Topographic mapping in planetary environments relies on accurate 3D scan registration methods. However, most global registration algorithms relying on features such as FPFH and Harris-3D show poor alignment accuracy in these settings due to the poor structure of the Mars-like terrain and variable resolution, occluded, sparse range data that is hard to register without some a-priori knowledge of the environment. In this paper, we propose an alternative approach to 3D scan registration using the curvelet transform that performs multi-resolution geometric analysis to obtain a set of coefficients indexed by scale (coarsest to finest), angle and spatial position. Features are detected in the curvelet domain to take advantage of the directional selectivity of the transform. A descriptor is computed for each feature by calculating the 3D spatial histogram of the image gradients, and nearest neighbor based matching is used to calculate the feature correspondences. Correspondence rejection using Random Sample Consensus identifies inliers, and a locally optimal Singular Value Decomposition-based estimation of the rigid-body transformation aligns the laser scans given the re-projected correspondences in the metric space. Experimental results on a publicly available data-set of planetary analogue indoor facility, as well as simulated and real-world scans from Neptec Design Group's IVIGMS 3D laser rangefinder at the outdoor CSA Mars yard demonstrates improved performance over existing methods in the challenging sparse Mars-like terrain.
Unlike the shape-fixed rectangles in the frequency domain of conventional FFT, multi-scale transforms such as the discrete wavelet transform (DWT) use dilated shape varying rectangles to find directional elements such as edges and ridge features in the laser scans. However, many wavelet coefficients are needed to account for singularities along lines or curves. To overcome this problem, other directional wavelets such as wedgelets @cite_44 , beamlets @cite_24 , contourlets @cite_14 , surfacelets @cite_45 , etc. have been proposed, however the detected features are less prominent especially at the edges. In order to account for curve-singularities, the curvelet transform has previously been proposed which generates a sparse representation of the points within the scan and employs angled polar wedges in the frequency domain to find directional features. Previously, applied the curvelet transform for the problem of image fusion @cite_20 where only an approximate sub-band of the coefficients was used for registration. The algorithm relied on the assumption that the curvelet coefficients were normally distributed.
@cite_14 @cite_44 @cite_24 @cite_45 @cite_20
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This paper introduces new tight frames of curvelets to address the problem of finding optimally sparse representations of objects with discontinuities along piecewise C 2 edges. Conceptually, the curvelet transform is a multiscale pyramid with many directions and positions at each length scale, and needle-shaped elements at fine scales. These elements have many useful geometric multiscale features that set them apart from classical multiscale representations such as wavelets. For instance, curvelets obey a parabolic scaling relation which says that at scale 2 -j , each element has an envelope that is aligned along a ridge of length 2 -j 2 and width 2 -j . We prove that curvelets provide an essentially optimal representation of typical objects f that are C 2 except for discontinuities along piecewise C 2 curves. Such representations are nearly as sparse as if f were not singular and turn out to be far more sparse than the wavelet decomposition of the object. For instance, the n-term partial reconstruction f C n obtained by selecting the n largest terms in the curvelet series obeys ∥f - f C n ∥ 2 L2 ≤ C . n -2 . (log n) 3 , n → ∞. This rate of convergence holds uniformly over a class of functions that are C 2 except for discontinuities along piecewise C 2 curves and is essentially optimal. In comparison, the squared error of n-term wavelet approximations only converges as n -1 as n → ∞, which is considerably worse than the optimal behavior. We discuss a Continuous Curvelet Transform (CCT), a transform f → Γf (a, b, θ) of functions f(x1, x2) on R^2, into a transform domain with continuous scale a > 0, location b ∈ R^2, and orientation θ ∈ [0, 2π). The transform is defined by Γf (a, b, θ) = f, γabθ where the inner products project f onto analyzing elements called curvelets γ_(abθ) which are smooth and of rapid decay away from an a by √a rectangle with minor axis pointing in direction θ. We call them curvelets because this anisotropic behavior allows them to ‘track’ the behavior of singularities along curves. They are continuum scale space orientation analogs of the discrete frame of curvelets discussed in Candes and Donoho (2002). We use the CCT to analyze several objects having singularities at points, along lines, and along smooth curves. These examples show that for fixed (x0, θ0), Γf (a, x0, θ0) decays rapidly as a → 0 if f is smooth near x0, or if the singularity of f at x0 is oriented in a different direction than θ_0. Generalizing these examples, we state general theorems showing