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Zaman-frekans analizinde yeni dönüşümler ve uygulama alanları

New transforms in time-frequency analysis and their applications

  1. Tez No: 39120
  2. Yazar: YAZGAN ERER
  3. Danışmanlar: PROF.DR. AHMET H. KAYRAN
  4. Tez Türü: Yüksek Lisans
  5. Konular: Elektrik ve Elektronik Mühendisliği, Electrical and Electronics Engineering
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1993
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Belirtilmemiş.
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 86

Özet

ÖZET İşaret işleme konusunda temel problem, işareti mümkün olduğunca basit ve az katsayıyla modelleyecek bir gösterilim bulmaktır. Modelleme işlemi bir dönüşüm yardımıyla işaret domeninden dönüşüm domenine geçilerek yapılır. En tanınmış dönüşüm tekniği olan Fourier analizinde frekans domeninde sınırlı trigonometrik fonksiyonlar kullanılması nedeniyle işaretin frekans domenindeki bileşenlerinin dağılımı saptanabilir. Ancak bu fonksiyonlar zamanda sınırsız olduklarından işaretin zamanla değişiminin analizinde yetersiz kalır. Zamanda sınırlı fonksiyonların kullanılması durumunda ise işaretin frekans spektrumu belirlenemeyecektir. Dönüşüm tekniklerinde kullanılan analiz fonksiyonlarının kısıtlamalarının yolaçtığı bu problem araştırmacıları hem zaman, hem de frekans domeninde sınırlı kalacak fonksiyonların kullanımına itmiştir. Bir domende sınırlı bir fonksiyon diğer bir domende sınırlı kalamaz. Ancak frekans domeninde sınırlı olan bir fonksiyonun enerjisi zaman domeninde belirli bir bölgede yoğunlaşmış ise analiz için optimum kabul edilebilir. Son yıllarda işaret işlemecilerin büyük ilgisini çeken Wavelet dönüşümünün temel prensibi budur. 2 domende sınırlı ve teoride belirtilen koşulları sağlayan fonksiyonlar Wavelet fonksiyonları olarak tanımlanır ve dönüşüm işaretin bu fonksiyonlar üzerine izdüşümü alınarak gerçekleştirilir. Wavelet teorisinden bağımsız olarak geliştirilen Fi dönüşüm metodunda da sınırlı fonksiyonlar kullanılır, ancak fonksiyon seçimindeki koşullar daha esnektir. 2 dönüşümde de kullanılan analiz fonksiyonları band geçiren özellik gösterdiklerinden işaret tekilliklerini incelemek için uygundur. Dönüşüm domeninde işaret az sayıda katsayıyla modelleneceğinden veri sıkıştırma uygulamaları için olumlu sonuçlar beklenebilir. Tezde, zaman-frekans teknikleri kısaca tanıtıldıktan sonra, incelenecek dönüşümlerin matematiksel temelleri verilmiş, algoritmaları kurulmuş ve 1 ve 2 boyutlu işaretler için denenmiştir. İlerleyen bölümlerde dönüşümlerin uygulama alanları araştırılmış, kenar kestirimi ve veri sıkıştırma problemlerine uygulanarak, dönüşümler bu problemlerin çözümünde kullanılan diğer tekniklerle karşılaştırılarak performansları araştırılmıştır. V -

Özet (Çeviri)

