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5
Compression
Equation (123) is used to obtain the following result:
[β]"
1
®
[ß]"
1
=([Φ][Π[Φ])®([Φ][Γ][Φ])
= ([Φ] ® [Φ])(([Γ][Φ] ® [Γ][Φ])) (124)
where we have used the identity
(M[B])
® ([C][D]) = (M ® [C])([ß ] ® [£>] )
Using th e sam e identit y agai n fo r th e secon d lef t direc t produc t i n (124) , w e
get
[βΓ
1
®
[βΓ
1
= ([Φ] ® [Φ])([Π ® [Γ])([Φ] ® [Φ]) (125)
which shows that the matrix of the eigenvectors of [β] ~
1
® [β] "
1
is given
by [Φ] ® [Φ], since [Γ] ® [Γ] is an Ν
2
χ Ν
2
diagonal matrix.
Let Η be the Karhunen-Loeve transform of the modified data vector h.
If Ψ„, is the mth eigenvector of [β]"
1
® [β]"
1
(i.e., the N
2
element ...