Linear feature extraction based on complex ridgelet transform

Available online at www.sciencedirect.com

Wear 264 (2008) 428–433

Linear feature extraction based on complex ridgelet transform Xiangqian Jiang a,∗ , Wenhan Zeng a , Paul Scott b , Jianwei Ma a , Liam Blunt a a

Centre for Precision Technologies, University of Huddersfield, Huddersfield HD1 3DH, UK b Taylor Hobson Ltd., 2 New Star Road, Leicester LE4 9JQ, UK Accepted 28 August 2006 Available online 23 February 2007

Abstract In our previous work, a new dual-tree complex wavelet transform (DT-CWT) model for surface analysis has been built, which solved the problem of the lack of shift-invariance that existed in the first and second generation wavelet models. Unfortunately, the DT-CWT model still has the same problem as the previous wavelet models in the lack of ability to detect line singularities or higher dimensional singularities, which causes the edges not to be smooth when extracting the directional features from engineering surfaces. In this paper, a complex finite ridgelet transform (CFRIT), which provides approximate shift invariance and analysis of line singularities, is proposed by taking the DT-CWT on the projections of the finite Radon transform (FRAT). The Numerical experiments show the remarkable potential of the methodology to analyse engineering and bioengineering surfaces with linear scratches in comparison to wavelet-based methods developed in our pervious work. © 2007 Elsevier B.V. All rights reserved. Keywords: Complex finite ridgelet transform; Finite Radon transform; Shift invariance; Linear singularity; Surface topography

1. Introduction Surface topography is one of the most important factors affecting the functional performance of components. For engineering and bioengineering surfaces, their topographies are usually composed of roughness, waviness, form error, and multiscalar features, such as random peaks/pits and ridges/valleys. These functional topographical features will impact directly on mechanical and physical properties of the whole system such as wear, friction, lubrication, corrosion, fatigue, coating, paintability, etc. The highly accurate characterization of surface topography such as the extraction of multi-scalar features is a challenging and important issue in the initial stages of engineering systems. In the last decade, a wide range of wavelet based multi scalar analysis methods for surface characterization have been investigated and proposed. The orthogonal wavelets have been used for analysis of multi-scalar surfaces in engineering by Chen and Raja, in which the phase distortions were neglected [1,2]. The first and second generation biorthogonal wavelet filtration for the

∗

Corresponding author. E-mail address: [email protected] (X. Jiang).

0043-1648/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.wear.2006.08.040

extraction of morphological features was proposed by our previous work [3,4]. The main advantages of biorthogonal wavelets are that it is possible to have linear phase (leading to real outputs without aliasing and phase distortion) and a traceable location property. The model for second generation biorthogonal wavelet has helped the steel industry to identify multi-scalar surfaces; the pump industry to diagnose pump failures; and the bioengineering industry to reconstruct isolated morphological features. In spite of the development of wavelet technologies for surface characterization, there are still no available techniques to extract morphological features such as linear scratches and plateau with direction/objective properties. Ridgelets were introduced by Cand`es and Donoho to deal effectively with line singularities by mapping a line singularity into a point singularity through the Radon transform, and then using the wavelet transform on each projection in the Radon transform domain [5–7]. The weakness of the traditional ridgelets transform is the lack of shift invariance due to the real DWT used. Recently, a novel dual-tree complex wavelet transform (DT-CWT) model for surface filtering was proposed [8–11]. The detailed investigation has shown that the DT-CWT filters have very good transmission characteristics for the separation of roughness, waviness and form, and most importantly, it can provide approximate shift-invariance property.

X. Jiang et al. / Wear 264 (2008) 428–433

In this paper, we bring together the ideas of complex wavelet pyramids and the geometric features of ridgelets, proposes a complex finite ridgelet transform (CFRIT) for the shift invariant extraction of line scratches from engineering and bioengineering surface topography by applying the finite Radon transform (FRAT) to a DT-CWT. 2. Complex ridgelet transform

as:

c A bivariate complex ridgelet ψa,b,θ in R2 space can be defined

c (x) ψa,b,θ

=a

−1/2

 ψ

c

x1 cos θ + x2 sin θ − b a

(1)

