Geometric Vision Research Articles

Andrew M. Cooper, A Bastard Kind of Reasoning: William Blake and Geometry

Andrew Cooper’s A Bastard Kind of Reasoning: William Blake and Geometry is a thorough and complex examination of the illuminated books alongside geometry, Enlightenment science, and phenomenology. It claims that “Blake did not simply reject Newton, geometry, and science” (2) and positions itself alongside Donald Ault’s Visionary Physics: Blake’s Response to Newton (1974). Cooper, with his provocatively tongue-in-cheek title, is eager to differentiate his views from Ault’s and offers “a less binary, more interinvolved, yin-yang or Blakean-Contrary view of the relation of Eternity to three-dimensional existence” (12). His main argument hinges on the suggestion that geometry is in fact “cosmology,” a term that will be familiar to scholars working on the illuminated books. Cooper establishes that “Blake’s engagements with geometry were not incidental but key to how he understood the workings of the universe” (6), and that prophecy is a transmission of a geometric vision (235).

Switchable THz wave absorber based on disks and its complement graphene surfaces

A new geometrical vision for a graphene-based THz wave absorber is investigated in this work. A high-performance THz absorber is proposed to exploit complementary conventional periodic arrays of graphene disks. The equivalent circuit model is modified to take account of the complement pattern. The absorber works in two different operational modes based on chemical potential values. One mode shows absorption around 1 THz while the other one expresses perfect absorption at 6 THz and 9 THz. Such a considerable shifting ability via gate stimulations makes the proposed absorber an ideal block for reconfigurable metasurface. The comparison between the circuit model and full-wave simulation verifies an excellent match while ample sensitivity analysis are reported to show the reliability of the proposed switchable THz absorber.

Accelerating Globally Optimal Consensus Maximization in Geometric Vision.

Branch-and-bound-based consensus maximization stands out due to its important ability of retrieving the globally optimal solution to outlier-affected geometric problems. However, while the discovery of such solutions caries high scientific value, its application in practical scenarios is often prohibited by its computational complexity growing exponentially as a function of the dimensionality of the problem at hand. In this work, we convey a novel, general technique that allows us to branch over an n-1 dimensional space for an n-dimensional problem. The remaining degree of freedom can be solved globally optimally within each bound calculation by applying the efficient interval stabbing technique. While each individual bound derivation is harder to compute owing to the additional need for solving a sorting problem, the reduced number of intervals and tighter bounds in practice lead to a significant reduction in the overall number of required iterations. Besides an abstract introduction of the approach, we present applications to four fundamental geometric computer vision problems: camera resectioning, relative camera pose estimation, point set registration, and rotation and focal length estimation. Through our exhaustive tests, we demonstrate significant speed-up factors at times exceeding two orders of magnitude, thereby increasing the viability of globally optimal consensus maximizers in online application scenarios.

Intuition in Mathematics: from Racism to Pluralism

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians (Felix Klein, Henri Poincaré, Ludwig Bieberbach, Arend Heyting) who interacted with Luitzen Egbertus Jan Brouwer (the father of the intuitionist foundational school) in order to compare their conceptions of intuition. We will see how to the same word “intuition” (in German Anschauung) very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty (shared by everybody) of considering concepts that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it.

Monocular spherical depth estimation with explicitly connected weak layout cues

Spherical cameras capture scenes in a holistic manner and have been used for room layout estimation. Recently, with the availability of appropriate datasets, there has also been progress in depth estimation from a single omnidirectional image. While these two tasks are complementary, few works have been able to explore them in parallel to advance indoor geometric perception, and those that have done so either relied on synthetic data, or used small scale datasets, as few options are available that include both layout annotations and dense depth maps in real scenes. This is partly due to the necessity of manual annotations for room layouts. In this work, we move beyond this limitation and generate a 360° geometric vision (360V) dataset that includes multiple modalities, multi-view stereo data and automatically generated weak layout cues. We also explore an explicit coupling between the two tasks to integrate them into a single-shot trained model. We rely on depth-based layout reconstruction and layout-based depth attention, demonstrating increased performance across both tasks. By using single 360° cameras to scan rooms, the opportunity for facile and quick building-scale 3D scanning arises. The project page is available at https://vcl3d.github.io/ExplicitLayoutDepth/.