SUMMARY NEW TRANSFORMS IN TIME-FREQUENCY ANALYSIS AND THEIR APPLICATIONS The purpose of the Time-frequency analysis is to obtain simultaneous information about the frequency content of a signal and about its evolution in time. The traditional Fourier analysis gives perfect information about the frequency content of a signal, but no idea about its representation in time domain. A signal can be represented as weighted sum of certain base functions. The weights in this sum operation are called transform coefficients and are thought as the projections of the signal into base functions. If these functions are localized in time and frequency domains the representation gives time-frequency representation of the signal analized and is considered to be optimum. Fourier method employs as basis, trigonometric functions which are perfectly localized in frequency domain, but completely global in time, in fact Fourier representation gives no information about the behavior of the signal in time domain. In order to obtain both frequency and time information in one representation, Gabor suggested to perform Fourier analisis on the signal as it appears when seen through a set of identical windows which are translated with respect to each other in time. In Gabor method, Gaussian functions which are localized in time and frequency domains are used as window functions. The use of other functions has caused Short- time- Fourier transform to come out.[l] In these methods, which give time-frequency representation, the bandwidht of analyzing functions is constant. If the analyzing functions are not wide enough, they. are unable to capture the low frequency information and wider they get, they lose short time duration changes in the signal. So it is felt that the analyzing functions should have a constant bandwidht-to-center-frequency ratio. The Wavelet Transform introduced in 1986 by Lemarie and Meyer [4] and The Fi Transform developed by Frazier and Jawert [16] in 1985 have this property. Both Wavelet and Fi functions are localized in time and frequency domains and are generated from a single function by dilatation and translation. - VIThe time domain and the frequency domain are covered by dilatation and translation processes. So in these methods one function will be enough to create the basis functions employed to perform the time-frequency analysis. The transforms are considered as the projection of the signal into basis functions and can be computed as inner-products of the signal with these basis functions. So every inner-product will carry information about the behavior of the signal in those regions of the time and frequency domains where the corresponding analyzing functions are located. This is the principal idea of Wavelet and Fi Transform techniques. A function must satisfy two conditions to be a Wavelet function. p(w)\ 2 \w\ _1 ı 2*b [1/ II 2 o) with 0 < A < B < «= 2 ~in f f(x)p{ 21 x-k)d: (6) gives transform coefficients. For each resolution step defined as r-2', The Wavelet Transform is defined as V 2>f = P m > (7) The Wavelet function y is produced with the help of scaling function 0. (2x-n) (8. a) nsZ *>(*)= I (-irc(_“.n*(2x-fi) (8.b) ft£Z CB coefficients are employed in the analysis of discrete time signals. The transforn process is seen as multiresolution signal decomposition using multirate filter bank with equal bandwidths in logarithmic scale. CR coefficients are impuls responses of the filters in the filter bank. - VIII -In each resolution step, the low frequency information is modeled with the scaling function and the detail information which is lost between two scales is recovered with the help of Wavelet functions. i> /- Y. 4>,-l.k >P,-X.t + keZ Y. V /-i.* >P ;-!.* (9) keZ ?d| Figure 1. Analysis step in Wavelet Transform Figure 2. Synthesis step in Wavelet Transform 4> and ¥ functions satisfy orthonormality conditions; so for perfect reconstruction, analyzing functions can be used also as synthesis functions. This sort of functions constitues orthogonal Wavelet families; each member is orthogonal to other members. If the orthonormality condition is not satisfied, but analysis and synthesis families are orthogonal, this is the case of biortogonal Wavelet families. In Fi Transform method also, basis functions are computed as dilatations and translations of one single function and transform coefficients are defined as projection of the signal into this analysis functions. - IX -/(o- I </>p *.* >*»*.» (o+ I I p..t do) *eZ* »-m*l teZ”gives the Psi-decoraposition of the signal f(t) with analysis functions 4> and synthesis functions $». The inner-products .».» > are called Phi Transform coefficients. The analysis functions i>m has a low-pass nature in the frequency domain, while its translated versions #" I s.mM i have bandpass nature. The decomposition process results from a lowpass and and a series of bandpass filter operations; so the Phi Transform method decomposes a signal into subspectrums, therefore it is able to zoom in the singularities of the signal. fix) *#> ** j,-. «4*0 y-i- - /i /: o- fj Figure 3. Analysis step in Fi Transform A Î2 'F, - - f(x) Figure 4. Synthesis step in Fi Transform - X -The Phi functions are easier to construct than the Wavelet functions. Once the frequency axis is covered with the set of analysis functions and the Parseval relation is satisfied, the signal can be reconstructed from its Phi coefficients, so the transform is complete. The number of Phi Transform coefficients at frequency level m is half at level(m+l). The high frequency information required to pass from last levels to precedent levels in the reconstruction step is contained in a small number of numerically significant coefficients. This indicates that for signals with localized high-frequency components, it is reasonable to expect the Phi Transform to yield data compression. Time-frequency transforms can be used to solve edge detection and data compression problems. [4, 10] The sharp variation points of an image intensity are important, because they are generally located at the boundaries of the image components. In order to detect these variation points, the concept of multiscale edge detection is introduced. The signal is smoothed by a convolution with a smoothing function which is dilated by a scale factor and variation points are detected with a first or second order differentiation operator. The smoothing function is often chosen to be a Gaussian. First and second order derivatives of this function are accepted as Wavelet functions, because the admissibility condition (2) is satisfied. In two dimensions, the image is smoothed at different scales 2' by a convolution with a smoothing kernel {x,y) dilated by 2' the gradient vector 3{t*o ^ fx.y)) is computed at each point of the smoothed image and edges are defined as points where the modulus of the gradient vector is maximum in the direction where the gradient vector points to. When the Wavelet function is the first order derivative of Gaussian function, the method is equivalent to the Gradient method and if the second order derivative is used the method is equivalent to the Laplacian method. For noisy images, better results are obtained after the first step, because the spectrum is localized by the use of smoothing function. In Phi method, the decomposition is realized by analyzing filters whose frequency responses cover the whole frequency axis. By choosing bandpass nature filters from the Phi filter set, the edge detection operation can be performed. Using bandpass filters is equivalent to the LOG filter of Marr and Hildreth. In the multiscale edge detection method of Marr and Hildreth, the image is first smoothed and then edge representation is obtained by computing the Laplacian of the image. These operations are equivalent to convolving the image by a bandpass filter and have good performance in noisy environments - XI -In chapter 4, several examples using these two methods are given and results are compared with the results of conventional techniques such as Gradient, Laplacian and LOG filter methods. [6] Wavelet and Phi transform methods are appropriate to paralel processing algorithms; the signal is decomposed into sub spectrums and processed separately and then reconstructed by inverse transform. Most of the operators and matrix became sparse in transform domains. Matrix product, matrix inversion operations, solution of linear equations take advantage of the sparcity. In chapter 5, some results of data compression in 1-D signals are investigated. - XII

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