Gao R.X., Yan R. Wavelets: Theory and Applications for Manufacturing

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12.3 Curvelet Transform

Ridgelet transform is a relatively new system of representation and analysis, which

has been shown to be effective in resolving edges (Candes and Donoho 1999;Do

and Vetterli 2003; Dettori and Semler 2007). However, ridgel ets are limited to

resolving only straight edges; curved edges cannot be represented with as few

coefﬁcients as required for a straight edge (Do and Vett erli 2003; Starck et al.

2003). The inadequacy of wavelets and ridgelets in resolving edges has been the

primary driving force behind the search for better representation and analysis.

Curvelet analysis, introduced in the year 2000 (Candes and Donoho 2000), holds

potential in addressing the shortcomings of wavelets and ridgelets. A brief intro-

duction to the fundamentals of curvelets is given in this section.

12.3.1 Curvelet Transform

The curvelet transform is deﬁned as the inner product of the function f to be

analyzed and a family of curvelets g

aby

(Candes and Donoho 2000, 2005a, b):

ða; b; yÞ¼hf ; g

aby

i (12.14)

where a>0 is the scale parameter, b is the translation parameter, and y 2 0; 2pÞ½ is

the orientation parameter. The symbol G

represents the curvelet transform. The

family of curvelets is explained by starting with two smooth, nonnegative, real

valued windowing functions called the radial window W(r) and the angular

window V(t), respectively (Candes and Donoho 2005a, b). The two windowing

functions are subject to the following two admission conditions:

WðarÞ

da ¼ 1; 8 r>0 (12.15)

VðtÞ

dt ¼ 1 (12.16)

In (12.5), r 2ð1=2; 2Þ is the radial coordinate and in (12.6) t 2½1; 1 denotes the

time variable. According to the deﬁnition in (12.14 ), at a given scale a, a family of

curvelets can be generated by translation and rotation of a basic element g

a00

shown in (12.17) (Candes and Donoho 2005a, b):

aby

¼ g

a00

ðR

ðx  bÞÞ (12.17)

where R

is a 2  2 rotation matrix that is related to planar rotation by an angle y.

214 12 Beyond Wavelets

The basic element itself is expressed mathematically as:

a00

ðr; oÞ¼WðarÞV



3=4

(12.18)

where r and o are polar coordinates deﬁned in the frequency domain.

Generally, the discrete curvelet transform is often used to process the function f,

which also starts with two window functions: the radial window W(r) and the

angular window V(t) (Candes et al. 2006). The transform is subject to the condi-

tions expressed in (12.19) (12.21) as:

j¼ 1

Wð2

rÞ

¼ 1 (12.19)

l¼ 1

Vðt  lÞ

¼ 1 (12.20)

ðr; yÞ¼2

3j=4

Wð2

rÞV

j=2



(12.21)

In (12.21), the window function U

is derived from the radial window W(r) and the

angular window V(t) and expressed in the Fourier domain. The symbols

r 2ð3= 4 ; 3=2Þ and y denote polar coordinates, and t 2ð1=2; 1=2Þ is the time

variable.

Based on these notations, a family of curvelets at a ﬁxed scale of 2

is deﬁned as:

’

j;l;k

ðxÞ¼’

ðR

ðx  x

ðj;lÞ

ÞÞ (12.22 )

where x

ðj;lÞ

¼ R

ðk

; k

j=2

Þ represents the position information, and

cos y sin y

sin y cos y

represents the rotation information in terms of y radians,

respectively.

Accordingly, the inner product between a curvelet ’

j;l;k

and a function f results in

the curvelet coefﬁcients as:

cðj; l; kÞ¼hf ;’

j;l;k

i¼

f ðxÞ ’

j;l;k

ðxÞdx (12.23)

where cðj; l; kÞ are the curvelet coef ﬁcients.

