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.

Treatment of Noise in Experimental Transport Measurements Plots With Discrete Wavelet Transforms

2009 ◽  
Author(s):  
Rube´n Panta Pazos
Keyword(s):  
Wavelet Transform ◽  
Wavelet Transforms ◽  
Transport Theory ◽  
Discrete Wavelet ◽  
Transform Method ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Thresholding Algorithm ◽  
High Pass

In this work it is applied the wavelet transform method [2] in order to reduce diverse type of noises of experimental measurement plots in transport theory. First, suppose that a problem is governed by the transport equation for neutral particles, and an unknown perturbation occurs. In this case, the perturbation can be associated to the source, or even to the flux inside the domain X. How is the behavior of the perturbed flux in relation to the flux without the perturbation? For that, we employ the wavelet transform method in order to compress the angular flux considered as a 1D, or n-th dimensional signal ψ. The compression of this signal can be performed up to some a convenient order (that depends of the length of the signal). Now, the transport signal is decomposed as [9, 11]: ψ=〈am|dm|dm−1|dm−2|⋯|d2|d1〉 where ak represents the sub signal of k-th level generated by the low-pass filter associated to the discrete wavelet transform (DWT) chosen, and dk the sub signal of k-th level generated by the high-pass filter associated to the same DWT. It is applied basically the Haar, Daub4 and Coiflet wavelets transforms. Indeed, the sub signal am cumulates the energy, for this work of order 96% of the original signal ψ. A thresholding algorithm provides treatment for the noise, with significant reduction in the compressed signal. Then, it is established a comparison with a base of data in order to identify the perturbed signal. After the identification, it is recomposed the signal applying the inverse DWT. Many assumptions can be established: the rate signal-to-noise is properly high, the base of data must contain so many perturbed signals all with the same level of compression. The problem considered is for perturbations in the signal. For measurements the problem is similar, but in this case the unknown perturbations are generated by the apparatus of measurements, problems in experimental techniques, or simply by random noises. With the same above assumptions, the DWT is applied. For the identification, it is used a method evolving statistical and metric techniques. It is given some results obtained with an algebraic computer system.

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.

Lossy to lossless image coding based on wavelets using a complex allpass filter

2014 ◽  
Vol 12 (04) ◽  
pp. 1460002 ◽  
Author(s):  
Xi Zhang ◽  
Noriaki Fukuda
Keyword(s):  
Image Coding ◽  
Wavelet Transforms ◽  
Filter Banks ◽  
Wavelet Filter ◽  
Biorthogonal Wavelets ◽  
Jpeg 2000 ◽  
Effective Implementation ◽  
Lossless Coding ◽  
Integer Wavelet ◽  
Allpass Filter

Wavelet-based image coding has been adopted in the international standard JPEG 2000 for its efficiency. It is well-known that the orthogonality and symmetry of wavelets are two important properties for many applications of signal processing and image processing. Both can be simultaneously realized by the wavelet filter banks composed of a complex allpass filter, thus, it is expected to get a better coding performance than the conventional biorthogonal wavelets. This paper proposes an effective implementation of orthonormal symmetric wavelet filter banks composed of a complex allpass filter for lossy to lossless image compression. First, irreversible real-to-real wavelet transforms are realized by implementing a complex allpass filter for lossy image coding. Next, reversible integer-to-integer wavelet transforms are proposed by incorporating the rounding operation into the filtering processing to obtain an invertible complex allpass filter for lossless image coding. Finally, the coding performance of the proposed orthonormal symmetric wavelets is evaluated and compared with the D-9/7 and D-5/3 biorthogonal wavelets. It is shown from the experimental results that the proposed allpass-based orthonormal symmetric wavelets can achieve a better coding performance than the conventional D-9/7 and D-5/3 biorthogonal wavelets both in lossy and lossless coding.

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.

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