Regularization parameter-free convolutional sparse coding via projections onto the ℓ₁-Ball and the discrepancy principle

Given a set of dictionary filters, the most widely used formulation of the convolutional sparse coding (CSC) problem is convolutional basis pursuit denoising (CBPDN), in which an image is represented as a sum over a set of convolutions of coefficient maps. When the input image is noisy, CBPDN's regularization parameter greatly influences the quality of the reconstructed image. Results for an automatic and sensible selection of this parameter are very limited for the CSC / CBPDN case.In this paper we propose a regularization parameter-free method to solve the CSC problem via its projection onto the ℓ₁-Ball formulation coupled with a warm-start like strategy, which, driven by the Morozov's discrepancy principle, adaptively increases/decreases its constrain at each major iteration. While the time performance of our proposed method is slower than that measured when solving CSC for a fixed regularization parameter, our computational results also show that our method's reconstruction quality is, in average, very close (within 0.16 SNR, 0.16 PSNR, 0.003 SSIM) to that obtained when the regularization parameter for CBPDN is selected to produce the best (SNR) quality result.