Wicksell's Problem Research Articles

Adaptive and efficient isotonic estimation in Wicksell's problem

We consider nonparametric estimation in Wicksell's problem, which has applications in astronomy for estimating the distribution of star positions in a galaxy and in material sciences for determining a material's 3D microstructure from 2D cross sections. We focus on the isotonised version of the plug-in estimator (IIE) for the cdf F of the spheres' squared radii. This estimator is fully automatic, requiring no tuning parameters, and we show it is adaptive to local smoothness properties of the distribution function F to be estimated. We also prove a local asymptotic minimax lower bound in this non-standard setting, with log ⁡ n / n -asymptotics and where the functional F to be estimated is not regular. Combined, our results prove that the isotonic estimator (IIE) is an adaptive, easy-to-compute, and efficient estimator.

Nonparametric confidence bands in Wicksell's problem

Nonparametric confidence bands in Wicksell's problem

Wicksell's Problem in Local Stereology

Wicksell's classical corpuscle problem deals with the retrieval of the size distribution of spherical particles from planar sections. We discuss the problem in a local stereology framework. Each particle is assumed to contain a reference point and the individual particle is sampled with an isotropic random plane through this reference point. Both the size of the section profile and the position of the reference point inside the profile are recorded and used to recover the distribution of the corresponding particle parameters. Theoretical results concerning the relationship between the profile and particle parameters are discussed. We also discuss the unfolding of the arising integral equations, uniqueness issues, and the domain of attraction relations. We illustrate the approach by providing reconstructions from simulated data using numerical unfolding algorithms.

Wicksell's Problem in Local Stereology

Wicksell's classical corpuscle problem deals with the retrieval of the size distribution of spherical particles from planar sections. We discuss the problem in a local stereology framework. Each particle is assumed to contain a reference point and the individual particle is sampled with an isotropic random plane through this reference point. Both the size of the section profile and the position of the reference point inside the profile are recorded and used to recover the distribution of the corresponding particle parameters. Theoretical results concerning the relationship between the profile and particle parameters are discussed. We also discuss the unfolding of the arising integral equations, uniqueness issues, and the domain of attraction relations. We illustrate the approach by providing reconstructions from simulated data using numerical unfolding algorithms.

Thinning and harvesting in stochastic forest models

This paper analyzes a stochastic forest growth model in which the manager is able to first thin the forest to promote better growth before harvesting. Both Wicksell single thinning and harvesting cycle and Faustmann on-going rotation problems are considered. The Wicksell problem is analyzed by first restricting the class of decision times to (thinning, harvesting) pairs that bound the growth away from infinity and imbedding the problem in an infinite-dimensional linear program on a space of triplets of measures. These measures capture the thinning and harvesting decisions along with the behavior of the growth process prior to harvest. An auxiliary linear program then leads to a nonlinear optimization problem for which an optimal value and solution are determined. The values of all the problems are be related through a set of inequalities. The solution of the nonlinear problem determines (random) thinning and harvesting times for the single thinning and harvesting cycle which demonstrate the equality of the values of these various problems. Finally for the Wicksell problem, the unrestricted class of thinning and harvest times is shown to give the same value as the restricted class. The Faustmann on-going thinning and harvesting rotation problem is reduced to a Wicksell problem which then allows for the characterization of the value as the solution to a different nonlinear optimization problem. The effects of the opportunity to thin the forest are illustrated on a mean-reverting stochastic model.

Needlet algorithms for estimation in inverse problems

We provide a new algorithm for the treatment of inverse problems which combines the traditional SVD inversion with an appropriate thresholding technique in a well chosen new basis. Our goal is to devise an inversion procedure which has the advantages of localization and multiscale analysis of wavelet representations without losing the stability and computability of the SVD decompositions. To this end we utilize the construction of localized frames (termed "needlets") built upon the SVD bases. We consider two different situations: the "wavelet" scenario, where the needlets are assumed to behave similarly to true wavelets, and the "Jacobi-type" scenario, where we assume that the properties of the frame truly depend on the SVD basis at hand (hence on the operator). To illustrate each situation, we apply the estimation algorithm respectively to the deconvolution problem and to the Wicksell problem. In the latter case, where the SVD basis is a Jacobi polynomial basis, we show that our scheme is capable of achieving rates of convergence which are optimal in the $L_2$ case, we obtain interesting rates of convergence for other $L_p$ norms which are new (to the best of our knowledge) in the literature, and we also give a simulation study showing that the NEED-D estimator outperforms other standard algorithms in almost all situations.

