Daniilidis, Kostas

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  • Publication
    Hybrid Control for Visibility-Based Pursuit-Evasion Games
    (2004-09-28) Isler, Volkan; Daniilidis, Kostas; Belta, Calin; Pappas, George J
    Pursuit-evasion games in complex environments have a rich but disconnected history. Continuous or differential pursuit-evasion games focus on optimal control methods, and rely on very intense computations in order to provide locally optimal controls. Discrete pursuit-evasion games on graphs are algorithmically much more appealing, but completely ignore the physical dynamics of the players, resulting in possibly infeasible motions. In this paper, we present a provable and algorithmically feasible solution for visibility-based pursuit-evasion games in simply-connected environments, for players with dynamic constraints. This is achieved by combining two recent but distant results.
  • Publication
    3D-Orientation Signatures with Conic Kernel Filtering for Multiple Motion Analysis
    (2001-12-08) Yu, Weichuan; Sommer, Gerald; Daniilidis, Kostas
    In this paper we propose a new 3D kernel for the recovery of 3D-orientation signatures. The kernel is a Gaussian function defined in local spherical coordinates and its Cartesian support has the shape of a truncated cone with its axis in the radial direction and very small angular support. A set of such kernels is obtained by uniformly sampling the 2D space of polar and azimuth angles. The projection of a local neighborhood on such a kernel set produces a local 3D-orientation signature. In the case of spatiotemporal analysis, such a kernel set can be applied either on the derivative space of a local neighborhood or on the local Fourier transform. The well known planes arising from single or multiple motion produce maxima in the orientation signature. Due to the kernel's local support spatiotemporal signatures possess higher orientation resolution than 3D steerable filters and motion maxima can be detected and localized more accurately. We describe and show in experiments the superiority of the proposed kernels compared to Hough transformation or EM-based multiple motion detection.
  • Publication
    Rotation Recovery from Spherical Images without Correspondences
    (2006-07-01) Makadia, Ameesh; Daniilidis, Kostas
    This paper addresses the problem of rotation estimation directly from images defined on the sphere and without correspondence. The method is particularly useful for the alignment of large rotations and has potential impact on 3D shape alignment. The foundation of the method lies in the fact that the spherical harmonic coefficients undergo a unitary mapping when the original image is rotated. The correlation between two images is a function of rotations and we show that it has an SO(3)-Fourier transform equal to the pointwise product of spherical harmonic coefficients of the original images. The resolution of the rotation space depends on the bandwidth we choose for the harmonic expansion and the rotation estimate is found through a direct search in this 3D discretized space. A refinement of the rotation estimate can be obtained from the conservation of harmonic coefficients in the rotational shift theorem. A novel decoupling of the shift theorem with respect to the Euler angles is presented and exploited in an iterative scheme to refine the initial rotation estimates. Experiments show the suitability of the method for large rotations and the dependence of the method on bandwidth and the choice of the spherical harmonic coefficients.
  • Publication
    Properties of the Catadioptric Fundamental Matrix
    (2002-05-28) Geyer, Christopher; Daniilidis, Kostas
    The geometry of two uncalibrated views obtained with a parabolic catadioptric device is the subject of this paper. We introduce the notion of circle space, a natural representation of line images, and the set of incidence preserving transformations on this circle space which happens to equal the Lorentz group. In this space, there is a bilinear constraint on transformed image coordinates in two parabolic catadioptric views involving what we call the catadioptric fundamental matrix. We prove that the angle between corresponding epipolar curves is preserved and that the transformed image of the absolute conic is in the kernel of that matrix, enabling thus euclidean reconstruction from two views. We establish the necessary and sufficient conditions for a matrix to be a catadioptric fundamental matrix.
  • Publication
    VC-Dimension of Exterior Visibility of Polyhedra
    (2001-01-01) Kannan, Sampath; Isler, Volkan; Daniilidis, Kostas
    In this paper, we address the problem of finding the minimal number of viewpoints outside a polyhedron in two or three dimensions such that every point on the exterior of the polyhedron is visible from at least one of the chosen viewpoints. This problem which we call the minimum fortress guard problem (MFGP) is the optimization version of a variant of the art-gallery problem (sometimes called the fortress problem with point guards) and has practical importance in surveillance and image-based rendering. Solutions in the vision and graphics literature are based on image quality constraints and are not concerned with the number of viewpoints needed. The corresponding question for art galleries (minimum number of viewpoints in the interior of a polygon to see the interior of the polygon) which we call the minimum art-gallery guard problem (MAGP) has been shown to be NP-complete. A simple reduction from this problem shows the NP-completeness of MFGP. Instead of relying on heuristic searches, we address the approximability of the camera placement problem. It is well known (and easy to see) that this problem can be cast as a hitting set problem. While the approximability of generic instances of the hitting set problem is well understood, Brönnimann and Goodrich[3] presented improved approximation algorithms for the problem in the case that the input instances have bounded Vapnik-Chervonenkis (VC) dimension. In this paper we explore the VC-dimension of set systems associated with the camera placement problem described above. We show a constant bound for the VC dimension in the two dimensional case but a tight logarithmic bound in the three dimensional case. In the two dimensional case we are also able to present an algorithm that uses at most one more viewpoint than the optimal in the case that the viewpoints are restricted to be on a circumscribing circle - a restriction that is justified in practice.
