- Open Access
Apply hyperanalytic shearlet transform to geometric separation
© Liu et al.; licensee Springer. 2014
- Received: 28 February 2014
- Accepted: 21 April 2014
- Published: 9 May 2014
This paper first proposes a novel image separation method based on the hyperanalytic shearlet. By combining the advantages of both the hyperanalytic wavelet transform and the shear operation, hyperanalytic shearlet is easy to implement and also has a low redundancy. By using such transform and the orthonormal wavelet, a new geometric separation dictionary is obtained which can sparsely represent points and curviline singularities, respectively. In order to get the different components of image faster and more accurate, a fast alternating direction method (FADM) is used to train the dictionary. Our algorithm can greatly improve the computational efficiency without causing damage to the accuracy of image separation. Furthermore, a proper measure to evaluate the separation performance called sep-degree is defined. The experimental results have demonstrated the proposed method’s effectiveness and superiority.
- Hyperanalytic shearlet
- Geometric separation
- Sparse approximation
- Separation degree
Astronomical images of the galaxy can be modeled as a superposition of pointlike and curvelike structures. In the further analysis, astronomers typically face the problem of extracting the stars from filaments which mostly are trajectory of the particle and hence separating pointlike from curvelike structures. Thus, this area is greatly attracting scholars’ attention in order to find efficient methodologies for accurately conquering this task.
Although this problem seems unsolvable - the problem is underdetermined, as there are two unknown (the images should be extracted) and only one known data (the given image) - experimental results using morphological component analysis (MAC)[1, 2] suggest that such a problem is possibly solvable when we get the prior information about the type of decomposed features and enough morphological difference between those features. For the separation of pointlike and curvelike features, perfect results have been achieved by employing a dictionary consisting of wavelet and curvelet with combination with l1 minimization techniques. We know that pointlike structures can be optimally sparse representation in wavelet dictionary, and curvelike structures also can be optimally sparse representation in curvelet dictionary. Thus, the pointlike structures can be expressed by wavelet and the curvelike structures can be expressed by curvelet with applying l1 minimization to the expansion coefficients, and then the image can be separated automatically.
The current papers[3, 4] give a newly combined dictionary of orthonormal wavelet and shearlet for separating pointlike and curvelike features. And numerical results give evidence that the shearlet-based decomposition algorithms have a superior behavior than curvelet-based algorithms in. In[3, 4], the authors introduce a nonsubsampled shearlet transform to design a separation dictionary which greatly increase the redundancy. And they choose an old method called block coordinate relaxation (BCR) to solve the l1 problem which is not only time-consuming but also unable to get an accurate result. Thus, in this paper, a novel approach to the separation of pointlike and curvelike features based on hyperanalytic shearlet is proposed.
Shearlet transform is a new multi-scale geometric analysis algorithm which inherits the advantages of the contourlet and curvelet transforms. It is also an optimal approximation presentation[5, 6] for singular curve or surface that contains C2 high-dimensional signals. A shift invariant shearlet is achieved by nonsubsampled Laplace pyramid in. This construction not only greatly increases the redundancy of the transform but also slows down the calculation sharply. From, we can know that hyperanalytic wavelet (HWT) just has limited redundancy to achieve multi-scale decomposition without shift sensitivity, which is similar to dual tree complex wavelet (DTCWT), but differently, this method is more easily achievable than dual tree complex wavelet. And the construction of hyperanalytic wavelet also greatly accelerates the computation. So we propose a new fast discrete shearlet called hyperanalytic shearlet transform. It is implemented by using hyperanalytic wavelet[3, 4] to achieve multi-scale decomposition, and then shear filter bank is applied to the high-pass coefficients. This new algorithm holds the advantages of simpler structure and higher sparsity. The hyperanalytic shearlet addresses the problem of the aliasing phenomenon and greatly reduces the redundancy and computing time compared to nonsubsampled shearlet transform. In our paper, we apply it to separate an image into its morphologically different contents. In order to get the different components of image faster and more accurate, we also use fast alternating direction method (FADM) instead of BCR to train dictionary. Our algorithm can greatly improve the computational efficiency without damage to the accuracy of image separation. To compare the performance difference between this new algorithm and the method in, we define a new measure of the separation called sep-degree. The experiment results will demonstrate that our scheme has a better separation effect.
