High-resolution spatial data are essential for characterizing and monitoring surface quality in manufacturing. However, the measurement of high-resolution spatial data is generally expensive and time-consuming. Interpolation based on spatial models is a typical approach to cost-effectively acquire high-resolution data. Conventional modeling methods fail to adequately model the spatial correlation induced by periodicity, and thus their interpolation precision is limited. In this paper, we propose using a Bessel additive periodic variogram model to capture such spatial correlation. When combined with kriging, a geostatistical interpolation method, accurate interpolation performance can be achieved for common periodic surfaces. In addition, parameters of the proposed model provide valuable insights for the characterization and monitoring of spatial processes in manufacturing. Both simulated and real-world case studies are presented to demonstrate the effectiveness of the proposed method.

References

References
1.
Shao
,
C.
,
Ren
,
J.
,
Wang
,
H.
,
Jin
,
J. J.
, and
Hu
,
S. J.
,
2016
, “
Improving Machined Surface Shape Prediction by Integrating Multi-Task Learning With Cutting Force Variation Modeling
,”
ASME J. Manuf. Sci. Eng.
,
139
(
1
), p.
011014
.
2.
Shao
,
C.
,
Jin
,
J. J.
, and
Hu
,
S. J.
,
2017
, “
Dynamic Sampling Design for Characterizing Spatiotemporal Processes in Manufacturing
,”
ASME J. Manuf. Sci. Eng.
,
139
(
10
), p.
101002
.
3.
Shao
,
C.
,
Hyung Kim
,
T.
,
Jack Hu
,
S.
,
(Judy) Jin
,
J.
,
Abell
,
J. A.
, and
Patrick Spicer
,
J.
,
2015
, “
Tool Wear Monitoring for Ultrasonic Metal Welding of Lithium-Ion Batteries
,”
ASME J. Manuf. Sci. Eng.
,
138
(
5
), p.
051005
.
4.
Shao
,
C.
,
Paynabar
,
K.
,
Kim
,
T. H.
,
Jin
,
J. J.
,
Hu
,
S. J.
,
Spicer
,
J. P.
,
Wang
,
H.
, and
Abell
,
J. A.
,
2013
, “
Feature Selection for Manufacturing Process Monitoring Using Cross-Validation
,”
J. Manuf. Syst.
,
32
(
4
), pp.
550
555
.
5.
Lee
,
S. S.
,
Shao
,
C.
,
Kim
,
T. H.
,
Hu
,
S. J.
,
Kannatey-Asibu
,
E.
,
Cai
,
W. W.
,
Spicer
,
J. P.
, and
Abell
,
J. A.
,
2014
, “
Characterization of Ultrasonic Metal Welding by Correlating Online Sensor Signals With Weld Attributes
,”
ASME J. Manuf. Sci. Eng.
,
136
(
5
), p.
051019
.
6.
Cai
,
W. W.
,
Kang
,
B.
, and
Hu
,
S. J.
,
2017
,
Ultrasonic Welding of Lithium-Ion Batteries
,
ASME
,
New York
.
7.
Xi
,
L.
,
Banu
,
M.
,
Hu
,
S. J.
,
Cai
,
W.
, and
Abell
,
J.
,
2016
, “
Performance Prediction for Ultrasonically Welded Dissimilar Materials Joints
,”
ASME J. Manuf. Sci. Eng.
,
139
(
1
), p.
011008
.
8.
Zhao
,
D.
,
Zhao
,
K.
,
Ren
,
D.
, and
Guo
,
X.
,
2017
, “
Ultrasonic Welding of Magnesium–Titanium Dissimilar Metals: A Study on Influences of Welding Parameters on Mechanical Property by Experimentation and Artificial Neural Network
,”
ASME J. Manuf. Sci. Eng.
,
139
(
3
), p.
031019
.
9.
Yang
,
T.-H.
, and
Jackman
,
J.
,
2000
, “
Form Error Estimation Using Spatial Statistics
,”
ASME J. Manuf. Sci. Eng.
,
122
(
1
), pp.
262
272
.
10.
Zhao
,
H.
,
Jin
,
R.
,
Wu
,
S.
, and
Shi
,
J.
,
2011
, “
Pde-Constrained Gaussian Process Model on Material Removal Rate of Wire Saw Slicing Process
,”
ASME J. Manuf. Sci. Eng.
,
133
(
2
), p.
021012
.
11.
Suriano
,
S.
,
Wang
,
H.
,
Shao
,
C.
