Signal Processing - November 2017 - 87
problem, while in the learning approach, the solution of a reguContinuing the CT example, H would be a discretization of the
larized minimization problem is a parametric function that can
X-ray transform (such as MATLAB's radon), and f could
be used to solve the inverse problem. The learning formulation
be the Euclidean distance, H {x} - y 2 . For many appliis attractive because it overcomes many of the limitations of
cations, decades of engineering have gone into developing
the objective function approach: there is no need to handcraft
a fast and reasonably accurate inverse operator, Hu -1, so (1)
the forward model, cost function, regularizer, and optimizer
is easily approximated by R obj {y} = Hu -1 {y}; for CT, Hu -1
from (2). On the other hand, the learning approach requires a
is the filtered back projection (FBP) algorithm. An important,
training set, and the minimization (3) is typically more difrelated operator is the back projection, H T : Y " X, which can
ficult than (2) and requires a problem-dependant choice of f,
be interpreted as the simplest way to put measurements back
into the image domain (see Figure 1).
g, and the class of functions described by R and H.
These direct inverses begin to show significant artifacts
Finally, we note that the learning and objective function
when the number or quality of the measurements decreases,
approaches describe a spectrum rather than a dichotomy. In
either because the underlying discretization breaks down or
fact, the learning formulation is strictly more general, includbecause the inversion of (1) becomes ill
ing the objective function formulation as a
posed (lacking a solution, lacking a unique
special case. As we will discuss further in
In the objective
solution, or being unstable with respect to the
the section "Network Architecture," which
function approach, the
measurements). Unfortunately, in many real(if any) aspects of the objective formulaworld problems, measurements are costly (in
tion approach to retain is a critical choice
reconstruction function
terms of time, or, e.g., X-ray damage to the
in the design of learning-based approaches
is itself a regularized
patient), which motivates us to collect as few
to inverse problems in imaging.
minimization problem,
as possible. To reconstruct from sparse or
while in the learning
noisy measurements, it is often better to use a
CNNs
approach, the solution of
regularized formulation,
Our focus here is the formulation of (3) using
a regularized minimization CNNs. Using a CNN means, roughly, fixing
R reg " y , = argmin f ^ H " x ,, yh + g ^ x h, (2)
the set of functions, R i , to be a sequence of
problem is a parametric
x!X
(linear) filtering operations alternating with
function that can be
simple nonlinear operations. This class of
where g : X " R + is a regularization funcused to solve the
functions is parametrized by the values of the
tional that promotes solutions that match our
inverse problem.
filters used (also known as filter weights),
prior knowledge of x and, simultaneously,
and these filter weights are the parameters
makes the problem well posed. For CT, g
over which the minimization occurs. For illustration, Figure 2
could be the total variation (TV) regularization, which penalizes
shows a typical CNN architecture.
large gradients in x.
We will discuss the theoretical motivations for using CNNs
From this perspective, the challenge of solving an inverse
as the learning architecture for inverse problems in the secproblem is designing and implementing (2) for a specific applition "Theory," but we mention some practical advantages
cation. Much effort has gone into designing general-purpose
here. First, the forward operation of a CNN consists of (usuregularizers and minimization algorithms. For example, comally small) convolutions and simple, pointwise nonlinear funcpressed sensing [29] provides sparsity-promoting regularizers.
tions. This means that, once training is complete, the execution
Nonetheless, in the worst case, a new application necessitates
of R learn is very fast and amenable to hardware acceleration
developing accurate and efficient H , g, and f, along with a
minimization algorithm.
on GPUs. Second, the gradient of (3) is computable via the
An alternative to the objective function approach is called
chain rule, and these gradients again involve small convoluthe learning approach, where a training set of ground-truth
tions, meaning that the parameters can be learned efficiently
images and their corresponding measurements, {(x n, y n)} nN= 1,
via gradient descent.
When the first CNN-based method entered the ImageNet
is known. A parametric reconstruction algorithm, R learn, is
Large-Scale Visual Recognition Challenge in 2012 [5], its
then learned by solving
error rate on the object localization and classification task was
N
15.3%, as compared to an error rate 26.2% for the next closest
R learn = argmin / f ^ x n, R i " y n ,h + g (i), (3)
method and 25.8% for the 2011 winner. In subsequent comR ,i ! H n =1
petitions (2013-2016), the majority of the entries (and all of
the winners) were CNN based and continued to improve subwhere H is the set of all possible parameters, f : X # X " R +
stantially, with the 2016 winner achieving an error rate of just
is a measure of error, and g: H " R + is a regularizer on the
2.99%. Can we expect such large gains in inverse problems?
parameters with the aim of avoiding overfitting. Once the
That is, can we expect denoising results to improve by an order
learning step is complete, R learn can then be used to reconstruct
of magnitude (20 dB) in the next few years? Next, we answer
a new image from its measurements.
this question by surveying the results reported by recent CNNTo summarize, in the objective function approach, the
based approaches to image reconstruction.
reconstruction function is itself a regularized minimization
i
IEEE SIGNAL PROCESSING MAGAZINE
|
November 2017
|
87
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
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Signal Processing - November 2017 - 17
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Signal Processing - November 2017 - 20
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Signal Processing - November 2017 - 28
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Signal Processing - November 2017 - 30
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Signal Processing - November 2017 - 86
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Signal Processing - November 2017 - 88
Signal Processing - November 2017 - 89
Signal Processing - November 2017 - 90
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Signal Processing - November 2017 - 92
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Signal Processing - November 2017 - 100
Signal Processing - November 2017 - 101
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Signal Processing - November 2017 - 103
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Signal Processing - November 2017 - 105
Signal Processing - November 2017 - 106
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Signal Processing - November 2017 - 110
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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 - 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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