Multiresolution analysis

Multiresolution analysis (MRA) [7,8,9] provides a concise framework for explaining many aspects of wavelet theory such as how wavelets can be constructed [1,10]. MRA provides greater insight into the representation of functions using wavelets and helps establish a link between the discrete wavelet transform of continuous functions and discrete signals. MRA also allows for an efficient algorithm for implementing the discrete wavelet transform. This is called the fast wavelet transform and follows a pyramidal 

Consider the following example which presents some concepts that we will use when we explain the idea of a multiresolution analysis in greater detail. 

Let Vo be a subspace that consists of all functions which are constant on unit intervals k t k -f- 1 for any k G Z. These intervals are denoted by 

You will notice that if we shift f(t) along by 1, then this function still remains in the same space, Vq. Hence, if fit) G Vq, then fit + 1) is also in Vq. This property is called a shift invariance or a translation invariance property. Integer translates of any function remain in the same space - this is more generally stated if fit) G Vq, then fit - k) G Vq. 

Notice that if we rescale fit) by a factor of 2, then this function will be constant on. The function f(2t) is then in Vi. If we translate f(2t) by 

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Multiresolution analysis conforms with the definition of structure (8). More importantly, it guarantees that by moving to higher subspaces (scales), better approximations of the unknown functions can potentially be obtained, which is the additional property sought. 

The framework, however, as introduced so far is of little help for our purpose since the shift from any subspace to its immediate in hierarchy would require to change entirely the set of basis functions. Although j x) are all created by the same function, they are different functions and, consequently, the approximation problem has to be solved from scratch with any change of subspace. The theory of wavelets and its relation to multiresolution analysis provides the ladder that allows the transition from one space to the other. 

Feauveau, J. C., Nonorthogonal multiresolution analysis using wavelets. In Wavelets—A Tutorial in Theory and Applications (C. K. Chui, ed.). Academic Press, San Diego, CA, p. 153. 1992. 

Takei, S., Multiresolution analysis of data in process operations and control. M.S. Thesis, Massachusetts Institute of Technology, Dept. Chem. Eng., Cambridge, MA, 1991. 

B. A. Hess Dr. Gaspard has introduced the vibrogram as a tool to extract periodic orbits from the spectrum by means of a windowed Fourier transform. This raises the question whether other recent techniques of signal analysis like multiresolution analysis or wavelet transforms of the spectrum could be used to separate the time scales and thus to disentangle the information pertinent to the quasiclassical, semiclassical, and long-time regimes. Has this ever been tried ... 

Verrecchia, E.P. (2004) Multiresolution analysis of shell growth increments to detect natural cycles. In Francus, P. (Ed.) Image Analysis, Sediments and Paleoen-vironments, Vol. 7, Developments in Paleoenvironmental Research. Dordrecht Kluwer Academic, pp. 273-293. 

Ismail, A.E., Rutledge, G.C. and Stephanopoulos, G. (2003a) Multiresolution Analysis in Statistical Mechanics — I. Using Wavelets to Calculate Thermodynamic Properties. 

The dual scaling function and dual wavelet function v / generate a dual multiresolution analysis such that... 

A biorthogonal scaling and wavelet function are semiorthogonal if they generate an orthogonal multiresolution analysis... 

H.L. Shen, J.H. Wang, Y.Z. Liang, K. Pettersson, M. Josefson, J. Gottfries and F. Lee, Chemical Rank Estimation by Multiresolution Analysis for Two-way Data in the Presence of Background, Chemometrics Intelligent Laboratory Systems 37 (1997), 261-269. 

Fig. 3 Multiresolution Analysis plot of a one-dimensional DWT noisy data (top left) wavelet shrinkage reconstruction (top right) MRA plot of the DWT of the noisy data (bottom left) MRA plot of the thresholded wavelet coefficients (bottom right). Figure courtesy of Prof. David Donoho, Stanford University.

T.A. Wilson, S.K. Rogers and L.R. Myers, Perceptual Based Hyperspectral Image Fusion Using Multiresolution Analysis, Optical Engineering. 34 (11) (1995). 3154-3164. 

Orthogonal wavelets are related with theory of multiresolution analysis and usually cannot be expressed in an informal context they must fulfill stringent orthogonal conditions, on the other hand, wavelet frames are constructed by simple operations of translation and dilation and are the easiest to use (Akay 1997, Heil 1989, Gutes et al. 

Wilson, T. A., Rogers, S. K. Myers, L. R. (1995) Perceptual-based hyperspectral image fusion using multiresolution analysis. Optical Engineering 34, 3154-3164. 

M. Desco, J. A. Hernandez, A. Santos, and M. Brammer, Multiresolution analysis in fMRl Sensitivity and specificity in the detection of brain activation. Hum. Brain Mapp. 14 16-27 (2001). 

K. Wang, H. Begleiter, and B. Potjesz, Spatial enhancement of event-related potentials using multiresolution analysis. Brain Topogr. 10 191-200 (1998). 

Under certain conditions on the generator [Pg.248]

If Wj is the orthogonal complement we obtain orthogonal wavelets [9]. In this case the transformation T , which relates the single and the multiscale coefficients is the Fast Wavelet Transform [5]. If V is a multiresolution analysis, then a refinement equation is valid for < ... 

Figure 8 Overview of the DWT and multiresolution analysis scheme known as the pyramid algorithm/ The original signal is separated into low-frequency and high-frequency components, which comprise the signal approximation and detail information, respectively. Each level decomposes the approximation information further, making each level of detail (dj) a separate frequency band.

Figure 9 This series illustrates multiresolution analysis, separating out the high-frequency information at each level of transformation in the pyramid algorithm (illustrated in Figure 8). Note that the approximation (a,) signal in higher level iterations contain much less detailed information, because this has been removed and encoded into wavelet detail coefficients at each DWT deconstruction step. To reconstruct the original signal, the inverse DWT needs the wavelet coefficients of a given approximation level / (ai) and all detail information leading to that level (di i).

The restrictions placed on the mother wavelets for multiresolution analysis do not limit the variety of shapes that can be used as mother wavelets different researchers have proposed several different wavelet functions, each with benefits and drawbacks. The wavelet shape tradeoff is between how compactly it can be localized in space and its level of smoothness. For example, the Haar wavelet, which is the simplest wavelet and was identified almost 100 years ago, is well localized in space, but it has an unnatural square-wave oscillation (see Figure 10). Many related wavelets exist, collectively referred to as wavelet families some of these families include the Meyer wavelet, Coiflet wavelet, spline wavelet, orthogonal wavelet, symmlet wavelet, and local cosine basis. Figure 10 depicts several of these wavelets and... 

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