HTML> Image Processing and Recognition WAVELET SUBSAMPLING THEOREM

The wavelet subsampling theorem arise from the observation that if we are given the values of a function F in equally spaced points, F(n T), then we know that (B=π/T):

The subsampling theorem can be put in the wavelet framework by letting B => 2m B. The scaling function f(x) = sinc(x) satisfies the dilation equation

f( x ) = ∑k sinc(π k/2) f(2x - k)

The corresponding wavelet is
w( x ) = [ sin(π (x-1/2)) - sin( 2 π (x-1/2) ) ] / [ π (x-1/2) ]

The multiresolution analysis corresponds to Paley-Wiener subspaces of L2(R).

 

This situation holds in general for any wavelet-based M.R.A.
Let F in Vj then

F(x) = ∑ cn s(2jx - n)

We compute the Forier transform of F and its sampled values:
F^(v) = ∑ cn s^(v/2j) e(-i v n / 2j)
F(m/2j) = m cn s(m-n)
m s(m-n) e(-1 v m/2j) = m Int s^(u) e(i u (m-n)) du e(-1 v m/2j)
  = Int du 2 π ∑k D(u - v/2j - 2 π k) s^(u) e(-i u k)
  = 2 π ∑k s^(v/2j + 2 π k) e(-i v n / 2j)
where we have used Poisson summation formula (assuming that we can interchange the sum and the integral).
Thus
m F(m/2j) e(-i v m/2j) = m,n cn s(m-n) e(-i v m/2j)
  = k s^(v/2j + 2 π k) ∑n cn e(-i v n/2j)
F^( v ) = n cn e(-i v n/2j)
  = m F(m/2j) 1/2j X^(v/2j) e(-i v m/2j)
F(x) = m F(m/2j) X(2jx - m )
where X is defined in terms of its Fourier transfor:
X^(v) = s^(v) / ∑ s^(v + 2 π k)

This is particularly interesting when the denominator is constant, and X is proportional to the scaling function.



Marco Corvi - Page hosted by geocities.com.