A Shearlet Approach to Edge Analysis and Detection Sheng Yi, Demetrio Labate, Glenn R

IEEE TRANS. IMAGE PROCESSING 1 A Shearlet Approach to Edge Analysis and Detection Sheng Yi, Demetrio Labate, Glenn R. Easley, and Hamid Krim Abstract—It is well known that the wavelet transform tracting and classifying basic edge features such as cor- provides a very effective framework for analysis of multi- ners and junctions. scale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional Our shearlet approach has similarities with a num- transform with a greater ability to localize distributed ber of other methods in applied mathematics and en- discontinuities such as edges. Indeed, unlike traditional gineering to overcome the limitations of traditional wavelets, shearlets are theoretically optimal in represent- ing images with edges and, in particular, have the ability wavelets. These methods include contourlets [6], [7] to fully capture directional and other geometrical fea- complex wavelets [8], ridgelets [9] and curvelets [10]. tures. Numerical examples demonstrate that the shear- In contrast to all these methods, the shearlet frame- let approach is highly effective at detecting both the loca- work provides a unique combination of mathematical tion and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. rigidness and computational efficiency when addressing Furthermore, the shearlet approach is useful to design edges, optimal efficiency in dealing with edges, and com- simple and effective algorithms for the detection of cor- putational efficiency. In addition, its continuous formu- ners and junctions. lation is particularly well-suited for designing an imple- Index Terms—Curvelets, edge detection, feature ex- traction, shearlets, singularities, wavelets. mentation, presented in this work, for the purpose of edge analysis and detection. I. Introduction A. Edge detection using wavelets. Edges are prominent features in images and their In the classic Canny edge detection algorithm [11], an analysis and detection are an essential goal in computer image u is smoothed by a convolution with a Gaussian vision and image processing. Indeed, identifying and filter: localizing edges are a low level task in a variety of ap- ua = u Ga, (I.1) plications such as 3D reconstruction, shape recognition, ∗ 1 1 image compression, enhancement and restoration. where Ga(x)= a− G(a− x), for a> 0, and G(x) is the In this paper, we apply a novel directional multiscale Gaussian function. Edges are then recognized as the mathematical framework which is especially adapted to local maxima of the magnitude of the gradient of ua. identification and analysis of distributed discontinuities The adjustable scaling factor a determines the amount such as edges occurring in natural images. Multiscale of smoothing: as a increases, the detector’s sensitivity methods based on wavelets, have been successfully ap- to noise decreases; however, as a increases, the local- plied to the analysis and detection of edges [1], [2], ization error in the detection of edges also increases. [3], [4], [5]. Despite their success, wavelets are how- As a result, the performance of the algorithm heavily ever known to have a limited capability in dealing with depends on the scaling factor a [12], [13]. directional information. By contrast, the shearlet ap- It was observed by Mallat et al. [1], [2] that, at a proach, which we propose here, is particularly designed single scale, the Canny edge detector is equivalent to to deal with directional and anisotropic features typi- the detection of the local maxima of the wavelet trans- cally present in natural images, and has the ability to form of u, for some particular choices of the analyzing wavelet. In fact, the function ψ = G is a wavelet accurately and efficiently capture the geometric infor- ∇ mation of edges. As a result, the shearlet framework known as the first derivative Gaussian wavelet. Thus, each image u L2(R2) satisfies: provides highly competitive algorithms, for detecting ∈ both the location and orientation of edges, and for ex- u(x)= Wψu(a,y) ψa(x y) dy, D. Labate, S. Yi and H. Krim are with North Carolina State − University, Raleigh, NC 27695, USA ([email protected]; Z [email protected]; [email protected]) 1 1 where ψa(x) = a− ψ(a− x), and Wψu(a, x) is the S.Yi and H.Krim were in part funded by AFOSR, AF9550-07- 1-0104.DL acknowledges partial support from NSF DMS 0604561 wavelet transform of u, defined by and DMS (Career) 0746778 G.R. Easley is with System Planning Corporation 1000 Wilson W u(a, x)= u(y) ψ (x y) dy = u ψ (x). Boulevard, Arlington, VA 22209, USA ([email protected]) ψ a − ∗ a Z 2 IEEE TRANS. IMAGE PROCESSING The significance of this representation is that the This transform, introduced by the authors and their col- wavelet transform provides a space-scale decomposition laborators in [19], [20], is a genuinely multidimensional of the image u, where u L2(R2) is mapped into ∈ version of the traditional wavelet transform, and is es- the