Surface Reconstruction This work describes a method for reconstructing watertight surfaces from an input of oriented points. It shows that the surface reconstruction algorithm presented by the FFT method can be expressed as a solution to a Poisson equation. Thus, by adapting an octree to the point set and solving the Poisson equation on the octree (rather than on a regular voxel grid) the algorithm provides a method for reconstructing much higher resolution models without incurring the prohibitive memory overhead exhibited by prior methods. The latest version also supports the incorporation of point constraints as a screening term, allowing for the reconstruction of more detailed surfaces within the same adapted Poisson framework. (SGP 2006, ToG 2013, CGF 2018, SGP 2020, CGF 2023) 

Exterior Poisson Reconstruction This work provides an extension of the standard Poisson Reconstruction algorithm that supports reconstruction of manifolds with codimension larger than one from framed samples. The approach computes a multivalued implicit function by formulating the reconstruction problem as a multilinear optimization problem expressed in terms of the exterior products of the gradients of the function and returns the zero levelset of the multivalued function. (SGP 2023) 

Variational Polynomial Shape Functions This work provides a way for computing finiteelement basis functions on polygonal and polyhedral domains. In addition to defining the basis functions, it provides applications including simulation of linear elasticity, numerical evaluation via the Franke test, geodesic computation using the Heat method, gradientdomain smoothing/sharpening of signals on the surface, and visualization of tangent vector fields through lineintegral convolution. (SIGGRAPH 2022) 

Euclidean Distance Transform This code provides implementations of the Euclidean Distance Transformation described in [Danielsson, 1990] and [Saito and Toriwaki, 1994]. The former produces higher quality results due to its use of subvoxel distance sampling. The latter computes distances at voxel precision (i.e. after rasterizing the mesh into a binary grid) but is noticeably faster. 

ECHO: Extended Convolution Histogram of Orientations for Local Surface Description This work provides a way for computing highly distinctive and robust local surface feature descriptor. An intrinsic signal and frame is associated with every point on the mesh. Then, the descriptor at a feature point is obtained by having each of the feature point's neighbor vote into the bin corresponding to the position of the feature point within the frame of the neighbor, with a weight equal to the value of the signal at the neighbor's position. (CGF 2021) 

Dense PointtoPoint Correspondences for GenusZero Surfaces This work provides a way for computing dense pointtopoint correspondences between two genuszero surfaces by decomposing the processing into a permodel parametrization phase and a perpairofmodels registration phase. In the parametrization phase surfaces are conformally mapped to the sphere, Möbius centered to remove ambiguity due to inversions, and authalically evolved to make the scale factors uniform. In the registration phase the surfaces are initially aligned for rotation and the alignment is then refined using optical flow. (SGP 2019) 

Texture Signal Processing This work provides a way for performing gradientdomain processing of signals represented over surface in 3D. To this end, we define finiteelements systems in texture spaces, addresing the problems of (1) metricawareness and (2) crossseam continuity. The advantages of this approach are twofold. First, it enables the processing of texture maps without resampling to mesh vertices. Second, the regularity of the texturespacde presentation enables an efficient hierarchical solver that supports processing of highresolution textures at interactive framerates. (SIGGRAPH 2018) 

Moebius Registration This work shows that the Moebius transformation registering two surfaces can be found efficiently. Breaking the transformation into two parts, inversion and rotation, we address the registration problem by handling these components separately. For inversion, we provide a new technique that canonically poses the parameterization. For rotation, we leverage earlier work on efficient spherical correlation to find the optimal alignment. (SGP 2018) 

Shape Gradient Domain This work extends the classical imagebased gradientdomain processing paradigm to the processing of signals on meshes, supporing the Laplacian smoothing/sharpening of color values and vertex positions. By formulating the Poisson equation in an intrinsic manner that only depends on the pertriangle metric tensor, the implementation supports edgeaware filtering by allowing the user to adjust the metric tensor using an esimate of the total curvature. (SGP 2009), (SIGGRAPH 2016), (JCGT 2016) 

Surface Optical Flow This work extends the classical imagebased optical flow comptuation to support processing of textures/signals on meshes. The code provides an application for interpolating between source and target textures/signals. Given the source and target, we compute the vector field whose forward flow takes the source to the target and whose backward flow takes the target to the source. We then define the inbetween texture/signal at time τ∈[0,1] by flowing the source forward for time τ, flowing the target backward for time 1τ, and then crossdisolving the advected textures/signals using weights 1τ and τ. (SIGGRAPH 2016) 

Gradient Domain Fusion This work provides an approach for performing colorcorrection of 3D EM data. The processing is decomposed into two phases: An initial smoothing pass to obtain a 3D dataset that is coherent along the zaxis, followed by independent 2D gradient domain fusion to obtain slices that preserve the highfrequency content of the input slices while exhibiting the coherence of the smoothed data. Separating the color correction into these two phases makes the approach trivially parallelizable anb provides the space and timeefficiency required for processing large 3D volumes. (ArXiv 2015) 

IsoSurface Extraction This work provides an extended implementation of the classical MarchingCubes algorithm, supporting a fullcase table for resolving ambiguities, polygon / minimialareatriangulation output, and Hermite data interpolation. (3DV 2015) 

Shock Filter Sharpening This work revisits the Shock Filters proposed by Osher and Rudin (1995) and shows how the PDE can be interpreted as the advction of the input signal. We show how this interpretation facilitates the theoretical analysis of the processing, as well provide extensions to the processing of signals on triangle meshes. Applying the technique to the normal field, we obtain a simple technique for meshsharpening. (SGP 2015) 

