15.11: Haar Wavelet Basis

Introduction

Fourier series is a useful orthonormal representation (Section 15.9) on \(L^2([0,T])\) especially for inputs into LTI systems. However, it is ill suited for some applications, i.e. image processing (recall Gibb's phenomena (Section 6.7)).

Wavelets, discovered in the last 15 years, are another kind of basis for \(L^2([0,T])\) and have many nice properties.

Basis Comparisons

Fourier series - \(c_n\) give frequency information. Basis functions last the entire interval.

Figure \(\PageIndex{1}\): Fourier basis functions

Wavelets - basis functions give frequency info but are local in time.

Figure \(\PageIndex{2}\): Wavelet basis functions

In Fourier basis, the basis functions are harmonic multiples of \(e^{j \omega_0 t}\)

Figure \(\PageIndex{3}\): \(\text { basis }=\left\{\frac{1}{\sqrt{T}} e^{j \omega_{0} n t}\right\}\)

In Haar wavelet basis, the basis functions are scaled and translated versions of a "mother wavelet" \(\psi(t)\).

Figure \(\PageIndex{4}\)

Basis functions \(\left\{\psi_{j, k}(t)\right\}\) are indexed by a scale j and a shift k.

Let \(\phi(t)=1\), \(0 \leq t<T\) Then \(\left\{\phi(t), 2^{\frac{j}{2}} \psi\left(2^{j} t-k\right), \phi(t), 2^{\frac{j}{2}} \psi\left(2^{j} t-k\right) \mid j \in \mathbb{Z} \text { and }\left(k=0,1,2, \ldots, 2^{j}-1\right)\right\}\).

Figure \(\PageIndex{5}\)

\[\psi(t)=\left\{\begin{array}{l}
1 \text { if } 0 \leq t<\frac{T}{2} \\
-1 \text { if } 0 \leq \frac{T}{2}<T
\end{array}\right. \nonumber \]

Figure \(\PageIndex{6}\)

Let \(\psi_{j, k}(t)=2^{\frac{j}{2}} \psi\left(2^{j} t-k\right)\).

Figure \(\PageIndex{7}\)

Larger \(j\) → "skinnier" basis function, \(j=\{0,1,2, \ldots\}\), \(2^j\) shifts at each scale: \(k=0,1, \ldots, 2^{j}-1\)

Check: each \(\psi_{j, k}(t)\) has unit energy

Figure \(\PageIndex{8}\)

\[\left(\int \psi_{j, k}^{2}(t) \mathrm{d} t=1\right) \Rightarrow\left(\left\|\psi_{j, k}(t)\right\|_{2}=1\right) \nonumber \]

Any two basis functions are orthogonal.

(a) Same scale

(b) Different Scale

Figure \(\PageIndex{9}\): Integral of product = 0

Also, \(\left\{\psi_{j, k}, \phi\right\}\) span \(L^2([0,T])\).

Haar Wavelet Transform

Using what we know about Hilbert spaces (Section 15.4): For any \(f(t) \in L^{2}([0, T])\), we can write

Synthesis

\[f(t)=\sum_{j} \sum_{k} w_{j, k} \psi_{j, k}(t)+c_{0} \phi(t) \nonumber \]

Analysis

\[w_{j, k}=\int_{0}^{T} f(t) \psi_{j, k}(t) d t \nonumber \]

\[c_{0}=\int_{0}^{T} f(t) \phi(t) d t \nonumber \]

Note

the \(w_{j,k}\) are real

The Haar transform is super useful especially in image compression

Haar Wavelet Demonstration

Figure \(\PageIndex{10}\): Interact (when online) with a Mathematica CDF demonstrating the Haar Wavelet as an Orthonormal Basis.