Dictionary learning for analysis-synthesis thresholding

Thresholding is a classical technique for signal denoising. In this process, a noisy signal is decomposed over an orthogonal or overcomplete dictionary, the smallest coefficients are nullified, and the transform pseudo-inverse is applied to produce an estimate of the noiseless signal. The dictionaries used is this process are typically fixed dictionaries such as the DCT or Wavelet dictionaries. In this work, we propose a method for incorporating adaptive, trained dictionaries in the thresholding process. We present a generalization of the basic process which utilizes a pair of overcomplete dictionaries, and can be applied to a wider range of recovery tasks. The two dictionaries are associated with the analysis and synthesis stages of the algorithm, and we thus name the process analysis-synthesis thresholding. The proposed training method trains both the dictionaries and threshold values simultaneously given examples of original and degraded signals, and does not require an explicit model of the degradation. Experiments with small-kernel image deblurring demonstrate the ability of our method to favorably compete with dedicated deconvolution processes, using a simple, fast, and parameterless recovery process.