Let f be a given signal, b the basis functions of a transform, and c the transform coefficients. The transform of f is with . The inverse transform is . For the signal description only those basis function are used that contribute to a sparse description. A direct implementation of this idea would lead to an NP-hard combinatorial problem.
Experiments show the approximation to be very close to an optimal solution.
The atomic image decomposition allows to efficiently represent ALL signal components in images. I therefore expect atomic image decomposition to have a great potential for image analysis, reconstruction, and compression as well as for feature detection and image indexing. This is joint work with X. Huo.