The physical interpretation of (12.23) in the Fourier domain at a scale of 2

can

be illustrated in Fig. 12.9. The concentric circles represent the family of curvelets

’

j;l;k

ðxÞ and the shaded portion represents one curvelet from this family.

12.3 Curvelet Transform 215

While the curvelets described above are expressed in the polar coordinates,

Cartesian coordinates are desirable to implement the curvelet transform (Donoho

and Duncan 2000 ; Candes et al. 2006). Consequently, the window functions

expressed in (12.19) (12.21) are expressed in the Cartesian coordinates as

ðoÞ¼ F

jþ1

ðoÞF

ðoÞ

; j>0 (12.24)

ðoÞ¼V

j=2



(12.25)

ðoÞ¼W

ðoÞV

ðoÞ (12.26)

where Fðo

Þ¼fð2

Þfð2

Þ; 2

o

2

jþ1

; 2

j=2

ðo

Þ2

j=2

and 0 f 1; f ¼

1; ½1=2; 1=2

0; ½2; 2

(

Currently, there are two prevalent methods of calculating the curvelet coefﬁ-

cients. The ﬁrst method, called the digital curvelet transform via unequispaced fast

Fourier transform (Candes et al. 2006), involves the following four steps:

1. Calculate the 2D FT of the function of interest (f [t

]), to obtain its Fourier

samples,

f ½n

; n

:

2. Resample

f ½n

; n

 for each scale/angle pair (j, l) to obtain sampled values

f ½n

; n

 n

tan y

.

3. Multiply the resampled

f with the window function U

, to obtain

j;l

½n

; n

¼

f ½n

; n

 n

tan y

 U

½n

; n

, where tan y

¼ l 2

j=2

4. Calculate the inverse of the 2D FT of every

j;l

to obta in the discrete curvelet

coefﬁcients, cðj; l; kÞ.

Fig. 12.9 A family of

curvelets in polar coordinates,

with the shaded area

representing support for a

single curvelet (Candes et al.

2006)

216 12 Beyond Wavelets

The second method is called digital curvelet transform via wrapping (Candes

et al. 2006), and involves the following computational steps:

1. Calculation of 2D FT of the function of interest (f [t

]), to obtain its Fourier

samples,

f ½n

; n

.

2. Calculation of the product U

j;l

½n

; n



f ½n

; n

.

3. Wrapping the product U

j;l

½n

; n



f ½n

; n

 around the origin to obtain

j;l

½n

; n

¼WðU

j;l

f Þ½n

; n

.

4. Calculation of inverse 2D FFT of every

j;l

to obtain discrete curve let

coefﬁcients, cðj; l; kÞ.

12.3.2 Application of the Curvelet Transform

Because of its multiscale nature, most of the curvelet applications are related

to image processing. Speciﬁcally, the curvelet transform has been applied to

image compression, contrast enhancement, feature extraction from noisy images,

pattern detection, nois e ﬁltration, edge detection, etc. As an example, a

denoising operation on a test image “Lena” is shown in Fig. 12.10. This image,

which is 512  512 pixels in size (Fig. 12.10a), was contaminated with random

noise (peak signal-to-noise ratio, PSNR: 22.1), as shown in Fig. 12.10b. Curvelet

transform is then applied for image denoising via thresholding of its curvelet

coefﬁcients. The result is a ﬁltered image as seen in Fig. 12.10c, which has an

improved PSNR value of 31.1.

In manufacturing, the curvelet has been used for surface characterization.

Fig 12.11a shows a surface of a worn metallic femoral head. When additive white

noise is added into it, the surface has a signal-to-noise ratio (SNR) of 56.43 dB.

When the wavelet transform is used to remove the noise, the SNR of the surface has

increased to 58.72 dB. Finally, the performance of the denoisi ng operation was

further improved when the curvelet transform is applied to process the surface,

where the SNR has increased to 61.62 dB.