A Fast and Robust Rotation Search and Point Cloud Registration Method for 2D Stitching and 3D Object Localization

Rotation search and point cloud registration are two fundamental problems in robotics, geometric vision, and remote sensing, which aim to estimate the rotation and transformation between the 3D vector sets and point clouds, respectively. Due to the presence of outliers (probably in very large numbers) among the putative vector or point correspondences in real-world applications, robust estimation is of great importance. In this paper, we present Inlier searching using COmpatible Structures (ICOS), a novel, efficient, and highly robust solver for both the correspondence-based rotation search and point cloud registration problems. Specifically, we (i) propose and construct a series of compatible structures for the two problems, based on which various invariants can be established, and (ii) design time-efficient frameworks to filter out outliers and seek inliers from the invariant-constrained random sampling based on the compatible structures proposed. In this manner, even with extreme outlier ratios, inliers can be effectively sifted out and collected for solving the optimal rotation and transformation, leading to our robust solver ICOS. Through plentiful experiments over standard datasets, we demonstrated that: (i) our solver ICOS is fast, accurate, and robust against over 95% outliers with nearly a 100% recall ratio of inliers for rotation search and both known-scale and unknown-scale registration, outperforming other state-of-the-art methods, and (ii) ICOS is practical for use in real-world application problems including 2D image stitching and 3D object localization.

Vision-Based Pose Optimization Using Learned Metrics

Camera pose optimization is the basis of geometric vision works, such as 3D reconstruction, structure from motion, and visual odometry. A pose optimization method based on learned metrics is proposed to improve the optimization convexity. The neural network was designed and trained based on the collected datasets, respectively. The network inputs pairwise patches and outputs the Euclidean distance of its center. This distance is involved in the residual calculation of Gauss-Newton, and the Jacobian corresponding to this distance can be analytically solved. The simulation verified convergence and generalization of the designed network. The accuracy and robustness of the proposed pose optimization compared with intensity- and feature-based optimizations are also verified.

Matera in Many Dimensions

In this paper, the city of Matera is described from a mathematical point of view. Previous papers on this subject have concentrated on seeing Matera as a fractal city. Here, this analysis is also dealt with as an extension of Euclidean dimensions. The idea is to create a double presentation narrative which is useful for the promotion of cultural heritage and also for the popularizing of mathematics. Those who like geometrical vision will discover some aspects of one of the most ancient cities in Italy. Those who like travelling will have new words to describe the wonders of this country. We reach this objective by using a combinatoric puzzle and suitable story telling.

Model Assisted Multi-band Fusion for Single Image Enhancement and Applications to Robot Vision

This paper presents a fast single image enhancement † † https://github.com/irapkaist/multi-band-enhancement . that is applicable regardless of channels in various environments. The main idea of the paper is combining model-based and fusion-based dehazing methods, thereby presenting balanced image enhancement while elaborating image details. The proposed method enhances both color and grayscale images without any prior information. Multiband decomposition is utilized to extract the base and detail layers for intensity and Laplacian modules. The proposed ambient map and transmission estimation for the intensity module are effective in restoring the true intensity. Adaptive nonlinear mapping functions adjust details on each residual layer. Through color-corrected reconstruction, our results demonstrate outstanding performance on various types of hazy images. The proposed method is thoroughly validated in terms of conventional image quality comparison. We also provide the evaluation at the application phase from both the semantic (segmentation) and geometric (direct odometry) vision based robotics application. The overall algorithm is presented in https://youtu.be/3Fk3kbaPkXQ .

