Signal Processing - November 2016 - 137
^E @per h) =
R
0
S 0
S
1
S I Ll # Ll
0
S2
I N - 2L l # N - 2L l
S 0
S 0
0
S
S1
0
S I Ll # Ll
T2
V
1I
l
l
2 L # L WW
0 W
W
0 W.
1I
W
l
l
2 L #L W
W
0 W
X
Again, in the unusual case that N # 2Ll ,
the form of the pseudoinverse changes
slightly. By viewing other signal extension operators as matrices acting on signal vectors, one can readily find the
appropriate (E @) ) .
The adjoint for orthogonal
wavelets
For orthogonal wavelets (e.g., Haar and
more generally Daubechies wavelets),
the adjoint W )zpd is exactly the analysis
operator W @zpd. For orthogonal wavelets
with general boundary conditions, we
can use the splitting (2) and imple)
ment ^E @h .
For an example image, consider a
standard resolution test chart [6], shown
in Figure 1. We prescribe symmetric
boundary conditions on the image,
which will affect both W, R, and their
adjoints. Symmetric boundary conditions (compared to periodic or zeropadded boundary conditions) produce
significantly fewer edge effects, and so
they are a natural choice.
Fortuitously, for orthogonal wavelets, the adjoint of W with symmetric
boundary conditions is "very close to"
the analysis operator W @. This fact is
often used (e.g., [2]), and as we can see
from the splitting (2), the error is intro)
duced through E . ^E @h . Again, by
)
@
merely implementing ^E h , we can use
)
the true adjoint W .
Figure 2 shows the deblurred image
after 2,500 iterations of a fast iterative
shrinkage-thresholding algorithm
(FISTA), a fast proximalgradient method,
developed by Beck and Teboulle [2]. The
wavelet reconstruction operator W is
taken to be a three-stage Haar discrete
wavelet transform with symmetric
boundary conditions. We set m = 2 #
10 -5. We use both the true adjoint W )
and the approximation W @. Using the
true adjoint, the image reconstruction relative error (versus the unblurred image)
is 8.91 # 10 - 4 and 31.8% of the coefficients are nonzero. Using the pseudoinverse approximation, the relative error is
8.91 # 10 - 4 and 32.1% of the coefficients are nonzero. In this case the
pseudoinverse approximation works
extremely well.
The adjoint for biorthogonal
wavelets
For biorthogonal wavelets, we no longer
have W )zpd = W @zpd. Since we have
relaxed ourselves to biorthogonal wavelets (which are nice for symmetric
boundary conditions [7]), it is perhaps
too much to ask that the adjoint of the
primal wavelet reconstruction operator
involves only the primal wavelets. Let us
recall briefly the pertinent aspects of
frames of R d . These facts are described
in more detail and generality in [4].
Let y ! R N be an arbitrary signal
p
vector and {z i} i = 1, p $ N be a set of
N
vectors in R . Define the analysis operator U as the p # N matrix
(a)
(b)
Figure 1. (a) The left half of the original, unblurred image and (b) the left half of the blurred
image. To create the blurred image b, the blurring operator R is applied to the original image
and small amount of Gaussian noise is added.
T
> h H.
z1
U=
T
zp
If U has full rank, we say that {z i} is a
frame and we call U the frame analysis
operator. Henceforth we assume {z i} is
a frame.
The product Uu computes the expansion coefficients of the signal u ! R N in
the frame {z i} . The frame synthesis
operator U ) constructs a vector in R N
given some expansion coefficients. Since
{z i} is assumed to be a frame, U ) U is
invertible and we may define the Moore-
Penrose pseudoinverse U @, which implements reconstruction in the frame as
-1
U @ = ^ U ) U h U.
A dual frame can be associated with
a (primal) frame by defining the dual
-1
frame vectors zu i = ^U ) Uh z i for
i = 1, ..., p. We may define analogously
u and
the dual frame analysis operator U
u ).
dual frame synthesis operator U
A fundamental relationship between
primal and dual frame operators is
u ) = U @ (see [4, Theorem 5.5]). We
U
can directly relate the frame operators
IEEE Signal Processing Magazine
|
November 2016
|
Figure 2. The deblurred image after 2,500
iterations of FISTA with W a three-stage Haar
transform with symmetric boundary conditions.
The left half of the figure shows the deblurred
image using the true adjoint W ). The top right
shows a zoomed-in view using W ); the bottom
right shows the same zoomed-in view using
W @ and 2,500 iterations of FISTA.
and the wavelet operators under zeropadded boundary conditions. We can
write zero-padded wavelet reconstruction as frame reconstruction:
W zpd = U @. Then using the above
frame relations, we have
)
@
u.
W )zpd = ^U @h = ^U )h = U
(3)
We present other relationships
between the primal and dual frames
in Table 1. Note that U @ U = I but
UU @ ! I in general.
