17,117 research outputs found
Spline Subdivision Schemes for Compact Sets. A Survey
Dedicated to the memory of our colleague Vasil Popov January 14, 1942 β May 31, 1990
* Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, IsraelAttempts at extending spline subdivision schemes to operate
on compact sets are reviewed. The aim is to develop a procedure for
approximating a set-valued function with compact images from a finite set of
its samples. This is motivated by the problem of reconstructing a 3D object
from a finite set of its parallel cross sections. The first attempt is limited to
the case of convex sets, where the Minkowski sum of sets is successfully applied
to replace addition of scalars. Since for nonconvex sets the Minkowski
sum is too big and there is no approximation result as in the case of convex
sets, a binary operation, called metric average, is used instead. With the
metric average, spline subdivision schemes constitute approximating operators
for set-valued functions which are Lipschitz continuous in the Hausdorff
metric. Yet this result is not completely satisfactory, since 3D objects are
not continuous in the Hausdorff metric near points of change of topology,
and a special treatment near such points has yet to be designed
Reconstructing Compact Metrizable Spaces
The deck, , of a topological space is the set
, where denotes the
homeomorphism class of . A space is (topologically) reconstructible if
whenever then is homeomorphic to . It is
known that every (metrizable) continuum is reconstructible, whereas the Cantor
set is non-reconstructible.
The main result of this paper characterises the non-reconstructible compact
metrizable spaces as precisely those where for each point there is a
sequence of pairwise disjoint
clopen subsets converging to such that and are homeomorphic
for each , and all and .
In a non-reconstructible compact metrizable space the set of -point
components forms a dense . For -homogeneous spaces, this condition
is sufficient for non-reconstruction. A wide variety of spaces with a dense
set of -point components are presented, some reconstructible and
others not reconstructible.Comment: 15 pages, 2 figure
Three dimensional structure from intensity correlations
We develop the analysis of x-ray intensity correlations from dilute ensembles
of identical particles in a number of ways. First, we show that the 3D particle
structure can be determined if the particles can be aligned with respect to a
single axis having a known angle with respect to the incident beam. Second, we
clarify the phase problem in this setting and introduce a data reduction scheme
that assesses the integrity of the data even before the particle reconstruction
is attempted. Finally, we describe an algorithm that reconstructs intensity and
particle density simultaneously, thereby making maximal use of the available
constraints.Comment: 17 pages, 9 figure
KMOS Data Flow: Reconstructing Data Cubes in One Step
KMOS is a multi-object near-infrared integral field spectrometer with 24
deployable pick-off arms. Data processing is inevitably complex. We discuss
specific issues and requirements that must be addressed in the data reduction
pipeline, the calibration, the raw and processed data formats, and the
simulated data. We discuss the pipeline architecture. We focus on its modular
style and show how these modules can be used to build a classical pipeline, as
well as a more advanced pipeline that can account for both spectral and spatial
flexure as well as variations in the OH background. A novel aspect of the
pipeline is that the raw data can be reconstructed into a cube in a single
step. We discuss the advantages of this and outline the way in which we have
implemented it. We finish by describing how the QFitsView tool can now be used
to visualise KMOS data.Comment: Contribution to "Ground-based and Airborne Instrumentation for
Astronomy III', SPIE 7735-254 (June 2010). High resolution version can be
found at http://spiedl.or
Numerical methods for coupled reconstruction and registration in digital breast tomosynthesis.
Digital Breast Tomosynthesis (DBT) provides an insight into the fine details of normal fibroglandular tissues and abnormal lesions by reconstructing a pseudo-3D image of the breast. In this respect, DBT overcomes a major limitation of conventional X-ray mam- mography by reducing the confounding effects caused by the superposition of breast tissue. In a breast cancer screening or diagnostic context, a radiologist is interested in detecting change, which might be indicative of malignant disease. To help automate this task image registration is required to establish spatial correspondence between time points. Typically, images, such as MRI or CT, are first reconstructed and then registered. This approach can be effective if reconstructing using a complete set of data. However, for ill-posed, limited-angle problems such as DBT, estimating the deformation is com- plicated by the significant artefacts associated with the reconstruction, leading to severe inaccuracies in the registration. This paper presents a mathematical framework, which couples the two tasks and jointly estimates both image intensities and the parameters of a transformation. Under this framework, we compare an iterative method and a simultaneous method, both of which tackle the problem of comparing DBT data by combining reconstruction of a pair of temporal volumes with their registration. We evaluate our methods using various computational digital phantoms, uncom- pressed breast MR images, and in-vivo DBT simulations. Firstly, we compare both iter- ative and simultaneous methods to the conventional, sequential method using an affine transformation model. We show that jointly estimating image intensities and parametric transformations gives superior results with respect to reconstruction fidelity and regis- tration accuracy. Also, we incorporate a non-rigid B-spline transformation model into our simultaneous method. The results demonstrate a visually plausible recovery of the deformation with preservation of the reconstruction fidelity
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