scholarly journals COMPUTATION OF THE PERIMETER OF MEASURABLE SETS VIA THEIR COVARIOGRAM. APPLICATIONS TO RANDOM SETS

2011 ◽  
Vol 30 (1) ◽  
pp. 39 ◽  
Author(s):  
Bruno Galerne
Keyword(s):  
Unit Volume ◽  
Lipschitz Constant ◽  
Directional Derivative ◽  
Random Sets ◽  
Directional Derivatives ◽  
Boolean Models ◽  
Closed Set ◽  
Finite Perimeter ◽  
Measurable Set ◽  
Grain Distribution

The covariogram of a measurable set A ⊂ Rd is the function gA which to each y ∈ Rd associates the Lebesgue measure of A ∩ (y + A). This paper proves two formulas. The first equates the directional derivatives at the origin of gA to the directional variations of A. The second equates the average directional derivative at the origin of gA to the perimeter of A. These formulas, previously known with restrictions, are proved for any measurable set. As a by-product, it is proved that the covariogram of a set A is Lipschitz if and only if A has finite perimeter, the Lipschitz constant being half the maximal directional variation. The two formulas have counterparts for mean covariogram of random sets. They also permit to compute the expected perimeter per unit volume of any stationary random closed set. As an illustration, the expected perimeter per unit volume of stationary Boolean models having any grain distribution is computed.

scholarly journals Nearest points to closed sets and directional derivatives of distance functions

1989 ◽  
Vol 39 (2) ◽  
pp. 233-238 ◽  
Author(s):  
Simon Fitzpatrick
Keyword(s):  
Distance Function ◽  
Directional Derivative ◽  
Distance Functions ◽  
Directional Derivatives ◽  
Closed Set ◽  
Closed Sets ◽  
Set Of Points ◽  
Derivatives Of ◽  
Nearest Points ◽  
Dense Set

We investigate the circumstances under which the distance function to a closed set in a Banach space having a one-sided directional derivative equal to 1 or −1 implies the existence of nearest points. In reflexive spaces we show that at a dense set of points outside a closed set the distance function has a directional derivative equal to 1.

Generalized Gradients, Lipschitz behavior and Directional Derivatives

1985 ◽  
Vol 37 (6) ◽  
pp. 1074-1084 ◽  
Author(s):  
Jay S. Treiman
Keyword(s):  
Optimization Problems ◽  
Lower Semicontinuous ◽  
Directional Derivative ◽  
Directional Derivatives ◽  
Value Functions ◽  
Generalized Gradients ◽  
Closed Set ◽  
Nonconvex Functions ◽  
Closed Sets ◽  
Optimal Value

In the study of optimization problems it is necessary to consider functions that are not differentiable. This has led to the consideration of generalized gradients and a corresponding calculus for certain classes of functions. Rockafellar [16] and others have developed a very strong and elegant theory of subgradients for convex functions. This convex theory gives point-wise criteria for the existence of extrema in optimization problems.There are however many optimization problems that involve functions which are neither differentiable nor convex. Such functions arise in many settings including optimal value functions [15]. In order to deal with such problems Clarke [3] defined a type of subgradient for nonconvex functions. This definition was initially for Lipschitz functions on R”. Clarke extended this definition to include lower semicontinuous (l.s.c.) functions on Banach spaces through the use of a directional derivative, the distance function from a closed set and tangent and normal cones to closed sets.

scholarly journals ON THE ESTIMATION OF DISTANCE DISTRIBUTION FUNCTIONS FOR POINT PROCESSES AND RANDOM SETS

2011 ◽  
Vol 20 (1) ◽  
pp. 65 ◽  
Author(s):  
Dietrich Stoyan ◽  
Helga Stoyan ◽  
André Tscheschel ◽  
Torsten Mattfeldt
Keyword(s):  
Distribution Function ◽  
Point Processes ◽  
Distribution Functions ◽  
Distance Distribution ◽  
Random Sets ◽  
Boolean Models ◽  
Closed Set ◽  
Stationary Point Process ◽  
Contact Distribution ◽  
Nearest Neighbour Distance Distribution

This paper discusses various estimators for the nearest neighbour distance distribution function D of a stationary point process and for the quadratic contact distribution function Hq of a stationary random closed set. It recommends the use of Hanisch's estimator of D, which is of Horvitz-Thompson type, and the minussampling estimator of Hq. This recommendation is based on simulations for Poisson processes and Boolean models.

