M.Sc Thesis
M.Sc Student Cohen Erez Wavelet Based Image Restoration Using Cross-Band Operators Department of Electrical Engineering Professor Israel Cohen

Abstract

Image restoration deals with recovery of an image from blurred and noisy observed measurements. Restoration algorithms often rely on statistical models for the observation process and for the ideal image formation. In the last decade, there has been a considerable interest in wavelet domain image restoration. Wavelet bases are known to have superior time-frequency localization, which enables a sparse representation for a wide class of stationary and nonstationary signals, including real-world images .

The sparsity enables an attractive compromise between local details preservation and noise suppression, compared to Fourier domain restoration . However, transform domain restoration requires an explicit representation of the convolutional blurring kernel. Differently from the Fourier domain, linear time-invariant (LTI) systems cannot be represented as scalar multiplication in the wavelet domain. As a result, wavelet-based image restoration involves calculation and storage of a large-scale convolution matrix, which doesn't have any sparse structure .

In this work, we address the problem of LTI systems representation in the wavelet packet domain .

We develop an explicit representation of LTI systems, based on the filter-bank interpretation of the wavelet packet transform (WPT) . In this representation, a time-domain convolution is transformed into a set of cross-band filters between each pair of the filter-bank frequency subbands. We show that the cross-band filters are time-varying in the general case, and express them as multirate filters. From this notation , it is possible to derive a sparse wavelet domain convolution matrix, whose sparsity depends on the number of WPT decomposition levels. The sparsity is exploited for efficient optimization of wavelet-based image restoration criterions. As an example, we develop a MAP criterion, based on the cross-band filtering notation and on Laplace prior density of the wavelet coefficients. In any case, the cross-band filtering notation can be employed with any image prior in the wavelet packet domain .

Besides the perfect representation with cross-band filters discussed above, we define multiplicative operators and examine their usage for LTI systems approximation. Multiplicative operators are defined in the wavelet packet domain by replacing the filtering terms with scalar multiplications. The advantage of multiplicative operators compared to the cross-band filtering notation is computational. However, multiplicative operators are not time-invariant, and averaging over their translation is needed for LTI systems approximation. We compare between the multiplicative transfer function (MTF) and the cross-MTF (CMTF), and propose two design methods for determining their parameters. We find that the additional cross-multiplicative terms in the CMTF significantly improve the approximation quality .