Here, a > 0 is a scale parameter, θ an orientation parameter, and b is a location scalar parameter. This function is constant along const, while its transverse is a comlines: x1 cos θ + x2 sin θ = √ plex wavelet ψc = ψr + −1ψi , ψr and ψi are themselves real wavelets. If the real and imaginary part of the complex wavelet can be viewed as two ‘fat’ points, then the complex ridgelet can be interpreted as two ‘fat’ lines so that it is especially adaptive to analyse line ridges/valleys contained in surface topography. The continuous complex ridgelet transform (CRIT) for an integrable bivariate function f(x) ∈ L2 (R2 ) is defined as:  c (x)f (x) dx (2) f (a, b, θ) = ψa,b,θ

The FRAT is defined as summations of surface pixels over a certain set of lines. These lines are defined by a finite geometry in a similar way as the lines for the continuous Radon transform in Euclidean geometry [12,13]. Denote the group G = Zp2 to be the Cartesian product Zp × Zp of two exemplars from the cyclic group Zp = {0, 1, . . ., p − 1} with addition modulo p, where p is a prime number, and let N = {0, 1, . . ., p}. This group has p + 1 non-trivial subgroups: 0 ≤ i < p,

Hp = {(k, 0) ∈ G; k ∈ Zp }.

(6)

The cosets of the factor group G/Hi are indexed by j ∈ Zp in the following way: j

Hi = {(k, l) ∈ G; li + j = k(mod p)}, Hpj

0 ≤ i < p,

= {(k, j) ∈ G; k ∈ Zp }.

(7)

The Radon projection of a function f on G is given by: j

p−1p−1

Λi f (Hi ) =

1  f (k, l)δj (πi (k, l)), p

i ∈ N, j ∈ Zp

(8)

k=0 l=0

Here, the function πi is a variance of the factor mapping of G on G/Hi . We have: πi (k, l) = k − li(mod p),

0 ≤ i < p,

πp (k, l) = l

(9)

j

(3)

Point and line singularities are related by the Radon transform. Comparing Eq. (2) with the application of the 1D DT-CWT on the projections of the Radon transform we obtain:    t−b f (a, b, θ) = Rf (θ, t)a−1/2 ψc dt (4) a The Radon transform is denoted as:  Rf (θ, t) = f (x1 , x2 )δ(x1 cos θ + x2 sin θ − t) dx1 dx2

3.1. Finite Radon transform

Hi = {(k, l) ∈ G; li = k(mod p)},



The reconstruction formula is given as:  2π  ∞  ∞ da dθ c f (x) = f (a, b, θ)ψa,b,θ (x) 3 db a 4π 0 −∞ 0

429

(5)

where δ is the Dirac delta function.

For surface topography or for an image, the coset Hi denotes the set of points that make up a line on the lattice G. Particularly j j H0 and Hp denote the horizontal and vertical lines respectively. πi denotes the set of lines that go through a point (k, l) ∈ G. As j in the Euclidean geometry, the line Hi on the affine plane G is uniquely represented by its slope i ∈ N and its intercept j ∈ Zp . It has p2 points, p2 + p lines, every point (k, l) ∈ G lies on p + 1 lines, every line contains p points. Moreover, any two distinct points on G lie on just one line. For any given slope i ∈ N, there are p parallel lines to provide a complete cover of the plane G. From the finite geometry property, for zero-average functions, the forward and inverse formula can be written as: 1  F = Λi f = √ f (k, l) p j (k,l) ∈ Hi

  1 1  f (k, l)δH j (k, l) = f, √ δH j = √ i p p i

3. Digital complex ridgelet transform From Eqs. (4) and (5), one can see that the basic strategy for computing the CRIT is to first calculate the Radon transform Rf (θ, t), then to calculate the 1D-CWT of the projections Rf (θ, ·). For the calculation of the Radon transform, numerous digital methods have been devised. However, most of them were not designed to be invertible transforms for digital surfaces or images. Alternatively, the finite Radon transform theory provided an interesting solution for finite length signals. According to the practical requirements of surface characterization, we use the digital form of the CRIT based on the FRAT and DT-CWT.

