sparse image representation via combined transforms - Convex ...
sparse image representation via combined transforms - Convex ... sparse image representation via combined transforms - Convex ...
78 CHAPTER 3. IMAGE TRANSFORMS AND IMAGE FEATURES Equations (3.38) (3.39) and (3.40) together are equivalent to the matrix multiplication ( H(z) H(−z) )( H(z −1 ) H(−z −1 ) ) ( 2 0 ) G(z) G(−z) G(z −1 ) G(−z −1 ) = 0 2 . Taking the determinant of both sides, we have [H(z)G(−z) − G(z)H(−z)][H(z −1 )G(−z −1 ) − G(z −1 )H(−z −1 )] = 4. (3.41) If both sequence {h(n) :n ∈ Z} and sequence {g(n) :n ∈ Z} have finite length (thinking of the impulse responses of FIR filters), then the polynomial H(z)G(−z) − G(z)H(−z) must have only one nonzero term. Since if the polynomial H(z)G(−z) − G(z)H(−z) has more than two terms and simultaneously has finite length, (3.41) can never be equal to a constant. On the other hand, we have [H(z −1 )+H(−z −1 )][H(z)G(−z) − G(z)H(−z)] = H(z −1 )H(z)G(−z)+H(−z −1 )H(z)G(−z) −H(z −1 )G(z)H(−z) − H(−z −1 )G(z)H(−z) (3.40) = H(z −1 )H(z)G(−z) − H(z −1 )H(z)G(z) +H(−z −1 )G(−z)H(−z) − H(−z −1 )G(z)H(−z) (3.38) = 2[G(−z) − G(z)]. We know that H(z)G(−z) − G(z)H(−z) only has one nonzero term. The polynomial H(z −1 )+H(−z −1 ) can only have terms with even exponents and polynomial G(−z) − G(z) can only have terms with odd exponents, so from the previous equation, there must exist an integer k, such that H(z)G(−z) − G(z)H(−z) =2z 2k+1 , and [H(z −1 )+H(−z −1 )]z 2k+1 = G(−z) − G(z). (3.42)
3.7. PROOFS 79 Similarly, we have [H(z −1 ) − H(−z −1 )][H(z)G(−z) − G(z)H(−z)] = 2[G(−z)+G(z)]. Consequently, [H(z −1 ) − H(−z −1 )]z 2k+1 = G(−z)+G(z). (3.43) From (3.42) and (3.43), we have G(z) =−z 2k+1 H(−z −1 ). The above equation is equivalent to relation (3.30) in the Theorem.
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3.7. PROOFS 79<br />
Similarly, we have<br />
[H(z −1 ) − H(−z −1 )][H(z)G(−z) − G(z)H(−z)] = 2[G(−z)+G(z)].<br />
Consequently,<br />
[H(z −1 ) − H(−z −1 )]z 2k+1 = G(−z)+G(z). (3.43)<br />
From (3.42) and (3.43), we have<br />
G(z) =−z 2k+1 H(−z −1 ).<br />
The above equation is equivalent to relation (3.30) in the Theorem.
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