Image restoration: A wavelet frame based model for piecewise smooth functions and beyond

Abstract

Recently, mapping a signal/image into a low rank Hankel/Toeplitz matrix has become an emerging alternative to the traditional sparse regularization, due to its ability to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we introduce a novel structured low rank matrix framework to restore piecewise smooth functions. Inspired by the total generalized variation to use sparse higher order derivatives, we derive that the Fourier samples of higher order derivatives satisfy an annihilation relation, resulting in a low rank multi-fold Hankel matrix. We further observe that the SVD of a low rank Hankel matrix corresponds to a tight wavelet frame system which can represent the image with sparse coefficients. Based on this observation, we also propose a wavelet frame analysis approach based continuous domain regularization model for the piecewise smooth image restoration. Finally, numerical results on image restoration tasks are presented as a proof-of-concept study to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low rank matrix approaches.

Tight wavelet frames in low dimensions with canonical filters

“…Some new applications of tight wavelet frames can be also found in [14,24]. Furthermore, the connections of wavelet frame based, especially spline tight wavelet frames based, approach for image restoration to PDE based methods are established in [1] for the total variational method and extension, in [7] for the nonlinear diffusion partial differential equation based methods, and in [2] for variational models on the space of piecewise smooth functions.…”

“…For a given FIR p, we will provide a constructive method to construct q (2 s ) , · · · , q (2 s+1 −1) such that they, together with p and the canonical filters q (1) , q (2) , · · · , q (2 s −1) , form a tight frame filter bank. All the three cases s = 1, s = 2 and s = 3 are considered.…”

“…together with p, q (1) , q (2) , · · · , q (2 s −1) , form a semi-canonical tight frame filter bank with the minimal number of highpass filters. In the first subsection of this section we will provide a constructive method to construct q (2 s ) , · · · , q (2 s+1 −1) for the cases s = 1, s = 2 and s = 3.…”

“…We will also consider the construction of semi-canonical tight frame filter banks with exact 2 s+1 − 1 highpass filters. In the case s = 1, for an FIR filter p satisfying (1.7), there always exists an FIR filter q (2) such that with q (1) given by (1.2) and with q (3) defined by…”

“…Such a filter bank will be called a double-canonical tight frame filter bank since q (3) is generated by q (2) as the canonical filter associated with q (2) . For s = 2, 3, two sets of constructive conditions for the existences of semi-canonical frame filter banks with exact 2 s+1 − 1 highpass filters are provided.…”