This paper is organized as follows. Section 2 introduces the construction of hyperanalytic shearlet transform. Section 3 describes the mathematical theory of geometric separation of pointlike and curvelike features and applies a novel algorithm to separate an image into morphologically different contents. Section 4 illustrates the numerical results. Finally, Section 5 presents some conclusions.
where,,. Then, you know.
Let, that means ∀(ξ1,ξ2) ∈ D0, the function group forms a tiling of D0 as shown in Figure1, notes that D0 is illustrated in solid line.
The discrete shearlet in uses nonsubsampled Laplace pyramid to achieve multi-scale decomposition, which greatly increases the redundancy of the transform. Also, it greatly increases the computing time.
Due to the high redundancy and massive calculation of nonsampled wavelet, DTCWT, one that is constructed through a pair of wavelet trees, is proposed in. Although DTCWT is an invertible quasi shift invariant and its 1-D case a redundancy of 2, the design of these quadrature wavelet pairs is so complicated that it can be done only through approximations. It means that the DTCWT requires special mother wavelet function. To overcome this restraint, in, Firoiu has proposed a new shift invariant called HWT using Hilbert transform and a two-stage mapping-based complex wavelet transform (MBCWT) in soft space. And she also gives the proof that HWT is equivalent to DTCWT. That means that HWT’s redundancy ratio is 2, the same as that of DTCWT. Moreover, HWT can be realized through classical mother wavelet function like those conceived by Daubechies. Using this method, we can get a higher degree of shift invariance and a better directional selectivity.
So we use HWT to achieve multi-scale decomposition and apply shear filter bank to the HWT coefficients. For short, we name this new algorithm hyperanalytic shearlet transform, which holds the advantages of a simpler structure and a higher sparseness. What is more, it has greatly improved the redundancy compared with shearlet transform and greatly reduced the computing time. We first introduce HWT and then construct hyperanalytic shearlet.
where ψ is wavelet function of hyperanalytic shearlet, W is a window function localized on a pair of trapezoid. And V(2-2jξ) is the Fourier coefficients of the multi-scale analysis. The 2-D discrete Fourier transform (DFT) of image is. Here, we adopt the convention that brackets [,] denote arrays of indices, parentheses (,) denote function evaluations, and N × N denotes the image size.
Here, J is the final scale.
The hyperanalytic shearlet eliminates the aliasing phenomenon and has better direction selectivity and lower redundancy than the nonsubsampled shearlet. Moreover, hyperanalytic shearlet realizes the multi-scale decomposition by using addition, subtraction, and FFT. All the analysis shows that the calculation speed of hyperanalytic shearlet is quite fast; at the mean time, it can have a good visual effect as nonsubsampled shearlet does. Then, we apply it to separate an image into its morphologically different contents.
It is an important technique to separate an image into its morphologically different contents using MCA from prior information about the type of features to be decomposed. Recently, Donoho and Kutyniok presented a mathematical framework in for image separation and constructed geometric separation mathematical theory of separating pointlike from curvelike features. And Kutyniok and Lim gave a new wavelet-shearlet dictionary based on their analysis in. After analyzing the advantages of hyperanalytic shearlet in Section 2, it is natural to construct a new combined dictionary of wavelet and hyperanalytic shearlet. In order to improve the computational efficiency, we first apply the new dictionary to solve the problem about the separation of pointlike and curvelike features to an l1 optimization model. And then, we use fast alternating direction method in to obtain the optimal solution. Compared with separation algorithms using wavelet and shearlet in various ways, such approach is proved superior by numerical results especially when it comes to the speed. In the following, we briefly review this theoretical approach to the geometric separation problem and present our method.
The geometric separation problem now is turned into recovering P and C from the observed signal f. Since curvilinear singularities can be sparsely represented by shearlet, they can also be sparsely represented by hyperanalytic shearlet, while point singularities can be optimally sparsely represented by wavelet. So we choose the orthonormal separable Meyer wavelet and hyperanalytic shearlet to construct a dictionary to separate pointlike from curvelike structures.
