Signal Processing - November 2017 - 88
Rlearn {x } = c3 ° T (c2 ° T (c1 ° x + b1) + b2) + b3
Architecture
Objective Function
25
6
25
25
6
25
W
C
T
3
64
3
64
c2
64
3
H
3
64
3
C
b1
1
1
W
Filters and Biases
64
3
c3
64
3
2
f (.) = .2
T
64
c1
256
256
256
256
-
x
H
6
6
Rθ
3
3
b2 1
1
3
64
b3
1 3
1
y
FIGURE 2. An illustration of a typical CNN architecture for 2562 pixel RGB images, including the objective function used for training. T ($) is the rectified
linear unit f-unction (point-wise nonlinear function). The symbol % denotes a 2-D convolution. The convolutions in each layer are described by a fourdimensional tensor representing a stack of 3-D filters.
Current state of performance
Of the inverse problems we review here, denoising provides the
best look at recent trends in results because there are standard
experiments that appear in most papers. Work on CNN-based
denoising from 2009 [6] showed an average peak signal-to-noise
ratio (PSNR) of 28.5 on the Berkeley segmentation data set, a
less than 1-dB improvement over contemporary wavelet and
Markov random field-based approaches. For comparison, one
very recent denoising work [11] reported a 0.7-dB improvement
on a similar experiment, which remains less than 1 dB better than
contemporary non-CNN methods (including block-matching and
3-D filtering, which had remained the state of the art for years).
As another point of reference, in 2012, one CNN approach [7]
reported an average PSNR of 30.2 dB on a set of standard test
images (Lena, peppers, etc.), less than 0.1 dB better than comparisons, and another [8] reported an average of 30.5 dB on the
same experiment. Recently, [11] achieved an average of 30.4 dB
under the same conditions. One important perspective on these
denoising results is that the CNN is learning the distribution of
natural images (or, equivalently, is learning a regularization).
Such a CNN could be reused inside an iterative optimization as a
proximal operator to enforce this learned regularization for any
inverse problem.
The trends are similar in deblurring and superresolution,
although experiments are more varied and therefore harder to
compare. For deblurring, [12] showed around a 1-dB PSNR
improvement over comparison methods, and [13] showed a
88
further improvement of approximately 1 dB. For superresolution, work from 2014 [15] reported a less than 0.5-dB improvement in PSNR over comparisons. During the next two years,
[16] and [19] both reported a 0.5-dB PSNR increase over this
baseline. Even more recent work, [30], improves on the 2014
work by around 1.5 dB in PSNR. For video superresolution,
[18] improves on non-CNN-based methods by about 0.5 dB
PSNR and [20] improves upon that result by another 0.5 dB.
For inverse problems in medical imaging, direct comparison between works is impossible due to the wide variety of experimental setups. A 2013 CNN-based work [24]
shows improvement in limited-view CT reconstruction over
direct methods and unregularized iterative methods but does
not compare to regularized iterative methods. In 2015, [25]
showed (in full-view CT) an improvement of several decibels
in signal-to-noise ratio (SNR) over direct reconstruction and
around 1-dB improvement over regularized iterative reconstruction. Recently, [26] showed about 0.5-dB improvement in
PSNR over TV-regularized reconstruction, while [27] showed
a larger (1-4 dB) improvement in SNR over a different TVregularized method (Figure 3). In magnetic resonance imaging
(MRI), [22] demonstrates performance equal to the state of the
art, with advantages in running time.