Modified Rodrigues Parameters: An Efficient Representation of Orientation in 3D Vision and Graphics

Modified Rodrigues parameters (MRPs) are triplets in {mathbb {R}}^3 bijectively and rationally mapped to quaternions through stereographic projection. We present here a compelling case for MRPs as a minimal degree-of-freedom parameterization of orientation through novel solutions to prominent problems in the fields of 3D vision and computer graphics. In our primary contribution, we show that the derivatives of a unit quaternion in terms of its MRPs are simple polynomial expressions of its scalar and vector part. Furthermore, we show that updates to unit quaternions from perturbations in parameter space can be computed without explicitly invoking the parameters in the computations. Based on the former, we introduce a novel approach for designing orientation splines by configuring their back-projections in 3D space. Finally, in the general topic of nonlinear optimization for geometric vision, we run performance analyses and provide comparisons on the convergence behavior of MRP parameterizations on the tasks of absolute orientation, exterior orientation and large-scale bundle adjustment of public datasets.

Geometrical Vision Measurement in Mechanical Engineering

Geometrical Vision Measurement in Mechanical Engineering

Non locality, Topology, Formal Languages: New Global Tools to Handle Large Data Sets

The basic idea that stems out of this work is that large sets of data can be handled through an organized set of mathematical and computational tools rooted in a global geometric vision of data space allowing to explore the structure and hidden information patterns thereof. Based on this perspective, the objective is naturally that of discovering and letting emerge, directly from probing the data space, the manifold hidden relations (patterns), e.g. correlations among facts, interactions among entities, relations among concepts and formally describing, in a semantic mining context, the discovered information. In this note, we propose an approach that exploits topological methods for classifying global information into equivalence classes and regular languages for describing the corresponding automaton as element an of hidden complex system.

Primordial Form: Anne Tyng's Geometrical Vision: Review of Anne Tyng: Inhabiting Geometry at the Graham Foundation

Primordial Form: Anne Tyng's Geometrical Vision: Review of <i>Anne Tyng: Inhabiting Geometry at the Graham Foundation</i>

Generalized Convexity in Multiple View Geometry

Recent work on geometric vision problems has exploited convexity properties in order to obtain globally optimal solutions. In this paper we give an overview of these developments and show the tight connections between different types of convexity and optimality conditions for a large class of multiview geometry problems. We also show how the convexity properties are closely linked to different types of optimization algorithms for computing the solutions. Moreover, it is also demonstrated how convexity can be used for detection and removal of outliers. The theoretical findings are accompanied with illustrative examples and experimental results on real data.

On Optimal Dynamic Sequential Search for Matching in Real-Time Machine Vision

In the matching tasks of tracking and geometrical vision, there are usually priors available on the absolute and/or relative image locations of features of interest. In this paper, we use these priors dynamically to guide a feature by feature matching search that can achieve global matching with much fewer image processing operations and lower overall computational cost. First, the concept of dynamic sequential search (DSS) is presented. Then, the problem of determining an optimal search order for DSS is investigated, when the probabilistic distribution of the features can be described by a multivariate Gaussian model. Based on the general formulas for sequentially updating the predicted positions of the features as well as their innovation covariance, the theoretic lower bound for the sum of the areas of the features search-regions is derived, and the necessary and sufficient condition for the optimal search order to approach this lower bound is presented. After that, an algorithm for dynamically determining a suboptimal search order is presented, with a computational complexity of O(n3), which is two magnitudes lower than those of the state-of-the-art algorithms. The effectiveness of the proposed method is validated by both statistical simulation and real-world experiments with a monocular visual SLAM (simultaneous localization and mapping) system. The results verify that the performance of the proposed method is better than the state-of-the-art algorithms, with both fewer image processing operations and lower overall computational cost.

