Extracting 3D parametric curves from 2D images of Helical objects

Helical objects occur in medicine, biology, cosmetics, nanotechnology, and engineering. Extracting a 3D parametric curve from a 2D image of a helical object has many practical applications, in particular being able to extract metrics such as tortuosity, frequency, and pitch. We present a method that is able to straighten the image object and derive a robust 3D helical curve from peaks in the object boundary. The algorithm has a small number of stable parameters that require little tuning, and the curve is validated against both synthetic and real-world data. The results show that the extracted 3D curve comes within close Hausdorff distance to the ground truth, and has near identical tortuosity for helical objects with a circular profile. Parameter insensitivity and robustness against high levels of image noise are demonstrated thoroughly and quantitatively

Curve shortening-straightening flow for non-closed planar curves with infinite length

We consider a motion of non-closed planar curves with infinite length. The motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature. We call the flow shortening-straightening flow. In this paper, first we prove a long time existence result for the shortening-straightening flow for non-closed planar curves with infinite length. Then we show that the solution converges to a stationary solution as time goes to infinity. Moreover we give a classification of the stationary solution

More Cappell-Shaneson spheres are standard

Akbulut has recently shown that an infinite family of Cappell-Shaneson homotopy 4-spheres is diffeomorphic to the standard 4-sphere. In the present paper, a strictly larger family is shown to be standard by a simpler method. This new approach uses no Kirby calculus except through the relatively simple 1979 paper of Akbulut and Kirby showing that the simplest example with untwisted framing is standard. Instead, hidden symmetries of the original Cappell-Shaneson construction are exploited. In the course of the proof, we give an example showing that Gluck twists can sometimes be undone using symmetries of fishtail neighborhoods.Comment: 11 pages, 2 figures. This (v2) is essentially the published version, with minor mathematical improvements over v

An index for Brouwer homeomorphisms and homotopy Brouwer theory

We use the homotopy Brouwer theory of Handel to define a Poincar{\'e} index between two orbits for an orientation preserving fixed point free homeomorphism of the plane. Furthermore, we prove that this index is almost additive.Comment: To appear in Ergodic Theory and Dynamical System

Numerical Schubert calculus

We develop numerical homotopy algorithms for solving systems of polynomial equations arising from the classical Schubert calculus. These homotopies are optimal in that generically no paths diverge. For problems defined by hypersurface Schubert conditions we give two algorithms based on extrinsic deformations of the Grassmannian: one is derived from a Gr\"obner basis for the Pl\"ucker ideal of the Grassmannian and the other from a SAGBI basis for its projective coordinate ring. The more general case of special Schubert conditions is solved by delicate intrinsic deformations, called Pieri homotopies, which first arose in the study of enumerative geometry over the real numbers. Computational results are presented and applications to control theory are discussed.Comment: 24 pages, LaTeX 2e with 2 figures, used epsf.st

Hierarchical nanomechanics of collagen microfibrils

Collagen constitutes one third of the human proteome, providing mechanical stability, elasticity and strength to connective tissues. Collagen is also the dominating material in the extracellular matrix (ECM) and is thus crucial for cell differentiation, growth and pathology. However, fundamental questions remain with respect to the origin of the unique mechanical properties of collagenous tissues, and in particular its stiffness, extensibility and nonlinear mechanical response. By using x-ray diffraction data of a collagen fibril reported by Orgel et al. (Proceedings of the National Academy of Sciences USA, 2006) in combination with protein structure identification methods, here we present an experimentally validated model of the nanomechanics of a collagen microfibril that incorporates the full biochemical details of the amino acid sequence of the constituting molecules. We report the analysis of its mechanical properties under different levels of stress and solvent conditions, using a full-atomistic force field including explicit water solvent. Mechanical testing of hydrated collagen microfibrils yields a Young’s modulus of ≈300 MPa at small and ≈1.2 GPa at larger deformation in excess of 10% strain, in excellent agreement with experimental data. Dehydrated, dry collagen microfibrils show a significantly increased Young’s modulus of ≈1.8 to 2.25 GPa (or ≈6.75 times the modulus in the wet state) owing to a much tighter molecular packing, in good agreement with experimental measurements (where an increase of the modulus by ≈9 times was found). Our model demonstrates that the unique mechanical properties of collagen microfibrils can be explained based on their hierarchical structure, where deformation is mediated through mechanisms that operate at different hierarchical levels. Key mechanisms involve straightening of initially disordered and helically twisted molecules at small strains, followed by axial stretching of molecules, and eventual molecular uncoiling at extreme deformation. These mechanisms explain the striking difference of the modulus of collagen fibrils compared with single molecules, which is found in the range of 4.8±2 GPa or ≈10-20 times greater. These findings corroborate the notion that collagen tissue properties are highly scale dependent and nonlinear elastic, an issue that must be considered in the development of models that describe the interaction of cells with collagen in the extracellular matrix. A key impact the atomistic model of collagen microfibril mechanics reported here is that it enables the bottom-up elucidation of structure-property relationships in the broader class of collagen materials such as tendon or bone, including studies in the context of genetic disease where the incorporation of biochemical, genetic details in material models of connective tissue is essential

An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry - Theory and Application