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.

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.

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.

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.

A direct design approach for two-dimensional low-pass and high-pass filters

1990 ◽  
Vol 327 (4) ◽  
pp. 551-571 ◽  
Author(s):  
J.J. Soltis ◽  
T.J. Kent ◽  
M.A. Sid-Ahmed
Keyword(s):  
Design Approach ◽  
Two Dimensional ◽  
Direct Design ◽  
Low Pass ◽  
High Pass

Two‐dimensional circularly symmetric filter design via coefficient mapping

Geophysics ◽  
1989 ◽  
Vol 54 (3) ◽  
pp. 392-401 ◽  
Author(s):  
Huseyin Özdemir
Keyword(s):  
Frequency Domain ◽  
Filter Design ◽  
Pass Band ◽  
Two Dimensional ◽  
Remotely Sensed Data ◽  
Mapping Algorithm ◽  
Complete Mapping ◽  
Low Pass ◽  
High Pass ◽  
Frequency Axis

Two‐dimensional (2-D) filters are used in geophysics for processing seismic, potential field, and remotely sensed data, and in other applied sciences for purposes such as image processing. These filters can be designed using optimum one‐dimensional (1-D) algorithms via mapping techniques: the 2-D desired response is mapped into the 1-D response either in the time domain or the frequency domain and mapped back after an approximation is obtained. A coefficient mapping algorithm is presented here for designing 2-D circularly symmetric filters. The 2-D frequency‐domain coefficients are mapped to the 1-D frequency axis after being sorted according to their distances from the origin. This kind of sorting brings together the coefficients of a particular passband, reject band, or transition band of the circularly symmetric filter before the coefficients are mapped into the 1-D frequency axis. As a result, there are as many prescribed bands in the 1-D domain as in the corresponding 2-D domain, which leads to optimal approximations in the 1-D domain and, consequently, in the 2-D domain. The mapping is performed between the Nyquist regions. A Chebychev or “min‐max” algorithm has been used for 1-D approximations. Due to a complete mapping between the 2-D and the 1-D domains, the 2-D filters designed via mapping show equiripple behavior similar to that of the 1-D filters. The new mapping algorithm is suitable for designing low‐pass, high‐pass, band‐pass, and band reject filters with multiple bands. The circular symmetries of the approximated responses improve with an increased number of filter coefficients. For a 2-D filter with [Formula: see text] coefficients, [Formula: see text] and [Formula: see text] must be odd; but they need not be equal. Applications of low‐pass and high‐pass circularly symmetric filters to data from a regional gravity field survey demonstrate that these filters can effectively separate anomalies of different wavelengths when there is no spectral overlap.

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