(10)

(k,l) ∈ G

1 f = Vi F = √ p =

1 p





F (i, j)

(i,j) ∈ πi (k,l)



f (k , l )

(i,j) ∈ πi (k,l)(k ,l ) ∈ H j i

1  f (k , l ) + f (k, l) = f (k, l) = p   (k ,l ) ∈ G

(11)

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X. Jiang et al. / Wear 264 (2008) 428–433

Here, the Dirac delta function δH j : G → R defined as:  δH j (k, l) = i

i

j

1

(k, l) ∈ Hi

0

elsewhere

(12)

(k,l) ∈ Hi

1  f (k + k, l + l)δH j (k + k, l + l) = √ i p

then

⎧ i = i , j = j   ⎪  ⎨1 1 1 i = i , j = j  √ δ H j , √ δH j  = 0 ⎪ p i p i ⎩ 1/p i =

i

(k,l) ∈ G

(13)

Due to the fact that the operator V is the adjoint of Λ, the inverse Radon transform algorithm has the same structure and is symmetric with the algorithm of the forward transform. The actual computational time of the designed forward transform is comparable with other O(p2 log(p2 )) transforms, such as the 2D FFT. 3.2. Digital ridgelet transform

(14)

The digital FRIT can be integrated as: CFRITf (i, m) =

Λi f, w(i) m



=



=



(k ,l )

 =

1 f, √ δH j−i l+k p i

 (19)

Consider an “horizontal” shift f = f(k + k, l), then:   1  F = f, √ δH j+k = F (i, j + k) p i

ΛTτ f = Tτ Λf

i∈N

1  = √ f (k , l )δH j−i l+k (k , l ) p   i

(20)

If let Tτ be the shift operator, we have:

The complex wavelet basis is defined as: {w(i) m , m ∈ Zp },

Combining Eq. (10), the finite Radon transform of the shift function becomes: 1  f (k + k, l + l) F  = Λi f  = √ p j

j ∈ Zp

 w(i) m (j)

1  (i) f, √ wm (j)δH j i p

1 f, √ δH j p i



(21)

For the ridgelet transform of the shift function, we have: 

CFRITf  (i, m) = Λi f  , w(i) m   1  (i) = f, √ wm (j)δH j−i l+k p i

(22)

j ∈ Zp

 (15)

j ∈ Zp

Therefore the basis functions of the discrete complex ridgelet transform can be written as: 1  (i) wm (j)δH j (16) ρi,m = √ i p j ∈ Zp

√ Here, although the {(1/ p)δH j } is not an orthonormal sysi

tem, if we take the p + 1 orthonormal bases for l2 (Zp ), (i) {w(i) m , m ∈ Zp } with w0 ≡ const, the system {ρi,m : i = 0, . . ., p; m = 1, . . ., p − 1} ∪ {ρ0 } is an orthonormal base for l2 (Zp ) where ρ0 (k, l) = 1/p, ∀(k, l) ∈ G. The rest of this paper we concentrate on the shift properties of the ridgelet transform. 4. Shift invariance analysis of the CFRIT If we rewrite the Radon transform (5) into a more general form:  ៝ = f (៝r )δ(ς − ξ៝ · r៝) d៝r ˆf (ς, ξ) (17) then the Radon transform of the shift function f  = f (៝r − τ៝ ) becomes:  ៝ f (៝r − τ៝ )δ(ς − ξ៝ · r៝) d៝r = fˆ (ς − ξ៝ · τ៝ , ξ) (18)

According to Eqs. (22) and (20), a direct way to design a ridgelet with the shift invariant property is to choose the {w(i) m } to be also shift invariant. Using the DT-CWT basis, the newly proposed CFRIT is a shift invariant transform. 5. Experiments 5.1. Shift invariance This test demonstrates the shift invariance of the CFRIT by an artificial image with a stepped edge. Fig. 1 shows 16 shifted versions of the image (at the top) and their subspace reconstructed components in turn from the coefficients at levels j ≤ j0 = 4 using the CFRIT (left) and real FRIT (right). In order to see the effects clearly, only the centre of the profiles of these images is shown. Each shift is displaced down a little to give a waterfall style display. The output of CFRIT is the modulus of the complex coefficients. Note that summing these components the input image can be reconstructed perfectly. Good shift invariance is seen from the fact that the shape and amplitude of each of the reconstructed components by CFRIT hardly varies as the input is shifted. In contrast, the reconstructed components using FRIT vary considerable with each shift. 5.2. Line singularity and denoising Our next test shows the good performance of CFRIT for denoising an image with line singularities. We consider an

