Fourier descriptors are used to parametrically represent closed contours. In practice, a finite set of Fourier descriptors can model a large class of smooth contours. In this paper, we propose a method for estimating the Fourier descriptors of a given contour from its partial samples. We take a sampling-theoretic approach to model the x and y coordinate functions of the shape and express them as a sum of weighted complex exponentials, which belong to the class of finite-rate-of-innovation (FRI) signals. The weights represent the Fourier descriptors of the shape. We use the FRI framework to estimate the shape parameters reliably from noisy and partial measurements. We model non-uniformities in sampling using the sampling jitter model and employ a prefiltering process to reduce the effect of measurement noise and jitter. The average sampling interval is estimated by a block annihilating filter, which is then followed by estimation of Fourier descriptors using least-squares fitting. We demonstrate the robustness of the proposed algorithm to noise and sampling jitter. Monte Carlo performance analysis shows that the variances of the estimators are close to the Cramér-Rao lower bounds. We present results for outlining shapes in synthetic as well as real images.