In (3), we have W )zpd in terms of
dual frame analysis. For biorthogonal
wavelets, dual frame analysis corresponds
137
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
Signal Processing - November 2016 - 10
Signal Processing - November 2016 - 11
Signal Processing - November 2016 - 12
Signal Processing - November 2016 - 13
Signal Processing - November 2016 - 14
Signal Processing - November 2016 - 15
Signal Processing - November 2016 - 16
Signal Processing - November 2016 - 17
Signal Processing - November 2016 - 18
Signal Processing - November 2016 - 19
Signal Processing - November 2016 - 20
Signal Processing - November 2016 - 21
Signal Processing - November 2016 - 22
Signal Processing - November 2016 - 23
Signal Processing - November 2016 - 24
Signal Processing - November 2016 - 25
Signal Processing - November 2016 - 26
Signal Processing - November 2016 - 27
Signal Processing - November 2016 - 28
Signal Processing - November 2016 - 29
Signal Processing - November 2016 - 30
Signal Processing - November 2016 - 31
Signal Processing - November 2016 - 32
Signal Processing - November 2016 - 33
Signal Processing - November 2016 - 34
Signal Processing - November 2016 - 35
Signal Processing - November 2016 - 36
Signal Processing - November 2016 - 37
Signal Processing - November 2016 - 38
Signal Processing - November 2016 - 39
Signal Processing - November 2016 - 40
Signal Processing - November 2016 - 41
Signal Processing - November 2016 - 42
Signal Processing - November 2016 - 43
Signal Processing - November 2016 - 44
Signal Processing - November 2016 - 45
Signal Processing - November 2016 - 46
Signal Processing - November 2016 - 47
Signal Processing - November 2016 - 48
Signal Processing - November 2016 - 49
Signal Processing - November 2016 - 50
Signal Processing - November 2016 - 51
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Signal Processing - November 2016 - 53
Signal Processing - November 2016 - 54
Signal Processing - November 2016 - 55
Signal Processing - November 2016 - 56
Signal Processing - November 2016 - 57
Signal Processing - November 2016 - 58
Signal Processing - November 2016 - 59
Signal Processing - November 2016 - 60
Signal Processing - November 2016 - 61
Signal Processing - November 2016 - 62
Signal Processing - November 2016 - 63
Signal Processing - November 2016 - 64
Signal Processing - November 2016 - 65
Signal Processing - November 2016 - 66
Signal Processing - November 2016 - 67
Signal Processing - November 2016 - 68
Signal Processing - November 2016 - 69
Signal Processing - November 2016 - 70
Signal Processing - November 2016 - 71
Signal Processing - November 2016 - 72
Signal Processing - November 2016 - 73
Signal Processing - November 2016 - 74
Signal Processing - November 2016 - 75
Signal Processing - November 2016 - 76
Signal Processing - November 2016 - 77
Signal Processing - November 2016 - 78
Signal Processing - November 2016 - 79
Signal Processing - November 2016 - 80
Signal Processing - November 2016 - 81
Signal Processing - November 2016 - 82
Signal Processing - November 2016 - 83
Signal Processing - November 2016 - 84
Signal Processing - November 2016 - 85
Signal Processing - November 2016 - 86
Signal Processing - November 2016 - 87
Signal Processing - November 2016 - 88
Signal Processing - November 2016 - 89
Signal Processing - November 2016 - 90
Signal Processing - November 2016 - 91
Signal Processing - November 2016 - 92
Signal Processing - November 2016 - 93
Signal Processing - November 2016 - 94
Signal Processing - November 2016 - 95
Signal Processing - November 2016 - 96
Signal Processing - November 2016 - 97
Signal Processing - November 2016 - 98
Signal Processing - November 2016 - 99
Signal Processing - November 2016 - 100
Signal Processing - November 2016 - 101
Signal Processing - November 2016 - 102
Signal Processing - November 2016 - 103
Signal Processing - November 2016 - 104
Signal Processing - November 2016 - 105
Signal Processing - November 2016 - 106
Signal Processing - November 2016 - 107
Signal Processing - November 2016 - 108
Signal Processing - November 2016 - 109
Signal Processing - November 2016 - 110
Signal Processing - November 2016 - 111
Signal Processing - November 2016 - 112
Signal Processing - November 2016 - 113
Signal Processing - November 2016 - 114
Signal Processing - November 2016 - 115
Signal Processing - November 2016 - 116
Signal Processing - November 2016 - 117
Signal Processing - November 2016 - 118
Signal Processing - November 2016 - 119
Signal Processing - November 2016 - 120
Signal Processing - November 2016 - 121
Signal Processing - November 2016 - 122
Signal Processing - November 2016 - 123
Signal Processing - November 2016 - 124
Signal Processing - November 2016 - 125
Signal Processing - November 2016 - 126
Signal Processing - November 2016 - 127
Signal Processing - November 2016 - 128
Signal Processing - November 2016 - 129
Signal Processing - November 2016 - 130
Signal Processing - November 2016 - 131
Signal Processing - November 2016 - 132
Signal Processing - November 2016 - 133
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
Signal Processing - November 2016 - 145
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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