12.10 (a) Original image, (b) image contaminated with noise, and (c) image after denoising

12.3 Curvelet Transform 217

12.4 Summary

This chapter brieﬂy presents development in signal processing that goes

beyond the classical wavelet transform. The SGW T base d on the lifting

scheme is ﬁrst introduced to allow for the design of the base wavelet functions

to better ﬁt the signal to be analyzed. The ridgelet and curvelet transforms

are then introduced from the point of view of overcoming the limitations

in detecting edges (straight and curved) of images when applying the classical

wavelet transform. These techniques, toge ther with furt her advancement

reported in the literature, such as multiwavelet transform (Cotronei et al. 1998),

dual-tree wavelet transform (Selesnick et al. 2005), and contourlet transform

(Do and V etterli 2005), promise to continually push the envelope of signal

and image processing to better serve the needs for a wide range of engineering

problems.

Fig. 12.11 Denoising of a microscalar surface (Ma 2007). (a) Original surface (scale unit: mm),

(b) noisy surface (signal to noise ratio, SNR 56.43 dB), (c) denoised surface using wavelet

(SNR 58.72 dB), and (d) denoised surface using curvelet (SNR 61.62 dB)

218 12 Beyond Wavelets

12.5 References

Candes EJ (1998) Ridgelets: theory and applications. Ph.D. Dissertation, Stanford University

Candes EJ, Donoho DL (1999) Ridgelets: a key to higher dimensional intermittency. Philos Trans

R Soc Math, Phys Eng Sci 357:2495 2509

Candes EJ, Donoho DL (2000) Curvelets a surprisingly effective nonadaptive representation for

objects with edges. In: Rabut C, Cohen A, Schumaker LL (eds) Curves and surfaces.

Vanderbilt University Press, Nashville, TN

Candes EJ, Donoho DL (2005a) Continuous curvelet transform: 1. resolution of the wavefront set.

Appl Comput Harmon Anal 19:162 197

Candes EJ, Donoho DL (2005b) Continuous curvelet transform: 2. discretization and frames.

Appl Comput Harmon Anal 19:198 222

Candes EJ, Demanet L, Donoho DL, Ying L (2006) Fast discrete curvelet transforms.

SIAM Multiscale Model Simul 5:861 899

Claypoole R (1999) Adaptive wavelet transforms via lifting. Thesis: computer engineering,

Rice University

Claypoole R, Davis G, Sweldens W (2003) Nonlinear wavelet transform for image coding via

lifting. IEEE Trans Image Process 12(12):1449 1459

Cotronei M, Montefusco LB, Puccio L (1998) Multiwavelet analysis and signal processing. IEEE

Trans Circuits Syst II Analog Digital Signal Process 45(8): 970 987

Dettori L, Semler L (2007) A comparison of wavelet, ridgelet and curvelet based texture classiﬁ

cation algorithms in computed tomography. Comput Biol Med 37:486 498

Do MN, Vetterli M (2003) The ﬁnite ridgelet transform for image representation. IEEE Trans

Image Process 12:16 28

Do MN, Vetterli M (2005) The contourlet transform: an efﬁcient directional multiresolution image

representation. IEEE Trans Image Process 14(12):2091 2106

Donoho DL, Duncan MR (2000) Digital curvelet transform: strategy, implementation and experi

ments. Proc SPIE 4056:12 29

Jiang HK, Wang ZS, He ZJ (2006) Wavelet design for extracting weak fault feature based on

lifting scheme. Front Mech Eng China 1(2):199 203

Jiang X, Blunt L, Stout KJ (2001a) Application of the lifting wavelet to rough surfaces. J Int Soc

Precision Eng Nanotechnol 25:83 89

Jiang X, Blunt L, Stout KJ (2001b) Lifting wavelet for three dimensional surface analysis. Int J

Mach Tools Manuf 41:2163 2169

Jiang X, Scott P, Whitehouse D (2008) Wavelets and their application in surface metrology. CIRP