that decay properties of Γf (a, x0, θ0) for fixed (x0, θ0), as a → 0 can precisely identify the wavefront set and the H^m- wavefront set of a distribution. In effect, the wavefront set of a distribution is the closure of the set of (x0, θ0) near which Γf (a, x, θ) is not of rapid decay as a → 0; the H^m-wavefront set is the closure of those points (x0, θ0) where the ‘directional parabolic square function’ S^m(x, θ) = ( ʃ|Γf (a, x, θ)|^2 ^(da) _a^3+^(2m))^(1 2) is not locally integrable. The CCT is closely related to a continuous transform used by Hart Smith in his study of Fourier Integral Operators. Smith’s transform is based on strict affine parabolic scaling of a single mother wavelet, while for the transform we discuss, the generating wavelet changes (slightly) scale by scale. The CCT can also be compared to the FBI (Fourier-Bros-Iagolnitzer) and Wave Packets (Cordoba-Fefferman) transforms. We describe their similarities and differences in resolving the wavefront set. Registration is a prerequisite for fusion of geometrically distorted images. Traditionally, intensity-based image registration methods are preferred to feature-based ones due to higher accuracy of the former than that of the latter. To reduce computational load, image registration is often carried out using the approximate-level coefficients of a wavelet-like transform. Directional selectivity of the transform and the objective function used for the coefficients play vital roles in the alignment process of images. This paper introduces an image registration algorithm that uses the approximate-level coefficients of the curvelet transform, directional selectivity of which is better than many wavelet-like transforms. A conditional entropy-based objective function is developed for registration using a suitable probabilistic model of the curvelet coefficients of images. Suitability of the probability distribution of the coefficients is validated using a standard method to assess goodness of fit. To align the distorted images, the affine transformation that possesses parameters related to the translation, rotation, scaling, and shearing is used. Extensive experimentations are carried out to test the performance of the proposed registration method considering that the images are synthetically or naturally distorted. Experimental results show that performance of the proposed registration method is superior to existing methods in terms of commonly used performance metrics. The limitations of commonly used separable extensions of one-dimensional transforms, such as the Fourier and wavelet transforms, in capturing the geometry of image edges are well known. In this paper, we pursue a "true" two-dimensional transform that can capture the intrinsic geometrical structure that is key in visual information. The main challenge in exploring geometry in images comes from the discrete nature of the data. Thus, unlike other approaches, such as curvelets, that first develop a transform in the continuous domain and then discretize for sampled data, our approach starts with a discrete-domain construction and then studies its convergence to an expansion in the continuous domain. Specifically, we construct a discrete-domain multiresolution and multidirection expansion using nonseparable filter banks, in much the same way that wavelets were derived from filter banks. This construction results in a flexible multiresolution, local, and directional image expansion using contour segments, and, thus, it is named the contourlet transform. The discrete contourlet transform has a fast iterated filter bank algorithm that requires an order N operations for N-pixel images. Furthermore, we establish a precise link between the developed filter bank and the associated continuous-domain contourlet expansion via a directional multiresolution analysis framework. We show that with parabolic scaling and sufficient directional vanishing moments, contourlets achieve the optimal approximation rate for piecewise smooth functions with discontinuities along twice continuously differentiable curves. Finally, we show some numerical experiments demonstrating the potential of contourlets in several image processing applications. We present a new method for contrast enhancement based on the curvelet transform. The curvelet transform represents edges better than wavelets, and is therefore well-suited for multiscale edge enhancement. We compare this approach with enhancement based on the wavelet transform, and the multiscale retinex. In a range of examples, we use edge detection and segmentation, among other processing applications, to provide for quantitative comparative evaluation. Our findings are that curvelet based enhancement out-performs other enhancement methods on noisy images, but on noiseless or near noiseless images curvelet based enhancement is not remarkably better than wavelet based enhancement. In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are—unlike more traditional transforms like wavelets—able to efficiently handle data with features along edges. The main result in Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009) confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions ψ with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution f with respect to the