The ideal of the trifocal variety

Techniques from representation theory, symbolic computational algebra, and numerical algebraic geometry are used to find the minimal generators of the ideal of the trifocal variety. An effective test for determining whether a given tensor is a trifocal tensor is also given

A Hilbert Scheme in Computer Vision

Multiview geometry is the study of two-dimensional images of three-dimensional scenes, a foundational subject in computer vision. We determine a universal Groebner basis for the multiview ideal of n generic cameras. As the cameras move, the multiview varieties vary in a family of dimension 11n-15. This family is the distinguished component of a multigraded Hilbert scheme with a unique Borel-fixed point. We present a combinatorial study of ideals lying on that Hilbert scheme.Comment: 26 page

A qcqp approach to triangulation

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Haptic fMRI: Combining Functional Neuroimaging with Haptics for Studying the Brain’s Motor Control Representation

Abstract — A challenging problem in motor control neuroimaging studies is the inability to perform complex human motor tasks given the Magnetic Resonance Imaging (MRI) scanner’s disruptive magnetic fields and confined workspace. In this paper, we propose a novel experimental platform that combines Functional MRI (fMRI) neuroimaging, haptic virtual simulation environments, and an fMRI-compatible haptic device for real-time haptic interaction across the scanner workspace (above torso ~.65x.40x.20m 3). We implement this Haptic fMRI platform with a novel haptic device, the Haptic fMRI Interface (HFI), and demonstrate its suitability for motor neuroimaging studies. HFI has three degrees-of-freedom (DOF), uses electromagnetic motors to enable high-fidelity haptic rendering (>350Hz), integrates radio frequency (RF) shields to prevent electromagnetic interference with fMRI (temporal SNR>100), and is kinematically designed to minimize currents induced by the MRI scanner’s magnetic field during motor displacement (<2cm). HFI possesses uniform inertial and force transmission properties across the workspace, and has low friction (.05–.30N). HFI’s RF noise levels, in addition, are within a 3 Tesla fMRI scanner’s baseline noise variation (~.85 ±.1%). Finally, HFI is haptically transparent and does not interfere with human motor tasks (tested for.4m reaches). By allowing fMRI experiments involving complex three-dimensional manipulation with haptic interaction, Haptic fMRI enables—for the first time—non-invasive neuroscience experiments involving interactive motor tasks, object manipulation, tactile perception, and visuo-motor integration. I