Ann Manuf Technol 57:555 558

Li Z, He ZJ, Zi YY, Jiang HK (2008) Rotating machinery fault diagnosis using signal adapted

lifting scheme. Mech Syst Signal Process 22(3):542 556

Logan BF, Shepp LA (1975) Optimal reconstruction of a function from its projections. Duke Math

J 42:645 659

Ma J, Jiang X, Scott P (2005) Complex ridgelets for shift invariant characterization of surface

topography with line singularities. Phys Lett A 344:423 431

Ma J (2007) Curvelets for surface characterization. Appl Phys Lett 90: 054109 1 3

Selesnick IW, Baraniuk RG, Kingsbury NG (2005) The dual tree complex wavelet transform.

IEEE Signal Process Mag 22(6): 123 151

Starck JL, Donoho DL, Candes EJ (2003) Astronomical image representation by the curvelet

transform. Astron Astrophys 398:785 800

Starck JL, Moudden Y, Abrial P, Nguyen M (2006) Wavelets, ridgelets and curvelets on the

sphere. Astron Astrophys 446:1191 1204

Sweldens W (1996) The lifting scheme: a custom design construction of biorthogonal wavelets.

Appl Comput Harmon Anal 3:186 200

12.5 References 219

Sweldens W (1997) Second generation wavelets: theory and application. http://www.ima.umn.

edu/industrial/97 98/sweldens/fourth.html. Accessed 30 June 2009

Sweldens W (1998) The lifting scheme: a construction of second generation wavelets. SIAM

J Math Anal 29(2):511 546

Uytterhoeven G, Dirk R, Adhemar B (1997) Wavelet transforms using the lifting scheme.

Department of Computer Science, Katholieke Universiteit Leuven, Belgium

Zhou R, Bao W, Li N, Huang X, Yu DR (2010) Mechanical equipment fault diagnosis based on

redundant second generation wavelet packet transform. Digit Signal Process 20(1):276 288

