Preface	6
	Acknowledgment	7
	Contents	8
	Notation	15
	Introduction	17
	1.1 Surface and Roundness Metrology	17
	1.2 Scope and Objectives	17
	1.3 Organization	19
	References	20
	Part I Filtering	21
	A Brief History of Filtering	22
	2.1 Introduction	22
	2.2 Electrical Filters	22
	2.3 Digital Filters	23
	2.4 The Envelope Method	23
	2.5 The Gaussian Filter	23
	2.6 Overlap of Measurement Techniques	24
	2.7 Recent Advances in Filtering	24
	2.8 Summary	25
	References	25
	Filtering in the Frequency Domain	27
	3.1 Surface Wavelengths	27
	3.2 Fourier Transform, Discrete Fourier Transform, and Fast Fourier Transform	28
	3.3 Filtering in the Frequency Domain	30
	3.4 Wrap-Around Effect	33
	3.5 Amplitude Transmission and Phase Characteristics	33
	3.6 Summary	34
	Exercises	34
	References	35
	Filtering in the Time Domain	36
	4.1 Filtering as an Averaging Process	36
	4.2 Relationship Between Frequency-Domain and Time- Domain Filtering	39
	4.3 Filtering Profiles: Putting It All Together	40
	4.4 Summary	42
	Exercises	43
	References	44
	Gaussian Filter	45
	5.1 Introduction	45
	5.2 High-Pass and Low-Pass Filters	47
	5.3 Roughness, Waviness, and Form Using the Gaussian Filter	47
	5.4 Effect of Cutoff	48
	5.5 Phase Characteristics	48
	5.6 Summary	49
	Exercises	49
	References	50
	The 2RC Filter	51
	6.1 Introduction	51
	6.2 The 2RC High-Pass Filter	51
	6.3 More on the 2RC Filter	54
	6.4 Comparison of the 2RC and Gaussian Filters	56
	6.5 Summary	56
	Exercises	57
	References	58
	Filtering Roundness Profiles	59
	7.1 Introduction	59
	7.2 Gaussian Filter for Roundness	59
	7.3 Amplitude Transmission Characteristics	61
	7.4 Filtering Roundness Profiles in the Time Domain	62
	7.5 Circular Convolution in the Frequency Domain	62
	7.6 Summary	64
	Exercises	64
	References	65
	Filtering 3D Surfaces	66
	8.1 Areal Surface Texture Analysis	66
	8.2 2D Convolution	66
	8.3 Gaussian Filter in 3D	67
	8.4 A Note on Indices	67
	8.5 Frequency-Domain Filtering in 3D	70
	8.6 Summary	73
	Exercises	73
	References	74
	Part II Advanced Filtering	75
	Gaussian Regression Filters	76
	9.1 Introduction	76
	9.2 Zero-Order Gaussian Regression Filter	77
	9.3 Second-Order Gaussian Regression Filter	80
	9.4 3D Zero-Order Gaussian Regression Filter	82
	9.5 3D Second-Order Gaussian Regression Filter	83
	9.6 Summary	84
	Exercises	84
	References	85
	Spline Filter	86
	10.1 Introduction	86
	10.2 Amplitude Transmission Characteristics	87
	10.3 Implementation of the Non-periodic Spline Filter	88
	10.4 Implementation of the Periodic Spline Filter	91
	10.5 Summary	95
	Exercises	95
	References	95
	Robust Filters	96
	11.1 Rk Filter	96
	11.2 Robust Gaussian Regression Filter	99
	11.3 Summary	100
	Exercises	101
	References	101
	Envelope and Morphological Filters	102
	12.1 Envelope Filters	102
	12.2 Stylus Tip Convolution	103
	12.3 Morphological Filters	105
	12.4 A Word on the Fundamentals	107
	12.5 3D Morphological Filtering	107
	12.6 Summary	109
	Exercises	109
	References	110
	Multi-scale Filtering	111
	13.1 Introduction	111
	13.2 Alternate Sequence Filters or Scale Space Analysis	111
	13.3 Wavelet-Based Filters	115
	13.4 Summary	117
	Exercises	118
	References	119
	Part III Fitting	120
	Introduction to Fitting Substitute Geometry	121
	14.1 Introduction	121
	14.2 Fitting Criteria	122
	14.3 Solution Methodologies	123
	References	124
	Least-Squares Best-Fit Line and Plane	126
	15.1 Introduction	126
	15.2 Closed-Form Solution for LS Best-Fit Line	126
	15.3 Matrix Formulation	127
	15.4 Centroid as a Point on the LS Line	128
	15.5 Normal LS in Parametric Form	129
