Linear feature extraction based on complex ridgelet transform
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 multi-scalar 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 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 Candès 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], [6], [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], [9], [10], [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.
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.
Section snippets
Complex ridgelet transform
A bivariate complex ridgelet in R2 space can be defined as:Here, a > 0 is a scale parameter, θ an orientation parameter, and b is a location scalar parameter. This function is constant along lines: x1cos θ + x2 sin θ = const, while its transverse is a complex wavelet , 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
Digital complex ridgelet transform
From Eqs. (4), (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
Shift invariance analysis of the CFRIT
If we rewrite the Radon transform (5) into a more general form:then the Radon transform of the shift function becomes:Combining Eq. (10), the finite Radon transform of the shift function becomes:Consider an “horizontal” shift f′ = f(k + Δk, l), then:If let Tτ be the shift
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
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
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.
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2018, CIRP AnnalsCitation Excerpt :A complex finite ridgelet transform (CFRIT), which provides approximate shift invariance and analysis of line singularities, has been proposed, taking the DTCWT on the projections of the finite Radon transform (FRAT). The numerical experiments show the remarkable potential of the methodology to analyze engineering and bioengineering surfaces with linear scratches in comparison to wavelet-based methods developed in previous work [90]. As pointed out by Jiang et al. [89], engineering surfaces have undergone a significant development, and more and more complicated freeform surfaces are being produced.
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2014, OptikCitation Excerpt :To overcome the limitation of the wavelets, the authors have reported the ridgelet representation of multidimensional signals which is used to map the line singularities to the point singularities using radon transform and after that the wavelet transform is used to provide better performance for characterizing the point singularities in radon transform [16]. In literature, ridgelet transform (RT) has been applied to various image processing applications, such as image denoising [16–18], feature extraction [19] and texture classification [20]. Ridgelet transform provides better results and breaks the limitation of the wavelets.