Uniform Discrete Curvelet Transform

An implementation of the discrete curvelet transform is proposed in this work. The transform is based on and has the same order of complexity as the Fast Fourier Transform (FFT). The discrete curvelet functions are defined by a parameterized family of smooth windowed functions that satisfies two conditions: i) <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2π</i> periodic; ii) their squares 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 positioned on a uniform lattice. The forward and inverse transform form a tight and self-dual frame, in the sense that they are the exact transpose of each other. Generalization to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> dimensional version of the UDCT is also presented. The novel discrete transform has several advantages over existing transforms, such as lower redundancy ratio, hierarchical data structure and ease of implementation.