A novel implementation of the discrete curvelet transform is proposed in this work. The transform is based on the Fast Fourier Transform (FFT) and has the same order of complexity as the FFT. The discrete curvelet functions are defined by a parameterized family of smooth windowed functions that are 2-pi periodic and form a partition of unity. The transform is named the Uniform Discrete Curvelet Transform (UDCT) because the centers of the curvelet functions at each resolution are located on a uniform grid. The forward and inverse transforms form a tight frame, in the sense that they are the exact transpose of each other. The novel discrete transform has several advantages over existing transforms, such as lower redundancy, hierarchical data structure, ease of implementation and possible extension to N dimension. Finally, we present a simple initial application of the UDCT in sparseness constraint seismic data interpolation to recover missing traces.