Signal Processing - September 2017 - 175

NSS into the sparse representation
framework and shows competitive per-
formance in different IR applications,
including denoising, debluring, and
superresolution [4].

Low-rank minimization
The sparse representation models stretch
an image patch to a vector and encode
it over a dictionary of one-dimensional
(1-D) atoms. With the NSS prior, we can
have a group of similar patches as input.
Group sparsity models have been pro-
posed to encode a group of correlated
patches, whereas they are still a type of
1-D sparse coding model. An alternative
way is to format those similar patches
as a matrix with each column being a
stretched patch vector, and exploit the
low-rank prior of this matrix for IR.
The rank of a data matrix X counts
the number of nonzero singular values
of it, which is NP-hard to minimize.
Alternatively, the nuclear norm of X,
defined as the , 1-norm of its singular
values X ) = R i v i (X) 1, is a convex
relaxation of matrix rank function. The
low-rankness of X can be viewed as a
2-D sparsity prior. It encodes the input
2-D data matrix over a set of rank-1
basis matrices and assumes its singular
values to be sparsely distributed, i.e., it
has only a few nonzero or significant
singular values.
Let Y be a matrix of degraded image
patches. The latent low-rank matrix X
can be estimated form Y via the follow-
ing nuclear norm minimization (NNM)
problem:
Xt = arg min X Y - X

2
F

lar values equally by the threshold m,
ignoring the different significances of
matrix singular values. It is known that
the larger singular values can be more
important to represent the latent data in
many applications. In [6], a weighted
nuclear norm is defined
X

w, )

= / i w i v i (X) 1,

(10)

where w i is the weight assigned on sin-
gular value v i (X). A weighted NNM
(WNNM) model is then presented [6] to
recover the latent data matrix X from Y
Xt = arg min X Y - X

2
F

+ X

w, ) .

(11)

Different from the convex NNM
model in (8), the WNNM model in (11)
becomes nonconvex. Fortunately-
and interestingly-it is proved in [6]
that WNNM still has a globally opti-
mal solution.
It is also shown in [6] that, if the
weights satisfy 0 # w 1 # g # w n,
the nonconvex WNNM problem has a
closed form optimal solution
Xt = US w ^R h V T ,
2

(12)

where Y = URV T is the SVD of Y and
S w2 ^R hii = max ^R ii - ^w i /2h, 0h.
The above conclusion is very useful.
In many IR applications, the weights
can be easily set as nonascending [6],
and, therefore, WNNM has a closed-

form solution, which makes the mini-
mization process efficient. Figure 3
illustrates the WNNM-based image
denoising scheme. For each noisy patch,
we search its nonlocal similar patches to
form the matrix Y. Then we solve the
WNNM problem in (11) to estimate the
clean patches X. Once the estimated
clean patch is put back into the image,
the noise is reduced. Such procedures
are repeated several times to obtain the
denoised image. WNNM has shown
state-of-the-art denoising results.

Deep prior learning
The sparse representation and low-rank
minimization-based IR methods dis-
cussed above are model-based optimiza-
tion schemes, where a model (objective
function) is built based on the image
degradation process and the available
image priors, and the desired image is
reconstructed by finding the optimal
solution of the model. Such models can
be generally written as
xt = arg min x F (x, y) + mR (x),

(13)

where F (·) is the data fidelity term
^e.g., F (x, y) = y - Hx 22 h and R (·) is
the regularization term (or prior term).
Another category of IR methods
is the so-called discriminative learn-
ing methods, which learn a compact
inference or a mapping function from
a training set of degraded-latent image

+ m X ) . (8)

Cai et al. [2] showed that (8) has a
closed-form solution
Xt = US m2 ^R h V T ,

(9)

where Y = URV T is the SVD of Y and
S m2 ^R hii = max ^R ii - ^m/2h, 0h is the
singular value thresholding operator.

Weighted NNM
The NNM mentioned previously has
shown interesting results on image and
video denoising. As can be seen from
(9), however, it shrinks all the singu-

WNNM

FIGURE 3. An example of WNNM-based image denoising.
IEEE SIGNAL PROCESSING MAGAZINE

|

September 2017

|

175



Table of Contents for the Digital Edition of Signal Processing - September 2017

Signal Processing - September 2017 - Cover1
Signal Processing - September 2017 - Cover2
Signal Processing - September 2017 - 1
Signal Processing - September 2017 - 2
Signal Processing - September 2017 - 3
Signal Processing - September 2017 - 4
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Signal Processing - September 2017 - Cover3
Signal Processing - September 2017 - Cover4
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