CHAPTER Sparse Representations 1 Signals carry overwhelming amounts of data in which relevant information is often more difficult to find than a needle in a haystack. Processing is faster and simpler in a sparse representation where few coefficients reveal the information we are looking for. Such representations can be constructed by decomposing signals over elementary waveforms chosen in a family called a dictionary. But the search for the Holy Grail of an ideal sparse transform adapted to all signals is a hopeless quest. The discovery of wavelet orthogonal bases and local time-frequency dictionaries has opened the door to a huge jungle of new transforms. Adapting sparse representa- tions to signal properties, and deriving efficient processing operators, is therefore a necessary survival strategy.