text: 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.
  @cite_14: 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.
  @cite_44: We study a simple Model" for the problem of recovering an image from noisy data; in this model the image has an edge with fi-Holder regularity. Adopting the viewpoint of computational harmonic analysis, we develop an overcomplete collection of atoms called wedgelets, dyadically organized indicator functions with a variety of locations, scales, and orientations. The wedgelet representation provides nearly-optimal representations of objects in the Horizon model, as measured by minimax description length. We show how to rapidly compute a wedgelet approximation to noisy data by flnding a special edgelet-decorated recursive partition which minimizes a complexity-penalized sum of squares. This estimate, using su-cient sub-pixel resolution, achieves nearly the minimax mean-squared error in the Horizon Model. In fact, the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the Horizon, 1• fi• 2. Wedgelet analysis and de-noising may be used successfully outside the Horizon model. We study images modelled as indicators of star-shaped sets with smooth bound- aries and show that complexity-penalized wedgelet partitioning achieves nearly the minimax risk in that setting also.
  @cite_24: We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis.
  @cite_45: In 1992, Bamberger and Smith proposed the directional filter bank (DFB) for an efficient directional decomposition of 2-D signals. Due to the nonseparable nature of the system, extending the DFB to higher dimensions while still retaining its attractive features is a challenging and previously unsolved problem. We propose a new family of filter banks, named NDFB, that can achieve the directional decomposition of arbitrary N-dimensional (Nges2) signals with a simple and efficient tree-structured construction. In 3-D, the ideal passbands of the proposed NDFB are rectangular-based pyramids radiating out from the origin at different orientations and tiling the entire frequency space. The proposed NDFB achieves perfect reconstruction via an iterated filter bank with a redundancy factor of N in N-D. The angular resolution of the proposed NDFB can be iteratively refined by invoking more levels of decomposition through a simple expansion rule. By combining the NDFB with a new multiscale pyramid, we propose the surfacelet transform, which can be used to efficiently capture and represent surface-like singularities in multidimensional data
  @cite_20: 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.
summary: 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.