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Publication# Sampling Curves with Finite Rate of Innovation

Abstract

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.

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Related concepts (2)

Sampling (signal processing)

In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or space; this definition differs from the term's usage in statistics, which refers to a set of such values. A sampler is a subsystem or operation that extracts samples from a continuous signal. A theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous signal at the desired points.

Sampling (statistics)

In statistics, quality assurance, and survey methodology, sampling is the selection of a subset or a statistical sample (termed sample for short) of individuals from within a statistical population to estimate characteristics of the whole population. Statisticians attempt to collect samples that are representative of the population. Sampling has lower costs and faster data collection compared to recording data from the entire population, and thus, it can provide insights in cases where it is infeasible to measure an entire population.