5,034 research outputs found
Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples
In this paper, we consider the problem of recovering a compactly supported
multivariate function from a collection of pointwise samples of its Fourier
transform taken nonuniformly. We do this by using the concept of weighted
Fourier frames. A seminal result of Beurling shows that sample points give rise
to a classical Fourier frame provided they are relatively separated and of
sufficient density. However, this result does not allow for arbitrary
clustering of sample points, as is often the case in practice. Whilst keeping
the density condition sharp and dimension independent, our first result removes
the separation condition and shows that density alone suffices. However, this
result does not lead to estimates for the frame bounds. A known result of
Groechenig provides explicit estimates, but only subject to a density condition
that deteriorates linearly with dimension. In our second result we improve
these bounds by reducing the dimension dependence. In particular, we provide
explicit frame bounds which are dimensionless for functions having compact
support contained in a sphere. Next, we demonstrate how our two main results
give new insight into a reconstruction algorithm---based on the existing
generalized sampling framework---that allows for stable and quasi-optimal
reconstruction in any particular basis from a finite collection of samples.
Finally, we construct sufficiently dense sampling schemes that are often used
in practice---jittered, radial and spiral sampling schemes---and provide
several examples illustrating the effectiveness of our approach when tested on
these schemes
An Algebraic Perspective on Multivariate Tight Wavelet Frames. II
Continuing our recent work we study polynomial masks of multivariate tight
wavelet frames from two additional and complementary points of view: convexity
and system theory. We consider such polynomial masks that are derived by means
of the unitary extension principle from a single polynomial. We show that the
set of such polynomials is convex and reveal its extremal points as polynomials
that satisfy the quadrature mirror filter condition. Multiplicative structure
of such polynomial sets allows us to improve the known upper bounds on the
number of frame generators derived from box splines. In the univariate and
bivariate settings, the polynomial masks of a tight wavelet frame can be
interpreted as the transfer function of a conservative multivariate linear
system. Recent advances in system theory enable us to develop a more effective
method for tight frame constructions. Employing an example by S. W. Drury, we
show that for dimension greater than 2 such transfer function representations
of the corresponding polynomial masks do not always exist. However, for wavelet
masks derived from multivariate polynomials with non-negative coefficients, we
determine explicit transfer function representations. We illustrate our results
with several examples
- …