high precision boundary length estimation by utilizing gray level information we present a novel method that provides an accurate and precise estimate of the length of the boundary perimeter of an object by taking into account gray levels on the boundary of the digitization of the same object assuming a model where pixel intensity is proportional to the coverage of a pixel we show that the presented method provides error free measurements of the length of straight boundary segments in the case of nonquantized pixel values for a more realistic situation where pixel values are quantized we derive optimal estimates that minimize the maximal estimation error we show that the estimate converges toward a correct value as the number of gray levels tends toward infinity the method is easy to implement we provide the complete pseudocode since the method utilizes only a small neighborhood it is very easy to parallelize we evaluate the estimator on a set of concave and convex shapes with known perimeters digitized at increasing resolution in addition we provide an example of applicability of the method on real images by suggesting appropriate preprocessing steps and presenting results of a comparison of the suggested method with other local approaches

rectilinearity measurements for polygons the paper introduces a shape measure intended to describe the extent to which a closed polygon is rectilinear other than somewhat obvious measures of rectilinearity e g the sum of the differences of each corner s angle from multiples of 90 spl deg there has been little work in deriving a measure that is straightforward to compute is invariant under scale rotation and translation and corresponds with the intuitive notion of rectilinear shapes there are applications in a number of different areas of computer vision and photogrammetry rectilinear structures often correspond to human made objects and are therefore justified as attentional cues for further processing for instance in aerial image processing and reconstruction where building footprints are often rectilinear on the local ground plane building structures once recognized as rectilinear can be matched to corresponding shapes in other views for stereo reconstruction perceptual grouping algorithms may seek to complete shapes based on the assumption that the object in question is rectilinear using the proposed measure such systems can verity this assumption

kodaira dimension and zeros of holomorphic one forms we show that every holomorphic one form on a smooth complex projective variety of general type must vanish at some point the proof uses generic vanishing theory for hodge modules on abelian varieties

fractal dimension of time indexed paths abstract measuring animal track or other trajectories for example the track of a flowing particle by divider method to obtain the fractal dimension without knowing the chronology of the points of the path one can obtain erroneous results the importance of time indexing of the track s point will be demonstrated by a simply geometrical example neglecting these indices the error can be high enough to mask relevant differences between various movement paths

logarithmic kodaira dimension and zeros of holomorphic log one forms in this paper we prove that the zero locus of any global holomorphic log one form on a projective log smooth pair x d of log general type must be non empty

a hu moment invariant as a shape circularity measure in this paper we propose a new circularity measure which defines the degree to which a shape differs from a perfect circle the new measure is easy to compute and being area based is robust e g with respect to noise or narrow intrusions also it satisfies the following desirable properties it ranges over 0 1 and gives the measured circularity equal to 1 if and only if the measured shape is a circle it is invariant with respect to translations rotations and scaling compared with the most standard circularity measure which considers the relation between the shape area and the shape perimeter the new measure performs better in the case of shapes with boundary defects which lead to a large increase in perimeter and in the case of compound shapes in contrast to the standard circularity measure the new measure depends on the mutual position of the components inside a compound shape also the new measure performs consistently in the case of shapes with very small i e close to zero measured circularity it turns out that such a property enables the new measure to measure the linearity of shapes in addition we propose a generalisation of the new measure so that shape circularity can be computed while controlling the impact of the relative position of points inside the shape an additional advantage of the generalised measure is that it can be used for detecting small irregularities in nearly circular shapes damaged by noise or during an extraction process in a particular image processing task

curvature estimation along noisy digital contours by approximate global optimization in this paper we introduce a new curvature estimator along digital contours which we called global min curvature gmc estimator as opposed to previous curvature estimators it considers all the possible shapes that are digitized as this contour and selects the most probable one with a global optimization approach the gmc estimator exploits the geometric properties of digital contours by using local bounds on tangent directions defined by the maximal digital straight segments the estimator is then adapted to noisy contours by replacing maximal segments with maximal blurred digital straight segments experiments on perfect and damaged digital contours are performed and in both cases comparisons with other existing methods are presented

projectivity of the moduli space of stable log varieties and subadditivity of log kodaira dimension reference epfl article 231100doi 10 1090 jams 871 url http dx doi org 10 1090 jams 871 record created on 2017 09 19 modified on 2017 10 02

a unified curvature definition for regular polygonal and digital planar curves in this paper we propose a new definition of curvature called visual curvature it is based on statistics of the extreme points of the height functions computed over all directions by gradually ignoring relatively small heights a multi scale curvature is obtained the theoretical properties and the experiments presented demonstrate that multi scale visual curvature is stable even in the presence of significant noise to our best knowledge the proposed definition of visual curvature is the first ever that applies to regular curves as defined in differential geometry as well as to turn angles of polygonal curves moreover it yields stable curvature estimates of curves in digital images even under sever distortions we also show a relation between multi scale visual curvature and convexity of simple closed curves

