

40

40

20

20

70

100

70

20

10

From Eq. (1) The authors are grateful to Dr. George Miliaresis, Department of Geology, University of Patras, Greece, Dr. Stephen Wise, Department of Geography, University of Sheffield, United Kingdom, and an anonymous reviewer for their suggestions. Appendix. A simple worked example of the lifting scheme The following signal is employed: c0 ½n ¼ x½n ¼ 10 20 30 40 50 40 30 20 10 20 40 70 90 100 90 70 40 20 For 4 iterations of the lifting scheme

P1 ðxe ½nÞ ¼

40

40

20

25

65

90

65

25

0

25

15

0

From Eq. (2) d1 ½n ¼

10

10

10

10

15

25

From Eq. (3) U1 ðd1 ½nÞ ¼

10

20

4

4

4

6

9

4

13

4

6

4

From Eq. (4) c1 ½n ¼

23

44

44

26

29

74

113

74

26

10

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Fig. A1. Plot of iterations of lifting scheme on given worked example.

Iteration 2 x½n ¼ xo ½n ¼ xe ½n ¼

23

44

44

26

29

23

44

29

113

26

44

26

74

74

P2 ðxe ½nÞ ¼ d2 ½ n ¼

34

37

71

70

26

10

13

42

43

0

U2 ðd2 ½nÞ ¼ c2 ½n ¼

6

13

14

11

11

50

39

88

85

88

85

74

113

74

26

Iteration 3 x½n ¼

50

39

xo ½n ¼ xe ½n ¼ P3 ðxe ½nÞ ¼

50

88

39

85

69

88

d3 ½ n ¼ U3 ðd3 ½nÞ ¼

30

0

8

0

c3 ½n ¼

48

85

x½n ¼ xo ½n ¼ xe ½n ¼

48 48

85

P4 ðxe ½nÞ ¼ D4 ½n ¼ U4 ðd4 ½nÞ ¼

48

C4 ½ n ¼

85

Iteration 4

85 0 0

c1[n]–c4[n] are the multiscale signals for iterations 1–4 respectively, (Fig. A1). At each iteration, as the odd points are removed, leaving the even points, the resolution of the generated multiscale signal is reduced by half (c0[n] has 19 items, c1[n] has 9, c2[n] has 4, c3[n] has 2, and c4[n] has 1). References Bennett, J.M., 1992. Recent developments in surface roughness characterization. Measurement Science and Technology 3 (12), 1119–1127.

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