Do these improvements matter? CNN-based methods have
not, so far, had the profound impact on inverse problems that
they have had for object classification. The difference between
30 and 30.5 dB is impossible to see by eye. On the other hand,
IEEE SIGNAL PROCESSING MAGAZINE
|
November 2017
|
Table of Contents for the Digital Edition of Signal Processing - November 2017
Signal Processing - November 2017 - Cover1
Signal Processing - November 2017 - Cover2
Signal Processing - November 2017 - 1
Signal Processing - November 2017 - 2
Signal Processing - November 2017 - 3
Signal Processing - November 2017 - 4
Signal Processing - November 2017 - 5
Signal Processing - November 2017 - 6
Signal Processing - November 2017 - 7
Signal Processing - November 2017 - 8
Signal Processing - November 2017 - 9
Signal Processing - November 2017 - 10
Signal Processing - November 2017 - 11
Signal Processing - November 2017 - 12
Signal Processing - November 2017 - 13
Signal Processing - November 2017 - 14
Signal Processing - November 2017 - 15
Signal Processing - November 2017 - 16
Signal Processing - November 2017 - 17
Signal Processing - November 2017 - 18
Signal Processing - November 2017 - 19
Signal Processing - November 2017 - 20
Signal Processing - November 2017 - 21
Signal Processing - November 2017 - 22
Signal Processing - November 2017 - 23
Signal Processing - November 2017 - 24
Signal Processing - November 2017 - 25
Signal Processing - November 2017 - 26
Signal Processing - November 2017 - 27
Signal Processing - November 2017 - 28
Signal Processing - November 2017 - 29
Signal Processing - November 2017 - 30
Signal Processing - November 2017 - 31
Signal Processing - November 2017 - 32
Signal Processing - November 2017 - 33
Signal Processing - November 2017 - 34
Signal Processing - November 2017 - 35
Signal Processing - November 2017 - 36
Signal Processing - November 2017 - 37
Signal Processing - November 2017 - 38
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Signal Processing - November 2017 - 41
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Signal Processing - November 2017 - 43
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Signal Processing - November 2017 - 48
Signal Processing - November 2017 - 49
Signal Processing - November 2017 - 50
Signal Processing - November 2017 - 51
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Signal Processing - November 2017 - 60
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Signal Processing - November 2017 - 63
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Signal Processing - November 2017 - 70
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Signal Processing - November 2017 - 73
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Signal Processing - November 2017 - 85
Signal Processing - November 2017 - 86
Signal Processing - November 2017 - 87
Signal Processing - November 2017 - 88
Signal Processing - November 2017 - 89
Signal Processing - November 2017 - 90
Signal Processing - November 2017 - 91
Signal Processing - November 2017 - 92
Signal Processing - November 2017 - 93
Signal Processing - November 2017 - 94
Signal Processing - November 2017 - 95
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Signal Processing - November 2017 - 97
Signal Processing - November 2017 - 98
Signal Processing - November 2017 - 99
Signal Processing - November 2017 - 100
Signal Processing - November 2017 - 101
Signal Processing - November 2017 - 102
Signal Processing - November 2017 - 103
Signal Processing - November 2017 - 104
Signal Processing - November 2017 - 105
Signal Processing - November 2017 - 106
Signal Processing - November 2017 - 107
Signal Processing - November 2017 - 108
Signal Processing - November 2017 - 109
Signal Processing - November 2017 - 110
Signal Processing - November 2017 - 111
Signal Processing - November 2017 - 112
Signal Processing - November 2017 - 113
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Signal Processing - November 2017 - 115
Signal Processing - November 2017 - 116
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Signal Processing - November 2017 - 118
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Signal Processing - November 2017 - 120
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Signal Processing - November 2017 - 125
Signal Processing - November 2017 - 126
Signal Processing - November 2017 - 127
Signal Processing - November 2017 - 128
Signal Processing - November 2017 - 129
Signal Processing - November 2017 - 130
Signal Processing - November 2017 - 131
Signal Processing - November 2017 - 132
Signal Processing - November 2017 - 133
Signal Processing - November 2017 - 134
Signal Processing - November 2017 - 135
Signal Processing - November 2017 - 136
Signal Processing - November 2017 - 137
Signal Processing - November 2017 - 138
Signal Processing - November 2017 - 139
Signal Processing - November 2017 - 140
Signal Processing - November 2017 - 141
Signal Processing - November 2017 - 142
Signal Processing - November 2017 - 143
Signal Processing - November 2017 - 144
Signal Processing - November 2017 - 145
Signal Processing - November 2017 - 146
Signal Processing - November 2017 - 147
Signal Processing - November 2017 - 148
Signal Processing - November 2017 - 149
Signal Processing - November 2017 - 150
Signal Processing - November 2017 - 151
Signal Processing - November 2017 - 152
Signal Processing - November 2017 - 153
Signal Processing - November 2017 - 154
Signal Processing - November 2017 - 155
Signal Processing - November 2017 - 156
Signal Processing - November 2017 - 157
Signal Processing - November 2017 - 158
Signal Processing - November 2017 - 159
Signal Processing - November 2017 - 160
Signal Processing - November 2017 - 161
Signal Processing - November 2017 - 162
Signal Processing - November 2017 - 163
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Signal Processing - November 2017 - 165
Signal Processing - November 2017 - 166
Signal Processing - November 2017 - 167
Signal Processing - November 2017 - 168
Signal Processing - November 2017 - 169
Signal Processing - November 2017 - 170
Signal Processing - November 2017 - 171
Signal Processing - November 2017 - 172
Signal Processing - November 2017 - 173
Signal Processing - November 2017 - 174
Signal Processing - November 2017 - 175
Signal Processing - November 2017 - 176
Signal Processing - November 2017 - Cover3
Signal Processing - November 2017 - Cover4
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