CONSTRUCTION OF COMPACTLY SUPPORTED CONJUGATE SYMMETRIC COMPLEX TIGHT WAVELET FRAMES

2010 ◽  
Vol 08 (06) ◽  
pp. 861-874 ◽  
Author(s):  
SHOUZHI YANG ◽  
YANMEI XUE
Keyword(s):  
Wavelet Frames ◽  
Refinable Function ◽  
Compactly Supported ◽  
Symmetric Complex ◽  
Low Pass ◽  
Tight Wavelet Frame ◽  
Free Parameters ◽  
Low Pass Filters ◽  
The Given ◽  
High Pass

Two algorithms for constructing a class of compactly supported complex tight wavelet frames with conjugate symmetry are provided. Firstly, based on a given complex refinable function ϕ, an explicit formula for constructing complex tight wavelet frames is presented. If the given complex refinable function ϕ is compactly supported conjugate symmetric, then we prove that there exists a compactly supported conjugate symmetric/anti-symmetric complex tight wavelet frame Ψ = {ψ1, ψ2, ψ3} associated with ϕ. Secondly, under the conditions that both the low-pass filters and high-pass filters are unknown, we give a parametric formula for constructing a class of smooth conjugate symmetric/anti-symmetric complex tight wavelet frames. Free parameters in the algorithm are explicitly identified, and can be used to optimize the result with respect to other criteria. Finally, two examples are given to illustrate how to use our method to construct conjugate symmetric complex tight wavelet frames.

scholarly journals Glass Patterns: Some Contrast Effects Re-Evaluated

Perception ◽  
1997 ◽  
Vol 26 (3) ◽  
pp. 253-268 ◽  
Author(s):  
Steven C Dakin
Keyword(s):  
Spatial Filtering ◽  
Contrast Effects ◽  
Visual Grouping ◽  
Relative Contrast ◽  
Low Pass ◽  
Band Pass ◽  
Opposite Contrast ◽  
Low Pass Filters ◽  
Glass Patterns ◽  
High Pass

The relative contrast of features is known to be important in determining if they can be grouped. Two manipulations of feature contrast have previously been used to criticise models of visual grouping based on spatial filtering: high-pass filtering and reversal of contrast polarity. The effects of these manipulations are considered in the context of the perception of Glass patterns. It is shown that high-pass filtering elements, whilst destroying structure in the output of low-pass filters, do not significantly disrupt the output of locally band-pass filters. The finding that subjects can perceive structure in Glass patterns composed of high-pass features therefore offers no evidence against such spatial filtering mechanisms. Band-pass filtering models are shown to explain the rotation of perceived structure in Glass patterns composed of opposite contrast features. However, structure is correctly perceived in patterns composed of two ‘interleaved’ opposite contrast patterns, which is problematic for oriented filtering mechanisms. Two possible explanations are considered: nonlinear contrast transduction prior to filtering, and integration of local orientation estimates from first-order and second-order mechanisms.

COMPACTLY SUPPORTED MULTIVARIATE, PAIRS OF DUAL WAVELET FRAMES OBTAINED BY CONVOLUTION

2008 ◽  
Vol 06 (02) ◽  
pp. 183-208 ◽  
Author(s):  
MARTIN EHLER
Keyword(s):  
Approximation Order ◽  
Refinable Functions ◽  
Wavelet Frames ◽  
Refinable Function ◽  
Vanishing Moments ◽  
Wavelet System ◽  
Compactly Supported ◽  
Mask Size ◽  
Dual Wavelet Frames ◽  
Dual Wavelet

In this paper, we present a construction of compactly supported multivariate pairs of dual wavelet frames. The approach is based on the convolution of two refinable distributions. We obtain smooth wavelets with any preassigned number of vanishing moments. Their underlying refinable function is fundamental. In the examples, we obtain symmetric wavelets with small support from optimal refinable functions, i.e. the refinable function has minimal mask size with respect to smoothness and approximation order of its generated multiresolution analysis. The wavelet system has maximal approximation order with respect to the underlying refinable function.

DIRECT DESIGN OF TWO‐DIMENSIONAL DIGITAL WAVENUMBER FILTERS

Geophysics ◽  
1970 ◽  
Vol 35 (6) ◽  
pp. 1073-1078 ◽  
Author(s):  
Peter M. Lavin ◽  
John F. Devane
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Weighting Coefficient ◽  
Basic Algorithm ◽  
Acceptable Error ◽  
Direct Design ◽  
Low Pass ◽  
Low Pass Filters ◽  
Weighting Coefficients ◽  
High Pass

A closed form solution is derived for generating the space domain weighting coefficients for phase‐distortionless low‐pass filters with flat pass regions, variable cutoff wavenumbers [Formula: see text] and variable cutoff rates (Δk). High‐pass and bandpass filters can be designed using the same basic algorithm. The error in the spectrum of the filters, due to truncation of the weighting coefficient set, depends on both Δk and [Formula: see text]. An empirical error analysis yielded criteria for estimating the size of the coefficient array for an acceptable error level in the spectrum.

