The interplay between the brain’s function and structure has been of immense interest to the neuroscience and connectomics communities. In this work we develop a simple linear model relating the structural network and the functional network. We propose that the two networks are related by the structural network’s Laplacian up to a shift. The model is simple to implement and gives accurate prediction of function’s eigenvalues at the subject level and its eigenvectors at group level.
We introduce new 5-band dyadic sibling frames with dense time-frequency grid. Given a lowpass filter satisfying certain conditions, the remaining filters are obtained using spectral factorization. The analysis and synthesis filterbanks share the same lowpass and bandpass filters but have different and oversampled highpass filters. This leads to wavelets approximating shift-invariance. The filters are FIR, have linear phase, and the resulting wavelets have vanishing moments. The filters are designed using spectral factorization method. The proposed method leads to smooth limit functions with higher approximation order, and computationally stable filterbanks.
We consider the design of 6-channel tight frame symmetric wavelet with scaling factor M = 4. The lowpass filter
is designed using either Grobner bases or truncated Taylor series methods. The bandpass and highpass filters are
then designed using Grobner bases. The resulting filters have linear phase, smooth limit functions, and reduced
redundancy. It was possible to obtain filterbanks with K0 (zeros at z = 1 and z = ±j for the lowpass filter) up
to 5 and Kmin (zeros at z = 1 for bandpass/highpass filters) of 1 or greater. The proposed filterbanks generate
five wavelets and a scaling function with the underlying filters related as follows: H5-i(z) = Hi(-z), i = 0... 5.
In this paper we explore the design of 5-band dual frame (overcomplete) wavelets with a dilation factor M = 4.
The resulting limit functions are significantly smoother than their orthogonal counterparts at the same dilation
factor. An advantage of the proposed filters over the dyadic filterbanks (M = 2) is that the proposed filterbanks
result in a reduced redundancy when compared with dyadic frames, while maintaining smoothness. The proposed
filterbanks are symmetric and generate four wavelets and a scaling function for each the synthesis and analysis
limit functions. All wavelets are equipped with at least one vanishing moment each.
Wavelets based on Hilbert pairs have appealing properties when applied to image denoising and feature detection
due to their directional sensitivity. In this paper we propose dual-tree tight frame wavelets and scaling functions
{φh, ψh1, ψh2, ψh3} and {φg, ψg1, ψg2, ψg3} based on FIR filterbanks of four filters, and downsampling by 2. Such
wavelets closely approximate shift invariance. Moreover, the resulting complex wavelets are smooth and lead to
exactly symmetric envelope. The filters in this paper enjoy vanishing moments property.
The paper presents new tight frame dyadic limit functions with dense time-frequency grid. The filterbank consists
of one lowpass filter and three bandpass and/or highpass filters. We add the requirement that the bandpass and
highpass filters be all of equal norms. All the filters in the paper are FIR and enjoy vanishing moments.
As is well known, wavelet filterbanks in general do not allow for shift invariance due to the downsampling operation. In this paper we discuss the design of a tight frame symmetric filterbank {h0, h1, h2, h3}, with the requirement that h2 and h1 be identical within a shift by one sample, or we seek h2 (n) = h1(n-1). This results in the wavelets ψ1 and ψ2 related as ψ2(t) = ψ1(t - 1/2).
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