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.
Another transformation found in the literature relies on finding a translation invariant Fourier transform on two Extended Gaussian Images (EGI) @cite_19 of laser scans, where the surface normals of an object are mapped onto the unit sphere. However, this approach can only be applied to smooth surfaces and fails to match surfaces with constant EGI (such as a sphere). @cite_36 proposed another approach to scan matching that projects the two scans into the Hough Radon domain @cite_36 defined on the unit sphere. Similar to the work of , a translation invariant spectrum is computed to find the rotation and cross-correlations are used to find the translation. Both EGI approaches and the ones based on transformation to Hough Radon domain are sensitive to the measurement noise and sampling density during the calculation of normals.
@cite_36 @cite_19
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We propose a new approach to appearance-based loop detection for mobile robots, using three-dimensional (3D) laser scans. Loop detection is an important problem in the simultaneous localization and mapping (SLAM) domain, and, because it can be seen as the problem of recognizing previously visited places, it is an example of the data association problem. Without a flat-floor assumption, two-dimensional laser-based approaches are bound to fail in many cases. Two of the problems with 3D approaches that we address in this paper are how to handle the greatly increased amount of data and how to efficiently obtain invariance to 3D rotations. We present a compact representation of 3D point clouds that is still discriminative enough to detect loop closures without false positives (i.e., detecting loop closure where there is none). A low false-positive rate is very important because wrong data association could have disastrous consequences in a SLAM algorithm. Our approach uses only the appearance of 3D point clouds to detect loops and requires no pose information. We exploit the normal distributions transform surface representation to create feature histograms based on surface orientation and smoothness. The surface shape histograms compress the input data by two to three orders of magnitude. Because of the high compression rate, the histograms can be matched efficiently to compare the appearance of two scans. Rotation invariance is achieved by aligning scans with respect to dominant surface orientations. We also propose to use expectation maximization to fit a gamma mixture model to the output similarity measures in order to automatically determine the threshold that separates scans at loop closures from nonoverlapping ones. We discuss the problem of determining ground truth in the context of loop detection and the difficulties in comparing the results of the few available methods based on range information. Furthermore, we present quantitative performance evaluations using three real-world data sets, one of which is highly self-similar, showing that the proposed method achieves high recall rates (percentage of correctly identified loop closures) at low false-positive rates in environments with different characteristics. © 2009 Wiley Periodicals, Inc. In this paper we propose a novel approach for detecting interest points invariant to scale and affine transformations. Our scale and affine invariant detectors are based on the following recent results: (1) Interest points extracted with the Harris detector can be adapted to affine transformations and give repeatable results (geometrically stable). (2) The characteristic scale of a local structure is indicated by a local extremum over scale of normalized derivatives (the Laplacian). (3) The affine shape of a point neighborhood is estimated based on the second moment matrix. Our scale invariant detector computes a multi-scale representation for the Harris interest point detector and then selects points at which a local measure (the Laplacian) is maximal over scales. This provides a set of distinctive points which are invariant to scale, rotation and translation as well as robust to illumination changes and limited changes of viewpoint. The characteristic scale determines a scale invariant region for each point. We extend the scale invariant detector to affine invariance by estimating the affine shape of a point neighborhood. An iterative algorithm modifies location, scale and neighborhood of each point and converges to affine invariant points. This method can deal with significant affine transformations including large scale changes. The characteristic scale and the affine shape of neighborhood determine an affine invariant region for each point. We present a comparative evaluation of different detectors and show that our approach provides better results than existing methods. The performance of our detector is also confirmed by excellent matching resultss the image is described by a set of scale affine invariant descriptors computed on the regions associated with our points. We present a technique to estimate the egomotion of an RGB-D sensor based on rotations of functions defined on the unit sphere. In contrast to traditional approaches, our technique is not based on image features and does not require