220 12 Beyond Wavelets

Index

Admissibility condition, 33, 38, 47

Aliasing, 20

Angular window, 214, 215

Approximate coefﬁcient, 55, 60, 67

Axial load, 114, 115, 120, 122

Band pass ﬁlter, 83, 84, 86, 89, 125

Base template function, 104 106, 113

Basewavelet, 12,28 31,33, 34,39, 40,49,52, 53,

56,60 64,66, 67, 69, 72, 79, 83 100, 103,

108, 112, 113, 116, 165 186, 189 191,

197 199, 202, 205, 218

Basis function, 12, 26, 154, 166

Bearing, 1, 8, 9, 13, 35, 42, 62, 63, 65, 78,

79, 83, 87, 93 99, 103, 109, 110,

113 125, 129, 132, 134 146, 183 186,

190 195, 197 199, 202, 209, 210

Bearing defect diagnosis, 93 99, 100,

113 124, 198

Bearing vibration signal, 79, 97, 98, 110, 114,

183 186

Biorthogonal wavelet, 63 65, 167, 168, 184

Center frequency, 41, 43, 44, 46, 90 92, 96, 97,

176, 177, 180 182

Children node, 153, 154

Chirp function, 46 47

Class, 129 131, 149 155, 157, 158, 162, 190

Classiﬁcation, 1 5, 13, 125 146, 149 162,

167 169, 212

Coiﬂet wavelet, 62, 63

Compact support, 66, 166, 167, 189

Completeness, 53

Complex conjugate, 17, 34, 40, 50

Complex valued signal, 85

Complex valued wavelet, 86 87, 176, 179 185

Computation procedure, 39

Conditional entropy, 172 173

Continuous frame, 104

Continuous wavelet transform, 33 47, 49,

107, 112, 179, 184, 185

Convolution theorem, 40, 84

Correlation index, 151 152

Covariant, 36 37

Curvelet analysis, 214

Curvelet transform, 214 218

Customized wavelet, 189, 197, 199

Daubechies function, 106

Daubechies, I., 30, 31

Daubechies scaling function, 192 194

Daubechies wavelet, 30, 61 63, 165 168

Decomposition level, 60, 66, 67, 76, 116 119,

123, 131, 151 153, 155, 160, 176,

177, 183

Deconstruction, 194, 197

Defect feature extraction, 115 118, 120

Defect severity, 13, 125 146, 160, 162

Denoising, 13, 44, 49, 65 67, 79, 80, 168,

169, 212, 213, 217, 218

Derived template function, 104 106

Detailed coefﬁcient, 56, 66, 67, 208

Deterministic signal, 1 5

Diagnosis, 11 13, 62, 65, 93 100, 113 124,

130, 131, 134, 146, 149, 165, 167, 169,

171, 175, 183, 198

Dilation, 12, 28, 34 37, 205, 212

equation, 190, 192, 194 196

regularity, 53

R.X. Gao and R. Yan, Wavelets: Theory and Applications for Manufacturing,

DOI 10.1007/978 1 4419 1545 0,

Springer Science+Business Media, LLC 2011

221

Discrete Fourier transform, 19

Discrete frame, 104, 105

Discrete wavelet transform, 13, 34, 49 69, 107,

111, 112, 114, 116, 124, 166, 206

Discriminant power, 129, 130, 138, 139, 143,

144, 154 157

Discrimination, 129, 153, 154, 158, 168

Dissimilarity measure, 149 154

Dual scale equation, 56 58, 68

Eigen value, 132 134

Energy, 11, 19, 33, 34, 38, 65, 74, 93, 95,

97, 103, 104, 120, 123, 125 128,

130, 137, 139, 140, 143, 146, 150,

151, 160, 161, 170 172, 177, 178,

180 185, 190

Energy difference, 151

Energy to Shannon entropy ratio, 172, 178,

181, 183 185

Enveloping, 83 100

Error function, 196

Expectation operation, 127, 131

Feature, 11, 12, 33, 35, 49, 60, 93, 96, 97,

100, 105, 107 110, 112 118,

120 146, 149, 154, 156, 158, 160 162,

165 168, 171, 174, 175, 199, 209,

210, 217

Feature selection, 128 134, 136 138, 143

Fisher linear discriminant, 128 131, 140

Fourier function, 106

Fourier transform, 11, 12, 17 31, 33 35, 40,

74, 75, 77, 84, 85, 93, 103 124, 198,

199, 211, 216

Frequency B Spline wavelet, 43

Frequency shifted keying (FSK) signal,

108, 109

Gabor transform, 12

Gaussian modulated sinusoidal signal,

176 178, 182

Gaussian pulse function, 46

Gaussian wavelet, 42, 167, 180 182, 185

Gearbox, 10, 13, 83, 93, 100, 125, 149,

150, 158 162

Gearbox defect classiﬁcation, 158 161