shearlet ψ can resolve the wavefront set of f. We demonstrate that the same result can be verified under much weaker assumptions on ψ, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for ( L^2( R ^2) ) from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure. Abstract The analysis and detection of edges is a central problem in applied mathematics and image processing. A number of results in recent years have shown that directional multiscale methods such as continuous curvelet and shearlet transforms offer a powerful theoretical framework to capture the geometry of edge singularities, going far beyond the capabilities of the conventional wavelet transform. The continuous shearlet transform, in particular, provides a precise geometric characterization of edges in piecewise constant functions in R 2 and R 3 , including corner points. However, a question has been raised frequently: What happens if the function is piecewise smooth and not just piecewise constant? Clearly, a piecewise smooth function is a much more realistic model of images with edges. In this paper, we extend the characterization results previously known and show that, also in the case of piecewise smooth functions, the continuous shearlet transform can detect the location and orientation of edge points, including corner points, through its asymptotic decay at fine scales. The new proof introduces innovative technical constructions to deal with the more challenging problem. The new results set the theoretical groundwork for the application of the shearlet framework to a wider class of problems from image processing. It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SH ψ f(a,s,t) = (fψ ast ), where the analyzing elements ψ ast are dilated and translated copies of a single generating function ψ. The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements ψ ast form a system of smooth functions at continuous scales a > 0, locations t ∈ R 2 , and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. This paper introduces an expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain. Regularization is achieved by promoting a reconstruction with low-complexity, expressed in the wavelet coefficients, taking advantage of the well known sparsity of wavelet representations. Previous works have investigated wavelet-based restoration but, except for certain special cases, the resulting criteria are solved approximately or require demanding optimization methods. The EM algorithm herein proposed combines the efficient image representation offered by the discrete wavelet transform (DWT) with the diagonalization of the convolution operator obtained in the Fourier domain. Thus, it is a general-purpose approach to wavelet-based image restoration with computational complexity comparable to that of standard wavelet denoising schemes or of frequency domain deconvolution methods. The algorithm alternates between an E-step based on the fast Fourier transform (FFT) and a DWT-based M-step, resulting in an efficient iterative process requiring O(NlogN) operations per iteration. The convergence behavior of the algorithm is investigated, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration. Moreover, our new approach performs competitively with, in some cases better than, the best existing methods in benchmark tests. The present thesis deals with amplitude and sign decompositions of images via complex wavelets. We develop the monogenic curvelet transform for the directional decomposition of images into amplitude and sign. We use the amplitude information for the development and the analysis of an algorithm for the separation of edges in images, and the sign information for a new edge detection method. Eventually, we draw a connection between amplitude and sign representations and the perception of optical illusions. Recent advances have seen a surge of deep learning approaches for image super-resolution. Invariably, a network, e.g. a deep convolutional neural network (CNN) or auto-encoder is trained to learn the relationship between low and high-resolution image patches. Recognizing that a wavelet transform provides a "coarse" as well as "detail" separation of image content, we design a deep CNN to predict the "missing details" of wavelet coefficients of the low-resolution images to obtain the Super-Resolution (SR) results, which we name Deep Wavelet Super-Resolution (DWSR). Out network is trained in the wavelet domain with four input and output channels respectively. The input comprises of 4 sub-bands of the low-resolution wavelet coefficients and outputs are residuals (missing details) of 4 sub-bands of high-resolution wavelet coefficients. Wavelet coefficients and wavelet residuals are used as input and outputs of our network to further enhance the sparsity of activation maps. A key benefit of such a design is that it greatly reduces the training burden of learning the network that reconstructs low frequency details. The output prediction is added to the input to form the final SR wavelet coefficients. Then the inverse 2d discrete wavelet transformation is applied to transform the predicted details and generate the SR results. We show that DWSR is computationally simpler and yet produces competitive and often better results than state-of-the-art