	15.6 Lagrange Multiplier Method	131
	15.7 Back to the LS Line	132
	15.8 LS Best-Fit Plane	133
	15.9 Summary	134
	Exercises	134
	References	135
	Non-linear Least-Squares I: Introduction	136
	16.1 Introduction	136
	16.2 Formulating the Circle in a Plane Problem	136
	16.3 The Steepest Descent Algorithm	137
	16.4 The Gauss-Newton Algorithm	139
	16.5 The Levenberg-Marquardt Algorithm	141
	16.6 Summary	142
	Exercises	143
	References	143
	Non-linear Least-Squares II: Circle, Sphere, and Cylinder	144
	17.1 Introduction	144
	17.2 Initial Estimates for Center and Radius of Circle	144
	17.3 Best-Fit Sphere	147
	17.4 Best-Fit Cylinder	148
	17.5 Summary	149
	Exercises	149
	References	149
	Fitting Radius-Suppressed Circle Data	150
	18.1 Introduction	150
	18.2 The Lima°on Approximation	150
	18.3 LS Best-Fit Circle	151
	18.4 LS Best-Fit Cylinder	153
	18.5 Errors in the Lima°on Approximation	154
	18.6 Summary	155
	Exercises	156
	References	156
	Exchange Algorithms for Minimum Zone	157
	19.1 Introduction	157
	19.2 Exchange Algorithms	157
	19.3 Exchange Algorithm for Line	158
	19.4 Exchange Algorithm for Plane	161
	19.5 Exchange Algorithm for Circle	163
	19.6 Summary	166
	Exercises	167
	References	167
	Reference Circle-Fitting Using Linear Programming Simplex	168
	20.1 Introduction	168
	20.2 LP Simplex	168
	20.3 Formulating the MI Problem	170
	20.4 Formulating the MC Problem	174
	20.5 Duality	175
	20.6 LP and Exchange Algorithms	177
	20.7 Exchange Algorithms for MI and MC Circles	179
	20.8 Summary	179
	Exercises	179
	References	180
	Part IV Parameterization	181
	Surface Finish Parameters I: Amplitude, Spacing, Hybrid, and Shape	182
	21.1 Introduction	182
	21.2 Amplitude Parameters	182
	21.3 Spacing Parameters	184
	21.4 Hybrid and Shape Parameters	187
	21.5 Summary	189
	Exercises	190
	References	191
	Surface Finish Parameters II: Autocorrelation, Power Spectral Density, Bearing Area	192
	22.1 Autocovariance and Autocorrelation Function	192
	22.2 Power Spectral Density	194
	22.3 Amplitude Density Function and Bearing Area Curve	196
	22.4 Summary	199
	Exercises	200
	References	200
	3D Surface Texture Parameters	201
	23.1 Introduction	201
	23.2 Amplitude and Shape Parameters	201
	23.3 AACV, APSD, and BAC for 3D Surfaces	202
	23.4 Spacing and Hybrid Parameters	205
	23.5 Summary	206
	Exercises	206
	References	206
	Part V Errors and Uncertainty	207
	Uncertainty Considerations	208
	24.1 Introduction	208
	24.2 Random and Systematic Components	208
	24.3 Uncertainty Modeling	209
	24.4 Uncertainty Propagation	209
	24.5 Systematic Errors: an Example	209
	References	210
	Uncertainty Propagation in Computations	211
	25.1 Introduction	211
	25.2 Intervals at Some Level of Confidence	212
	25.3 The Central Limit Theorem	213
	25.4 Limitations of GUM	215
	25.5 The Monte Carlo Method	215
	25.6 Filtering Surface Profiles	216
	25.7 Fitting Substitute Geometry	219
	25.8 Summary	221
	Exercises	222
	References	222
	Error Separation Techniques in Roundness Metrology	223
	26.1 Introduction	223
	26.2 Full Reversal	223
	26.3 Two-Position Method	224
	26.4 Two-Probe Method	227
	26.5 Three-Probe FFT Method	227
	26.6 Three-Probe Sequential Method	229
	26.7 Three-Position FFT Method	231
	26.8 Error Separation by Solving Linear Equations	232
	26.9 Uncertainty Propagation	234
	26.10 Summary	234
	Exercises	234
	References	235
	Other Relevant Topics	237
	27.1 Introduction	237
	27.2 Parameter Extraction	237
	27.3 Tools for Correlation and Diagnostics	238
	27.4 Challenges	239
	References	239
	Answers to Selected Exercises	241
	Index	258