Global confidence regions in parametric shape estimation (original) (raw)

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.

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.

Shape Estimation from Support and Diameter Functions

Journal of Mathematical Imaging and Vision, 2006

We address the problem of reconstructing a planar shape from a finite number of noisy measurements of its support function or its diameter function. New linear and non-linear algorithms are proposed, based on the parametrization of the shape by its Extended Gaussian Image. This parametrization facilitates a systematic statistical analysis of the problem via the Cramér-Rao lower bound (CRLB), which provides a fundamental lower bound on the performance of estimation algorithms. Using CRLB, we also generate confidence regions which conveniently display the effect of parameters like eccentricity, scale, noise, and measurement direction set, on the quality of the estimated shapes, as well as allow a performance analysis of the algorithms.

Cramer-Rao bounds for parametric shape estimation in inverse problems

IEEE Transactions on Image Processing, 2003

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 Cramér-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 Cramér-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. As an illustration, we derive explicit formulas for computed tomography, Fourier imaging, and deconvolution problems. The bounds reveal that highly accurate parametric reconstructions are possible in these examples, using severely limited and noisy data.

On the uncertainty analysis of shape reconstruction from areas of silhouettes

2003

We address the inverse problem of shape reconstruction of a convex object from noisy measurements of the areas of its silhouettes (shadows) in several directions. Such data represent values of the brightness function of the object, which in the 2-D case is simply a phase-shifted version of its diameter (width) function. In the past, we have proposed non-linear and linear algorithms for reconstructing an ndimensional convex body using finitely many noisy measurements of its brightness function. Here we carry out a statistical uncertainty analysis of the problem for the 2-D case by generating asymptotic confidence regions around the underlying shape. Confidence regions conveniently display the effect of experimental parameters like eccentricity, scale, noise power, viewing direction set, on the quality of the estimated shape. We also present a statistical performance analysis of our proposed linear algorithm using global confidence regions.

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.

Shape from support-type functions: Algorithms and statistical analysis

We address the problem of reconstructing a shape from a finite number of noisy measurements of its support function or its diameter function. New linear and non-linear algorithms are proposed, based on the parametrization of the shape by its Extended Gaussian Image (EGI). A systematic statistical analysis of the algorithms via the Cramér-Rao lower bound is carried out and from this confidence regions are also generated. These confidence regions conveniently display the effect of parameters like eccentricity, scale, noise, and measurement direction set, on the quality of the estimated shapes, as well as allowing a performance analysis of the algorithms. A byproduct of the statistical analysis is the introduction of a new and better method for reconstructing a planar shape from its EGI, a problem of considerable interest in its own right. This paper is motivated by the problem of reconstructing an unknown planar shape from a finite number of noisy measurements of its support function or its diameter function. Given a measurement direction (i.e., a unit vector, or angle), the corresponding support function measurement gives the (signed) distance from some fixed reference point (usually taken to be the origin) to the support line to the shape orthogonal to the direction; see Fig. 2. The corresponding diameter function measurement provides the distance between the two support lines parallel to this direction; see Fig. 3. We shall refer to support and diameter functions collectively as support-type functions. In view of the data, it is natural to focus on convex bodies. Support function data arise in a variety of physical experiments and therefore have been studied by researchers with diverse interests. Prince and Willsky [31] used such data in computerized tomography as a prior to improve performance, particularly when only limited data is available. Lele, Kulkarni, and Willsky [21] applied support function measurements to target reconstruction from range-resolved and Doppler-resolved laser-radar data. The general approach in these papers is to fit a polygon to the data, in contrast to that of Fisher, Hall, Turlach, and Watson [2], who use spline interpolation and the so-called von Mises kernel to fit a smooth curve to the data. This method was taken up in [12] and [26], the former dealing with convex bodies with corners and the latter giving an example to show that the algorithm in [2] may fail for a given data set. Further studies and applications can be found in [9], [10], and [15]. Support function data has also featured in robotics via the notion of a line probe; however, the focus here has been on algorithmic complexity issues under the assumption of exact rather than noisy data (see, for example, [25] and [33]).

Bayesian estimation of the shape skeleton

Proceedings of The National Academy of Sciences, 2006

Biol 38:205-287], because of their potential to provide a compact, but meaningful, shape representation, suitable for both neural modeling and computational applications. But effective computation of the shape skeleton remains a notorious unsolved problem; existing approaches are extremely sensitive to noise and give counterintuitive results with simple shapes. In conventional approaches, the skeleton is defined by a geometric construction and computed by a deterministic procedure. We introduce a Bayesian probabilistic approach, in which a shape is assumed to have ''grown'' from a skeleton by a stochastic generative process. Bayesian estimation is used to identify the skeleton most likely to have produced the shape, i.e., that best ''explains'' it, called the maximum a posteriori skeleton. Even with natural shapes with substantial contour noise, this approach provides a robust skeletal representation whose branches correspond to the natural parts of the shape.