measuring linearity of planar point sets our goal is to design algorithms that give a linearity measure for planar point sets there is no explicit discussion on linearity in literature although some existing shape measures may be adapted we are interested in linearity measures which are invariant to rotation scaling and translation these linearity measures should also be calculated very quickly and be resistant to protrusions in the data set the measures of eccentricity and contour smoothness were adapted from literature the other five being triangle heights triangle perimeters rotation correlation average orientations and ellipse axis ratio the algorithms are tested on 30 sample curves and the results are compared against the linear classifications of these curves by human subjects it is found that humans and computers typically easily identify sets of points that are clearly linear and sets of points that are clearly not linear they have trouble measuring sets of points which are in the gray area in between although they appear to be conceptually very different approaches we prove theoretically and experimentally that eccentricity and rotation correlation yield exactly the same linearity measurements they however provide results which are furthest from human measurements the average orientations method provides the closest results to human perception while the other algorithms proved themselves to be very competitive

limit theorems for sums of random fuzzy sets we consider limit theorems for counting processes generated by minkowski sums of random fuzzy sets using support functions we prove almost sure convergence for a renewal process indexed by fuzzy sets in an inner product vector space we also get convergence for the associated containment renewal function

on shape of plane elastic curves we study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending stretching or compressing line segments non uniformly along their extensions shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape preserving transformations the elastic properties of the curves are encoded in riemannian metrics on these spaces geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques the shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated

the distance between shape centroids is less than a quarter of the shape perimeter in this paper we consider the distance between the shape centroid computed from the shape interior points and the shape centroid computed from the shape boundary points we show that the distance between those centroids is upper bounded by the quarter of the perimeter of the shape considered the obtained upper bound is sharp and cannot be improved next we introduce the shape centredness as a new shape descriptor which informally speaking should indicate to which degree a shape has a uniquely defined centre by exploiting the result mentioned above we give a formula for the computation of the shape centredness such a computed centredness is invariant with respect to translation rotation and scaling transformations

geometric moment invariants this paper derives moment invariants in an intuitive way by multiple integrals of invariant geometric primitives like distance area and volume many existing 2 d and 3 d moment invariants are re expressed to prove the correctness of this method furthermore this construction method can be easily extended to higher dimensional space and get higher order moment invariants explicit expressions of 3 d moment invariants are generated automatically with the help of symbolic computation software and can be regarded as a kind of shape descriptor for the representation of solid objects

a new convexity measure based on a probabilistic interpretation of images in this paper we present a novel convexity measure for object shape analysis the proposed method is based on the idea of generating pairs of points from a set and measuring the probability that a point dividing the corresponding line segments belongs to the same set the measure is directly applicable to image functions representing shapes and also to gray scale images which approximate image binarizations the approach introduced gives rise to a variety of convexity measures which make it possible to obtain more information about the object shape the proposed measure turns out to be easy to implement using the fast fourier transform and we would consider this in detail finally we illustrate the behavior of our measure in different situations and compare it to other similar ones

integral invariants for shape matching for shapes represented as closed planar contours we introduce a class of functionals which are invariant with respect to the euclidean group and which are obtained by performing integral operations while such integral invariants enjoy some of the desirable properties of their differential counterparts such as locality of computation which allows matching under occlusions and uniqueness of representation asymptotically they do not exhibit the noise sensitivity associated with differential quantities and therefore do not require presmoothing of the input shape our formulation allows the analysis of shapes at multiple scales based on integral invariants we define a notion of distance between shapes the proposed distance measure can be computed efficiently and allows warping the shape boundaries onto each other its computation results in optimal point correspondence as an intermediate step numerical results on shape matching demonstrate that this framework can match shapes despite the deformation of subparts missing parts and noise as a quantitative analysis we report matching scores for shape retrieval from a database

improved moment invariants for shape discrimination abstract moment invariants have been frequently used as features for shape recognition they are computed based on the information provided by both the shape boundary and its interior region although several fast algorithms for computing traditional moment invariants have been proposed none has ever shown the theoretical results of moment invariants computed based on the shape boundary only this paper proposes improved moment invariants computed using the shape boundary only which tremendously reduces computations the new moment invariants called improved moment invariants are mathematically proved to be invariant to scaling translation and rotation graphical plots of the first two improved moment invariants for six country maps and four industrial tools using improved moment invariants are also given the results suggest that improved moment invariants can be used as effective features for shape discrimination or recognition

viehweg s hyperbolicity conjecture for families with maximal variation we use the theory of hodge modules to construct viehweg zuo sheaves on base spaces of families with maximal variation and fibers of general type and more generally families whose geometric generic fiber has a good minimal model combining this with a result of campana p un we deduce viehweg s hyperbolicity conjecture in this context namely the fact that the base spaces of such families are of log general type this is approached as part of a general problem of identifying what spaces can support hodge theoretic objects with certain positivity properties