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3. Numerical Linear Algebra 121
The approximated co-efficients of the signal obtained in the first level
(Approximation 1) is further decomposed using Haar transformation to
obtain approximation and detail co-efficients of the second level using the
transformation matrix as given below.
½ ½ 0 0
TM2 = ½ -½ 0 0
0 0 ½ ½
0 0 ½ -½
Multiply the matrix [TM2] with the approximated data obtained in the
first level decomposition to obtain approximation and detail co-efficient of
the signal at the second level.
Thus second level approximation is given as
Approximation 2 = [½ ( ½ (d0+d1) + ½ (d4+d5) )]
= ¼ [ d0+d1+d4+d5]
Detail 2 = [½ ( ½ (d0+d1) - ½ (d4+d5) )]
= ¼ [d0 + d 1 - d4 - d5]
Note that approximation co-efficient in the first level and second level
are the low frequency information derived from the signal. Similarly the
detail co-efficient in the first level and the second level are the high
frequency information derived from the signal.
The Approximation 2, Detail 1 and Detail 2 completely describes the
signal. It is possible to reconstruct the signal using Approximation 2, Detail2
and Detail 2 co-efficients of the signal.
Reconstruction of the signal is obtained using the inverse Haar matrix
Formed with the diagonal matrices filled up with the matrix
1 1
1 -1
