Andrew R. Willis

Accurately Estimating Sherd 3D Surface Geometry with Application to Pot Reconstruction

This paper deals with the problem of precise automatic estimation of the surface geometry of pot sherds uncovered at archaeological excavation sites using dense 3D laser-scan data. Critical to ceramic fragment analysis is the ability to geometrically classify excavated sherds, and, if possible, reconstruct the original pots using the sherd fragments. To do this, archaelogists must estimate the pot geometry in terms of an axis and associated profile curve from the discovered fragments. In this paper, we discuss an automatic method for accurately estimating an axis/profile curve pair for each archeological sherd (even when they are small) based on axially symmetric implicit polynomial surface models. Our method estimates the axis/profile curve for a sherd by finding the axially symmetric algebraic surface which best fits the measured set of dense 3D points and associated normals. We note that this method will work on 3D point data alone and does not require any local surface computations such as differentiation. Axis/profile curve estimates are accompanied by a detailed statistical error analysis. Estimation and error analysis are illustrated with application to a number of sherds. These fragments, excavated from Petra, Jordan, are chosen as exemplars of the families of geometrically diverse sherds commonly found on an archeological excavation site. We then briefly discuss how the estimation results may be integrated into a larger pot reconstruction program.

Bayesian Pot-Assembly from Fragments as Problems in Perceptual-Grouping and Geometric-Learning

A heretofore unsolved problem of great archaeological importance is the automatic assembly of pots made on a wheel from the hundreds (or thousands) of sherds found at an excavation site. An approach is presented to the automatic estimation of mathematical models of such pots from 3D measurements of sherds. A Bayesian approach is formulated beginning with a description of the complete set of geometric parameters that determine the distribution of the sherd measurement data. Matching of fragments and aligning them geometrically into configurations is based on matching break-curves (curves on a pot surface separating fragments), estimated axis and profile curve pairs for individual fragments and configurations of fragments, and a number of features of groups of break-curves. Pot assembly is a bottom-up maximum likelihood performance-based search. Experiments are illustrated on pots which were broken for the purpose, and on sherds from an archaeological dig located in Petra, Jordan. The performance measure can also be an aposteriori probability, and many other types of information can be included, e.g., pot wall thickness, surface color, patterns on the surface, etc. This can also be viewed as the problem of learning a geometric object from an unorganized set of free-form fragments of the object and of clutter, or as a problem of perceptual grouping.

Computational reconstruction of ancient artifacts

In this article, we discuss the development of automatic artifact reconstruction systems capable of coping with the realities of real-world geometric puzzles that anthropologists and archaeologists face on a daily basis. Such systems must do more than find matching fragments and subsequently align these matched fragments; these systems must be capable of simultaneously solving an unknown number of multiple puzzles where all of the puzzle pieces are mixed together in an unorganized pile and each puzzle may be missing an unknown number of its pieces. Discussion has cast the puzzle reconstruction problem into a generic terminology that is formalized appropriately for the 2-D and 3-D artifact reconstruction problems. Two leading approaches for 2-D tablet reconstruction and four leading approaches for 3-D object reconstruction have been discussed in detail, including partial or complete descriptions for the numerous algorithms upon which these systems rely. Several extensions to the geometric matching problem that use patterns apparent on the fragment outer surface were also discussed that generalize the problem beyond that of matching strictly geometry. The models needed for solving these problems are new and challenging, and most involve 3-D that is largely unexplored by the signal processing community. This work is highly relevant to the new 3-D signal processing that is looming on the horizon for tele-immersion.

