Clearly, each
step of the averaging loses some information; the lost information is captured
by the detail coe?±cients shown in this table. Each detail coe?±cient is the
di?®erence between the computed average and the second of the pair of numbers
from which that average was computed. For example, vector V has two
pairs of elements: (5, 6) and (4, 4). The averages are 5.5 = ((5 + 6)/2) and
4 = ((4 + 4)/2). The detail coe?±cients are ??” 0.5 and 0. The overall average,
together with an ordered (starting from the lowest resolution) list of the detail
coe?±cients, constitutes the wavelet coe?±cients of V .
Table 4. [16].
Resolution Averages Detail Coe?±cients
2 [5, 6, 4, 4] ??’
1 [5.5, 4] [??’0.5, 0]
0 [4.75] [0.75]
There are several important properties of this coding technique. First, we
can reconstruct the original data given all the wavelet coe?±cients. As a corollary,
we can also e?±ciently compute averages of arbitrary subsequences of the
original data vector V by using the wavelet coe?±cients at the appropriate levels
of resolution. Second, we can compute wavelet coe?±cients at level i merely
from the wavelet coe?±cients at level i+1. Finally, if the detail coe?±cients are
small, they can be neglected with some loss of accuracy in computed averages.
In this sense, wavelet coding is multi-resolution.
304
Chapter 12 Data Management in Sensor Networks
Wavelet encoding can be used to support spatial-temporal queries.
Pages:
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489