alternatives. Multimodal medical image fusion is an important task for the retrieval of complementary information from medical images. Shift sensitivity, lack of phase information and poor directionality of real valued wavelet transforms motivated us to use complex wavelet transform for fusion. We have used Daubechies complex wavelet transform (DCxWT) for image fusion which is approximately shift invariant and provides phase information. In the present work, we have proposed a new multimodal medical image fusion using DCxWT at multiple levels which is based on multiresolution principle. The proposed method fuses the complex wavelet coefficients of source images using maximum selection rule. Experiments have been performed over three different sets of multimodal medical images. The proposed fusion method is visually and quantitatively compared with wavelet domain (Dual tree complex wavelet transform (DTCWT), Lifting wavelet transform (LWT), Multiwavelet transform (MWT), Stationary wavelet transform (SWT)) and spatial domain (Principal component analysis (PCA), linear and sharp) image fusion methods. The proposed method is further compared with Contourlet transform (CT) and Nonsubsampled contourlet transform (NSCT) based image fusion methods. For comparison of the proposed method, we have used five fusion metrics, namely entropy, edge strength, standard deviation, fusion factor and fusion symmetry. Comparison results prove that performance of the proposed fusion method is better than any of the above existing fusion methods. Robustness of the proposed method is tested against Gaussian, salt & pepper and speckle noise and the plots of fusion metrics for different noise cases established the superiority of the proposed fusion method. We introduce a complete parameterization of the family of two-dimensional steerable wavelets that are polar-separable in the Fourier domain under the constraint of self-reversibility. These wavelets are constructed by multiorder generalized Riesz transformation of a primary isotropic bandpass pyramid. The backbone of the transform (pyramid) is characterized by a radial frequency profile function @math , while the directional wavelet components at each scale are encoded by an @math shaping matrix @math , where @math is the number of wavelet channels and @math the order of the Riesz transform. We provide general conditions on @math and @math for the underlying wavelet system to form a tight frame of @math (with a redundancy factor @math ). The proposed framework ensures that the wavelets are steerable and provides new degrees of freedom (shaping matrix @math ) that can be exploited for designing specific wavelet systems. It encompasses many known transforms as part... This paper primarily researches on the feasibility of embedding watermark to relational databases in the Discrete Wavelet Transform (DWT) domain. Watermark embedding is known to be difficult for relational databases. This paper focuses on the analysis of the wavelet high frequency coefficients of corresponding data and gives the definition of the intensive factor. By employing the linear correlation detecting method, this paper proposes the watermarking algorithm, which can embed the watermark into relational database successfully in DWT domain. The watermark can be distributed to different parts of the relational database. Experiments show that the embedded digital watermarks in the proposed algorithm are invisible and some degree of the robustness of the proposed algorithm against the commonly attacks that are used in databases processing. Reduced-reference (RR) image quality measures aim to predict the visual quality of distorted images with only partial information about the reference images. In this paper, we propose an RR quality assessment method based on a natural image statistic model in the wavelet transform domain. In particular, we observe that the marginal distribution of wavelet coefficients changes in different ways for different types of image distortions. To quantify such changes, we estimate the Kullback-Leibler distance between the marginal distributions of wavelet coefficients of the reference and distorted images. A generalized Gaussian model is employed to summarize the marginal distribution of wavelet coefficients of the reference image, so that only a relatively small number of RR features are needed for the evaluation of image quality. The proposed method is easy to implement and computationally efficient. In addition, we find that many well-known types of image distortion lead to significant changes in wavelet coefficient histograms, and thus are readily detectable by our measure. The algorithm is tested with subjective ratings of a large image database that contains images corrupted with a wide variety of distortion types. A new formalism is derived for the analysis and exact reconstruction of band-limited signals on the sphere with directional wavelets. It represents an evolution of a previously developed wavelet formalism developed by Antoine & Vandergheynst and The translations of the wavelets at any point on the sphere and their proper rotations are still defined through the continuous three-dimensional rotations. The dilations of the wavelets are directly defined in harmonic space