Bayesian assembly of 3D axially symmetric shapes from fragments

We present a complete system for the purpose of automatically assembling 3D pots given 3D measurements of their fragments commonly called sherds. A Bayesian approach is formulated which, at present, models the data given a set of sherd geometric parameters. Dense sherd measurement data is obtained by scanning the outside surface of each sherd with a laser scanner. Mathematical models, specified by a set of geometric parameters, represent the sherd outer surface and break curves on the outer surface (where two sherds have broken apart). Optimal alignment of assemblies of sherds, called configurations, is implemented as maximum likelihood estimation (MLE) of the surface and break curve parameters given the measured sherd data for all sherds in a configuration. The assembly process starts with a fast clustering scheme which approximates the MLE solution for all sherd pairs, i.e., configurations of size 2, using a subspace of the geometric parameters, i.e., the sherd break curves. More accurate MLE values based on all parameters, i.e., sherd alignments, are computed when sherd pairs are merged with other sherd configurations. Merges take place in order of constant probability starting at the most probable configuration. This method is robust to missing sherds or groups of sherds which contain sherds from more than one pot. The system represents at least three significant advances over previous 3D puzzle solving approaches : (1) a Bayesian framework which allows for easily combining diverse types of information extracted from each sherd, (2) a search which reduces comparisons on unlikely configurations, and (3) a robust computationally reasonable method for aligning break curves and sherd outer surfaces simultaneously. In addition, a number of insights are given which have not previously been discussed and significantly reduce computation. Methods proposed for (1),(2), and (3) represent important contributions to the field of puzzle assembly, 3D geometry learning, and dataset alignment and are critical to making 3D puzzle solutions tractable to compute. Results are presented which include assembling a 13 sherd pot where only an incomplete set of 10 sherds is available.

Rapid prototyping 3D objects from scanned measurement data

It has become increasingly important to be able to generate free-form 3D shapes in commercial applications using rapid prototyping technologies. In many cases, the shapes of interest are taken from real-world objects that do not have pre-existing computer models. Constructing an accurate model for these objects by hand is extremely time consuming and difficult with even the latest 3D software packages. To aid in the modeling process, 3D scanners are used to capture the object shape and generate a high resolution model of the object. However, these models built from scans often have irregularities that prevent the construction of a useful prototype. This paper proposes a method for generating 3D models suitable for rapid prototyping from measurements of real-world objects taken by a 3D scanner. This is accomplished by taking a cloud of 3D point data as input and fitting a closed 3D surface to the data in such a way as to ensure accuracy in the representation of the object surface and compatibility with a rapid prototyping machine. We treat surface modeling and adaptation to the data in a new framework as 3D stochastic surface estimation.

A computational/experimental platform for investigating three-dimensional puzzle solving of comminuted articular fractures.

Reconstructing highly comminuted articular fractures poses a difficult surgical challenge, akin to solving a complicated three-dimensional (3D) puzzle. Preoperative planning using computed tomography (CT) is critically important, given the desirability of less invasive surgical approaches. The goal of this work is to advance 3D puzzle-solving methods towards use as a preoperative tool for reconstructing these complex fractures. A methodology for generating typical fragmentation/dispersal patterns was developed. Five identical replicas of human distal tibia anatomy were machined from blocks of high-density polyetherurethane foam (bone fragmentation surrogate), and were fractured using an instrumented drop tower. Pre- and post-fracture geometries were obtained using laser scans and CT. A semi-automatic virtual reconstruction computer program aligned fragment native (non-fracture) surfaces to a pre-fracture template. The tibiae were precisely reconstructed with alignment accuracies ranging from 0.03 to 0.4 mm. This novel technology has the potential to significantly enhance surgical techniques for reconstructing comminuted intra-articular fractures, as illustrated for a representative clinical case.

Cracks in Martian boulders exhibit preferred orientations that point to solar-induced thermal stress

The origins of fractures in Martian boulders are unknown. Here, using Mars Exploration Rover 3D data products, we obtain orientation measurements for 1,857 cracks visible in 1,573 rocks along the Spirit traverse and find that Mars rock cracks are oriented in statistically preferred directions similar to those compiled herein for Earth rock cracks found in mid-latitude deserts. We suggest that Martian directional cracking occurs due to the preferential propagation of microfractures favourably oriented with respect to repeating geometries of diurnal peaks in sun-induced thermal stresses. A numerical model modified here with Mars parameters supports this hypothesis both with respect to the overall magnitude of stresses as well as to the times of day at which the stresses peak. These data provide the first direct field and numerical evidence that insolation-related thermal stress potentially plays a principle role in cracking rocks on portions of the Martian surface.

