Asymptotic global confidence regions in parametric shape estimation problems (original) (raw)

Global confidence regions in parametric shape estimation

2000 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.00CH37100)

We introduce confidence region techniques for analyzing and visualizing the performance of two-dimensional parametric shape estimators. Assuming an asymptotically normal and efficient estimator for a finite parameterization of the object boundary, CramQ-Rao bounds are used to define a confidence region, centered around the true boundary. Computation of the probability that an entire boundary estimate lies within the confidence region is a challenging problem, because the estimate is a two-dimensional nonstationary random process. We derive lower bounds on this probability using level crossing statistics. The results make it possible to generate confidence regions for arbitrary prescribed probabilities. These global confidence regions conveniently display the uncertainty in various geometric parameters such as shape, size, orientation, and position of the estimated object, and facilitate geometric inferences. Numerical simulations suggest that the new bounds are quite tight.

Asymptotic Global Confidence Regions for 3-D Parametric Shape Estimation in Inverse Problems

IEEE Transactions on Image Processing, 2000

This paper derives fundamental performance bounds for statistical estimation of parametric surfaces embedded in 3 . Unlike conventional pixel-based image reconstruction approaches, our problem is reconstruction of the shape of binary or homogeneous objects. The fundamental uncertainty of such estimation problems can be represented by global confidence regions, which facilitate geometric inference and optimization of the imaging system. Compared to our previous work on global confidence region analysis for curves [two-dimensional (2-D) shapes], computation of the probability that the entire surface estimate lies within the confidence region is more challenging because a surface estimate is an inhomogeneous random field continuously indexed by a 2-D variable. We derive an asymptotic lower bound to this probability by relating it to the exceedence probability of a higher dimensional Gaussian random field, which can, in turn, be evaluated using the tube formula due to Sun. Simulation results demonstrate the tightness of the resulting bound and the usefulness of the three-dimensional global confidence region approach.

Parametric and semiparametric inference for shape: the role of the scale functional

Statistics & Decisions, 2006

We are considering the problem of efficient inference on the shape matrix of an elliptic distribution with unspecified location and either (a) fully specified radial density, (b) radial density specified up to a scale parameter, or (c) completely unspecified radial density. Bickel in [1] has shown that efficiencies under (b) and (c), while being strictly less than under (a), coincide: the efficiency loss caused by an unspecified radial density thus is entirely due to the non-specification of its scale (scale here is not necessarily measured by standard error, as second-order moments may be infinite). Defining scale however requires the choice of a particular scale functional, a choice which has an impact on efficiency bounds. We provide a closed form expression for this efficiency loss, both in hypothesis testing and in point estimation, as a function of the standardized radial density and the scale functional. We show that this loss is maximum at arbitrarily light tails whereas, under arbitrarily heavy tails, it is arbitrarily close to zero: hence, under very heavy tails, ignoring the scale (ignoring the exact density) asymptotically does not harm inference on shape. However, the same loss is nil, irrespective of the standardized radial density, when the scale functional (in dimension k) is the k-th root of the scatter determinant.

Cram er-Rao Bounds for Parametric Shape Estimation

We address the problem of computing fundamental performance bounds for estimation of object boundaries from noisy measurements in inverse problems, when the boundaries are parameterized by a finite number of unknown variables. Our model applies to multiple unknown objects, each with its own unknown gray level, or color, and boundary parameterization, on an arbitrary known background. While such fundamental bounds on the performance of shape estimation algorithms can in principle be derived from the Cramer-Rao lower bounds, very few results have been reported due to the difficulty of computing the derivatives of a functional with respect to shape deformation. In this paper, we provide a general formula for computing Cramer-Rao lower bounds in inverse problems where the observations are related to the object by a general linear transform, followed by a possibly nonlinear and noisy measurement system.

Estimation of a function under shape restrictions. Applications to reliability

The Annals of Statistics, 2005

This paper deals with a nonparametric shape respecting estimation method for U-shaped or unimodal functions. A general upper bound for the nonasymptotic L 1-risk of the estimator is given. The method is applied to the shape respecting estimation of several classical functions, among them typical intensity functions encountered in the reliability field. In each case, we derive from our upper bound the spatially adaptive property of our estimator with respect to the L 1-metric: it approximately behaves as the best variable binwidth histogram of the function under estimation.

Comparison of shape regression methods under landmark position uncertainty

2011

Despite the growing interest in regression based shape estimation, no study has yet systematically compared different regression methods for shape estimation. We aimed to fill this gap by comparing linear regression methods with a special focus on shapes with landmark position uncertainties. We investigate two scenarios: In the first, the uncertainty of the landmark positions was similar in the training and test dataset, whereas in the second the uncertainty of the training and test data were different. Both scenarios were tested on simulated data and on statistical models of the left ventricle estimating the end-systolic shape from end-diastole with landmark uncertainties derived from the segmentation process, and of the femur estimating the 3D shape from one projection with landmark uncertainties derived from the imaging setup. Results show that in the first scenario linear regression methods tend to perform similar. In the second scenario including estimates of the test shape landmark uncertainty in the regression improved results.

Robust estimation of shape parameters

1990

We investigate the use of Robust Estimation in an application requiring the accurate location of the centres of circular objects in an image. A common approach used throughout computer vision for extracting shape information from a data set is to fit a feature model using the Least Squares method. The well known sensitivity of this method to outliers is traditionally accommodated by outlier rejection methods. These usually consist of heuristic applications of model templates or data trimming.