THE DESIGN OF COMPLEX WAVELET PACKET TRANSFORMS BASED ON PERFECT TRANSLATION INVARIANCE THEOREMS

2010 ◽  
Vol 08 (04) ◽  
pp. 537-558 ◽  
Author(s):  
HIROSHI TODA ◽  
ZHONG ZHANG ◽  
TAKASHI IMAMURA
Keyword(s):  
Wavelet Transforms ◽  
Wavelet Packet ◽  
Wavelet Packets ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Discrete Wavelet Transforms ◽  
Wide Range ◽  
Different Shapes ◽  
Complex Wavelet ◽  
Meyer Wavelet

The useful theorems for achieving perfect translation invariance have already been proved, and based on these theorems, dual-tree complex discrete wavelet transforms with perfect translation invariance have been proposed. However, due to the complication of frequency divisions with wavelet packets, it is difficult to design complex wavelet packet transforms with perfect translation invariance. In this paper, based on the aforementioned theorems, novel complex wavelet packet transforms are designed to achieve perfect translation invariance. These complex wavelet packet transforms are based on the Meyer wavelet, which has the important characteristic of possessing a wide range of shapes. In this paper, two types of complex wavelet packet transforms are designed with the optimized Meyer wavelet. One of them is based on a single Meyer wavelet and the other is based on a number of different shapes of the Meyer wavelets to create good localization of wavelet packets.

Perfect-translation-invariant variable-density complex discrete wavelet transform

2014 ◽  
Vol 12 (04) ◽  
pp. 1460001 ◽  
Author(s):  
Hiroshi Toda ◽  
Zhong Zhang ◽  
Takashi Imamura
Keyword(s):  
Frequency Domain ◽  
Frequency Band ◽  
Time Domain ◽  
Wavelet Transforms ◽  
Filter Banks ◽  
Wavelet Packet ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Logarithmic Scale ◽  
The Time Domain

The theorems giving the conditions for discrete wavelet transforms (DWTs) to achieve perfect translation invariance (PTI) have already been proven, and based on these theorems, the dual-tree complex DWT and the complex wavelet packet transform, achieving PTI, have already been proposed. However, there is not so much flexibility in their wavelet density. In the frequency domain, the wavelet density is fixed by octave filter banks, and in the time domain, each wavelet is arrayed on a fixed coordinate, and the wavelet packet density in the frequency domain can be only designed by dividing an octave frequency band equally in linear scale, and its density in the time domain is constrained by the division number of an octave frequency band. In this paper, a novel complex DWT is proposed to create variable wavelet density in the frequency and time domains, that is, an octave frequency band can be divided into N filter banks in logarithmic scale, where N is an integer larger than or equal to 3, and in the time domain, a distance between wavelets can be varied in each level, and its transform achieves PTI.

A NOVEL DESIGN METHOD FOR DIRECTIONAL SELECTION BASED ON 2-DIMENSIONAL COMPLEX WAVELET PACKET TRANSFORM

2013 ◽  
Vol 11 (04) ◽  
pp. 1360010 ◽  
Author(s):  
TAKESHI KATO ◽  
ZHONG ZHANG ◽  
HIROSHI TODA ◽  
TAKASHI IMAMURA ◽  
TETSUO MIYAKE
Keyword(s):  
Wavelet Transforms ◽  
Design Method ◽  
Wavelet Packet ◽  
Wavelet Packet Transform ◽  
Directional Selection ◽  
Discrete Wavelet ◽  
Two Dimensional ◽  
Semiconductor Wafer ◽  
Directional Filters ◽  
Complex Wavelet

In this paper, we propose a design method for directional selection in the two-dimensional complex wavelet packet transform (2D-CWPT). Current two-dimensional complex discrete wavelet transforms (2D-CDWT) can extract directional components from images, but the number of directions is small, and the directions and resolutions are fixed. Thus the current 2D-CDWTs are not flexible enough. In this study, we propose a new design method of the directional filters that can detect desirable direction components. Additionally flexible directional selection is achieved because the directional filters are added to the 2D-CWPT. Finally, the proposed method is applied to defect detection in semiconductor wafer circuits and an encouraging result is obtained.

PERFECT-TRANSLATION-INVARIANT CUSTOMIZABLE COMPLEX DISCRETE WAVELET TRANSFORM

2013 ◽  
Vol 11 (04) ◽  
pp. 1360003 ◽  
Author(s):  
HIROSHI TODA ◽  
ZHONG ZHANG ◽  
TAKASHI IMAMURA
Keyword(s):  
Wavelet Transform ◽  
Discrete Wavelet Transform ◽  
Frequency Domain ◽  
Wavelet Transforms ◽  
Wavelet Packet ◽  
Variable Density ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Translation Invariant ◽  
High Degree

The theorems, giving the condition of perfect translation invariance for discrete wavelet transforms, have already been proven. Based on these theorems, the dual-tree complex discrete wavelet transform, the 2-dimensional discrete wavelet transform, the complex wavelet packet transform, the variable-density complex discrete wavelet transform and the real-valued discrete wavelet transform, having perfect translation invariance, were proposed. However, their customizability of wavelets in the frequency domain is limited. In this paper, also based on these theorems, a new type of complex discrete wavelet transform is proposed, which achieves perfect translation invariance with high degree of customizability of wavelets in the frequency domain.

PERFECT TRANSLATION INVARIANCE WITH A WIDE RANGE OF SHAPES OF HILBERT TRANSFORM PAIRS OF WAVELET BASES

2010 ◽  
Vol 08 (04) ◽  
pp. 501-520 ◽  
Author(s):  
HIROSHI TODA ◽  
ZHONG ZHANG
Keyword(s):  
Wavelet Transform ◽  
Hilbert Transform ◽  
Calculation Method ◽  
Discrete Wavelet ◽  
Translation Invariance ◽  
Wavelet Bases ◽  
Discrete Signal ◽  
Wide Range ◽  
Meyer Wavelet ◽  
The Hilbert Transform

It is well known that a Hilbert transform pair of wavelet bases improves the lack of translation invariance of the discrete wavelet transform. However, its shapes and improvement are limited by the difficulty in applying the Hilbert transform pair to a discrete signal. In this paper, novel Hilbert transform pairs of wavelet bases, which are based on a Meyer wavelet and have a wide range of shapes, are proposed to create perfect translation invariance, and their calculation method is designed to apply these wavelet bases to any discrete signal. Therefore, perfect translation invariance is achieved with a wide range of shapes of the Hilbert transform pairs of wavelet bases.

Image Weight Models Based on Discrete Wavelet Transforms

2021 ◽  
Author(s):  
Maya M. Lyasheva ◽  
Stella A. Lyasheva ◽  
Mikhail P. Shleymovich
Keyword(s):  
Wavelet Transforms ◽  
Discrete Wavelet ◽  
Discrete Wavelet Transforms
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