We present a new efficient edge-preserving filter-"tree filter"-to achieve strong image smoothing. The proposed filter can smooth out high-contrast details while preserving major edges, which is not achievable for bilateral-filter-like techniques. Tree filter is a weighted-average filter, whose kernel is derived by viewing pixel affinity in a probabilistic framework simultaneously considering pixel spatial distance, color/intensity difference, as well as connectedness. Pixel connectedness is acquired by treating pixels as nodes in a minimum spanning tree (MST) extracted from the image. The fact that an MST makes all image pixels connected through the tree endues the filter with the power to smooth out high-contrast, fine-scale details while preserving major image structures, since pixels in small isolated region will be closely connected to surrounding majority pixels through the tree, while pixels inside large homogeneous region will be automatically dragged away from pixels outside the region. The tree filter can be separated into two other filters, both of which turn out to have fast algorithms. We also propose an efficient linear time MST extraction algorithm to further improve the whole filtering speed. The algorithms give tree filter a great advantage in low computational complexity (linear to number of image pixels) and fast speed: it can process a 1-megapixel 8-bit image at ~ 0.25 s on an Intel 3.4 GHz Core i7 CPU (including the construction of MST). The proposed tree filter is demonstrated on a variety of applications.
We study energy spectra, eigenstates and quantum diffusion for one-and two-dimensional quasiperiodic tight-binding models. As our one-dimensional model system we choose the silver mean or "octonacci" chain. The twodimensional labyrinth tiling, which is related to the octagonal tiling, is derived from a product of two octonacci chains. This makes it possible to treat rather large systems numerically. For the octonacci chain, one finds singular continuous energy spectra and critical eigenstates which is the typical behaviour for one-dimensional Schrödinger operators based on substitution sequences. The energy spectra for the labyrinth tiling can, depending on the strength of the quasiperiodic modulation, be either band-like or fractallike. However, the eigenstates are multifractal. The temporal spreading of a wavepacket is described in terms of the autocorrelation function C(t) and the mean square displacement d(t). In all cases, we observe power laws C(t) ∼ t −δ and d(t) ∼ t β . For the octonacci chain, 0 < δ < 1, whereas for the labyrinth tiling a crossover is observed from δ = 1 to 0 < δ < 1 with increasing modulation strength. Corresponding to the multifractal eigenstates, we obtain anomalous diffusion with 0 < β < 1 for both systems. Moreover, we find that the behaviour of C(t) and d(t) is independent of the shape and the location of the initial wavepacket. We use our results to check several relations between the diffusion exponent β and the fractal dimensions of energy spectra and eigenstates that were proposed in the literature.
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