Michael M. Kazhdan

Poisson surface reconstruction

We show that surface reconstruction from oriented points can be cast as a spatial Poisson problem. This Poisson formulation considers all the points at once, without resorting to heuristic spatial partitioning or blending, and is therefore highly resilient to data noise. Unlike radial basis function schemes, our Poisson approach allows a hierarchy of locally supported basis functions, and therefore the solution reduces to a well conditioned sparse linear system. We describe a spatially adaptive multiscale algorithm whose time and space complexities are proportional to the size of the reconstructed model. Experimenting with publicly available scan data, we demonstrate reconstruction of surfaces with greater detail than previously achievable.

The Princeton Shape Benchmark

In recent years, many shape representations and geometric algorithms have been proposed for matching 3D shapes. Usually, each algorithm is tested on a different (small) database of 3D models, and thus no direct comparison is available for competing methods. We describe the Princeton Shape Benchmark (PSB), a publicly available database of polygonal models collected from the World Wide Web and a suite of tools for comparing shape matching and classification algorithms. One feature of the benchmark is that it provides multiple semantic labels for each 3D model. For instance, it includes one classification of the 3D models based on function, another that considers function and form, and others based on how the object was constructed (e.g., man-made versus natural objects). We find that experiments with these classifications can expose different properties of shape-based retrieval algorithms. For example, out of 12 shape descriptors tested, extended Gaussian images by B. Horn (1984) performed best for distinguishing man-made from natural objects, while they performed among the worst for distinguishing specific object types. Based on experiments with several different shape descriptors, we conclude that no single descriptor is best for all classifications, and thus the main contribution of this paper is to provide a framework to determine the conditions under which each descriptor performs best.

Rotation invariant spherical harmonic representation of 3D shape descriptors

One of the challenges in 3D shape matching arises from the fact that in many applications, models should be considered to be the same if they differ by a rotation. Consequently, when comparing two models, a similarity metric implicitly provides the measure of similarity at the optimal alignment. Explicitly solving for the optimal alignment is usually impractical. So, two general methods have been proposed for addressing this issue: (1) Every model is represented using rotation invariant descriptors. (2) Every model is described by a rotation dependent descriptor that is aligned into a canonical coordinate system defined by the model. In this paper, we describe the limitations of canonical alignment and discuss an alternate method, based on spherical harmonics, for obtaining rotation invariant representations. We describe the properties of this tool and show how it can be applied to a number of existing, orientation dependent descriptors to improve their matching performance. The advantages of this tool are two-fold: First, it improves the matching performance of many descriptors. Second, it reduces the dimensionality of the descriptor, providing a more compact representation, which in turn makes comparing two models more efficient.

A search engine for 3D models

As the number of 3D models available on the Web grows, there is an increasing need for a search engine to help people find them. Unfortunately, traditional text-based search techniques are not always effective for 3D data. In this article, we investigate new shape-based search methods. The key challenges are to develop query methods simple enough for novice users and matching algorithms robust enough to work for arbitrary polygonal models. We present a Web-based search engine system that supports queries based on 3D sketches, 2D sketches, 3D models, and/or text keywords. For the shape-based queries, we have developed a new matching algorithm that uses spherical harmonics to compute discriminating similarity measures without requiring repair of model degeneracies or alignment of orientations. It provides 46 to 245% better performance than related shape-matching methods during precision--recall experiments, and it is fast enough to return query results from a repository of 20,000 models in under a second. The net result is a growing interactive index of 3D models available on the Web (i.e., a Google for 3D models).

Modeling by example

In this paper, we investigate a data-driven synthesis approach to constructing 3D geometric surface models. We provide methods with which a user can search a large database of 3D meshes to find parts of interest, cut the desired parts out of the meshes with intelligent scissoring, and composite them together in different ways to form new objects. The main benefit of this approach is that it is both easy to learn and able to produce highly detailed geometric models -- the conceptual design for new models comes from the user, while the geometric details come from examples in the database. The focus of the paper is on the main research issues motivated by the proposed approach: (1) interactive segmentation of 3D surfaces, (2) shape-based search to find 3D models with parts matching a query, and (3) composition of parts to form new models. We provide new research contributions on all three topics and incorporate them into a prototype modeling system. Experience with our prototype system indicates that it allows untrained users to create interesting and detailed 3D models.

Screened poisson surface reconstruction

Poisson surface reconstruction creates watertight surfaces from oriented point sets. In this work we extend the technique to explicitly incorporate the points as interpolation constraints. The extension can be interpreted as a generalization of the underlying mathematical framework to a screened Poisson equation. In contrast to other image and geometry processing techniques, the screening term is defined over a sparse set of points rather than over the full domain. We show that these sparse constraints can nonetheless be integrated efficiently. Because the modified linear system retains the same finite-element discretization, the sparsity structure is unchanged, and the system can still be solved using a multigrid approach. Moreover we present several algorithmic improvements that together reduce the time complexity of the solver to linear in the number of points, thereby enabling faster, higher-quality surface reconstructions.

A Reflective Symmetry Descriptor for 3D Models

Abstract Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D model for all planes through the model’s center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. We show how to obtain a voxel grid from arbitrary 3D shapes and, using Fourier methods, we present an algorithm computes the symmetry descriptor in O(N4 log N) time for an N × N × N voxel grid and computes a multiresolution approximation in O(N3 log N) time. In our initial experiments, we have found that the symmetry descriptor is insensitive to noise and stable under point sampling. We have also found that it performs well in shape matching tasks, providing a measure of shape similarity that is orthogonal to existing methods.

Symmetry descriptors and 3D shape matching

In this paper, we present the Symmetry Descriptors of a 3D model. This is a collection of spherical functions that describes the measure of a models rotational and reflective symmetry with respect to every axis passing through the center of mass. We show that Symmetry Descriptors can be computed efficiently using fast signal processing techniques, and demonstrate the empirical value of Symmetry Descriptors by showing that they improve matching performance in a variety of shape retrieval experiments.

Streaming multigrid for gradient-domain operations on large images

We introduce a new tool to solve the large linear systems arising from gradient-domain image processing. Specifically, we develop a streaming multigrid solver, which needs just two sequential passes over out-of-core data. This fast solution is enabled by a combination of three techniques: (1) use of second-order finite elements (rather than traditional finite differences) to reach sufficient accuracy in a single V-cycle, (2) temporally blocked relaxation, and (3) multi-level streaming to pipeline the restriction and prolongation phases into single streaming passes. A key contribution is the extension of the B-spline finite-element method to be compatible with the forward-difference gradient representation commonly used with images. Our streaming solver is also efficient for in-memory images, due to its fast convergence and excellent cache behavior. Remarkably, it can outperform spatially adaptive solvers that exploit application-specific knowledge. We demonstrate seamless stitching and tone-mapping of gigapixel images in about an hour on a notebook PC.

A Reflective Symmetry Descriptor

Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper, we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D voxel model for all planes through the models center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. Using Fourier methods, our algorithm computes the symmetry descriptor in O(N4 logN) time for an N × N × N voxel grid, and computes a multiresolution approximation in O(N3 logN) time. In our initial experiments, we have found the symmetry descriptor to be useful for registration, matching, and classification of shapes.