correspondences to be generated between frames of data. Instead, consecutive functions are correlated using spherical harmonic analysis. An Extended Gaussian Image (EGI), created from the local normal estimates of a point cloud, defines each function. Correlations are efficiently computed using Fourier transformations, resulting in a 3 Degree of Freedom (3-DoF) rotation estimate. An Iterative Closest Point (ICP) process then refines the initial rotation estimate and adds a translational component, yielding a full 6-DoF egomotion estimate. The focus of this work is to investigate the merits of using spherical harmonic analysis for egomotion estimation by comparison with alternative 6-DoF methods. We compare the performance of the proposed technique with that of stand-alone ICP and image feature based methods. As with other egomotion techniques, estimation errors accumulate and degrade results, necessitating correction mechanisms for robust localization. For this report, however, we use the raw estimates; no filtering or smoothing processes are applied. In-house and external benchmark data sets are analyzed for both runtime and accuracy. Results show that the algorithm is competitive in terms of both accuracy and runtime, and future work will aim to combine the various techniques into a more robust egomotion estimation framework. The error sensitivity in the estimation of the 3-D motion and the normal of a planar surface from an instantaneous motion field is studied. The statistical theory or the Cramer-Rao lower bound is used for error covariance in the estimated motion and structure parameters. This enables the derivation of results valid for any unbiased estimator under the assumption of Gaussian noise in the motion field. The obtained lower-bound-matrix is studied analytically with respect to measurement noise, size of the field of view, and the motion-geometry configuration. The coupling between translation and rotation is exacerbated if the field of view and the slant of the plane become smaller, and the deviation of the translation from the viewing direction becomes larger. The relationships of the uncertainty bounds for every unknown motion parameter to the angle between translation and the plane-normal, the size of the field of view, and the distance from the perceived plane and the translation magnitude are discussed. > We present Spectral Registration with Multilayer Resampling (SRMR) as a 6 Degrees Of Freedom (DOF) registration method for noisy 3D data with partial overlap. The algorithm is based on decoupling 3D rotation from 3D translation by a corresponding resampling process of the spectral magnitude of a 3D Fast Fourier Transform (FFT) calculation on discretized 3D range data. The registration of all 6DOF is then subsequently carried out with spectral registrations using Phase Only Matched Filtering (POMF). There are two main aspects for the fast and robust registration of Euler angles from spherical information in SRMR. First of all, there is the permanent use of phase matching. Second, based on the FFT on a discrete Cartesian grid, not only one spherical layer but also a complete stack of layers are processed in one step. Experiments are presented with challenging datasets with respect to interference and overlap. The results include the fast and robust registration of artificially transformed data for ground-truth comparison, scans from the Stanford Bunny dataset, high end 3D laser range finder (LRF) scans of a city center, and range data from a low-cost actuated LRF in a disaster response scenario. We propose a novel framework to build descriptors of local intensity that are invariant to general deformations. In this framework, an image is embedded as a 2D surface in 3D space, with intensity weighted relative to distance in x-y. We show that as this weight increases, geodesic distances on the embedded surface are less affected by image deformations. In the limit, distances are deformation invariant. We use geodesic sampling to get neighborhood samples for interest points, and then use a geodesic-intensity histogram (GIH) as a deformation invariant local descriptor. In addition to its invariance, the new descriptor automatically finds its support region. This means it can safely gather information from a large neighborhood to improve discriminability. Furthermore, we propose a matching method for this descriptor that is invariant to affine lighting changes. We have tested this new descriptor on interest point matching for two data sets, one with synthetic deformation and lighting change, and another with real non-affine deformations. Our method shows promising matching results compared to several other approaches The generalised Hough transform (GHT) is useful for detecting or locating translated two-dimensional objects. However, a weakness of the GHT is its storage requirements and hence the increased computational complexity resulting from the four-dimensional parameter space. In this paper, we present the results of our work which involves investigation of the performance of several efficient GHT techniques including an extension of Thomas's rotation-invariant algorithm. It is shown that our extension of Thomas's algorithm has very low memory requirements and computational