Which is the difference between Fourier Transformation and discrete wavelet transformation? [closed]

Question

I'm sorry, I'm pretty new about these argoments. Is there a difference between Fourier Transformation and discrete wavelet transformation? I looked in Internet but I didn't understand if there is a difference between them.

Answer 1

For a meaningful comparison you want to compare the wavelet transform (WT) with the short-time Fourier transform (STFT or "windowed" Fourier transform). Both transformations give you information on the frequencies that are present in your signal and when these frequencies appear --- which is helpful if the signal changes over time.

Both are examples of filter banks, separating the input signal into multiple components, each one carrying a single band of frequencies. The difference then boils down to how you construct the bands. For the WT you can use a rectangular band (if you take Shannon wavelets). For the STFT the band edges decay less rapidly. Moreover, the band width for the STFT is typically kept fixed (but not always), while for the WT it is typically adapted to the frequency.

A comparison of both image processing techniques in real-world applications can be found in Comparison of wavelet and short time Fourier transform methods in the analysis of electromyographic signals and in A comparison of the wavelet and short-time fourier transforms for Doppler spectral analysis.

Answer 2

Redundancy. Let us start with the Fourier transformation: a fonction $f$, say of one real variable (which could be only a tempered distribution) can be written as $$ f(x)=\int_{\mathbb R}\hat f(\xi) e^{2iπ x\xi} d\xi, \quad\text{where the Fourier transform } \hat f(\xi)=\int_{\mathbb R}f(x) e^{-2iπ x\xi} dx. \tag 1$$ Now, Gabor wavelet: we have $$ f(x)=\iint_{\mathbb R^2}W_f(y,\eta)\phi_{y,\eta}(x) dy d\eta,\quad \text{ with }\quad W_f(y,\eta)=\langle f(x),(\tau_{y,\eta}\phi_0)(x) \rangle_{L^2(\mathbb R)}, \tag 2$$ with $ \phi_{y,\eta}(x)=(\tau_{y,\eta}\phi_0)(x)=\phi_0(x-y)e^{2iπ(x-\frac y2)\eta},\quad \phi_0(t)=2^{1/4} e^{-π t^2}. $

Although $(1)$ is an "orthonormal" decomposition since, a bit formally, we have $$ \int e^{2iπ x\xi} e^{-2iπ x\eta} dx=\delta_0 (\xi-\eta), $$ the wavelet decomposition (2) does not have this property but we have $$ \int \phi_{y,\eta}(x)\overline{\phi_{z,\zeta}(x)} dx=e^{-\frac π 2\vert Y-Z\vert^2}\times\text{oscillatory term}, \qquad Y=(y,\eta), Z=(z,\zeta), $$ so that we have some almost orthogonality with Gaussian decay off diagonal which is enough to manipulate these integrals. The gain that you have at using (Gabor) wavelet is for instance the anti-Wick quantization for which operators with non-negative symbols are non-negative operators, a property which is not fulfilled with "ordinary" or Weyl quantization. The family $\phi_{y,\eta}$ is redundant, but has some almost orthogonality and some new properties as the one mentioned above.