It was recently demonstrated that one can perform fast nonlocal means (NLM) denoising of one-dimensional (1-D) signals using a method called lifting. The cost of lifting is independent of the patch length, which dramatically reduces the run-time for large patches. Unfortunately, it is difficult to directly extend lifting for NLM denoising of images. To bypass this, the authors proposed a separable approximation in which the image rows and columns are filtered using lifting. The overall algorithm is significantly faster than NLM, and the results are comparable in terms of PSNR. However, the separable processing often produces vertical and horizontal stripes in the image. This problem was previously addressed using a bilateral filter-based postsmoothing, which was effective in removing some of the stripes. We demonstrate that stripes can be mitigated in the first place simply by involving the neighboring rows (or columns) in the filtering. In other words, we use a two-dimensional (2-D) search (similar to NLM), while still using 1-D patches (as in the previous proposal). The innovation is in the observation that one can use lifting for performing 2-D searches. The proposed approach produces artifact-free images, whose quality and PSNR are comparable to NLM, while being significantly faster.
We propose a simple and fast algorithm called PatchLift for computing distances between patches (contiguous block of samples) extracted from a given one-dimensional signal. PatchLift is based on the observation that the patch distances can be efficiently computed from a matrix that is derived from the one-dimensional signal using lifting; importantly, the number of operations required to compute the patch distances using this approach does not scale with the patch length. We next demonstrate how PatchLift can be used for patch-based denoising of images corrupted with Gaussian noise. In particular, we propose a separable formulation of the classical nonlocal means (NLM) algorithm that can be implemented using PatchLift. We demonstrate that the PatchLift-based implementation of separable NLM is a few orders faster than standard NLM and is competitive with existing fast implementations of NLM. Moreover, its denoising performance is shown to be consistently superior to that of NLM and some of its variants, both in terms of peak signal-to-noise ratio/structural similarity index and visual quality.
We propose an amplitude-phase representation of the dual-tree complex wavelet transform (DT-CWT) which
provides an intuitive interpretation of the associated complex wavelet coefficients. The representation, in particular,
is based on the shifting action of the group of fractional Hilbert transforms (fHT) which allow us to extend
the notion of arbitrary phase-shifts beyond pure sinusoids. We explicitly characterize this shifting action for a
particular family of Gabor-like wavelets which, in effect, links the corresponding dual-tree transform with the
framework of windowed-Fourier analysis.
We then extend these ideas to the bivariate DT-CWT based on certain directional extensions of the fHT. In
particular, we derive a signal representation involving the superposition of direction-selective wavelets affected
with appropriate phase-shifts.