Asymptotic properties of stereological estimators of volume fraction for stationary random sets

1982 ◽  
Vol 19 (1) ◽  
pp. 111-126 ◽  
Author(s):  
Shigeru Mase
Keyword(s):  
Volume Fraction ◽  
Asymptotic Variance ◽  
Asymptotic Properties ◽  
Parametric Estimation ◽  
Random Sets ◽  
Stationary Time Series ◽  
Boolean Models ◽  
Window Functions ◽  
Spectral Density Functions ◽  
Point Estimators

We shall discuss asymptotic properties of stereological estimators of volume (area) fraction for stationary random sets (in the sense of Matheron) under natural and general assumptions. Results obtained are strong consistency, asymptotic normality, and asymptotic unbiasedness and consistency of asymptotic variance estimators. The method is analogous to the non-parametric estimation of spectral density functions of stationary time series using window functions. Proofs are given for areal estimators, but they are also valid for lineal and point estimators with slight modifications. Finally we show that stationary Boolean models satisfy the relevant assumptions reasonably well.

Variance reducing modifications for estimators of standardized moments of random sets

2000 ◽  
Vol 32 (03) ◽  
pp. 682-700
Author(s):  
Jeffrey D. Picka
Keyword(s):  
Statistical Analysis ◽  
Point Process ◽  
Random Sets ◽  
Boolean Models ◽  
Asymptotic Regime ◽  
Estimation Strategy ◽  
Moment Estimation ◽  
Second Moments ◽  
Simulation Results ◽  
Variance Result

In the statistical analysis of random sets, it is useful to have simple statistics that can be used to describe the realizations of these sets. The cumulants and several other standardized moments such as the correlation and second cumulant can be used for this purpose, but their estimators can be excessively variable if the most straightforward estimation strategy is used. Through exploitation of similarities between this estimation problem and a similar one for a point process statistic, two modifications are proposed. Analytical results concerning the effects of these modifications are found through use of a specialized asymptotic regime. Simulation results establish that the modifications are highly effective at reducing estimator standard deviations for Boolean models. The results suggest that the reductions in variance result from a balanced use of information in the estimation of the first and second moments, through eliminating the use of observations that are not used in second moment estimation.

scholarly journals MILES FORMULAE FOR BOOLEAN MODELS OBSERVED ON LATTICES

2011 ◽  
Vol 28 (2) ◽  
pp. 77 ◽  
Author(s):  
Joachim Ohser ◽  
Werner Nagel ◽  
Katja Schladitz
Keyword(s):  
Mean Curvature ◽  
Surface Density ◽  
Euler Number ◽  
Volume Density ◽  
Random Sets ◽  
Boolean Models ◽  
Intrinsic Volumes ◽  
Binary Images ◽  
Density Surface ◽  
The Mean

The densities of the intrinsic volumes – in 3D the volume density, surface density, the density of the integral of the mean curvature and the density of the Euler number – are a very useful collection of geometric characteristics of random sets. Combining integral and digital geometry we develop a method for efficient and simultaneous calculation of the intrinsic volumes of random sets observed in binary images in arbitrary dimensions. We consider isotropic and reflection invariant Boolean models sampled on homogeneous lattices and compute the expectations of the estimators of the intrinsic volumes. It turns out that the estimator for the surface density is proved to be asymptotically unbiased and thusmultigrid convergent for Boolean models with convex grains. The asymptotic bias of the estimators for the densities of the integral of the mean curvature and of the Euler number is assessed for Boolean models of balls of random diameters. Miles formulae with corresponding correction terms are derived for the 3D case.

