Michael J. Johnson

Measurement of retinal vascular tortuosity and its application to retinal pathologies

The tortuosity of retinal blood vessels is an important diagnostic indicator for a number of retinal pathologies. We applied robust quantitative tortuosity metrics, which are well suited to automated detection and measurement, to retinal fluorescein images of normal and diseased vessels exhibiting background diabetic retinopathy, retinitis pigmentosa and retinal vasculitis. We established the validity of the mean tortuosity (M) and the normalized root-mean-square tortuosity (K) by their strong correlation with the ranking of tortuosity by an expert panel of ophthalmologists. The low prevalences of the diseased conditions in the general population affect the classification process, and preclude the use of tortuosity for screening for all of these conditions simultaneously in the general population. Tortuosity may be useful as a screening test for retinitis alone, and may be useful for distinguishing diabetic retinopathy or vasculitis from normal in a discretionary (i.e. referred) population.

Scattered data reconstruction by regularization in B-spline and associated wavelet spaces

The problem of fitting a nice curve or surface to scattered, possibly noisy, data arises in many applications in science and engineering. In this paper we solve the problem using a standard regularized least square framework in an approximation space spanned by the shifts and dilates of a single compactly supported function @f. We first provide an error analysis to our approach which, roughly speaking, states that the error between the exact (probably unknown) data function and the obtained fitting function is small whenever the scattered samples have a high sampling density and a low noise level. We then give a computational formulation in the univariate case when @f is a uniform B-spline and in the bivariate case when @f is the tensor product of uniform B-splines. Though sparse, the arising system of linear equations is ill-conditioned; however, when written in terms of a short support wavelet basis with a well-chosen normalization, the resulting system, which is symmetric positive definite, appears to be well-conditioned, as evidenced by the fast convergence of the conjugate gradient iteration. Finally, our method is compared with the classical cubic/thin-plate smoothing spline methods via numerical experiments, where it is seen that the quality of the obtained fitting function is very much equivalent to that of the classical methods, but our method offers advantages in terms of numerical efficiency. We expect that our method remains numerically feasible even when the number of samples in the given data is very large.

The L 2 -approximation order of surface spline interpolation

We show that if the open, bounded domain Ω C R d has a sufficiently smooth boundary and if the data function, f is sufficiently smooth, then the L p (Ω)-norm of the error between f and its surface spline interpolant is O (δ γp+1/2 ) (1 ≤ p ≤ ∞), where γp:= min{m,m - d/2 + d/p} and m is an integer parameter specifying the surface spline. In case p = 2, this lower bound on the approximation order agrees with a previously obtained upper bound, and so we conclude that the L 2 -approximation order of surface spline interpolation is m + 1/2.

An upper bound on the approximation power of principal shift-invariant spaces

An upper bound on theLp-approximation power (1 ≤p ≤ ∞) provided by principal shift-invariant spaces is derived with only very mild assumptions on the generator. It applies to both stationary and nonstationary ladders, and is shown to apply to spaces generated by (exponential) box splines, polyharmonic splines, multiquadrics, and Gauss kernel.

On a stochastic demand jump inventory model

This paper considers a Quasi-Variational Inequality (QVI) arising from a stochastic demand jump inventory model in a continuous review setting with a fixed ordering cost and where demand is made up of a deterministic part (which is a function of the stock level) punctuated by random jumps. Under some restrictions on the parameters, a solution to the QVI is found which corresponds to an (s,S) policy.

Optimality of ( s , S ) policies for jump inventory models

This paper is concerned with the optimality of (s, S) policies for a single-item inventory control problem which minimizes the total expected cost over an infinite planning horizon and where the demand is driven by a piecewise deterministic process. Our approach is based on the theory of quasi-variational inequality.

Assessment of scoliosis by direct measurement of the curvature of the spine

We present two novel metrics for assessing scoliosis, in which the geometric centers of all the affected vertebrae in an antero-posterior (A-P) radiographic image are used. This is in contradistinction to the existing methods of using selected vertebrae, and determining either their endplates or the intersections of their diagonals, to define a scoliotic angle. Our first metric delivers a scoliotic angle, comparable to the Cobb and Ferguson angles. It measures the sum of the angles between the centers of the affected vertebrae, and avoids the need for an observer to decide on the extent of component curvatures. Our second metric calculates the normalized root-mean-square curvature of the smoothest path comprising piece-wise polynomial splines fitted to the geometric centers of the vertebrae. The smoothest path is useful in modeling the spinal curvature. Our metrics were compared to existing methods using radiographs from a group of twenty subjects with spinal curvatures of varying severity. Their values were strongly correlated with those of the scoliotic angles (r = 0.850 - 0.886), indicating that they are valid surrogates for measuring the severity of scoliosis. Our direct use of positional data removes the vagaries of determining variably shaped endplates, and circumvented the significant interand intra-observer errors of the Cobb and Ferguson methods. Although we applied our metrics to two-dimensional (2- D) data in this paper, they are equally applicable to three-dimensional (3-D) data. We anticipate that they will prove to be the basis for a reliable 3-D measurement and classification system.

Scattered Date Interpolation from Principal Shift-Invariant Spaces

Under certain assumptions on the compactly supported function @f@?C(R^d), we propose two methods of selecting a function s from the scaled principal shift-invariant space S^h(@f) such that s interpolates a given function f at a scattered set of data locations. For both methods, the selection scheme amounts to solving a quadratic programming problem and we are able to prove error estimates similar to those obtained by Duchon for surface spline interpolation.

An improved order of approximation for thin-plate spline interpolation in the unit disc

Summary. We show that the

A constructive framework for minimal energy planar curves

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