coefficients Wψu(a,y) which depend on the loca- pecially designed to address anisotropic and directional tion y R2 and the scaling variable a > 0. Another useful observation∈ is that the wavelet transform of u is information at various scales. Indeed, the traditional proportional to the gradient of the smoothed image ua: wavelet approach, which is based on isotropic dilations, has a very limited capability to account for the geome- ∇ua(x)= u ∗∇Ga(x) = u ∗ ψa(x) = Wψu(a, x). (I.2) try of multidimensional functions. In contrast, the analyzing functions associated to the shearlet transform This shows that the maxima of the magnitude of the are highly anisotropic, and, unlike traditional wavelets, gradient of the smoothed image ua correspond exactly are defined at various scales, locations and orientations. to the maxima of the magnitude of the wavelet trans- As a consequence, this transform provides an optimally form Wψu(a, x); this provides a natural mathematical efficient representation of images with edges [21]. framework for the multiscale analysis of edges [1], [2]. The shearlet transform has similarities to the curvelet In particular, utilizing multiscale wavelet representa- transform, first introduced by Candès and Donoho [10], tion avoids the problem of finding the appropriate scale [22]. Shearlets and curvelets, in fact, are the only two a which produces an improved detection of the clas- systems which were mathematically known to provide sic Canny algorithm. Furthermore, there are very effi- optimally sparse representations of images with edges. cient numerical implementations of the wavelet trans- The spatial-frequency tilings of the two representations form [14]. are completely different, yet the implementations of the The difficulty of edge detection is particularly promi- curvelet transform corresponds to essentially the same nent in the presence of noise, and when several edges are tiling as that of the shearlet transform. In spite of this close together or cross each other, e.g., 2–dimensional similarity, it is not clear how the curvelet transform im- projections of 3–dimensional objects [15]. In such cases, plementations could be modified to do the applications the following limitations of the wavelet approach (and described in this paper. Yet an application of low-level other traditional edge detectors) become evident: vision using curvelets has been suggested in [23]. Difficulty in distinguishing close edges. The • Both systems are related to contourlets [6], [7] and isotropic Gaussian filtering causes edges running steerable filters [24], [25]. Contourlets, however, provide close together to be blurred into a single curve. a purely discrete approach which presents difficulties in Poor angular accuracy. In the presence of sharp • rigorously addressing the edge detection problem. We changes in curvature or crossing curves, the refer to [19] for more details about the comparison of isotropic Gaussian filtering leads to an inaccurate shearlets and other orientable multiscale transforms. detection of edge orientation. This affects the de- In this paper, we combine the shearlet framework tection of corners and junctions. with several well established ideas from the image pro- To address these difficulties one has to account for the cessing literature to obtain improved and computation- anisotropic nature of edge lines and curves. For exam- ally efficient algorithms for edge analysis and detection. ple, in [16], [17], [18] it is proposed to replace the scal- Our approach may be viewed as a truly multidimen- able collection of isotropic Gaussian filters Ga(x1, x2), sional refinement of the approach of Mallat et al., where a > 0 in (I.1) with a family of steerable and scalable the isotropic wavelet transform Wψu(a, x) is replaced by anisotropic Gaussian filters an anisotropic directional multiscale transform. Specif- 1/2 1/2 1 1 ically, for an image u, the shearlet transform is a map- Ga1,a2,θ(x1, x2)= a1− a2− Rθ G(a1− x1,a2− x2), ping u ψu(a,s,x), where a1,a2 > 0 and Rθ is the rotation matrix by θ. → SH Unfortunately, the design and implementation of such depending on the scale a > 0, the orientation s and filters is computationally demanding. In addition, the the location x. This provides a directional scale-space justification for this approach is essentially intuitive, decomposition of u and, as shown below, a theoretical and there is no proper theoretical model to indicate justifiable framework for the identification and analy- how to “optimize” such family of filters to best capture sis of the edges of u. The shearlet transform can be edges. expressed as B. The Shearlet Approach. u(a,s,x)= u(y) ψ (x y) dy = u ψ (x). SHψ as − ∗ as The approach proposed in this paper is based on a Z new multiscale transform called the shearlet transform. where the analyzing elements ψas are well localized A SHEARLET APPROACH TO EDGE ANALYSIS AND DETECTION 3 waveforms at various scales and orientations.

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