Irina Voiculescu

Comparison of interval methods for plotting algebraic curves

This paper compares the performance and efficiency of different function range interval methods for plotting f(x, y)=0 on a rectangular region based on a subdivision scheme, where f(x, y) is a polynomial. The solution of this problem has many applications in CAGD. The methods considered are interval arithmetic methods (using the power basis, Bernstein basis, Homer form and centred form), an affine arithmetic method, a Bernstein coefficient method, Taubins method, Rivlins method, Gopalsamys method, and related methods which also take into account derivative information. Our experimental results show that the affine arithmetic method, interval arithmetic using the centred form, the Bernstein coefficient method, Taubins method, Rivlins method, and their related derivative methods have similar performance, and generally they are more accurate and efficient than Gopalsamys method and interval arithmetic using the power basis, the Bernstein basis, and Horner form methods.

Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms

Implicit objects have gained increasing importance in geometric modeling, visualisation, animation, and computer graphics, because their geometric properties provide a good alternative to traditional parametric objects. This book presents the mathematics, computational methods and data structures, as well as the algorithms needed to render implicit curves and surfaces, and shows how implicit objects can easily describe smooth, intricate, and articulatable shapes, and hence why they are being increasingly used in graphical applications. Divided into two parts, the first introduces the mathematics of implicit curves and surfaces, as well as the data structures suited to store their sampled or discrete approximations, and the second deals with different computational methods for sampling implicit curves and surfaces, with particular reference to how these are applied to functions in 2D and 3D spaces.

Interval methods in geometric modeling

This paper is about using interval computations in location, simplification, and root-finding for multivariate implicit functions that are used as shape primitives in a set-theoretic (that is, a CSG) geometric modeller. Three problems are discussed, and solutions to them presented: the location and simplification of the surfaces of semialgebraic sets (surfaces involving some transcendental functions are dealt with as well); the generalization of Newton-Raphson using intervals; and interval ray-tracing. Examples are presented for both conventional three-dimensional geometric models and for CSG models in higher dimensions representing configuration-space maps for moving and colliding three-dimensional objects.

Interval and Affine Arithmetic for Surface Location of Power- and Bernstein-Form Polynomials

This paper describes a problem of interest in CSG modelling, namely the location of implicit polynomial surfaces in space. It is common for surfaces defined by implicits to be located using interval arithmetic. However, the method only gives conservative bounds for the values of the function inside a region of interest. This paper gives two possible ways of producing tighter bounds. One involves using a Bernstein–form representation of the implicit polynomials used as input to the method. The other fine–tunes the method itself by employing careful use of affine arithmetic—a more sophisticated version of interval arithmetic. As both methods contribute significant improvements, we speculate about combining the two into a fast and accurate method for surface location.

Two Tree−Based Methods for the Waterfall

Abstract The waterfall transform is a hierarchical segmentation technique based on the watershed transform from the field of mathematical morphology. Watershed-based techniques are useful in numerous fields ranging from image segmentation to cell-and-portal generation for games. The waterfall helps mitigate the problem of over-segmentation that commonly occurs when applying the basic watershed transform. It can also be used as a core part of a method for constructing image partition forests , a tree-based, multi-scale representation of an image. The best existing method for the waterfall is fast and effective, but our experience has been that it is not as straightforward to implement as might be desired. Furthermore, it does not deal consistently with the issue of non-minimal plateaux. This paper therefore proposes two new tree-based methods for the waterfall. Both are easier to implement than the existing state-of-the-art, and in our implementations, both were faster by a constant factor. The Simplified Waterfall (SW) method focuses on simplicity and ease of implementation; the Balanced Waterfall (BW) method focuses on robust handling of non-minimal plateaux. We perform experiments on both 2D and 3D images to contrast the new methods with each other and with the existing state-of-the-art, and show that both achieve a noticeable speed-up whilst producing similar results.

Empirical study of an evaluation-based subdivision algorithm for complex root isolation

We provide an empirical study of subdivision algorithms for isolating the simple roots of a polynomial in any desired box region B0 of the complex plane. One such class of algorithms is based on Newton-like interval methods (Moore, Krawczyk, Hansen-Sengupta). Another class of subdivision algorithms is based on function evaluation. Here, Yakoubsohn discussed a method that is purely based on an exclusion predicate. Recently, Sagraloff and Yap introduced another algorithm of this type, called Ceval. We describe the first implementation of Ceval in Core Library. We compare its performance to the above mentioned algorithms, and also to the well-known MPSolve software from Bini and Florentino. Our results suggest that certified evaluation-based methods such as Ceval are encouraging and deserve further exploration.

Region analysis of abdominal CT scans using image partition forests

The segmentation of medical scans (CT, MRI, etc.) and the subsequent identification of key features therein, such as organs and tumours, is an important precursor to many medical imaging applications. It is a difficult problem, not least because of the extent to which the shapes of organs can vary from one image to the next. One interesting approach is to start by partitioning the image into a region hierarchy, in which each node represents a contiguous region of the image. This is a well-known approach in the literature: the resulting hierarchy is variously referred to as a partition tree, an image tree, or a semantic segmentation tree. Such trees summarise the image information in a helpful way, and allow efficient searches for regions which satisfy certain criteria. However, once built, the hierarchy tends to be static, making the results very dependent on the initial tree construction process (which, in the case of medical images, is done independently of any anatomical knowledge we might wish to bring to bear). In this paper, we describe our approach to the automatic feature identification problem, in particular explaining why modifying the hierarchy at a later stage can be useful, and how it can be achieved. We illustrate the efficacy of our method with some preliminary results showing the automatic identification of ribs.

AFFINE INTERVALS IN A CSG GEOMETRIC MODELLER

Our CSG modeller, svLis, uses interval arithmetic to categorize implicit functions representing primitive shapes against boxes; this allows an efficient implementation of recursive spatial division to localize the primitives for a variety of purposes, such as rendering or the computation of integral properties.

Knowing when to give up: early-rejection stratagems in ligand docking

Virtual screening is an important resource in the drug discovery community, of which protein–ligand docking is a significant part. Much software has been developed for this purpose, largely by biochemists and those in related disciplines, who pursue ever more accurate representations of molecular interactions. The resulting tools, however, are very processor-intensive. This paper describes some initial results from a project to review computational chemistry techniques for docking from a non-chemistry standpoint. An abstract blueprint for protein–ligand docking using empirical scoring functions is suggested, and this is used to discuss potential improvements. By introducing computer science tactics such as lazy function evaluation, dramatic increases to throughput can and have been realized using a real-world docking program. Naturally, they can be extended to any system that approximately corresponds to the architecture outlined.

Automatic spine identification in abdominal CT slices using image partition forests

The identification of key features (e.g. organs and tumours) in medical scans (CT, MRI, etc.) is a vital first step in many other image analysis applications, but it is by no means easy to identify such features automatically. Using statistical properties of image regions alone, it is not always possible to distinguish between different features with overlapping greyscale distributions. To do so, it helps to make use of additional knowledge that may have been acquired (e.g. from a medic) about a patients anatomy. One important form this external knowledge can take is localization information: this allows a program to narrow down its search to a particular region of the image, or to decide how likely a feature candidate is to be correct (e.g. it would be worrisome were the aorta identified as running through the middle of a kidney). To make use of this information, however, it is necessary to identify a suitable frame of reference in which it can be specified. This frame should ideally be based on rigid structures, e.g. the spine and ribs. In this paper, we present a method for automatically identifying cross-sections of the spine in image partition forests of axial abdominal CT slices as a first step towards defining a robust coordinate system for localization.