SurfaceofRevolution Poisson This work considers the problem of solving linear systems on geometries with symmetries. In particular, when the linear system commutes with the symmetry group, the decomposition of the function space into irreducible representations results in a blockdiagonalization of the matrix, replacing the solution of one large system with the solutions of many small ones. For Poisson equations on surfaces of revolution, the decomposition into irreducibles is obtained by computing a set of FFTs along the parallels of the surface, and the diagonal blcoks are tridiagonally banded so that they can be solved in linear time. (SGP 2015) 

Conformalized Mean Curvature Flow This work presents a modification to the traditional meancurvature flow that appears to evolve genuszero surface to conformal mappings onto the unit sphere. By adapting the flow to use a conformalized metric, the flow avoids the numerical instabilities arising in implementations of the traditional flow, slowing down the evolution in cylindrical regions to allow the surface to evolve without forming neckpinches. We further adapt the approach to compute conformal spherical orbifold embeddings of sphere and disklike surfaces by introducing symmetry constraints within the lienar system. (SGP 2012) 

Interactive Geometry Editor This work develops an approach for performing anisotropic geometry processing. Formulating geometry processing as the solution to a screened Poisson equation, using an efficient multigrid solver to solve the linear system, and using piecewiseconstant elements to represent the anisotropic scale, we support the editing of large meshes at interactive rates. (SIGGRAPH 2011) 

Interactive Element Flow This work develops an approach for efficiently evolving meshes using meancurvature flow. Using an octreebased finiteelements system, we track the flow of quadrature points and their Jacobians in order to be able to correctly define the Poisson system over the evolved geometry. (CGF 2011) 

MetricAware Spherical Image Processing This work develops a metricaware, streaming multigrid solver for efficiently processing equirectangular spherical images. The solver uses the symmetry of the parameterization to efficiently define the linear system and hierarchicaly adapts the tesselation of the sphere near the poles to ensure that the linear system remains wellconditioned. (SIGGRAPH Asia 2010) 

Distributed Multigrid Image Processing This work develops a distributed and streaming multigrid solver to efficiently process large planar or spherical images. The solver partitions images into bands, streams through these bands in parallel within a networked cluster, and schedules computation to hide the necessary synchronization latency. Using the solver, we can process images ranging from tens of millions up to onetrillion pixels. (ToG 2010) 

Surface Texture Stitching This work explores a new formulation of finiteelements over meshes. By considering the reconstruction of 3D elements defined over a voxel grid to the suface of the mesh, we can define a function space that inherits the regularity of the voxel grid, facilitating the design of a multigrid solver for solving the Poisson equation. (SGP 2009) 

Streaming Multigrid Image Processing This work introduces a new tool for solving the large linear systems arising from gradientdomain image processing. It develops a streaming multigrid solver, which needs just two sequential passes over outofcore data. The resulting system can solve the huge linear systems associated with performing stitching and tonemapping on gigapixel images while maintaining a small incore memory footprint. Due to its fast convergence and excellent cache behavior, the streaming solver is also efficient for inmemory images. (SIGGRAPH 2008) 

OutofCore Poisson Surface Reconstruction This work describes an outofcore method for performing Poisson surface reconstruction. We introduce a novel multilevel streaming octree representation that enables solving the global reconstruction problem by performing only three passes through the data. Since each pass only requires that local data is maintained in core, our approach provides a method for reconstructing surfaces when the initial point set, the output mesh, and the intermediate data structures are themselves too large to fit into working memory. (SGP 2007) 

IsoSurface Extraction from Octrees This work describes a method for extracting a watertight surfaces from an octree representation of an implicit function. Using the topology of the octree to define a set of binary edgetree, the work shows that inconsistencies due to depth disparities between adjacent leaf nodes can be resolved, and a watertight isosurface can be extracted without restricting either the topology or the values associated to the octree. (SGP 2007) 

FFTBased Surface Reconstruction This work describes a method for reconstructing watertight surfaces from an input of oriented points. The method reduces the problem of surface reconstruction to convolution, and provides an efficient method for reconstruction that reduces the reconstruction process to three simple steps: (1) splatting the oriented points into a voxel grid, (2) efficiently convolving with a fixed filter using the FFT, and (3) extracting an isosurface use marching cubes. The additive nature of the reconstruction makes it stable in the presence of noise, and a simple heuristic allows to work well when the points are nonuniformly distributed. (SGP 2005) 

Symmetry Detection This work describes a method for efficiently computing the symmetries of a model with respect to every axis passing through the model's center of mass. The SGP '04 paper describes how fast signalprocessing over S^2 and SO(3) can be used to compute the symmetry descriptors, and describes how the symmetry values can be used to improve retrieval performance. (SGP 2004) 

Affine Shape Alignment This work presents an approach for registering two 3D models by representing each one by an implicit function in 3D and using the fast spherical harmonic transform and fast WignerD transform to compute the correllation between the. invariant representations of 3D shapes. The SIGGRAPH '04 paper describes an application of this approach in the context of modeling by example. performance. (SIGGRAPH 2004) 

Shape Matching This work considers the limitations of canonical alignment and presents a new mathematical tool, based on spherical harmonics, for obtaining rotation invariant representations of 3D shapes. The SGP '03 paper describes the properties of this tool and shows how it can be applied to a number of existing, orientation dependent, descriptors to improve their matching performance. (SGP 2003) 

Anisotropic Scale Normalization This work describes an iterative method for transforming anisotropic models (models whose surface point variance is a function of direction) into isotropic models (models whose covariance matrix is a constant multiple of the identity matrix). The SIGGRAPH '04 paper describes applications of anisotropic factorization to the domain of shape matching, where classes of models that vary across anisotropy are difficult to match, due to the fact that often the wrong correspondences are established between points on the two surfaces. (SIGGRAPH 2004) 