X. Jiang et al. / Wear 264 (2008) 428–433

431

Fig. 1. The reconstructed components (centre profiles) at levels 1–4 of 16 shifted image with a stepped edge using the CFRIT (left) and real FRIT (right). Each shift is displaced down a little to give a waterfall style of display.

artificial image with a deep scratch that is contaminated by an additive zero-mean Gaussian white noise of variance σ 2 . The denoising includes the following steps: (1) transform the noisy image using CFRIT; (2) hard-thresholding of the coeffi√ cients using the universal threshold T = σ 2 log N (where N is the number of pixels); (3) reconstruct the thresholded coefficients. For comparison, the same uniform threshold value is

also applied to the DWT and DT-CWT based algorithms (see Fig. 2). It can be seen from Fig. 2 that the CFRIT is effective in recovering straight edges, as well as in term of the signal to noise ratio (SNR). The CFRIT reconstruction does not contain the undesirable artefacts along edge that one finds in the wavelet reconstruction. The simple thresholding scheme for CFRIT is

Fig. 2. Denoising an image with line singularities using the DWT (upper right), DT-CWT (lower left), and CFRIT (lower right). The upper left image is a noisy image.

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X. Jiang et al. / Wear 264 (2008) 428–433

Fig. 3. Feature extraction of hone surface using the DWT (upper right), FRIT (lower left), and CFRIT (lower right). The upper left image is the original data (form removed).

effective in denoising the piecewise smooth image with line singularities. This is because the linear singularities are represented by a few significant coefficients in the CFRIT domain, whereas random noisy singularities are unlikely to produce the similar amplitude coefficients. The DT-CWT is relatively superior to the DWT in reducing the artefacts due to its shift invariance and good directional selectively. From the view of the hybrid approach, the FRIT-C just combines the multiresolution analysis of DT-CWT with the anisotropy of the Radon transform. Furthermore, the complex ridgelet transform can provide better phase information that is useful for future research. 5.3. Extraction of linear scratches from surface topography Fig. 3 shows a honed surface from an engine cylinder. As is well know, the most important features that influence the performance of cylinders are the deep scratches, the distribution and amplitude will considerably influence the flow of gas or air in the pressure balance of an engine. It can be seen that the DWT still exhibits numerical embedded blemishes (i.e., pits/peaks) in the extracted honed surface. Setting higher thresholds to remove these would cause even more of the intrinsic linear scratches to be destroyed or missed. In addition, the non-smooth aliasing along the scratches is clearly visible. In the FRIT result, although the pits/peaks have been removed efficiently, the edges of the linear scratches are not very smooth with aliasing due to the lack of the shift invariance property. In the CFRIT result, not only are the peaks/pits in the honed surface removed effectively, but also the shapes of the extracted scratches are well retained. There is also no affine aliasing, and the edges of the deep valleys are preserved perfectly.

6. Conclusion CFRIT was proposed by taking DT-CWT on the projections of the FRAT. It brings together the ideas of complex wavelet pyramids and the geometric features of ridgelets to solve problems that exist in previous wavelet-based methods. By mapping a line singularity into a point singularity through the FRAT, and then using the DT-CWT on each projection in the Radon transform domain, the CFRIT can efficiently represent functions with shift-invariant property. Numerical experiments and the practical feature extraction have verified the ability of the linear feature extraction of this method. Overall, it is clear that the CFRIT has great potential to extract linear scratches from surface topography. Acknowledgement The authors would like to thank the Engineering and Physical Sciences Research Council of UK for support funding to carry out this research under its programme: GR/S13316/01. References [1] X. Chen, J. Raja, S. Simanapalli, Multi-scale analysis of engineering surfaces, Int. J. Mach. Tools Manufact. 35 (1995) 231–238. [2] S.H. Lee, H. Zahouani, R. Caterini, T.G. Mathia, Multi-scale analysis of engineering surfaces, Int. J. Mach. Tools Manufact. 38 (1998) 581–589. [3] X. Jiang, L. Blunt, K.J. Stout, Development of a lifting wavelet representation for surface characterization, Proc. Roy. Soc. Lond. A 456 (2000) 1–31. [4] X. Jiang, L. Blunt, Morphological assessment of in vivo wear of orthopedic implants using multiscale wavelets, Wear 250 (2001) 217–221.

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