Stable Signal Reconstruction via $\ell^1$-Minimization in Redundant, Non-Tight Frames

In many signal and image processing applications, a desired clean signal is distorted from blur and noise. Reconstructing the clean signal usually yields to a high dimensional ill-conditioned system of equations, where a direct solution would severely amplify the noise. Stable signal reconstruction requires the use of regularization techniques, which incorporate a priori knowledge about the signal. A particular successful property for that purpose is the sparsity of the analysis coefficients of the clean signal in a suitable frame or dictionary, which can be implemented via
l
¹
-minimization. Most existing stable recovery results for
l
¹
-analysis minimization require the frame to be an orthonormal basis. This contrasts practical applications, where redundant frames often perform better than bases. In this paper we address this issue and derive stable recovery results for
l
¹
-analysis minimization in redundant, possibly non-tight frames.