  • Publication
    Structure and Motion From Uncalibrated Catadioptric Views
    (2001-05-25) Geyer, Christopher; Daniilidis, Kostas
    In this paper we present a new algorithm for structure from motion from point correspondences in images taken from uncalibrated catadioptric cameras with parabolic mirrors. We assume that the unknown intrinsic parameters are three: the combined focal length of the mirror and lens and the intersection of the optical axis with the image. We introduce a new representation for images of points and lines in catadioptric images which we call the circle space. This circle space includes imaginary circles, one of which is the image of the absolute conic. We formulate the epipolar constraint in this space and establish a new 4x4 catadioptric fundamental matrix. We show that the image of the absolute conic belongs to the kernel of this matrix. This enables us to prove that Euclidean reconstruction is feasible from two views with constant parameters and from three views with varying parameters. In both cases, it is one less than the number of views necessary with perspective cameras.
  • Publication
    Compression of Stereo Disparity Streams Using Wavelets and Optical Flow
    (2001-01-01) Bülow, Thomas; Mulligan, Jane; Bonnafos, Geraud de; Chibane, Alexandre; Daniilidis, Kostas
    Recent advances in computing have enabled fast reconstructions of dynamic scenes from multiple images. However, the efficient coding of changing 3D-data has hardly been addressed. Progressive geometric compression and streaming are based on static data sets which are mostly artificial or obtained from accurate range sensors. In this paper, we present a system for efficient coding of 3D-data which are given in forms of 2 + 1/2 disparity maps. Disparity maps are spatially coded using wavelets and temporally predicted by computing flow. The resulted representation of a 3D-stream consists then of spatial wavelet coefficients, optical flow vectors, and disparity differences between predicted and incoming image. The approach has also very useful by-products: disparity predictions can significantly reduce the disparity search range and if appropriately modeled increase the accuracy of depth estimation.
  • Publication
    Robust Invariants From Functionally Constrained Motion
    (1998-07-01) Hicks, Andrew R.; Daniilidis, Kostas; Bajcsy, Ruzena; Pettey, David
    We address in the problem of control-based recovery of robot pose and the environmental lay-out. Panoramic sensors provide us with an 1D projection of characteristic features of a 2D operation map. Trajectories of these projections contain the information about the position of a priori unknown landmarks in the environment. We introduce here the notion of spatiotemporal signatures of projection trajectories. These signatures are global measures, like area, characterized by considerably higher robustness with respect to noise and outliers than the commonly applied point correspondence. By modeling the 2D motion plane as the complex plane we show that by means of complex analysis our method can be embedded in the well-known affine reconstruction paradigm.
  • Publication
    Visual Servoing of Quadrotors for Perching by Hanging From Cylindrical Objects
    (2016-01-01) Thomas, Justin; Daniilidis, Kostas; Kumar, Vijay; Loianno, Giuseppe
    This paper addresses vision-based localization and servoing for quadrotors to enable autonomous perching by hanging from cylindrical structures using only a monocular camera. We focus on the problems of relative pose estimation, control, and trajectory planning for maneuvering a robot relative to cylinders with unknown orientations. We first develop a geometric model that describes the pose of the robot relative to a cylinder. Then, we derive the dynamics of the system, expressed in terms of the image features. Based on the dynamics, we present a controller which guarantees asymptotic convergence to the desired image space coordinates. Finally, we develop an effective method to plan dynamically-feasible trajectories in the image space, and we provide experimental results to demonstrate the proposed method under different operating conditions such as hovering, trajectory tracking, and perching.
  • Publication
    Catadioptric Projective Geometry
    (2001-12-01) Geyer, Christopher; Daniilidis, Kostas
    Catadioptric sensors are devices which utilize mirrors and lenses to form a projection onto the image plane of a camera. Central catadioptric sensors are the class of these devices having a single effective viewpoint. In this paper, we propose a unifying model for the projective geometry induced by these devices and we study its properties as well as its practical implications. We show that a central catadioptric projection is equivalent to a two-step mapping via the sphere. The second step is equivalent to a stereographic projection in the case of parabolic mirrors. Conventional lens-based perspective cameras are also central catadioptric devices with a virtual planar mirror and are, thus, covered by the unifying model. We prove that for each catadioptric projection there exists a dual catadioptric projection based on the duality between points and line images (conics). It turns out that planar and parabolic mirrors build a dual catadioptric projection pair. As a practical example we describe a procedure to estimate focal length and image center from a single view of lines in arbitrary position for a parabolic catadioptric system.