The decomposition of signals using the classical Daubechies wavelets can equivalently be described as a decomposition
using a two channel filterbank with the scaling function corresponding to the low pass and the mother
wavelet to the high pass channel.
This classical two channel approach was extended to also comprise filterbanks with more than two channels,
corresponding to one scaling function and two or more mother wavelets.1 Although these newly found
wavelets fitted in very well with the theory of the existing Daubechies wavelets, the frequency selectivity of the
corresponding filterbank was in general not satisfying.
We presented a new method2 to improve the selectivity of the corresponding filterbank by introducing additional
elementary building blocks in the design process, leaving the low pass filter and the polynomial degree of
the impulse responses of the other filters of the filterbank untouched.
In this paper we take the new method one step further and not only show how the improvement could be
achieved but also what possibilities there are to modify the design and what patterns they adhere to. Furthermore
we introduce a figure of merit of the total filterbank and its individual filters and show the improvements to the design process.
Wavelet decomposition of signals using the classical
Daubechies-Wavelets could also be considered as a decomposition
using a filter bank with two channels, a low pass and a high pass channel, represented by the father and
mother wavelet, respectively. By generalizing this two channel approach filter banks with N ≥ 2 channels can
be constructed. They possess one scaling function or father wavelet representing the low pass filter and one, two
or more mother wavelets representing band pass filters.
The resulting band pass filters do not show a satisfactory selective behavior, in general. Hence, a modification
of the generalized design seems appropriate. Based on Vaidyanathan's procedure we developed a method to
modify the modulation matrix under the condition that the low pass is unchanged and the degree of the band
pass filters is not increased. This can be achieved by introducing one or more additional elementary building
blocks under certain orthogonality constraints with respect to their generating vectors. While the (polynomial)
degree of the modulation matrix remains unchanged, its complexity increases due to its increased McMillan degree.
In Computed Tomography, axial resolution is determined by the slice collimation and the spiral algorithm, while in-plane resolution is determined by the reconstruction kernel. Both choices select a tradeoff between image resolution (sharpness) and pixel noise. We investigated an alternative approach using default settings for image reconstruction which provide narrow reconstructed slice-width and high in-plane resolution. If smoother images are desired, we filter the original (sharp) images, instead of performing a new reconstruction with a smoother kernel. A suitable filter function in the frequency domain is the ratio of smooth and original (sharp) kernel. Efficient implementation was achieved by a Fourier transform of this ratio to the spatial domain. Separating the 2D spatial filtering into two subsequent 1D filtering stages in x- and y-direction further reduces computational complexity. Using this approach, arbitrarily oriented multi-planar reformats (MPRs) can be treated in exactly the same way as axial images. Due to efficient implementation, interactive modification of the filter settings becomes possible, which completely replace the variety of different reconstruction kernels. We implemented a further promising application of the method to thorax imaging, where different regions of the thorax (lungs and mediastinum) are jointly presented in the same images using different filter settings and different windowing.
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