Simultaneous Localization and Mapping using RGB-D cameras has been a fertile research topic in the latest decade, due to the suitability of such sensors for indoor robotics. In this paper we propose a direct RGB-D SLAM algorithm with state-of-the-art accuracy and robustness at a los cost. Our experiments in the RGB-D TUM dataset [34] effectively show a better accuracy and robustness in CPU real time than direct RGB-D SLAM systems that make use of the GPU. The key ingredients of our approach are mainly two. Firstly, the combination of a semi-dense photometric and dense geometric error for the pose tracking (see Figure 1), which we demonstrate to be the most accurate alternative. And secondly, a model of the multi-view constraints and their errors in the mapping and tracking threads, which adds extra information over other approaches. We release the open-source implementation of our approach 1 . The reader is referred to a video with our results 2 for a more illustrative visualization of its performance.
@cite_4 proposes to use the inverse depth in the minimization of the geometric reprojection error. We evaluate this parametrization in our system.
@cite_4
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We present a new parametrization for point features within monocular simultaneous localization and mapping (SLAM) that permits efficient and accurate representation of uncertainty during undelayed initialization and beyond, all within the standard extended Kalman filter (EKF). The key concept is direct parametrization of the inverse depth of features relative to the camera locations from which they were first viewed, which produces measurement equations with a high degree of linearity. Importantly, our parametrization can cope with features over a huge range of depths, even those that are so far from the camera that they present little parallax during motion---maintaining sufficient representative uncertainty that these points retain the opportunity to "come in'' smoothly from infinity if the camera makes larger movements. Feature initialization is undelayed in the sense that even distant features are immediately used to improve camera motion estimates, acting initially as bearing references but not permanently labeled as such. The inverse depth parametrization remains well behaved for features at all stages of SLAM processing, but has the drawback in computational terms that each point is represented by a 6-D state vector as opposed to the standard three of a Euclidean XYZ representation. We show that once the depth estimate of a feature is sufficiently accurate, its representation can safely be converted to the Euclidean XYZ form, and propose a linearity index that allows automatic detection and conversion to maintain maximum efficiency---only low parallax features need be maintained in inverse depth form for long periods. We present a real-time implementation at 30 Hz, where the parametrization is validated in a fully automatic 3-D SLAM system featuring a handheld single camera with no additional sensing. Experiments show robust operation in challenging indoor and outdoor environments with a very large ranges of scene depth, varied motion, and also real time 360deg loop closing. The concept of depth from focus involves calculating distances to points in an observed scene by modeling the effect that the camera's focal parameters have on images acquired with a small depth of field. This technique is passive and requires only a single camera. The most difficult segment of calculating depth from focus is deconvolving the defocus operator from the scene and modeling it. Most current methods for determining the defocus operator employ inverse filtering. The authors reveal some fundamental problems with inverse filtering: inaccuracies in finding the frequency domain representation, windowing effects, and border effects. A general, matrix-based method using regularization is presented, which eliminates these problems. The new method is confirmed experimentally, with the results showing an RMS error of 1.3 . In this paper, we revisit the classical perspective-n-point (PnP) problem, and propose the first non-iterative O(n) solution that is fast, generally applicable and globally optimal. Our basic idea is to formulate the PnP problem into a functional minimization problem and retrieve all its stationary points by using the Gr"obner basis technique. The novelty lies in a non-unit quaternion representation to parameterize the rotation and a simple but elegant formulation of the PnP problem into an unconstrained optimization problem. Interestingly, the polynomial system arising from its first-order optimality condition assumes two-fold symmetry, a nice property that can be utilized to improve speed and numerical stability of a Grobner basis solver. Experiment results have demonstrated that, in terms of accuracy, our proposed solution is definitely better than the state-of-the-art O(n) methods, and even comparable with the reprojection error minimization method. In this paper we present a dense visual odometry system for RGB-D cameras performing both photometric and geometric error minimisation to estimate the camera motion between frames. Contrary to most works in the literature, we parametrise the geometric error by the inverse depth instead of the depth, which translates into a better fit of the distribution of the geometric error to the used robust cost functions. We also provide a unified evaluation under the same framework of different estimators and ways of computing the scale of the residuals which can be found spread along the related literature. For the comparison of our approach with state-of-the-art approaches we use the popular dataset from the TUM for RGB-D benchmarking. Our approach shows to be competitive with state-of-the-art methods in terms of drift in meters per second, even compared to methods performing loop closure too. When comparing to approaches performing pure odometry like ours, our method outperforms them in the majority of the tested datasets. Additionally we show that our approach is able to work in real time and we provide a qualitative evaluation on our own sequences showing a low drift in the 3D reconstructions. The classical approach to depth from defocus (DFD) uses lenses with circular apertures for image capturing. We show in this paper that the use of a circular aperture severely restricts the accuracy of DFD. We derive a criterion for