New Approaches To the Design of Active Filters

SIMULATION ◽  
1966 ◽  
Vol 6 (5) ◽  
pp. 323-336 ◽  
Author(s):  
Peter D. Hansen
Keyword(s):  
High Performance ◽  
Impedance Matching ◽  
Analog Computer ◽  
Active Filters ◽  
Pass Band ◽  
Design Data ◽  
Low Pass ◽  
Band Pass ◽  
Low Pass Filters ◽  
High Pass

Operational amplifiers can greatly simplify the design of high performance signal filters because they elimi nate the need for inductors and for impedance matching. Furthermore, use of active filters can result in reduc tion of weight, size, and cost. Filters designed to satisfy sophisticated mathematical criteria can be realized without resort to "equalization" or trimming. In this issue we discuss the design of operational amplifier and analog computer circuits suitable for use as low pass filters. We also discuss the commonly used mathematically designed filters, i.e. Butterworth, Chebyshev, and Bessel. In addition, we present two new types of theoretical filters, the Paynter and the Aver aging filters. Design data necessary for realizing these theoretical filters with amplifier circuits is provided. In the next issue we shall discuss the design of band pass, band reject, high pass and all pass active filter circuits.

TRIANGULAR BIORTHOGONAL WAVELETS WITH EXTENDED LIFTING

2013 ◽  
Vol 11 (04) ◽  
pp. 1360002 ◽  
Author(s):  
KENSUKE FUJINOKI ◽  
SHUNSUKE ISHIMITSU
Keyword(s):  
Wavelet Decomposition ◽  
Two Dimensional ◽  
Biorthogonal Wavelets ◽  
Energy Distributions ◽  
Edge Structure ◽  
Wavelet Filters ◽  
New Family ◽  
Low Pass ◽  
Low Pass Filters ◽  
High Pass

We present a new family of triangular biorthogonal wavelets that is defined on a triangular lattice by introducing a new operation to generalize two-dimensional lifting, which we call twist. The resulting filters inherit several remarkable features of the early triangular biorthogonal wavelet filters such as the hexagonal symmetry of low-pass filters, symmetrical arrangement of three high-pass filters on the lattice, and that the wavelet decomposition produces uniform energy distributions over three detail components, preserving the isotropy of decomposed images. Additionally, these filters are a biorthogonal set of truly nonseparable two-dimensional wavelet filters that have much larger support, which provides much larger portions of the total energy to three detail components of decomposed images. We show that this plays a crucial role when extracting the edge structure of an image.

Construction of symmetric fractional over-complete wavelets and applications in image restoration

2016 ◽  
Vol 14 (04) ◽  
pp. 1650020
Author(s):  
Zhengwei Shen
Keyword(s):  
Image Restoration ◽  
Wavelet Transforms ◽  
Filter Banks ◽  
Minimum Length ◽  
Wavelet Filter ◽  
Split Bregman ◽  
Low Pass ◽  
Low Pass Filters ◽  
High Pass ◽  
Novel Design

In this work, a novel design scheme is proposed for the construction of symmetric fractional over-complete wavelet filter banks. We first provide solutions to the open problem of designing low-pass filters that are symmetric and of minimum-length. We then obtain the high high-pass filters via Toeplitz matrix factorization which is of less computational complexity than existing methods. The resulting filter banks are approximately shift-invariant. The designed filter banks are applied in image restoration that uses an analysis based model solved by split Bregman algorithms. The experiments show the constructed symmetric fractional over-complete wavelet transforms (FOWTs) allow better restoration results than some other wavelet transforms in the literature.

CONSTRUCTION FOR A CLASS OF SMOOTH TIGHT FRAMES IN L2(ℝ2)

2007 ◽  
Vol 05 (04) ◽  
pp. 613-625 ◽  
Author(s):  
HAIHUI WANG ◽  
LIZHONG PENG
Keyword(s):  
Tight Frames ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Compactly Supported ◽  
Dilation Matrix ◽  
Low Pass ◽  
High Pass

We generalize the inequality |P(z)|2 + |P(-z)|2 ≤ 1 (Ref. 1) to the inequality |P(z1, z2)|2 + |P(-z1, z2)|2 + |P(z1, -z2)|2 + |P(-z1, -z2)|2 ≤ 1 in L2(ℝ2); and construct compactly supported tight frames with good symmetry and dilation matrix 2I under the condition that both the low-pass filter and high-pass filters are unknown.

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