14 research outputs found
Sampling and Reconstruction of Shapes with Algebraic Boundaries
We present a sampling theory for a class of binary images with finite rate of
innovation (FRI). Every image in our model is the restriction of
\mathds{1}_{\{p\leq0\}} to the image plane, where \mathds{1} denotes the
indicator function and is some real bivariate polynomial. This particularly
means that the boundaries in the image form a subset of an algebraic curve with
the implicit polynomial . We show that the image parameters --i.e., the
polynomial coefficients-- satisfy a set of linear annihilation equations with
the coefficients being the image moments. The inherent sensitivity of the
moments to noise makes the reconstruction process numerically unstable and
narrows the choice of the sampling kernels to polynomial reproducing kernels.
As a remedy to these problems, we replace conventional moments with more stable
\emph{generalized moments} that are adjusted to the given sampling kernel. The
benefits are threefold: (1) it relaxes the requirements on the sampling
kernels, (2) produces annihilation equations that are robust at numerical
precision, and (3) extends the results to images with unbounded boundaries. We
further reduce the sensitivity of the reconstruction process to noise by taking
into account the sign of the polynomial at certain points, and sequentially
enforcing measurement consistency. We consider various numerical experiments to
demonstrate the performance of our algorithm in reconstructing binary images,
including low to moderate noise levels and a range of realistic sampling
kernels.Comment: 12 pages, 14 figure
An Inner-Product Calculus for Periodic Functions and Curves
Our motivation is the design of efficient algorithms to process closed curves represented by basis functions or wavelets. To that end, we introduce an inner-product calculus to evaluate correlations and distances between such curves. In particular, we present formulas for the direct and exact evaluation of correlation matrices in the case of closed (i.e., periodic) parametric curves and periodic signals. We give simplifications for practical cases that involve B-splines. To illustrate this approach, we also propose a least-squares approximation scheme that is able to resample curves while minimizing aliasing artifacts. Another application is the exact calculation of the enclosed area
The Generalized Operator Based Prony Method
The generalized Prony method introduced by Peter & Plonka (2013) is a
reconstruction technique for a large variety of sparse signal models that can
be represented as sparse expansions into eigenfunctions of a linear operator
. However, this procedure requires the evaluation of higher powers of the
linear operator that are often expensive to provide.
In this paper we propose two important extensions of the generalized Prony
method that simplify the acquisition of the needed samples essentially and at
the same time can improve the numerical stability of the method. The first
extension regards the change of operators from to , where
is an analytic function, while and possess the same
set of eigenfunctions. The goal is now to choose such that the powers
of are much simpler to evaluate than the powers of . The second
extension concerns the choice of the sampling functionals. We show, how new
sets of different sampling functionals can be applied with the goal to
reduce the needed number of powers of the operator (resp. ) in
the sampling scheme and to simplify the acquisition process for the recovery
method.Comment: 31 pages, 2 figure
Sampling curves with finite rate of innovation
In this paper, we extend the theory of sampling signals with finite rate of innovation (FRI) to a specific class of two-dimensional curves, which are defined implicitly as the zeros of a mask function. Here the mask function has a parametric representation as a weighted summation of a finite number of complex exponentials, and therefore, has finite rate of innovation . An associated edge image, which is discontinuous on the predefined parametric curve, is proved to satisfy a set of linear annihilation equations. We show that it is possible to reconstruct the parameters of the curve (i.e., to detect the exact edge positions in the continuous domain) based on the annihilation equations. Robust reconstruction algorithms are also developed to cope with scenarios with model mismatch. Moreover, the annihilation equations that characterize the curve are linear constraints that can be easily exploited in optimization problems for further image processing (e.g., image up-sampling). We demonstrate one potential application of the annihilation algorithm with examples in edge-preserving interpolation. Experimental results with both synthetic curves as well as edges of natural images clearly show the effectiveness of the annihilation constraint in preserving sharp edges, and improving SNRs
FRI Sampling With Arbitrary Kernels
This paper addresses the problem of sampling non-bandlimited signals within the Finite Rate of Innovation (FRI) setting. We had previously shown that, by using sampling kernels whose integer span contains specific exponentials (generalized Strang-Fix conditions), it is possible to devise non-iterative, fast reconstruction algorithms from very low-rate samples. Yet, the accuracy and sensitivity to noise of these algorithms is highly dependent on these exponential reproducing kernels â actually, on the exponentials that they reproduce. Hence, our first contribution here is to provide clear guidelines on how to choose the sampling kernels optimally, in such a way that the reconstruction quality is maximized in the presence of noise. The optimality of these kernels is validated by comparing with CrameÌr-Raoâs lower bounds (CRB). Our second contribution is to relax the exact exponential reproduction requirement. Instead, we demonstrate that arbitrary sampling kernels can reproduce the âbest â exponentials within quite a high accuracy in general, and that applying the exact FRI algorithms in this approximate context results in near-optimal reconstruction accuracy for practical noise levels. Essentially, we propose a universal extension of the FRI approach to arbitrary sampling kernels. Numerical results checked against the CRB validate the various contributions of the paper and in particular outline the ability of arbitrary sampling kernels to be used in FRI algorithms