,
Hu
,
S. J.
, and
Sekhar
,
P.
,
2015
, “
Progressive Measurement and Monitoring for Multi-Resolution Data in Surface Manufacturing Considering Spatial and Cross Correlations
,”
IIE Trans.
,
47
(
10
), pp.
1033
1052
.
12.
Du
,
S.
, and
Fei
,
L.
,
2015
, “
Co-Kriging Method for Form Error Estimation Incorporating Condition Variable Measurements
,”
ASME J. Manuf. Sci. Eng.
,
138
(
4
), p.
041003
.
13.
Schabenberger
,
O.
, and
Gotway
,
C. A.
,
2005
,
Statistical Methods for Spatial Data Analysis
,
CRC Press
,
Boca Raton, FL
.
14.
Rasmussen
,
C. E.
, and
Williams
,
C. K.
,
2006
,
Gaussian Processes for Machine Learning
, Vol.
1
,
MIT Press
,
Cambridge, UK
.
15.
Ajil Jassim
,
F.
, and
Hasan Altaany
,
F.
,
2013
, “
Image Interpolation Using Kriging Technique for Spatial Data
,”
Can. J. Image Process. Comput. Vision
,
4
(
2
), pp. 16–21.https://arxiv.org/ftp/arxiv/papers/1302/1302.1294.pdf
16.
Zhang
,
Q.
, and
Wu
,
J.
,
2015
, “
Image Super-Resolution Using Windowed Ordinary Kriging Interpolation
,”
Opt. Commun.
,
336
, pp.
140
145
.
17.
Dai
,
F.
,
Zhou
,
Q.
,
Lv
,
Z.
,
Wang
,
X.
, and
Liu
,
G.
,
2014
, “
Spatial Prediction of Soil Organic Matter Content Integrating Artificial Neural Network and Ordinary Kriging in Tibetan Plateau
,”
Ecol. Indic.
,
45
, pp.
184
194
.
18.
Murphy
,
R. R.
,
Curriero
,
F. C.
, and
Ball
,
W. P.
,
2010
, “
Comparison of Spatial Interpolation Methods for Water Quality Evaluation in the Chesapeake Bay
,”
J. Environ. Eng.
,
136
(
2
), pp.
160
171
.
19.
Emery
,
X.
,
2005
, “
Simple and Ordinary Multigaussian Kriging for Estimating Recoverable Reserves
,”
Math. Geology
,
37
(
3
), pp.
295
319
.
20.
Kuntz
,
M.
, and
Helbich
,
M.
,
2014
, “
Geostatistical Mapping of Real Estate Prices: An Empirical Comparison of Kriging and Cokriging
,”
Int. J. Geogr. Inf. Sci.
,
28
(
9
), pp.
1904
1921
.
21.
Atkinson
,
P. M.
, and
Lewis
,
P.
,
2000
, “
Geostatistical Classification for Remote Sensing: An Introduction
,”
Comput. Geosci.
,
26
(
4
), pp.
361
371
.
22.
Cressie
,
N.
,
2015
,
Statistics for Spatial Data
,
Wiley
,
New York
.
23.
Miller
,
T. F.
,
Mladenoff
,
D. J.
, and
Clayton
,
M. K.
,
2002
, “
Old-Growth Northern Hardwood Forests: Spatial Autocorrelation and Patterns of Understory Vegetation
,”
Ecol. Monogr.
,
72
(
4
), pp.
487
503
.
24.
Péron
,
G.
,
Fleming
,
C. H.
,
de Paula
,
R. C.
, and
Calabrese
,
J. M.
,
2016
, “
Uncovering Periodic Patterns of Space Use in Animal Tracking Data With Periodograms, Including a New Algorithm for the Lomb-Scargle Periodogram and Improved Randomization Tests
,”
Mov. Ecol.
,
4
(
1
), p.
19
.
25.
Baeumer
,
C.
,
Saldana-Greco
,
D.
,
Martirez
,
J. M. P.
,
Rappe
,
A. M.
,
Shim
,
M.
, and
Martin
,
L. W.
,
2015
, “
Ferroelectrically Driven Spatial Carrier Density Modulation in Graphene
,”
Nat. Commun.
,
6
(
1
), p.
6136
.
26.
Bush
,
D.
,
Barry
,
C.
, and
Burgess
,
N.
,
2014
, “
What Do Grid Cells Contribute to Place Cell Firing?
,”
Trends Neurosci.
,
37
(
3
), pp.