An algebraic model for fast corner detection

This paper revisits the classical problem of detecting interest points, popularly known as “corners,” in 2D images by proposing a technique based on fitting algebraic shape models to contours in the edge image. Our method for corner detection is targeted for use on structural images, i.e., images that contain man-made structures for which corner detection algorithms are known to perform well. Further, our detector seeks to find image regions that contain two distinct linear contours that intersect. We define the intersection point as the corner, and, in contrast to previous approaches such as the Harris detector, we consider the spatial coherence of the edge points, i.e., the fact that the edge points must lie close to one of the two intersecting lines, an important aspect to stable corner detection. Comparisons between results for the proposed method and that for several popular feature detectors are provided using input images exhibiting a number of standard image variations, including blurring, affine transformation, scaling, rotation, and illumination variation. A modified version of the repeatability rate is proposed for evaluating the stability of the detector under these variations which requires a 1-to-1 mapping between matched features. Using this performance metric, our method is found to perform well in contrast to several current methods for corner detection. Discussion is provided that motivates our method of evaluation and provides an explanation for the observed performance of our algorithm in contrast to other algorithms. Our approach is distinct from other contour-based methods since we need only compute the edge image, from which we explicitly solve for the unknown linear contours and their intersections that provide image corner location estimates. The key benefits to this approach are: (1) performance (in space and time); since no image pyramid (space) and no edge-linking (time) is required and (2) compactness; the estimated model includes the corner location, and direction of the incoming contours in space, i.e., a complete model of the local corner geometry.

Surface sculpting with stochastic deformable 3D surfaces

This paper introduces a new stochastic surface model for deformable 3D surfaces and demonstrates its utility for the purpose of 3D sculpting. This is the problem of simple-to-use and intuitively interactive 3D free-form model building. A 3D surface is a sample of a Markov random field (MRF) defined on the vertices of a 3D mesh where MRF sites coincide with mesh vertices and the MRF cliques consist of subsets of sites. Each site has 3D coordinates (x,y,z) as random variables and is a member of one or more clique potentials which are functions of the vertices in a clique and describe stochastic dependencies among sites. Data, which is used to deform the surface can consist of, but is not limited to, an unorganized set of 3D points and is modeled by a conditional probability distribution given the 3D surface. A deformed surface is a MAP (maximum a posteriori probability) estimate of the joint distribution of the MRF surface model and the data. The generality and simplicity of the MRF model provides the ability to incorporate unlimited local and global deformation properties. Included in our development is the introduction of new data models, new anisotropic clique potentials, and cliques, which involve sites that are spatially far apart. Other applications of these models are possible, e.g., stereo reconstruction.

Alignment of multiple non-overlapping axially symmetric 3D datasets

An axially-symmetric surface is broken into disjoint pieces along a set of break-curves, i.e., the curves along which the surface locally breaks into two pieces. A subset of the pieces is available and for each of them we obtain noisy 3D measurements of its surface and break-curves. Using the piece measurements and knowledge of which pieces share a common break-curve, we propose a stochastic method for automatically estimating the unknown axially-symmetric global surface. Surface and break-curve estimation is then an alignment problem where we must estimate the unknown axially-symmetric surface and break-curves while simultaneously estimating the Euclidean transformation that positions each measured piece with respect to the a-priori unknown surface. Parameter estimation is implemented as maximum likelihood estimation where we seek the global pot geometry which best explains the measured fragment data. This new approach is robust, fast, and accurate. Experimental results are presented which solves an application of interest, specifically the reconstruction of archaeological pots from subsets of their surface pieces.