Removing outliers by minimizing the sum of infeasibilities

This paper shows that we can classify latent outliers efficiently through the process of minimizing the sum of infeasibilities (SOI). The SOI minimization has been developed in the area of convex optimization to find an initial solution, solve a feasibility problem, or check out some inconsistent constraints. It was also adopted recently as an approximation method to minimize a robust error function under the framework of the L ∞ norm minimization for geometric vision problems. In this paper, we show that the SOI minimization is practically effective in collecting outliers when it is applied to geometric vision problems. In particular, this method is useful in structure and motion reconstruction where methods such as RANSAC are not applicable. We demonstrate the effectiveness of the method through experiments with synthetic and real data sets.

Ezekiel's Geometric Vision of the Restored Temple: From the Rod of His Wrath to the Reed of His Measuring

“I am the man who has seen affliction under the rod of His wrath.” (Lam 3:1)“Now there was a wall all around the outside of the temple area. The length of the measuring reed in the man's hand was six long cubits, each being a cubit and a handbreadth in length; so he measured the thickness of the wall.” (Ezek 40:5)

The Divine Feminine in Geometric Consciousness

Plato spoke of geometry as an extremely rare natural gift of consciousness, but also as a mode of perception accessible to anyone initiated into the tradition by a master. Such teachers were the shamans of his day, and it was these individuals who convened the mystery schools and sponsored initiations into the esoteric philosophies of geomancy that have grown from their vision. Though the history of women in ancient mystery traditions is largely lost to us, Greek mythology holds that our human capacity for geometric vision is a gift of the divine feminine—energetic sources of wisdom conceptualized as a lineage of goddesses. Born from primal Chaos is Gaia, from whose name comes "geometry"—geo (earth) + metr (measure, mother). She gives birth to Mnemosyne, goddess of Memory, from whose name comes "mnemonic." The daughters of Mnemosyne are the Muses—the arts and sciences. Memory is the legacy of the sacred Earth, and the arts enable humans to actively remember. The essence of this divine feminine lineage is sustainability of sacred place (Chora) through enduring values of order, proportion, a universal aesthetic, and connectivity. In this essay, I explore the ways in which I learned to embrace myself as a physical and spiritual geomantic consciousness and how I have used geometric vision as an interdisciplinary teaching and learning process.

Orden y caos en la traducción creativa (aspectos de coherencia textual)

The main aim of the present article is to approach the problems of stability, prototypicality, reorganization and virtuous self-organization in the text of creative translation from a semantic-topological view. The structural shape of texture relationships between the source language text and the target language text shows a chain of regulating mechanisms apart from the context. Finally, the topological analysis of the translation of Lewis Carroll's Jabberwocky into Spanish has allowed us to have a geometric vision of the underlying morphological stability and its sense attractors.

Revolutions in Mathematics.

Ten laws concerning patterns of change in the history of mathematics 1975, Michael Crowe T.S. Kuhn's theories and mathematics - a discussion paper on the new historiography of mathematics 1976, Herbert Mehrtens appendix 1992 - revolutions reconsidered, Herbert Mehrtens conceptual revolutions and the history of mathematics - two studies in the growth of knowledge 1984, Joseph Dauben appendix 1992 - revolutions revisisted, Joseph Dauben Descartes' geometrie and revolutions in mathematics, Paolo Mancosu was Leibniz a mathematical revolutionary?, Emily Grosholz the fine structure of mathematical revolutions - metaphysics, legitimacy, and rigour - the case of the calculus from Newton to Berkeley and MacLaurin, Giulio Giorelio non-Euclidean geometry and revolutions in mathematics, Yuxin Zheng the in the geometrical vision of space in the 19th century and the hermeneutical epistemology of mathematics, Luciano Boi meta-level revolutions in mathematics, Caroline Dunmore the 19th century-revolution in mathematical ontology, Jeremy Gray a restoration that failed - Paul Finsler's theory of sets, Herbert Breger the Fregean revolution in logic, Donald Gillies afterword 1982 - a revolution in the historiography of mathematics?, Michael Crowe.