136
145
.
27.
Aptoula
,
E.
,
2014
, “
Remote Sensing Image Retrieval With Global Morphological Texture Descriptors
,”
IEEE Trans. Geosci. Remote Sens.
,
52
(
5
), pp.
3023
3034
.
28.
Prikockis
,
M.
,
Wijesinghe
,
H.
,
Chen
,
A.
,
VanCourt
,
J.
,
Roderick
,
D.
, and
Sooryakumar
,
R.
,
2016
, “
An On-Chip Colloidal Magneto-Optical Grating
,”
Appl. Phys. Lett.
,
108
(
16
), p.
161106
.
29.
Ding
,
W.
,
Li
,
H.
,
Zhang
,
L.
,
Xu
,
J.
,
Fu
,
Y.
, and
Su
,
H.
,
2017
, “
Diamond Wheel Dressing: A Comprehensive Review
,”
ASME J. Manuf. Sci. Eng.
,
139
(
12
), p.
121006
.
30.
Nguyen
,
H. T.
,
Wang
,
H.
, and
Hu
,
S. J.
,
2014
, “
Modeling Cutter Tilt and Cutter-Spindle Stiffness for Machine Condition Monitoring in Face Milling Using High-Definition Surface Metrology
,”
Int. J. Adv. Manuf. Technol.
,
70
(
5–8
), pp.
1323
1335
.
31.
Anand
,
P. S. P.
,
Arunachalam
,
N.
, and
Vijayaraghavan
,
L.
,
2017
, “
Performance of Diamond and Silicon Carbide Wheels on Grinding of Bioceramic Material Under Minimum Quantity Lubrication Condition
,”
ASME J. Manuf. Sci. Eng.
,
139
(
12
), p.
121019
.
32.
Ma
,
Y. Z.
, and
Jones
,
T. A.
,
2001
, “
Modeling Hole-Effect Variograms of Lithology-Indicator Variables
,”
Math. Geology
,
33
(
5
), pp. 631–648.
33.
Pyrcz
,
M. J.
, and
Deutsch
,
C. V.
,
2003
, “
The Whole Story on the Hole Effect
,”
Geostatistical Assoc. Australasia
,
18
, pp.
3
5
.http://www.gaa.org.au/pdf/gaa_pyrcz_deutsch.pdf
34.
Radeloff
,
V. C.
,
Miller
,
T. F.
,
He
,
H. S.
,
Mladenoff
,
D. J.
,
Radeloff
,
V. C.
,
Miller
,
T. F.
,
He
,
H. S.
, and
Mladenoff
,
D. J.
,
2000
, “
Periodicity in Spatial Data and Geostatistical Models: Autocorrelation Between Patches
,”
Ecography
,
23
(
1
), pp.
81
91
.
35.
Ye
,
J.
,
Lazar
,
N. A.
, and
Li
,
Y.
,
2015
, “
Nonparametric Variogram Modeling With Hole Effect Structure in Analyzing the Spatial Characteristics of FMRI Data
,”
J. Neurosci. Methods
,
240
, pp.
101
115
.
36.
Bates
,
D. M.
, and
Watts
,
D. G.
,
1988
,
Nonlinear Regression Analysis and Its Applications
, Vol.
2
,
Wiley
,
New York
.
37.
Ibaraki
,
S.
,
Kimura
,
Y.
,
Nagai
,
Y.
, and
Nishikawa
,
S.
,
2015
, “
Formulation of Influence of Machine Geometric Errors on Five-Axis On-Machine Scanning Measurement by Using a Laser Displacement Sensor
,”
ASME J. Manuf. Sci. Eng.
,
137
(
2
), p.
021013
38.
Li
,
J.
, and
Heap
,
A. D.
,
2011
, “
A Review of Comparative Studies of Spatial Interpolation Methods in Environmental Sciences: Performance and Impact Factors
,”
Ecol. Inf.
,
6
(3–4), pp.
137
145
.
39.
Zerehsaz
,
Y.
,
Shao
,
C.
, and
Jin
,
J.
,
2016
, “
Tool Wear Monitoring in Ultrasonic Welding Using High-Order Decomposition
,”
J. Intell. Manuf.
, epub.
40.
Bivand
,
R. S.
,
Pebesma
,
E. J.
,
Gomez-Rubio
,
V.
, and
Pebesma
,
E. J.
,
2008
,
Applied Spatial Data Analysis With R
,
Springer
,
New York
.
You do not currently have access to this content.