This work extends the classical image-based gradient-domain 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 per-triangle metric tensor, the implementation supports edge-aware filtering by allowing the user to adjust the metric tensor using an esimate of the total curvature.
(SGP 09), (SIGGRAPH 16), (JCGT 16)
This work extends the classical image-based 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 in-between texture/signal at time τ∈[0,1] by flowing the source forward for time τ, flowing the target backward for time 1-τ, and then cross-disolving the advected textures/signals using weights 1-τ and τ.
This work provides an approach for performing color-correction 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 z-axis, followed by independent 2D gradient domain fusion to obtain slices that preserve the high-frequency 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 time-efficiency required for processing large 3D volumes.
This work provides an extended implementation of the classical Marching-Cubes algorithm, supporting a full-case table for resolving ambiguities, polygon / minimial-area-triangulation output, and Hermite data interpolation.
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 mesh-sharpening.
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 block-diagonalization 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 tri-diagonally banded so that they can be solved in linear time.
This work describes a method for reconstructing water-tight 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.
(ToG 13) (SGP 2006)
This work presents a modification to the traditional mean-curvature flow that
appears to evolve genus-zero 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 neck-pinches.
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 piecewise-constant
elements to represent the anisotropic scale, we support the editing of large
meshes at interactive rates.
This work develops an approach for efficiently evolving meshes using
mean-curvature flow. Using an octree-based finite-elements 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.
(Computer Graphics Forum 2011)
This work develops a metric-aware, 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 well-conditioned.
(SIGGRAPH Asia 2010)
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 one-trillion pixels.
This work explores a new formulation of finite-elements 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.
This work introduces a new tool for solving the large linear systems arising
from gradient-domain image processing. It develops a streaming multigrid solver,
which needs just two sequential passes over out-of-core data. The resulting
system can solve the huge linear systems associated with performing stitching and
tone-mapping on gigapixel images while maintaining a small in-core memory footprint.
Due to its fast convergence and excellent cache behavior, the streaming solver is also
efficient for in-memory images.
This work describes an out-of-core 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
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 edge-tree, 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.
This work describes a method for reconstructing water-tight 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 iso-surface 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 non-uniformly distributed.
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 signal-processing
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.
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 Wigner-D 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.
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
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