Limit theorems for convex hulls of random sets

1993 ◽  
Vol 25 (02) ◽  
pp. 395-414 ◽  
Author(s):  
Ilya S. Molchanov
Keyword(s):  
Limit Theorems ◽  
Random Sets ◽  
Convex Hulls ◽  
Stable Sets ◽  
Limiting Distributions ◽  
Closed Set ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Random Closed Set ◽  
Simple Limit

Let , be i.i.d. random closed sets in . Limit theorems for their normalized convex hulls conv () are proved. The limiting distributions correspond to C-stable random sets. The random closed set A is called C-stable if, for any , the sets anA and conv ( coincide in distribution for certain positive an, compact Kn , and independent copies A 1, …, An of A. The distributions of C-stable sets are characterized via corresponding containment functionals.

scholarly journals Marginal Decomposition of Risk Measures

2006 ◽  
Vol 36 (2) ◽  
pp. 375-413
Author(s):  
Gary G. Venter ◽  
John A. Major ◽  
Rodney E. Kreps
Keyword(s):  
Value At Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Directional Derivative ◽  
Component Composition ◽  
Capital Allocation ◽  
Directional Derivatives ◽  
Total Risk ◽  
Business Decision ◽  
Marginal Risk

The marginal approach to risk and return analysis compares the marginal return from a business decision to the marginal risk imposed. Allocation distributes the total company risk to business units and compares the profit/risk ratio of the units. These approaches coincide when the allocation actually assigns the marginal risk to each business unit, i.e., when the marginal impacts add up to the total risk measure. This is possible for one class of risk measures (scalable measures) under the assumption of homogeneous growth and by a subclass (transformed probability measures) otherwise. For homogeneous growth, the allocation of scalable measures can be accomplished by the directional derivative. The first well known additive marginal allocations were the Myers-Read method from Myers and Read (2001) and co-Tail Value at Risk, discussed in Tasche (2000). Now we see that there are many others, which allows the choice of risk measure to be based on economic meaning rather than the availability of an allocation method. We prefer the term “decomposition” to “allocation” here because of the use of the method of co-measures, which quantifies the component composition of a risk measure rather than allocating it proportionally to something.Risk adjusted profitability calculations that do not rely on capital allocation still may involve decomposition of risk measures. Such a case is discussed. Calculation issues for directional derivatives are also explored.

scholarly journals Generalized contact distributions of inhomogeneous Boolean models

2002 ◽  
Vol 34 (01) ◽  
pp. 21-47 ◽  
Author(s):  
Daniel Hug ◽  
Günter Last ◽  
Wolfgang Weil
Keyword(s):  
Convex Body ◽  
Large Class ◽  
Random Vector ◽  
Boundary Point ◽  
Conditional Distribution ◽  
Boolean Model ◽  
Spatial Density ◽  
Boolean Models ◽  
Grain Distribution ◽  
Contact Distribution

The main purpose of this work is to study and apply generalized contact distributions of (inhomogeneous) Boolean models Z with values in the extended convex ring. Given a convex body L ⊂ ℝ d and a gauge body B ⊂ ℝ d , such a generalized contact distribution is the conditional distribution of the random vector (d B (L,Z),u B (L,Z),p B (L,Z),l B (L,Z)) given that Z∩L = ∅, where Z is a Boolean model, d B (L,Z) is the distance of L from Z with respect to B, p B (L,Z) is the boundary point in L realizing this distance (if it exists uniquely), u B (L,Z) is the corresponding boundary point of B (if it exists uniquely) and l B (L,·) may be taken from a large class of locally defined functionals. In particular, we pursue the question of the extent to which the spatial density and the grain distribution underlying an inhomogeneous Boolean model Z are determined by the generalized contact distributions of Z.

Integrating Feature Direction Information with a Level Set Formulation for Image Segmentation

2016 ◽  
Vol 6 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Meng Li ◽  
Yi Zhan
Keyword(s):  
Image Segmentation ◽  
Level Set ◽  
Directional Derivative ◽  
Image Features ◽  
Second Order ◽  
Directional Derivatives ◽  
Level Set Function ◽  
Variational Framework ◽  
Level Set Evolution ◽  
Level Set Formulation

AbstractA feature-dependent variational level set formulation is proposed for image segmentation. Two second order directional derivatives act as the external constraint in the level set evolution, with the directional derivative across the image features direction playing a key role in contour extraction and another only slightly contributes. To overcome the local gradient limit, we integrate the information from the maximal (in magnitude) second-order directional derivative into a common variational framework. It naturally encourages the level set function to deform (up or down) in opposite directions on either side of the image edges, and thus automatically generates object contours. An additional benefit of this proposed model is that it does not require manual initial contours, and our method can capture weak objects in noisy or intensity-inhomogeneous images. Experiments on infrared and medical images demonstrate its advantages.

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