Instrumentation & Measurement Magazine 24-2 - 49

as the regions corrupted by noise [13]. More specifically, for
an image fusion method using a shift invariant signal/image
representation approach, it is easy to develop simple fusion
strategies (e.g., window-based averaging) to achieve high robustness to mis-registration and noise. Further discussion on
this issue can be found in [43]. In the CSR-based method [43], to
increase its robustness to mis-registration and noise, a simple
window-based strategy is performed on the sparse coefficient maps to calculate the activity level measures owing to the
shift-invariance property of CSR. Vishwakarma and Bhuyan
proposed a hybrid transform based image fusion method by
combining an adjustable NSST and CSR [45]. In their method,
the CSR-based fusion approach presented in [43] is applied to
merge the low-frequency coefficients of NSST.

Fig. 4. The convolutional sparse representation model.

patch-based SR, and it can obtain single-valued representation
which is optimized over the entire image. A key issue in CSR
is to learn the dictionary filters. The dictionary learning model
for CSR [39] can be formulated as:

	

M
1 K
Yk   d m  X k ,m

d m ,X k ,m 2
m 1
k 1


2

min

s.t. d m

2

2

K

M

   X m
m 1
k 1 

1

Simultaneous Multi-component and Global SRbased Method
Liu et al. recently proposed a medical image fusion method by
combining multi-component SR and global SR into a unified
framework [46], which is formulated as:

,	(7)

 1, m

Mc
Mt
1
Y   d c ,m  X c ,m   d t ,m  X t ,m
X c , m ,Xt ,m 2

m 1
m 1

2

min

M
where {Yk }Kk1 is a set of K training images, {d m }m
is a set of M
1
dictionary filters to be learned and Xk,m denote the sparse coefficient maps. The constraint on the norm of dm aims to avoid
the scaling ambiguity between filters and coefficients. Several
efficient algorithms [39, 40, 41] using the alternating direction
method of multipliers (ADMM) framework [42] have been recently proposed to solve the problems in (7) and (6).
In [43], Liu et al. first introduced CSR into the field of image fusion to address the drawbacks of conventional SR-based
methods. For one thing, since the result of CSR is single-valued
and optimized over the entire image, the accumulating-averaging strategy is no longer required. As a consequence, the
ability of the image fusion method in detail preservation is improved. For another, CSR is actually a shift-invariant version
of SR as well, and the convolutional form is derived from the
shift-invariance principle [44]. It is known that shift invariance is a quite important property to ensure the fusion quality
of mis-registered regions between two source images as well

	



Mc

X



Mt

X

c
c ,m 1
t

m 1
m 1

	(8)

2

t ,m 1

Mt
Mc
where Y is the image to be decomposed. {d c ,m }m
and {d t ,m }m
1
1
denote two sets of dictionary filters for the SR of the cartoon
Mc
Mt
and {X t ,m }m
and the texture components, respectively. {X c ,m }m
1
1
are the corresponding sparse coefficient maps. It can be seen
that this model can simultaneously obtain multi-component
and global sparse representations for the input image. On one
hand, it can be viewed as the convolutional sparsity extended
version of the MCA model given in (5). On the other hand, it
can also be regarded as the cartoon-texture decomposition
extended version of the CSR model given in (6). In [46], the dicM
Mt
are independently learned
tionary filters {d c ,m }mc 1 and {d t ,m }m
1
form a number of cartoon and texture images with the CSR dictionary learning model given in (7). Fig. 5 shows an example of
Mc
Mt
and {d t ,m }m
. The problem
the learned dictionary filters {d c ,m }m
1
1

M

M

Fig. 5. An example of the learned dictionary filters in the simultaneous multi-component and global SR-based method [46]. (a) {d c ,m }mc 1. (b) {d t ,m }mt 1.
April 2021	

IEEE Instrumentation & Measurement Magazine	49



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