Page 128 - Algorithm Collections for Digital Signal Processing Applications using MATLAB

116 Chapter 3

There the transformation matrix which converts the co-efficients of the
particular basis into the co-efficients of the another set of basis belongs to
the same space is given as

0.5 0.5 0.5 0.5
0.5 0.5i -0.5 -0.5i
0.5 -0.5 0.5 -0.5
0.5 -0.5i -0.5 0.5i

For instant consider the vector [2 3 4 5] can be represented using the
basis u1, u2, u3, u4 with co-efficients [2 3 4 5]. The vector [2 3 4 5] can be
represented using the basis v1, v2, v3, v4 using the co-efficients computed
as

0.5 0.5 0.5 0.5 2 7
-
0.5 0.5i -0.5 -0.5i 3 = 1-i

-
0.5 -0.5 0.5 -0.5 4 1
0.5 -0.5i -0.5 0.5i 5 -1+i

(i.e.) 7*(v1)+(-1-I)*v2+(-1)*v3+(-1+i)*v4 = [2 3 4 5]

Thus the transformation matrix which converts the co-efficients of the
particular vector represented using the basis ‘u’ into the co-efficients of the
same vector represented using the basis ‘v’.
If there are only few significant co-efficients obtained when represented
using the particular set of basis, data compression is achieved. (See chapter
4). This property is called data compaction property of the transformation.
Discrete Cosine Transformation (DCT) is having very high data compaction
property. Hence JPEG (Joint Photographic expert group) is using DCT for
image compression. Also JPEG2000 standard is using DWT (Discrete
Wavelet Transformation) for image compression.