Let (F j ) j denote a family of filters including wavelet and hyperanalytic shearlet filters. Then, the function f can be decomposed into pieces f j with different scales j based on different sub-band filters. So we can get the equation f j = F j ∗ f, and the original function can be reconstructed by using.
In (17), the noise cannot be represented sparsely by either wavelet or hyperanalytic hearlet, and then it can be related with the residual f j - W j - S j .
where and f j = F j ∗ f.
- (a)First, fix S; then, we can get the solution of W :(20)
- (b)Then, fix W; then, we can get the solution of S :(21)
where k ≥ 1,,.
In order to verify the reliability and validity of the proposed algorithm, we introduce a measure named sep-degree to evaluate the separation performance of our transform. f denotes the noise image, f c denotes the separated image containing curvelike features, and f p denotes the separated image containing pointlike features. Let μ(f) denote the mathematical expectation of f. Then, we can define the sep-degree as follows:
where (f) G = Gradient(f), which denotes the gradient map of the image f, and f S = f p + f c ,(f S ) G = Gradient(f S ). a b s(·) denotes the absolute value of the function, and ϖ denotes the similarity between (f) G and (f S ) G .
The definition of sep-degree shows that the separation method’s performance is better when ς is larger. It is because that μ(f - f S ) should be smaller enough for a better separating method, which means we get a most approximate image of f. From Jensen’s inequality, we know ϖ ≤ 1 in (23), and the closer ϖ to 1 is, the more similar (f) G and (f S ) G are, which means that the extracted curvelike component is more perfect. For a better separation method, μ(f - f S ) should be smaller and ϖ should be larger, so sep-degree ζ should be larger based on (24). In the following, ζ is used to distinguish the numerical experiments.
In this section, we compare our scheme of wavelet and hyperanalytic shearlet dictionary (W-HSD) with FADM against the separation algorithm based on wavelet and shearlet dictionary (W-SD) with BCR in. And the scale of all is four; for hyperanalytic shearlet and nonsubsampled shearlet, the direction vector is [2 3 3 4]. All routines were run using the Matlab R2009a which is based on an Intel 2.00GHz CPU.
The performance of different methods
The dictionary using for image separating
The computing times (s)
In the experimental results, Figure4a is an artificial image consisting of a composition of pointlike and curvelike structures on a smooth background, and Figure4b is the same image added by some additive white Gaussian noise with variance 20. Figure4c shows the curvelike component of image separating by W-SD, and Figure4d shows the pointlike component of image separating by W-SD. Correspondingly, Figure4e,f shows the curvelike and pointlike components of image separating by W-HSD. Compared with Figure4c separated via W-SD method, Figure4e shows that W-HSD does not keep all circle-like contents, which makes the circle of our scheme a little pale. Though the visual effect of Figure4e is just a little worse than Figure4c, the visual effect of Figure4f is much better than Figure4d. From Figure4d, we know that the separating method W-SD can bring some visible artifacts, but our new algorithm will solve this problem. Finally, the comparison of the computing time shows that the separation can be much faster performed by our scheme because hyperanalytic shearlet has a low redundancy which greatly reduces the computational complexity. Table1 shows that our algorithm can reduce the computing time to one fifth of the time of separating image by W-SD. Table1 also shows that our algorithm can get a higher sep-degree than separating image by W-SD. It means that our algorithm can get a better performance in separating an artificial image consisting of a composition of pointlike and curvelike structures.
The performance of different methods apply to neuron image
The dictionary using for image separating
The computing times (s)
Table2 shows that our algorithm gets a better performance in separating a neuron image of a composition of pointlike and curvelike structures.
This paper proposes a novel approach of separation of pointlike and curvelike features exploiting a new combined dictionary of wavelet and hyperanalytic shearlet and defines a new objective measure called sep-degree to evaluate the separation performance. The experimental results demonstrate that the proposed method is more applicable to geometric separation. It has a better visual effect as wavelet and shearlet dictionary, and its computing time decreases sharply.
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