Porosity and pore size control in starch consolidation casting of oxide ceramics—Achievements and problems

The possibilities and limits of porosity and pore size control via starch consolidation casting (SCC) are discussed from a principal point of view. The results for alumina ceramics indicate that porosity control between 25 and 50% is feasible, while lower and higher porosities are difficult to achieve by SCC. The main factor of pore size control is the selected starch type, although swelling should be taken into account for a more precise size control. Of the starch types investigated here, potato starch is the largest (resulting in pore sizes of 50 μm and higher) and corn starch is the smallest (14 μm), while wheat starch is intermediate (20 μm). A quantitative comparison of pore size results, however, is complicated by Wicksell's problem and (in the case of potato and wheat starch) the anisometric shape (median aspect ratios of 1.3 and 2.0 for prolate and oblate shape, respectively).

Adaptive minimax estimation of a fractional derivative

This paper considers a problem of adaptation in estimating a fractional antiderivative of an unknown drift density from observations in Gaussian white noise. This problem is closely related to the Wicksell problem. Under the assumption that the drift density belongs to a Sobolev class with unknown smoothness, an adaptive estimator is constructed.

B-splines and discretization in an inverse problem for Poisson processes

Recent results on quasi-maximum likelihood histogram sieve estimators in inverse problems for Poisson processes are generalized to B-spline sieves. The impact of discretization effects on strong L 2 consistency and convergence rates are studied in detail. In particular, a “rates saturation effect”, caused by discretization, is demonstrated. Finite-sample implementation is proposed and tested in a Monte Carlo experiment with the Wicksell problem, which shows a superior performance of the new approach, when compared to other methods commonly used in that context. The proposed algorithm can also be used in cases with only approximately known folding kernel.

Doubly Smoothed EM Algorithm for Statistical Inverse Problems

Recent results on discretization effects in unfolding intensity function of an indirectly observed nonhomogeneous Poisson process show that a critical feasibility condition for the strong L2 consistency of the histogram sieve maximum likelihood estimates requires that the approximation error of the function of interest with step functions tend to 0 faster than the squared singular values of the discretized folding operator. This condition may not be fulfilled in some standard inverse problems like, for example, the deconvolution problem and the Wicksell problem discretized in a standard way. The condition may be satisfied, however, with suitably modified discrete operator and suitably constructed parametric sets for the discrete problems. Motivated by these results, an additional smoothing step is added to the EMS (expectation-maximization-smoothing) algorithm and an automatic procedure for the choice of a smoothing parameter is proposed. An application to the Wicksell problem is presented. In this context, the theory also suggests a special, nonuniform binning in the data space. A simulation study demonstrates that the new approach considerably reduces the L2 risk when compared to both the standard EMS algorithm and to a spectral procedure specially designed for the Wicksell problem. A priori knowledge of the value of the estimated function at 0 may further reduce the risk very significantly. Theoretical results on the strong L2 consistency and convergence rates are given.

A wavelet–vaguelet method for unfolding sphere size distributions

We consider the classical Wicksell problem of estimating an unknown probabilitydensity function of the radii of spheres in a medium, based on their observedrandom cross-sections. This problem is known as Wicksell’s corpuscle problem.We first convert this problem to a form suitable for the application of thresholdingvaguelet–wavelet methods for the solution of ill posed integral equations givennoisy data. We outline the disadvantages of using standard kernel-type orcross-validated spline smoothing methods to solve this problem. A detaileddescription of the wavelet–vaguelet method applied to the Wicksell problem isgiven. The method relies upon a specifically built family of vaguelets.

On the Minimax Estimation Problem of a Fractional Derivative

We consider the problem of minimax estimating the fractional derivative of the order~$-\frac{1}{2}$ of an unknown function in the Gaussian white noise model. This problem is closely related to the well-known Wicksell problem. In this paper the second-order minimax approach is developed.

Sieved Maximum Likelihood Estimation in Wicksell's Problem and Related Deconvolution Problems

It is shown that the classical Wicksell problem is related to a deconvolution problem where the convolution kernel is unbounded, convex and decreasing on (0, ∞). For that type of deconvolution problems, the usual non‐parametric maximum likelihood estimator of the distribution function is shown not to exist. A sieved maximum likelihood estimator is defined, and some algorithms are described that can be used to compute this estimator. Moreover, this estimator is proved to be strongly consistent.

Asymptotically Efficient Smoothing in the Wicksell Problem under Squared Losses

In the Wicksell problem, it is required to reconstruct a distribution function of radii of balls located in an opaque medium from measurements of radii of circles obtained by intersecting the medium with a certain plane. This problem is intimately bound up with estimating a fractional derivative of order 1/2 for a distribution function concentrated on the positive semi-axis. In this paper, the locally asymptotically minimax risk in the Wicksell problem is evaluated up to a constant. Estimators on which this risk is attained are also constructed.

