New Applications of (Old School) Projective Geometry

Since its origin as the underlying fundament of perspective drawings by Renaissance artists of the 15th–16th century, projective geometry has emerged as an elegant thought level for analyzing many geometric problems, getting rid of irrelevant details while focusing on the essential incidence structure.

During the previous century, PG was threatened by dust, suffering from an old-fashioned image, seemingly made redundant by the rise of differential and algebraic geometry. However, during the last decades of the 20th century, we observed a revival of PG in several disciplines:

(1) Geometric modeling and 3D-reconstruction,

(2) Analysis of mechanismes (generic/singular),

(3) Infinitesimal kinematics and statics (robot motion, rigidity of structures, etc.),

(4) Rendering for computer graphics/animation,

(5) Computer vision (motion-from-structure, structure-from-motion, intrinsic and extrinsic calibration, etc.).

Nevertheless, in recent years, the increasing computation power has enabled engineers to use deep learning techniques in mechanical control and advanced FEM in structural design and inspection and to process (2D/3D) digital images in real time. As a consequence, there was a tendancy to avoid feature-based approaches and to use less geometry, certainly in vision tasks such as people tracking or visual odometry.

Still, PG seems to have recovered from this (second) attack, realizing another revival. Indeed, modern types of range sensors and new techniques for sensor fusion using multicamera configurations require more general camera models and/or advanced calibration procedures (both intrinsic and extrinsic). As a consequence, more involved concepts of PG pop up in applications depending on 3D scanning and/or 3D reconstruction, often yielding a line-based approach rather than a point cloud perspective.

Equally important as the geometric concepts of PG, but not always recognized as such, is the algebra and the calculus of projective invariants, often providing exact, coordinate-free algorithms that can be effectively carried out by the current availability of powerful symbolic computation platforms. This tool appears to be of extreme importance for the study of the kinematics of mechanisms and the rigidity of structures, as it provides a higher level of qualitative reasoning in engineering and designing.

In this Special Issue, we encourage submissions presenting new results in or providing an inspiring overview of technological applications of projective geometry in fields belonging but not restricted to the following list:

• Computer vision: alternative calibration for classical models (spheres, incidence patterns, etc.) and general cameras (line congruences, etc.), 3D modeling and reconstruction (line interpolations, etc.), multicamera configurations, etc.;
• Robot kinematics, singularity analysis, rigidity of structures, etc.;
• Algebra of mechanics and computer vision performed in synthetic environments such as the bracket algebra, Grassmann algebra, etc.;
• Other engineering branches such as quantum mechanics, laser and nano technology, etc.

Prof. Dr. Rudi Penne
Guest Editor

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