Generalized frame, 104, 106 109

Haar, A., 26

Haar function, 106

Haar wavelet, 26, 61, 70, 71, 108

Hard thresholding, 66

Harmonic wavelet, 44, 180 182, 185

packet transform, 74 77

transform, 29, 74 76

Hilbert transform, 83 86, 125

Impulse response, 96, 191 199, 202

Impulse wavelet, 169, 190 202

Inclusion relationship, 53, 54

Inequality, 168

Injection molding, 6, 8, 87 90, 100

Inner product, 17, 18, 22, 28, 37, 40, 50, 51,

72, 75, 166, 214, 215

Inverse continuous wavelet transform, 38 39

Inverse discritized wavelet transform, 52

Isotropic, 210

Joint entropy, 172 175, 178, 179, 181,

184, 186

Kurtosis, 125, 127 128, 130, 137, 140,

143, 146

Leakage, 20

Levy, P., 26, 27

Lifting scheme, 205 207, 218

Linear discriminant analysis (LDA), 156,

157, 160

Local discriminant base, 149 162

Low pass ﬁlter, 60, 70, 83, 84

Mallat algorithm, 58 60, 68

Mallat, S., 30

Manufacturing, 1 13, 17, 21, 30, 61, 63, 65,

100, 103, 124, 149, 162, 167, 169,

185, 190, 209, 217

Maximum information, 175, 178, 182 183, 185

Measure function, 105, 107, 108, 110 112

Message, 108, 109

222 Index

Mexican hat wavelet, 41, 165, 167, 190

Meyer wavelet, 65, 177, 178

Meyer, Y., 30, 31, 33

Milling, 5, 6, 130

Modulus, 86, 87, 95, 100

Monotonicity, 53

Morlet, J., 27

Morlet wavelet, 28, 29, 41 42, 96, 165, 167,

180 182, 185, 189

Moyal principle, 37 38

Multiresolution analysis (MRA), 29, 30,

53 56, 58, 68

Multi scale, 83 100, 210, 217

Multi scale enveloping, 83 100

Multiscale enveloping spectrogram

(MuSEnS), 93, 95 100

Mutual information, 169, 173 175, 178, 179,

182, 184, 186

Neural network classiﬁer, 134 137, 140,

141, 145, 146, 156

Non deterministic signal, 1, 3 5

Non stationarity, 21, 152 153

Non stationary signal, 21

Orthogonal basis, 53, 54, 154

Orthogonality, 53, 72, 154, 166, 167

Orthogonal wavelet, 30, 53 56, 60, 68,

154, 166

Orthonormal eigenvectors, 132

Orthonormal system, 26

Parent node, 153, 154

Parseval’s theorem, 37

Periodic signal, 1 3, 18, 205

Prediction, 206 208

Pressure measurement, 87 93, 100

Principal component analysis (PCA),

128, 131 134, 139, 141 143,

145, 146

Probability density function, 127, 150, 168

Pruning approach, 153

Pulse differentiation, 87 93, 100

Quadrature mirror ﬁlters, 194

Radial load, 97 100, 113 115, 120 122,

130, 137, 142

Radial window, 214, 215

Real valued signal, 83 86

Real valued wavelet, 176 180, 184

Reconstruction, 66, 189, 190, 194, 196, 197,

207 209, 212

Rectifying, 84

Recursive algorithm, 73 74

Reﬂection, 91, 93

Regularity, 53, 167, 177

Relative entropy, 150, 153, 154, 160, 169,

175, 178, 180, 182, 184, 186

Ridge analysis, 210

Ridge function, 210, 211

Ridgelet transform, 210 214

Rotary machine, 21, 93 99

Rotational speed, 95, 97, 98, 110, 114, 120,

123, 142, 158

Sampling theorem, 20

Scale function, 53, 54, 56

Scaling coefﬁcient, 191

Scatter matrix, 131 133

Second generation wavelet transform

(SGWT), 205 210, 218

Shannon entropy, 73, 74, 149, 168, 170 172,

177, 178, 181, 183 185

Shannon wavelet, 43 44, 180 182, 185

Shift orthogonality, 72

Short time Fourier transform (STFT), 12,

21 28

Signal transformation, 103 109

Sinusoidal function, 45

Soft thresholding, 66, 67

Spalling, 9, 78, 83, 103, 109

Spectral post processing, 103, 109 113

Spindle, 9, 13, 93, 100, 130

Splitting, 206

Stamping, 5 7, 61

Stationary signal, 4, 5, 11, 12

Sub band, 125 130, 136, 137, 139 141,

143 146

Superposition property, 35 36

Surface, 9, 63, 78, 83, 95, 103, 109, 114,

158, 167 169, 209, 210, 212, 213,

217, 218

Symlet wavelet, 63, 64

Symmetry, 61 63, 166, 167

Synthetic signal, 95 97, 156

Index 223