Inverse wavelet transform

A desirable property of any transform is to be able to revert from the transformed function into the original function. An inverse transform exists for the continuous wavelet transform. The original function can be reconstructed using 

It is not necessary to perform the continuous wavelet transform for all values of a and b, since f(t) can be reconstructed from a much sparser set of (a, b) 

---

Figure 10.23 demonstrates one aspect of discrete wavelet transforms that shows similarity to discrete Fourier transforms. Typically, for an. V-point observed signal, the points available to decomposition to approximation and detail representations decrease by (about) a factor of 2 for each increase in scale. As the scale increases, the number of points in the wavelet approximation component decreases until, at very high scales, there is a single point. Also, like a Fourier transform, it is possible to reconstruct the observed signal by performing an inverse wavelet transform,... 

The general idea of all wavelet based image fusion schemes is that the wavelet transforms of the two registered input images L and I2 are computed and these transforms are combined utilising some kind ol fusion rule x (see Fig. 37). Then, the inverse wavelet transform W is computed, and the fused image I is reconstructed by... 

The CWT is compactly described by Eqs. [4] and [6], but this definition allows for infinitely redundant transformations.There is no limit to the number of dilated and translated wavelets (4 (a ), where a and b are real numbers) used in the transform. This unrestricted and unguided use of wavelets to convert a signal into wavelet space often prevents the use of an inverse wavelet transformation because of violations of the conditions required by Eq. [7]. Even though these transforms are redundant and nonreversible, they still reveal information about the character of a particular signal. 

S is a wavelets spectrum, W is an analyzing wavelets matrix, and is the transpose matrix of W. This study uses Coifman function of order 30 as the analyzing wavelets as shown in Fig. 2. This basie wavelet is expanded in the multi-scale to analyze the image D [9]. Because the wavelets transform is an orthonormal transformation, the inverse wavelets transform and its multiresolution is expressed by... 

Like the FFT, the fast wavelet transform (FWT) is a fast, linear operation that operates on a data vector in which the length is an integer power of two (i.e., a dyadic vector), transforming it into a numerically different vector of the same length. Like the FFT, the FWT is invertible and in fact orthogonal that is, the inverse transform, when viewed as a matrix, is simply the transpose of the transform. Both the FFT and the discrete wavelet transform (DWT) can be regarded as a rotation in function... 

We point out that this operation from two dimensions to one dimension is not the unique possible inverse transformation. Two properties of CWT will appear to be important for our study Not every function on the positive half-plane is a wavelet transformation. Thus the successive transformation... 

This ensures that the information stored in a wavelet coefficient is not repeated elsewhere. In some applications orthogonality is not required since redundancy can help to reduce the sensitivity of noise or improve the shift invariance of the transform. With this supporting theory discrete wavelet transform and inverse discrete wavelet transform can now be defined as... 

For each decomposition level f a faithful reconstruction of the original signal is possible using the inverse discrete wavelet transform (IDWT) and the set of approximation coefficients obtained at level altogether with all sets of detail coefficients from level f until level 1. IDWT is achievable by upsampling the coefficients obtained at level j and applying Eq. 9.18 ... 

This remark demands great care and consideration. Through the signal-wavelet inversion formula, derived later on, we can represent the (physical) wavefunction as a superposition of dual basis functions and wavelet transform coefficients. We symbolically denote this, for the dyadic representation (Sec. 1.3.2), by 9(b) = i J2j,i Thus at a given point b, the... 

There is an alternate representation for the signal-wavelet inversion relation in which the wavelet transform, W ia. b), defines the effective basis states (Handy and Murenzi (1999)) (6) =, b — j2 ). 

Given the wavelet transform, fe), the CWT signal-wavelet inversion... 

For every wavelet in a strict sense, g t), a reconstruction wavelet h t) fulfilling certain properties can be found [8]. Utilizing this, one can define an inverse transformation from the positive half plane H to the real axis,... 

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).

According to Mallat s multiresolution analysis (MRA), the discretising of b) that preserves all information about the decomposed function cannot be coarser than the critical sampling, and any coarse discretisation will not give a unique inverse transformation. Under the critical sampling, is assigned a value of 2 and the value of 1. By discretising (, b) - (2, " 2, we have the wavelet function... 

The scaling transform appears in two distinct ways in wavelet analysis. As implied above, it is the generator for the pointwise reconstruction of the wavefunction, at point b. This asymptotic, zero scale limit, reconstruction of (6) is exactly what the signal( )-wavelet inversion formula, reviewed in this work, achieves. The zero scale limit in Eq.(1.5), is exactly duplicated by transforming into a dual-wavelet basis representation (refer to Secs. 1.3.1 and 1.3.2). 

---