complexity, and produces the best results in various tests. Abstract Being motivated by the multilayer RECOS (REctified-COrrelations on a Sphere) transform, we develop a data-driven Saak (Subspace approximation with augmented kernels) transform in this work. The Saak transform consists of three steps: (1) building the optimal linear subspace approximation with orthonormal bases using the second-order statistics of input vectors, (2) augmenting each transform kernel with its negative, (3) applying the rectified linear unit (ReLU) to the transform output. The Karhunen-Loeve transform (KLT) is used in the first step. The integration of Steps 2 and 3 is powerful since they resolve the sign confusion problem, remove the rectification loss and allow a straightforward implementation of the inverse Saak transform at the same time. Multiple Saak transforms are cascaded to transform images of a larger size. All Saak transform kernels are derived from the second-order statistics of input random vectors in a one-pass feedforward manner. Neither data labels nor backpropagation is used in kernel determination. Multi-stage Saak transforms offer a family of joint spatial-spectral representations between two extremes; namely, the full spatial-domain representation and the full spectral-domain representation. We select Saak coefficients of higher discriminant power to form a feature vector for pattern recognition, and use the MNIST dataset classification problem as an illustrative example. In many vision problems, rotation-invariant analysis is necessary or preferred. Popular solutions are mainly based on pose normalization or brute-force learning, neglecting the intrinsic properties of rotations. In this paper, we present a rotation invariant detection approach built on the equivariant filter framework, with a new model for learning the filtering behavior. The special properties of the harmonic basis, which is related to the irreducible representation of the rotation group, directly guarantees rotation invariance of the whole approach. The proposed kernel weighted mapping ensures high learning capability while respecting the invariance constraint. We demonstrate its performance on 2D object detection with in-plane rotations, and a 3D application on rotation-invariant landmark detection in microscopic volumetric data. If a physical object has a smooth or piecewise smooth boundary, its images obtained by cameras in varying positions undergo smooth apparent deformations. These deformations are locally well approximated by affine transforms of the image plane. In consequence the solid object recognition problem has often been led back to the computation of affine invariant image local features. Such invariant features could be obtained by normalization methods, but no fully affine normalization method exists for the time being. Even scale invariance is dealt with rigorously only by the scale-invariant feature transform (SIFT) method. By simulating zooms out and normalizing translation and rotation, SIFT is invariant to four out of the six parameters of an affine transform. The method proposed in this paper, affine-SIFT (ASIFT), simulates all image views obtainable by varying the two camera axis orientation parameters, namely, the latitude and the longitude angles, left over by the SIFT method. Then it covers the other four parameters by using the SIFT method itself. The resulting method will be mathematically proved to be fully affine invariant. Against any prognosis, simulating all views depending on the two camera orientation parameters is feasible with no dramatic computational load. A two-resolution scheme further reduces the ASIFT complexity to about twice that of SIFT. A new notion, the transition tilt, measuring the amount of distortion from one view to another, is introduced. While an absolute tilt from a frontal to a slanted view exceeding 6 is rare, much higher transition tilts are common when two slanted views of an object are compared (see Figure hightransitiontiltsillustration). The attainable transition tilt is measured for each affine image comparison method. The new method permits one to reliably identify features that have undergone transition tilts of large magnitude, up to 36 and higher. This fact is substantiated by many experiments which show that ASIFT significantly outperforms the state-of-the-art methods SIFT, maximally stable extremal region (MSER), Harris-affine, and Hessian-affine. In this paper, we cast the problem of point cloud matching as a shape matching problem by transforming each of the given point clouds into a shape representation called the Schrodinger distance transform (SDT) representation. This is achieved by solving a static Schrodinger equation instead of the corresponding static Hamilton-Jacobi equation in this setting. The SDT representation is an analytic expression and following the theoretical physics literature, can be normalized to have unit 2 norm -- making it a square-root density, which is identified with a point on a unit Hilbert sphere, whose intrinsic geometry is fully known. The Fisher-Rao metric, a natural metric for the space of densities leads to analytic expressions for the geodesic distance between points on this sphere. In this paper, we use the well