New constraints on data-closeness and needle map consistency for shape-from-shading

This paper makes two contributions to the problem of needle-map recovery using shape-from-shading. First, we provide a geometric update procedure which allows the image irradiance equation to be satisfied as a hard constraint. This not only improves the data closeness of the recovered needle-map, but also removes the necessity for extensive parameter tuning. Second, we exploit the improved ease of control of the new shape-from-shading process to investigate various types of needle-map consistency constraint. The first set of constraints are based on needle-map smoothness. The second avenue of investigation is to use curvature information to impose topographic constraints. Third, we explore ways in which the needle-map is recovered so as to be consistent with the image gradient field. In each case we explore a variety of robust error measures and consistency weighting schemes that can be used to impose the desired constraints on the recovered needle-map. We provide an experimental assessment of the new shape-from-shading framework on both real world images and synthetic images with known ground truth surface normals. The main conclusion drawn from our analysis is that the data-closeness constraint improves the efficiency of shape-from-shading and that both the topographic and gradient consistency constraints improve the fidelity of the recovered needle-map

Force dipoles and stable local defects on fluid vesicles

An exact description is provided of an almost spherical fluid vesicle with a fixed area and a fixed enclosed volume locally deformed by external normal forces bringing two nearby points on the surface together symmetrically. The conformal invariance of the two-dimensional bending energy is used to identify the distribution of energy as well as the stress established in the vesicle. While these states are local minima of the energy, this energy is degenerate; there is a zero mode in the energy fluctuation spectrum, associated with area and volume preserving conformal transformations, which breaks the symmetry between the two points. The volume constraint fixes the distance $S$, measured along the surface, between the two points; if it is relaxed, a second zero mode appears, reflecting the independence of the energy on $S$; in the absence of this constraint a pathway opens for the membrane to slip out of the defect. Logarithmic curvature singularities in the surface geometry at the points of contact signal the presence of external forces. The magnitude of these forces varies inversely with $S$ and so diverges as the points merge; the corresponding torques vanish in these defects. The geometry behaves near each of the singularities as a biharmonic monopole, in the region between them as a surface of constant mean curvature, and in distant regions as a biharmonic quadrupole. Comparison of the distribution of stress with the quadratic approximation in the height functions points to shortcomings of the latter representation. Radial tension is accompanied by lateral compression, both near the singularities and far away, with a crossover from tension to compression occurring in the region between them.Comment: 26 pages, 10 figure

Balancing torques in membrane-mediated interactions: Exact results and numerical illustrations

Torques on interfaces can be described by a divergence-free tensor which is fully encoded in the geometry. This tensor consists of two terms, one originating in the couple of the stress, the other capturing an intrinsic contribution due to curvature. In analogy to the description of forces in terms of a stress tensor, the torque on a particle can be expressed as a line integral along any contour surrounding the particle. Interactions between particles mediated by a fluid membrane are studied within this framework. In particular, torque balance places a strong constraint on the shape of the membrane. Symmetric two-particle configurations admit simple analytical expressions which are valid in the fully nonlinear regime; in particular, the problem may be solved exactly in the case of two membrane-bound parallel cylinders. This apparently simple system provides some flavor of the remarkably subtle nonlinear behavior associated with membrane-mediated interactions.Comment: 16 pages, 10 figures, REVTeX4 style. The Gaussian curvature term was included in the membrane Hamiltonian; section II.B was rephrased to smoothen the flow of presentatio

Interface mediated interactions between particles -- a geometrical approach

Particles bound to an interface interact because they deform its shape. The stresses that result are fully encoded in the geometry and described by a divergence-free surface stress tensor. This stress tensor can be used to express the force on a particle as a line integral along any conveniently chosen closed contour that surrounds the particle. The resulting expression is exact (i.e., free of any "smallness" assumptions) and independent of the chosen surface parametrization. Additional surface degrees of freedom, such as vector fields describing lipid tilt, are readily included in this formalism. As an illustration, we derive the exact force for several important surface Hamiltonians in various symmetric two-particle configurations in terms of the midplane geometry; its sign is evident in certain interesting limits. Specializing to the linear regime, where the shape can be analytically determined, these general expressions yield force-distance relations, several of which have originally been derived by using an energy based approach.Comment: 18 pages, 7 figures, REVTeX4 style; final version, as appeared in Phys. Rev. E. Compared to v2 several minor mistakes, as well as one important minus sign in Eqn. (18a) have been cured. Compared to v1, this version is significantly extended: Lipid tilt degrees of freedom for membranes are included in the stress framework, more technical details are given, estimates for the magnitude of forces are mad

Minimal-area metrics on the Swiss cross and punctured torus

The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2\pi. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics in a given homotopy class form a band. The extremal metric is unknown when bands of geodesics cross, as it happens for surfaces of non-zero genus. We use recently proposed convex programs to numerically find the minimal-area metric on the square torus with a square boundary, for various sizes of the boundary. For large enough boundary the problem is equivalent to the "Swiss cross" challenge posed by Strebel. We find that the metric is positively curved in the two-band region and flat in the single-band regions. For small boundary the metric develops a third band of geodesics wrapping around it, and has both regions of positive and negative curvature. This surface can be completed to provide the minimal-area metric on a once-punctured torus, representing a closed-string tadpole diagram.Comment: 59 pages, 41 figures. v2: Minor edits and reference update

Terrain analysis using radar shape-from-shading

This paper develops a maximum a posteriori (MAP) probability estimation framework for shape-from-shading (SFS) from synthetic aperture radar (SAR) images. The aim is to use this method to reconstruct surface topography from a single radar image of relatively complex terrain. Our MAP framework makes explicit how the recovery of local surface orientation depends on the whereabouts of terrain edge features and the available radar reflectance information. To apply the resulting process to real world radar data, we require probabilistic models for the appearance of terrain features and the relationship between the orientation of surface normals and the radar reflectance. We show that the SAR data can be modeled using a Rayleigh-Bessel distribution and use this distribution to develop a maximum likelihood algorithm for detecting and labeling terrain edge features. Moreover, we show how robust statistics can be used to estimate the characteristic parameters of this distribution. We also develop an empirical model for the SAR reflectance function. Using the reflectance model, we perform Lambertian correction so that a conventional SFS algorithm can be applied to the radar data. The initial surface normal direction is constrained to point in the direction of the nearest ridge or ravine feature. Each surface normal must fall within a conical envelope whose axis is in the direction of the radar illuminant. The extent of the envelope depends on the corrected radar reflectance and the variance of the radar signal statistics. We explore various ways of smoothing the field of surface normals using robust statistics. Finally, we show how to reconstruct the terrain surface from the smoothed field of surface normal vectors. The proposed algorithm is applied to various SAR data sets containing relatively complex terrain structure

Nonlinear morphoelastic plates II: exodus to buckled states

Morphoelasticity is the theory of growing elastic materials. This theory is based on the multiple decomposition of the deformation gradient and provides a formulation of the deformation and stresses induced by growth. Following a companion paper, a general theory of growing nonlinear elastic Kirchhoff plate is described. First, a complete geometric description of incompatibility with simple examples is given. Second, the stability of growing Kirchhoff plates is analyzed