16,078 research outputs found
Euclidean distance geometry and applications
Euclidean distance geometry is the study of Euclidean geometry based on the
concept of distance. This is useful in several applications where the input
data consists of an incomplete set of distances, and the output is a set of
points in Euclidean space that realizes the given distances. We survey some of
the theory of Euclidean distance geometry and some of the most important
applications: molecular conformation, localization of sensor networks and
statics.Comment: 64 pages, 21 figure
Eigenvector Synchronization, Graph Rigidity and the Molecule Problem
The graph realization problem has received a great deal of attention in
recent years, due to its importance in applications such as wireless sensor
networks and structural biology. In this paper, we extend on previous work and
propose the 3D-ASAP algorithm, for the graph realization problem in
, given a sparse and noisy set of distance measurements. 3D-ASAP
is a divide and conquer, non-incremental and non-iterative algorithm, which
integrates local distance information into a global structure determination.
Our approach starts with identifying, for every node, a subgraph of its 1-hop
neighborhood graph, which can be accurately embedded in its own coordinate
system. In the noise-free case, the computed coordinates of the sensors in each
patch must agree with their global positioning up to some unknown rigid motion,
that is, up to translation, rotation and possibly reflection. In other words,
to every patch there corresponds an element of the Euclidean group Euc(3) of
rigid transformations in , and the goal is to estimate the group
elements that will properly align all the patches in a globally consistent way.
Furthermore, 3D-ASAP successfully incorporates information specific to the
molecule problem in structural biology, in particular information on known
substructures and their orientation. In addition, we also propose 3D-SP-ASAP, a
faster version of 3D-ASAP, which uses a spectral partitioning algorithm as a
preprocessing step for dividing the initial graph into smaller subgraphs. Our
extensive numerical simulations show that 3D-ASAP and 3D-SP-ASAP are very
robust to high levels of noise in the measured distances and to sparse
connectivity in the measurement graph, and compare favorably to similar
state-of-the art localization algorithms.Comment: 49 pages, 8 figure
Riemannian Optimization for Distance-Geometric Inverse Kinematics
Solving the inverse kinematics problem is a fundamental challenge in motion
planning, control, and calibration for articulated robots. Kinematic models for
these robots are typically parametrized by joint angles, generating a
complicated mapping between the robot configuration and the end-effector pose.
Alternatively, the kinematic model and task constraints can be represented
using invariant distances between points attached to the robot. In this paper,
we formalize the equivalence of distance-based inverse kinematics and the
distance geometry problem for a large class of articulated robots and task
constraints. Unlike previous approaches, we use the connection between distance
geometry and low-rank matrix completion to find inverse kinematics solutions by
completing a partial Euclidean distance matrix through local optimization.
Furthermore, we parametrize the space of Euclidean distance matrices with the
Riemannian manifold of fixed-rank Gram matrices, allowing us to leverage a
variety of mature Riemannian optimization methods. Finally, we show that bound
smoothing can be used to generate informed initializations without significant
computational overhead, improving convergence. We demonstrate that our inverse
kinematics solver achieves higher success rates than traditional techniques,
and substantially outperforms them on problems that involve many workspace
constraints.Comment: 20 pages, 14 figure
GPU-based Iterative Cone Beam CT Reconstruction Using Tight Frame Regularization
X-ray imaging dose from serial cone-beam CT (CBCT) scans raises a clinical
concern in most image guided radiation therapy procedures. It is the goal of
this paper to develop a fast GPU-based algorithm to reconstruct high quality
CBCT images from undersampled and noisy projection data so as to lower the
imaging dose. For this purpose, we have developed an iterative tight frame (TF)
based CBCT reconstruction algorithm. A condition that a real CBCT image has a
sparse representation under a TF basis is imposed in the iteration process as
regularization to the solution. To speed up the computation, a multi-grid
method is employed. Our GPU implementation has achieved high computational
efficiency and a CBCT image of resolution 512\times512\times70 can be
reconstructed in ~5 min. We have tested our algorithm on a digital NCAT phantom
and a physical Catphan phantom. It is found that our TF-based algorithm is able
to reconstrct CBCT in the context of undersampling and low mAs levels. We have
also quantitatively analyzed the reconstructed CBCT image quality in terms of
modulation-transfer-function and contrast-to-noise ratio under various scanning
conditions. The results confirm the high CBCT image quality obtained from our
TF algorithm. Moreover, our algorithm has also been validated in a real
clinical context using a head-and-neck patient case. Comparisons of the
developed TF algorithm and the current state-of-the-art TV algorithm have also
been made in various cases studied in terms of reconstructed image quality and
computation efficiency.Comment: 24 pages, 8 figures, accepted by Phys. Med. Bio
Computing quasiconformal folds
We propose a novel way of computing surface folding maps via solving a linear
PDE. This framework is a generalization to the existing quasiconformal methods
and allows manipulation of the geometry of folding. Moreover, the crucial
quantity that characterizes the geometry occurs as the coefficient of the
equation, namely the Beltrami coefficient. This allows us to solve an inverse
problem of parametrizing the folded surface given only partial data but with
known folding topology. Various interesting applications such as fold sculpting
on 3D models and self-occlusion reasoning are demonstrated to show the
effectiveness of our method
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