Database Reference
In-Depth Information
The Haar wavelets are the most elementary example of wavelets. The
mother wavelet
ψ
for the Haar wavelets is the following function:
⎧
⎨
⎩
1
,
if 0
<t<
0
.
5
,
ψ
Haar
(
t
)=
(2.21)
−
1
,
if 0
.
5
<t<
1
,
0
,
otherwise
.
Ganesan
et al.
[26, 25] proposed in-network storage of wavelet-based
summaries of sensor data. Recently, discrete wavelet transform (DWT)
was also proposed in [53, 7] for sensor data compression. For sustainable
storage and querying, they propose progressive aging of summaries and
load sharing techniques.
5.6.3 Discussion.
The basis functions of some wavelet trans-
forms are non-zero only on a finite interval. Therefore, wavelets may
be only able to capture local (time dependent) properties of the data,
as opposed to Fourier transforms, which can capture global properties.
The computational eciency of the wavelet transforms is higher than the
Fast Fourier transform (FFT). However, while the Fourier transform can
accurately approximate arbitrary signals, the Haar wavelet is not likely
to approximate a smooth function using few features.
The wavelet transform representation is intrinsically coupled with ap-
proximating sequences whose length is a power of two. Using wavelets
with sequences that have other lengths require ad-hoc measures that
reduce the fidelity of the approximation, and increase the complexity of
the implementation. DFT and DCT have been successfully adapted to
incremental computation [72]. However, as each DFT/DCT coecient
makes a global contribution to the entire data stream, assigning less
significance to the past data is not obvious with these transformations.
5.7 Lossless vs. Lossy Compression
While lossless compression is able to accurately reconstruct the origi-
nal data, lossy compression techniques approximate data streams within
a certain error bound. Most lossless compression schemes perform two
steps in sequence: the first step generates a statistical model for the
input data, and the second step uses this model to map input data to
bit sequences. These bit sequences are mapped in such a way that fre-
quently encountered data will produce shorter output than infrequent
data. General-purpose compression schemes include DEFLATE (em-
ployed by gzip, ZIP, PNG, etc.), LZW (employed by GIF, compress,
etc.), LZMA (employed by 7zip). The primary encoding algorithms used
to produce bit sequences are Huffman coding (also used by DEFLATE)
