We focus on a specific class of curves that can be parametrized using a finite number of variables in two dimensions. The corresponding indicator plane, which is a binary image, has infinite bandwidth and can not be sampled and perfectly reconstructed with classical sampling theory. In this paper, we illustrate that it is possible to recover parameters from finite samples of the indicator plane and have a perfect reconstruction of the indicator plane. The algorithm presented here extends the application of FRI signals to multi-dimensional cases and may find its application in field, like super-resolution.