3 research outputs found
Single-angle Radon samples based reconstruction of functions in refinable shift-invariant space
The traditional approaches to computerized tomography (CT) depend on the
samples of Radon transform at multiple angles. In optics, the real time imaging
requires the reconstruction of an object by the samples of Radon transform at a
single angle (SA). Driven by this and motivated by the connection between Bin
Han's construction of wavelet frames (e.g [13]) and Radon transform, in
refinable shift-invariant spaces (SISs) we investigate the SA-Radon sample
based reconstruction problem. We have two main theorems. The fist main theorem
states that, any compactly supported function in a SIS generated by a general
refinable function can be determined by its Radon samples at an appropriate
angle. Motivated by the extensive application of positive definite (PD)
functions to interpolation of scattered data, we also investigate the SA
reconstruction problem in a class of (refinable) box-spline generated SISs.
Thanks to the PD property of the Radon transform of such spline, our second
main theorem states that, the reconstruction of compactly supported functions
in these spline generated SISs can be achieved by the samples of Radon
transform at almost every angle. Numerical simulation is conducted to check the
result
Framelet perturbation and application to nouniform sampling approximation for Sobolev space
The Sobolev space , where , is an important
function space that has many applications in various areas of research.
Attributed to the inertia of a measuring instrument, it is desirable in
sampling theory to reconstruct a function by its nonuniform samples. In the
present paper, we investigate the problem of constructing the approximation to
all the functions in with nonuniform samples by
utilizing dual framelet systems for the Sobolev space pair
. We first establish the
convergence rates of the framelet series in , and then construct the framelet approximation
operator holding for the entire space . Using the
approximation operator, any function in can be
approximated at the exponential rate with respect to the scale level. We
examine the stability property for the perturbations of the framelet
approximation operator with respect to shift parameters, and obtain an estimate
bound for the perturbation error. Our result shows that under the condition , the approximation operator is robust to the shift perturbation. These
results are used to establish the nonuniform sampling approximation for every
function in . In particular, the new nonuniform sampling
approximation error is robust to the jittering of the samples
Nonuniform sampling and approximation in Sobolev space from the perturbation of framelet system
The Sobolev space , where ,
is an important function space that has many applications in various areas of
research. Attributed to the inertia of a measurement instrument, it is
desirable in sampling theory to recover a function by its nonuniform sampling.
In the present paper, based on dual framelet systems for the Sobolev space pair
, where , we
investigate the problem of constructing the approximations to all the functions
in by nonuniform sampling. We first establish
the convergence rate of the framelet series in , and then construct the framelet approximation
operator that acts on the entire space . We
examine the stability property for the framelet approximation operator with
respect to the perturbations of shift parameters, and obtain an estimate bound
for the perturbation error. Our result shows that under the condition
, the approximation operator is robust to shift perturbations.
Motivated by some recent work on nonuniform sampling and approximation in
Sobolev space (e.g., [20]), we don't require the perturbation sequence to be in
. Our results allow us to establish the
approximation for every function in by
nonuniform sampling. In particular, the approximation error is robust to the
jittering of the samples.Comment: arXiv admin note: text overlap with arXiv:1707.0132