through a new kernel dilation, which is a modification of an existing harmonic dilation. A family of factorized steerable functions with compact harmonic support which are suitable for this kernel dilation are first identified. A scale-discretized wavelet formalism is then derived, relying on this dilation. The discrete nature of the analysis scales allows the exact reconstruction of band-limited signals. A corresponding exact multi-resolution algorithm is finally described and an implementation is tested. The formalism is of interest notably for the denoising or the deconvolution of signals on the sphere with a sparse expansion in wavelets. In astrophysics, it finds a particular application for the identification of localized directional features in the cosmic microwave background data, such as the imprint of topological defects, in particular, cosmic strings, and for their reconstruction after separation from the other signal components. Abstract In spite of their remarkable success in signal processing applications, it is now widely acknowledged that traditional wavelets are not very effective in dealing multidimensional signals containing distributed discontinuities such as edges. To overcome this limitation, one has to use basis elements with much higher directional sensitivity and of various shapes, to be able to capture the intrinsic geometrical features of multidimensional phenomena. This paper introduces a new discrete multiscale directional representation called the discrete shearlet transform. This approach, which is based on the shearlet transform, combines the power of multiscale methods with a unique ability to capture the geometry of multidimensional data and is optimally efficient in representing images containing edges. We describe two different methods of implementing the shearlet transform. The numerical experiments presented in this paper demonstrate that the discrete shearlet transform is very competitive in denoising applications both in terms of performance and computational efficiency. This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar regions. This study is motivated by imaging applications, where such boundary curves represent edges of images. The shearlet approach is able to characterize both locations and orientations of the edge points, including corner points and junctions, where the edge curves exhibit abrupt changes in tangent or curvature. Our results encompass and greatly extend previous results based on the shearlet and curvelet transforms which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature. We consider the problem of characterizing the wavefront set of a tempered distribution (u S '( R ^ d ) ) in terms of its continuous wavelet transform, where the latter is defined with respect to a suitably chosen dilation group (H GL ( R ^ d ) ). In this paper we develop a comprehensive and unified approach that allows to establish characterizations of the wavefront set in terms of rapid coefficient decay, for a large variety of dilation groups. For this purpose, we introduce two technical conditions on the dual action of the group H, called microlocal admissibility and (weak) cone approximation property. Essentially, microlocal admissibility sets up a systematic relationship between the scales in a wavelet dilated by (h H ) on one side, and the matrix norm of h on the other side. The (weak) cone approximation property describes the ability of the wavelet system to adapt its frequency-side localization to arbitrary frequency cones. Together, microlocal admissibility and the weak cone approximation property allow the characterization of points in the wavefront set using multiple wavelets. Replacing the weak cone approximation by its stronger counterpart gives rise to single wavelet characterizations. We illustrate the scope of our results by discussing—in any dimension (d 2 )—the similitude, diagonal and shearlet dilation groups, for which we verify the pertinent conditions. As a result, similitude and diagonal groups can be employed for multiple wavelet characterizations, whereas for the shearlet groups a single wavelet suffices. In particular, the shearlet characterization (previously only established for (d=2 )) holds in arbitrary dimensions. Curvelets are plausible candidates for simultaneous compressionofseismicdata,theirimages,andtheimagingoperator itself. We show that with curvelets, the leading-order approximation in angular frequency, horizontal wavenumber, and migrated location to common-offset CO Kirchhoff depth migration becomes a simple transformation of coordinates of curvelets in the data, combined with amplitude scaling. This transformation is calculated using map migration, whichemploysthelocalslopesfromthecurveletdecomposition of the data. Because the data can be compressed using curvelets, the transformation needs to be calculated for relatively few curvelets only. Numerical examples for homogeneous media show that using the leading-order approximationonlyprovidesagoodapproximationtoCOmigrationfor moderate propagation times.As the traveltime increases and raysdivergebeyondthespatialsupportofacurvelet;however, the leading-order approximation is no longer accurate enough. This shows the need for correction beyond leading order,evenforhomogeneousmedia.