evaluating a pair of apertures with respect to the precision of depth recovery. This criterion is optimized using a genetic algorithm and gradient descent search to arrive at a pair of high resolution apertures. These two coded apertures are found to complement each other in the scene frequencies they preserve. This property enables them to not only recover depth with greater fidelity but also obtain a high quality all-focused image from the two captured images. Extensive simulations as well as experiments on a variety of real scenes demonstrate the benefits of using the coded apertures over conventional circular apertures. We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s. We prove near-optimal trade-offs for quantifier depth versus number of variables in first-order logic by exhibiting pairs of n-element structures that can be distinguished by a k-variable first-order sentence but where every such sentence requires quantifier depth at least nΩ(k log k). Our trade-offs also apply to first-order counting logic, and by the known connection to the k-dimensional Weisfeiler–Leman algorithm imply near-optimal lower bounds on the number of refinement iterations. A key component in our proof is the hardness condensation technique recently introduced by [Razborov ’16] in the context of proof complexity. We apply this method to reduce the domain size of relational structures while maintaining the quantifier depth required to distinguish them.Categories and Subject Descriptors F.4.1 [Mathematical Logic]: Computational Logic, Model theory; F.2.3 [Tradeoffs between Complexity Measures] In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–Łojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing. Abstract Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map everywhere on the low-dimensional image of the forward map. We demonstrate that the scale-free cubic RBF kernel performs better than the Gaussian kernel: it does not suffer from ill-conditioning, and does not require the choice of a scale. The proposed construction is shown to be similar to the Nystrom extension of the eigenvectors of the symmetric normalized graph Laplacian matrix. Based on this observation, we provide a new interpretation of the Nystrom extension with suggestions for improvement. We study an equivalence of (i) deterministic pathwise statements appearing in the online learning literature (termed regret bounds), (ii) high-probability tail bounds for the supremum of a collection of martingales (of a specific form arising from uniform laws of large numbers for martingales), and (iii) in-expectation bounds for the supremum. By virtue of the equivalence, we prove exponential tail bounds for norms of Banach space valued martingales via deterministic regret bounds for the online mirror descent algorithm with an adaptive step size. We extend these results beyond the linear structure of the Banach space: we define a notion of martingale type for general classes of real-valued functions and show its equivalence (up to a logarithmic factor) to various sequential complexities of the class (in particular, the sequential Rademacher complexity and its offset version). For classes with the general martingale type 2, we exhibit a finer notion of variation that allows partial adaptation to the function indexing the martingale. Our proof technique rests on sequential symmetrization and on certifying the existence of regret minimization strategies for certain online prediction problems. We study a class of ill-posed linear inverse problems in which the underlying model of interest has simple algebraic structure. We consider the setting in which we have access to a limited number of linear measurements of the underlying model, and we propose a general framework based on convex optimization in order to recover this model. This formulation generalizes previous methods based on 1-norm minimization and nuclear norm minimization for recovering sparse vectors and low-rank matrices from a small number of linear measurements. For example some problems to which our framework is applicable include (1) recovering an orthogonal matrix from limited linear measurements, (2) recovering a measure given random linear combinations of its moments, and (3) recovering a low-rank tensor from limited linear observations. Monocular depth estimation, which plays a crucial role in understanding 3D scene geometry, is an ill-posed problem. Recent methods have gained significant improvement by exploring image-level information and hierarchical features from deep convolutional neural networks (DCNNs). These methods model depth estimation as a regression problem and train the regression networks by minimizing mean squared error, which suffers from slow convergence and unsatisfactory local solutions. Besides, existing depth estimation networks employ repeated spatial pooling operations, resulting in undesirable low-resolution feature maps. To obtain high-resolution depth maps, skip-connections or multi-layer deconvolution networks are required, which complicates network training and consumes much more computations. To eliminate or at least largely reduce these problems, we introduce a spacing-increasing discretization (SID) strategy to discretize depth and recast depth network learning as an ordinal regression problem. By training the network using an ordinary regression loss, our method achieves much higher accuracy and faster convergence in synch . Furthermore, we adopt a multi-scale network structure which avoids unnecessary spatial pooling and captures multi-scale information in parallel. The method described in this paper achieves state-of-the-art results on four challenging benchmarks, i.e., KITTI [17], ScanNet [9], Make3D [50], and NYU Depth v2 [42], and win the 1st prize in Robust Vision Challenge 2018. Code has been made available at: this https URL In this paper, we consider nonlinear inverse problems where the solution is assumed to have a sparse expansion with respect to a preassigned basis or frame. We develop a scheme which allows to minimize a Tikhonov functional where the usual quadratic regularization term is replaced by a one-homogeneous (typically weighted l p ) penalty on the coefficients (or isometrically transformed coefficients) of such expansions. For (p < 2), the regularized solution will have a sparser expansion with respect to the basis or frame under consideration. The computation of the regularized solution amounts