Signal Processing - November 2016 - 138

Table 1. A few primal and dual frame
relationships.
Primal Frame

Dual Frame
u = ^U @ h)
U

@
W zpd

Analysis

U=

Synthesis

U ) = ` W zpd j

u ) = U@
U

Reconstruction

U @ = W zpd

u @ = U)
U

@

)

to wavelet analysis with the dual wavelets, which has a fast transform and is
typically implemented together with
the primal wavelets [3]. For biorthogonal (and orthogonal) wavelets, this
@
states W )zpd = R
W zpd , the latter of
which is a standard operation and has a
fast implementation. Combining this
with the signal extension operator, we
finally find
)

W =

)
W )zpd ^E @h

)
@
=R
W zpd ^E @h . (4)

From this we see that with a complete
wavelet software library, one only
needs to implement (E @) ) to use the
true adjoint instead of the approximation W ) . W @ .
Consider deblurring the resolution
chart from Figure 1 with 2,500 iterations of FISTA using a three-stage
Cohen-Daubechies-Feauveau (CDF)
9/7 discrete wavelet transform with
symmetric boundary conditions. The
results of deblurring are visually very
similar to Figure 2. Using the true
adjoint, the image reconstruction relative error (versus the unblurred
image) is 9.45 # 10 - 4 and 29.23% of
the coefficients are nonzero. Using

the pseudoinverse approximation,
the relative error is 9.67 # 10 - 4 and
29.74% of the coefficients are nonzero. Here the pseudoinverse approximation works, but not quite as well as
the true adjoint.
Figure 3 shows the structural similarity (SSIM) index [8] for the CDF
9/7 experiment using both the pseudoinverse approximation and true
adjoint. For the first few hundred iterations of FISTA, the pseudoinverse
approximation works well, but using
the true adjoint appears to recover an
image more similar to the original,
unblurred image.

Example 2: Blind channel
estimation
Another application that depends on a
fast adjoint in the gradient computation
is in some formulations of blind channel estimation. In blind channel estimation, a single source sends an unknown
signal over multiple channels with
unknown responses. Observers collect
the output of each channel and collectively attempt to determine the source
signal and the channel impulse response
from each channel. Let s be the
unknown source signal of length N and
h i the channel impulse response of the
ith channel, each of length K. Then the
output of the ith channel is
x i = h i ) s,
where * denotes a linear convolution.
Here we prescribe zero-padded bound-

0.85

SSIM

0.8
0.75
0.7
True Adjoint
pinv Approx

0.65
0.6

0

500

1,000
1,500
Iteration

2,000

2,500

Figure 3. The SSIM index over 2,500 iterations of FISTA with W a three-stage CDF 9/7 transform
and using the true adjoint and the pseudoinverse approximation. SSIM compares the deblurred image to the original, unblurred image.

138

IEEE Signal Processing Magazine

|

November 2016

|

ary conditions and take x i to be the
"full" convolution of length K + N - 1.
We hope to recover the source s and
channel responses h i from the output
signals x i . However, note that there is
both a magnitude and phase ambiguity
in h i and s, since we can multiply h i
by a ! 0 and divide s by a , leaving
x i unchanged.
For simplicity, assume we have a
single channel h with observed output
x; we will extend the problem to more
channels in an example to follow. We
can pose the blind deconvolution problem for a single channel as
1 


Table of Contents for the Digital Edition of Signal Processing - November 2016

Signal Processing - November 2016 - Cover1

Signal Processing - November 2016 - Cover2

Signal Processing - November 2016 - 1

Signal Processing - November 2016 - 2

Signal Processing - November 2016 - 3

Signal Processing - November 2016 - 4

Signal Processing - November 2016 - 5

Signal Processing - November 2016 - 6

Signal Processing - November 2016 - 7

Signal Processing - November 2016 - 8

Signal Processing - November 2016 - 9

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Signal Processing - November 2016 - 100

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Signal Processing - November 2016 - 134

Signal Processing - November 2016 - 135

Signal Processing - November 2016 - 136

Signal Processing - November 2016 - 137

Signal Processing - November 2016 - 138

Signal Processing - November 2016 - 139

Signal Processing - November 2016 - 140

Signal Processing - November 2016 - 141

Signal Processing - November 2016 - 142

Signal Processing - November 2016 - 143

Signal Processing - November 2016 - 144

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Signal Processing - November 2016 - 146

Signal Processing - November 2016 - 147

Signal Processing - November 2016 - 148

Signal Processing - November 2016 - Cover3

Signal Processing - November 2016 - Cover4


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