Methods for Density Estimation in Thick-Slice Versions of Wicksell's Problem

Abstract A new, implicit method is suggested for density estimation in inverse problems, where data are drawn not from the target distribution, but rather from its image under a transformation. The approach that we propose produces density estimators that are themselves densities, without the negativity problems known to plague more explicit inversion techniques. We also suggest a general empirical approach to selecting the smoothing parameter so as to optimize performance in the context of the target distribution, rather than its image after the transformation. We apply the new methods, and competing techniques, to a thick-section Wicksell-type problem, using data on the radii of nerve terminals from the electric organ of the electric ray Torpedo marmorata. It is shown that statistical properties of estimators in this problem are very different from those for the thin-slice, classical Wicksell problem, and so the two cases cannot be developed simply by analogy with one another.

Methods for Density Estimation in Thick-Slice Versions of Wicksell's Problem

A new, implicit method is suggested for density estimation in inverse problems, where data are drawn not from the target distribution, but rather from its image under a transformation. The approach that we propose produces density estimators that are themselves densities, without the negativity problems known to plague more explicit inversion techniques. We also suggest a general empirical approach to selecting the smoothing parameter so as to optimize performance in the context of the target distribution, rather than its image after the transformation. We apply the new methods, and competing techniques, to a thick-section Wicksell-type problem, using data on the radii of nerve terminals from the electric organ of the electric ray Torpedo marmorata. It is shown that statistical properties of estimators in this problem are very different from those for the thin-slice, classical Wicksell problem, and so the two cases cannot be developed simply by analogy with one another.

Some comments on Wicksell's problem

In this paper we derive some properties of inverse estimators in the Wicksell unfolding problem, based on a regularized version of the inverse of the preconditioned operator. We focus on the integrated mean square error, data-driven selection of the regularization parameter, and asymptotic efficiency of linear functionals in the Hájek–LeCam sense.

Stereological analysis and modelling of gradient structures

Gradient structures are inhomogeneous along a particular gradient direction but homogeneous perpendicular to that direction. Consequently, structural parameters such as volume fraction or surface area density are local characteristics which depend on the 'vertical' coordinate with respect to the 'vertical' gradient axis. Analogously, models for gradient structures have model parameters depending on the vertical coordinate z. For example, a Voronoi tessellation with a gradient is generated by a gradient point process with a local intensity which is a function of z. Similarly, a gradient germ grain model is obtained from a gradient point process where the grain size distribution may also depend on z. For a gradient Boolean model, local volume fraction VV(z) and local surface area density SV(z) can be calculated from the model parameters. Stereological methods for gradient structures are based on vertical sections parallel to the gradient direction. Estimation of VV(z), SV(z) and local length density LV(z) is done by lineal analysis using horizontal test lines with vertical coordinate z. Similarly, lineal analysis is used to estimate local mean cell volume of gradient tessellations. For the estimation of local particle number density and size in the spirit of the Wicksell problem the use of kernel methods and distributional assumptions is required.

Asymptotically efficient estimation in the Wicksell problem

We consider the classical Wicksell problem of estimating an unknown distribution function $G$of the radii of balls, based on their observed crosssections. It is assumed that the underlying distribution function$G$ belongs to a Holder class of smoothness $\gamma >1/2$. We prove that, for a suitable choice of the smoothing parameters, kernel-type estimators are asymptotically efficient for a large class of symmetric bowl-shaped loss functions.

A Method for Parametric Estimation of the Number and Size Distribution of Cell Clusters from Observations in a Section Plane

The problem of finding the number and size distribution of cell clusters that grow in an organ or tissue from observations of the number and sizes of transections of such cell clusters in a planar section is considered. This problem is closely related to the well-known corpuscle or Wicksell problem in stereology, which deals with transections of spherical objects. However, for most biological applications, it is unrealistic to assume that cell clusters have spherical shapes since they may grow in various ways. We therefore propose a method that allows for more general spatial configurations of the clusters. Under the assumption that a parametric growth model is available for the number and sizes of the cell clusters, expressions are obtained for the probability distributions of the number and sizes of transections of the clusters in a section plane for each point in time. These expressions contain coefficients that are independent of the parametric growth model and time but depend on which model is chosen for the configuration of the cell clusters in space. These results enable us to perform estimation of the parameters of the growth model by maximum likelihood directly on the data instead of having to deal with the inverse problem of estimation of three-dimensional quantities based on two-dimensional data. For realistic choices of the configuration model, it will not be possible to obtain the exact values of the coefficients, but they can easily be approximated by means of computer simulations of the spatial configuration. Monte Carlo simulations were performed to approximate the coefficients for two particular spatial configuration models. For these two configuration models, the proposed method is applied to data on preneoplastic minifoci in rat liver under the assumption of a two-event model of carcinogenesis as the parametric growth model.