in our setting to a Landweber-fixed-point iteration with a projection applied in each fixed-point iteration step. The performance of the resulting numerical scheme is demonstrated by solving the nonlinear inverse single photon emission computerized tomography (SPECT) problem. We describe the wavelet–vaguelette decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type—such as numerical differentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon transform. We propose to solve ill-posed linear inverse problems by nonlinearly "shrinking" the WVD coefficients of the noisy, indirect data. Our approach offers significant advantages over traditional SVD inversion in recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L2 loss, over the entire scale of Besov spaces. The important case of Besov spaces Bσp,q, p < 2, which model spatial inhomogeneity, is included. In comparison, linear procedures— SVD included—cannot attain optimal rates of convergence over such classes in the case p < 2. For example, our methods achieve faster rates of convergence for objects known to lie in the bump algebra or in bounded variation than any linear procedure. In their paper on the ''chasm at depth four'', Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o^(^m^) also admit arithmetic circuits of depth four and size 2^o^(^m^). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of nxn matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O^(^n^l^o^g^n^). If the original circuit uses only integer constants of polynomial size, then the same is true for the resulting depth four circuit. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also use our techniques to reprove two results on: -the existence of nontrivial boolean circuits of constant depth for languages in LOGCFL; -reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree. We propose an approach for dense semantic 3D reconstruction which uses a data term that is defined as potentials over viewing rays, combined with continuous surface area penalization. Our formulation is a convex relaxation which we augment with a crucial non-convex constraint that ensures exact handling of visibility. To tackle the non-convex minimization problem, we propose a majorize-minimize type strategy which converges to a critical point. We demonstrate the benefits of using the non-convex constraint experimentally. For the geometry-only case, we set a new state of the art on two datasets of the commonly used Middlebury multi-view stereo benchmark. Moreover, our general-purpose formulation directly reconstructs thin objects, which are usually treated with specialized algorithms. A qualitative evaluation on the dense semantic 3D reconstruction task shows that we improve significantly over previous methods. We study the basic problem of robust subspace recovery. That is, we assume a data set that some of its points are sampled around a fixed subspace and the rest of them are spread in the whole ambient space, and we aim to recover the fixed underlying subspace. We first estimate "robust inverse sample covariance" by solving a convex minimization procedure; we then recover the subspace by the bottom eigenvectors of this matrix (their number correspond to the number of eigenvalues close to 0). We guarantee exact subspace recovery under some conditions on the underlying data. Furthermore, we propose a fast iterative algorithm, which linearly converges to the matrix minimizing the convex problem. We also quantify the effect of noise and regularization and discuss many other practical and theoretical issues for improving the subspace recovery in various settings. When replacing the sum of terms in the convex energy function (that we minimize) with the sum of squares of terms, we obtain that the new minimizer is a scaled version of the inverse sample covariance (when exists). We thus interpret our minimizer and its subspace (spanned by its bottom eigenvectors) as robust versions of the empirical inverse covariance and the PCA subspace respectively. We compare our method with many other algorithms for robust PCA on synthetic and real data sets and demonstrate state-of-the-art speed and accuracy. We present a novel multi-baseline, multi-resolution stereo method, which varies the baseline and resolution proportionally to depth to obtain a reconstruction in which the depth error is constant. This is in contrast to traditional stereo, in which the error grows quadratically with depth, which means that the accuracy in the near range far exceeds that of the far range. This accuracy in the near range is unnecessarily high and comes at significant computational cost. It is, however, non-trivial to reduce this without also reducing the accuracy in the far range. Many datasets, such as video captured from a moving camera, allow the baseline to be selected with significant flexibility. By selecting an appropriate baseline and resolution (realized using an image pyramid), our algorithm computes a depthmap which has these properties: 1) the depth accuracy is constant over the reconstructed volume, 2) the computational effort is spread evenly over the volume, 3) the angle of triangulation is held constant w.r.t. depth. Our approach achieves a given target accuracy with minimal computational effort, and is orders of magnitude faster than traditional stereo. This note provides a condition under which lscr1 minimization (also known as basis pursuit) can recover short linear combinations of complex vectors chosen from fixed, overcomplete collection. This condition has already been established in the real setting by Fuchs, who used convex analysis. The proof given here is more direct Abstract This paper introduces a new method of registering point sets. The registration error is directly minimized using general-purpose non-linear optimization (the Levenberg–Marquardt algorithm). The surprising conclusion of the paper is that this technique is comparable in speed to the special-purpose Iterated Closest Point algorithm, which is most commonly used for this task. Because the routine directly minimizes an energy function, it is easy to extend it to incorporate robust estimation via a Huber kernel, yielding a basin of convergence that is many times wider than existing techniques. Finally, we introduce a data structure for the minimization based on the chamfer distance transform, which yields an algorithm that is both faster and more robust than previously described methods.