Two-dimensional wavelet transform

In conclusion, wavelet transforms have been employed by analytical chemists to solve various problems in chromatographic studies. Owing to the popularity of hyphenated instruments, more applications based on two-dimensional wavelet transform (2D-WT) will be developed. The 2D-WT technique is more suitable for processing data produced from such instruments. 

Step 1 Decompose the input medical image f x, y) through the two-dimensional wavelet transform to obtain wavelet decomposition of sub-bands ... 

Let the crystalline object be a thin foil characterized by a two-dimensional transmission function q (jc. y) that describes at each point of the exit surface of the specimen the amplitude and phase of the electron beams emerging from the column situated at (.r, y) after dynamic diffraction in the foil. The diffraction pattern can be described to a good approximation as the Fourier transform Q (if. ) of the object function q (x, y). This diffraction pattern acts in turn as a source of Huyghens wavelets, which interfere to form the image, after linear magnification by the optica) lens systems the image is, in turn, the Fourier transform ip (x, y) of the diffraction pattern. 

Defective parts between two particle surface images have been clearly extracted by means of Fourier-wavelets transform methods without threshold values even though the two particles are differently located in a two-dimensional space. This concept is applicable to inferior products classification of a complicated image such as an 1C pattern. The method consists of two steps the first is to acquire the difference between the two particle surface images in Fourier space. The second is to extract the feature of the difference image by means of wavelets transform and multiresolution. The low wavelets level indicates the whole image of the defective part. The high level indicates the outline position of the defective part. This technique contributes to automation of products classification. 

The basic concept of Fourier-wavelets transform is composed of two steps that are a different part extraction by Fourier transform and a clarification by wavelets transform as shown in Fig. 1. In the first step, a two-dimensional standard image matrix Xs and a test image matrix Xt are transformed to Fourier space images as. 

It is possible to obtain the defective part clearly without threshold values even though the two particles are differently located on a two-dimensional space. This is caused by the combination of Fourier transform and wavelets transform. 

Due to the non-periodic velocity fluctuations, wavelet analysis was used (Torrence Compo [8], Chui [3]) in order to identify the frequencies of the fluctuations. A Fourier transformation is not applicable because the frequencies are changing or do not occur continuously due to shocks or dampened motion of the bulk solid. In such cases it is necessary to get information about frequency changes with time. With a Fourier transformation, this is not possible because the transformation is only one dimensional. With a wavelet transformation, it is possible because this transformation is two dimensional. 

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