known Riemannian framework never before used for point cloud matching, and present a novel matching algorithm. We pose point set matching under rigid and non-rigid transformations in this framework and solve for the transformations using standard nonlinear optimization techniques. Finally, to evaluate the performance of our algorithm -- dubbed SDTM -- we present several synthetic and real data examples along with extensive comparisons to state-of-the-art techniques. The experiments show that our algorithm outperforms state-of the-art point set registration algorithms on many quantitative metrics. This paper constructs translation-invariant operators on L 2 .R d , which are Lipschitz-continuous to the action of diffeomorphisms. A scattering propagator is a path-ordered product of nonlinear and noncommuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is proved to be Lipschitz-continuous to the action of C 2 diffeomorphisms. As the window size increases, it converges to a wavelet scattering transform that is translation invariant. Scattering coefficients also provide representations of stationary processes. Expected values depend upon high-order moments and can discriminate processes having the same power spectrum. Scattering operators are extended on L 2 .G , where G is a compact Lie group, and are invariant under the action of G. Combining a scattering on L 2 .R d and on L 2 .SO.d defines a translation- and rotation-invariant scattering on L 2 .R d . © 2012 Wiley Periodicals, Inc. Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typically sought through data augmentation, but patch-wise equivariance is more difficult. We present Harmonic Networks or H-Nets, a CNN exhibiting equivariance to patch-wise translation and 360-rotation. We achieve this by replacing regular CNN filters with circular harmonics, returning a maximal response and orientation for every receptive field patch. H-Nets use a rich, parameter-efficient and fixed computational complexity representation, and we show that deep feature maps within the network encode complicated rotational invariants. We demonstrate that our layers are general enough to be used in conjunction with the latest architectures and techniques, such as deep supervision and batch normalization. We also achieve state-of-the-art classification on rotated-MNIST, and competitive results on other benchmark challenges. This paper introduces an algorithm for the registration of rotated and translated volumes using the three-dimensional (3-D) pseudopolar Fourier transform, which accurately computes the Fourier transform of the registered volumes on a near-spherical 3-D domain without using interpolation. We propose a three-step procedure. The first step estimates the rotation axis. The second step computes the planar rotation relative to the rotation axis. The third step recovers the translational displacement. The rotation estimation is based on Euler's theorem, which allows one to represent a 3-D rotation as a planar rotation around a 3-D rotation axis. This axis is accurately recovered by the 3-D pseudopolar Fourier transform using radial integrations. The residual planar rotation is computed by an extension of the angular difference function to cylindrical motion. Experimental results show that the algorithm is accurate and robust to noise A large amount of 2D drawings have been produced in engineering fields. To reuse and share the available drawings efficiently, we propose two methods in this paper, namely 2.5D spherical harmonics transformation and 2D shape histogram, to retrieve 2D drawings by measuring their shape similarity. The first approach represents a drawing as a spherical function by transforming it from a 2D space into a 3D space. Then a fast spherical harmonics transformation is employed to get a rotation invariant descriptor. The second statistics-based approach represents the shape of a 2D drawing using a distance distribution between two randomly sampled points. To allow users to interactively emphasize certain local shapes that they are interested in, we have adopted a flexible sampling strategy by specifying a bias sampling density upon these local shapes. The two proposed methods have many valuable properties, including transform invariance, efficiency, and robustness. In addition, their insensitivity to noise allows for the user's causal input, thus supporting a freehand sketch-based retrieval user interface. Experiments show that a better performance can be achieved by combining them together using weights. We present an extension of the BM3D filter to volumetric data. The proposed algorithm, BM4D, implements the grouping and collaborative filtering paradigm, where mutually similar d -dimensional patches are stacked together in a (d+1) -dimensional array and jointly filtered in transform domain. While in BM3D the basic data patches are blocks of pixels, in BM4D we utilize cubes of voxels, which are stacked into a 4-D “group.” The 4-D transform applied on the group simultaneously exploits the local correlation present among voxels in each cube and the nonlocal correlation between the corresponding voxels of different cubes. Thus, the spectrum of the group is highly sparse, leading to very effective separation of signal and noise through coefficient shrinkage. After inverse transformation, we obtain estimates of each grouped cube, which are then adaptively aggregated at their original locations. We evaluate the algorithm on denoising of volumetric data corrupted by Gaussian and Rician noise, as well as on reconstruction of volumetric phantom data with non-zero phase from noisy and incomplete Fourier-domain (k-space) measurements. Experimental results demonstrate the state-of-the-art denoising performance of BM4D, and its effectiveness when exploited as a regularizer in volumetric data reconstruction. Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typically sought through data augmentation, but patch-wise equivariance is more difficult. We present Harmonic Networks or H-Nets, a CNN exhibiting equivariance to patch-wise translation and 360-rotation. We achieve this by replacing regular CNN filters with circular harmonics, returning a maximal response and orientation for every receptive field patch. H-Nets use a rich, parameter-efficient and low computational complexity representation, and we show that deep feature maps within the network encode complicated rotational invariants. We demonstrate that our layers are general enough to be used in conjunction with the latest architectures and techniques, such as deep supervision and batch normalization. We also achieve state-of-the-art classification on rotated-MNIST, and competitive results on other benchmark challenges. This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset @math , associated with the Euclidean metric, with points in the cube @math and we associate the cube with a pseudo-metric that approximates Euclidean distance among points in @math . Our methods rely on quantizing fast Johnson-Lindenstrauss embeddings based on bounded orthonormal systems and partial circulant ensembles, both of which admit fast transforms. Our quantization methods utilize noise-shaping, and include Sigma-Delta schemes and distributed noise-shaping schemes. The resulting approximation errors decay polynomially and exponentially fast in @math , depending on the embedding method. This dramatically outperforms the current decay rates associated with binary embeddings and Hamming distances. Additionally, it is the first such binary embedding result that applies to fast Johnson-Lindenstrauss maps while preserving @math norms. Second, we again consider noise-shaping schemes, albeit this time to quantize compressed sensing measurements arising from bounded orthonormal ensembles and partial circulant matrices. We show that these methods yield a reconstruction error that again decays with the number of measurements (and bits), when using convex optimization for reconstruction. Specifically, for Sigma-Delta schemes, the error decays polynomially in the number of measurements, and it decays exponentially for distributed noise-shaping schemes based on beta encoding. These results are near optimal and the first of their kind dealing with bounded orthonormal systems. We present a novel method to obtain fine-scale detail in 3D reconstructions generated with low-budget RGB-D cameras or other commodity scanning devices. As the depth data of these sensors is noisy, truncated signed distance fields are typically used to regularize out the noise, which unfortunately leads to over-smoothed results. In our approach, we leverage RGB data to refine these reconstructions through shading cues, as color input is typically of much higher resolution than the depth data. As a result, we obtain reconstructions with high geometric detail, far beyond the depth resolution of the camera itself. Our core contribution is shading-based refinement directly on the implicit surface representation, which is generated from globally-aligned RGB-D images. We formulate the inverse shading problem on the volumetric distance field, and present a novel objective function which jointly optimizes for fine-scale surface geometry and spatially-varying surface reflectance. In order to enable the efficient reconstruction of sub-millimeter detail, we store and process our surface using a sparse voxel hashing scheme which we augment by introducing a grid hierarchy. A tailored GPU-based Gauss-Newton solver enables us to refine large shape models to previously unseen resolution within only a few seconds. We propose a novel technique for the registration of 3D point clouds which makes very few assumptions: we avoid any manual rough alignment or the use of landmarks, displacement can be arbitrarily large, and the two point sets can have very little overlap. Crude alignment is achieved by estimation of the 3D-rotation from two Extended Gaussian Images even when the data sets inducing them have partial overlap. The technique is based on the correlation of the two EGIs in the Fourier domain and makes use of the spherical and rotational harmonic transforms. For pairs with low overlap which fail a critical verification step, the rotational alignment can be obtained by the alignment of constellation images generated from the EGIs. Rotationally aligned sets are matched by correlation using the Fourier transform of volumetric functions. A fine alignment is acquired in the final step by running Iterative Closest Points with just few iterations.