Hindawi

Computational and Mathematical Methods in Medicine

Volume 2021, Article ID 1544955, 16 pages
https://doi.org/10.1155/2021/1544955

Research Article
Multimodal Medical Image Fusion Based on Multiple Latent
Low-Rank Representation

Xi-Cheng Lou and Xin Feng

School of Mechanical Engineering, Key Laboratory of Manufacturing Equipment Mechanism Design and Control of Chongqing,
Chongqing Technology and Business University, Chongqing 400067, China

Correspondence should be addressed to Xin Feng; 149495263@qq.com

Received 18 June 2021; Revised 31 August 2021; Accepted 2 September 2021; Published 29 September 2021

Academic Editor: Martti Juhola

Copyright © 2021 Xi-Cheng Lou and Xin Feng. This is an open access article distributed under the Creative Commons
Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.

A multimodal medical image fusion algorithm based on multiple latent low-rank representation is proposed to improve
imaging quality by solving fuzzy details and enhancing the display of lesions. Firstly, the proposed method decomposes the
source image repeatedly using latent low-rank representation to obtain several saliency parts and one low-rank part.
Secondly, the VGG-19 network identifies the low-rank part’s features and generates the weight maps. Then, the fused low-
rank part can be obtained by making the Hadamard product of the weight maps and the source images. Thirdly, the
fused saliency parts can be obtained by selecting the max value. Finally, the fused saliency parts and low-rank part are
superimposed to obtain the fused image. Experimental results show that the proposed method is superior to the traditional
multimodal medical image fusion algorithms in the subjective evaluation and objective indexes.

1. Introduction method is multimodal medical image fusion, which com-

bines multiple medical images from various modalities to
Medical imaging modalities are varied, each of which pro- improve quality and reduce redundancy of imaging.
vides different information about organs in the body. For Most medical image fusion algorithms are based on the
example, computerized tomography (CT) has excellent reso- multiscale transform (MST), which converts the source
lution, enabling examination of finer details in tissue, but it images into the transform domain and obtain the trans-
is weak in showing the global organ structure and patholog- formed coefficients through preset functions. Then, the
ical changes. Magnetic resonance imaging (MRI) reveals processed coefficients can be converted to the fused image
remarkable soft tissue definition with high spatial resolution, by inverse MST. According to the different decomposition
but it is limited in detecting fractures. Furthermore, anatom- methods of source images, MST can be divided into
ical imaging techniques such as CT and MRI can not reflect pyramid-based methods [2–4], wavelet-based methods
the body’s movement information, such as metabolism. Pos- [5–8], and multiscale geometric analysis- (MGA-) based
itron emission tomography (PET) and single-photon emis- methods [9–18]. Due to the limitation of preset functions
sion computed tomography (SPECT) can visualize in the MST-based algorithm, some essential features of the
metabolic processes and other physiological activities, such source images, such as edge and texture information, may
as blood flow and regional chemical composition absorption not be well expressed and extracted, which significantly
[1]. Nevertheless, functional imaging techniques such as reduces the fusion performance. Moreover, the MST-based
PET and SPECT have a low spatial resolution. In summary, algorithm is usually sensitive to misregistration.
it is impossible to obtain all the details of an organ from a Yang and Li [19] first applied sparse representation (SR)
separate imaging modality. For improving the clinical accu- [20] in image processing. SR decomposes the source images
racy of diagnosing based on medical images, an effective into several patches through a sliding window and
2 Computational and Mathematical Methods in Medicine

LatLRR

I LX XZ E

Figure 1: LatLRR decomposition diagram.

LatLRR

X LX XZ E
Superimpose

LatLRR

X1 LX1 X1Z E

Figure 2: Two-layer LatLRR decomposition diagram.

rearranges these patches to sparse coefficient vectors, which based algorithm’s dictionary matrix cannot fully include
are the linear combination of vectors in the dictionary matrix. source image data, it fails to extract the source image’s detailed
Then, the fused image’s sparse coefficient vectors can be deter- texture information. Some scholars applied MST [24–26] or
mined through maximal l1 -norm, rearrange these vectors to filter [27–29] to decompose the source images. And SR can
patches of fused image, and put these patches return to the seat be used to fuse the low-frequency subbands. Theoretically,
can obtain the fused image. In terms of edge feature extraction, such methods can preserve the edge information of the image
SR has certain advantages over MST. Many improved versions better than using SR [30] to decompose the source images. Liu
of SR have appeared in recent years to increase computational et al. [31] proposed the low-rank representation (LRR), which
efficiency or improve fusion quality. Liu and Wang [21] pro- applies the source image as the dictionary matrix, and can
posed adaptive sparse representation (ASR), with seven sub- solve dictionary completeness. Liu and Yan [32] proposed
dictionaries trained in advance to match the patches of the latent low-rank representation (LatLRR), an improved ver-
images categorized by the gradient. Liu et al. [22] proposed sion of LRR, which can decompose the source image to
convolutional sparse representation (CSR), which does not saliency part, low-rank part, and noise part. Li et al. [33] pro-
use the sliding window to decompose the source image but posed MDLatLRR, which integrated SR and LatLRR by using a
applies a globe process. Liu et al. [23] proposed convolutional sliding window to sample the saliency parts of LatLRR and
sparsity-based morphological component analysis (CSMCA), processed the sparse coefficient vectors just like in SR-based
which simultaneously achieves multicomponent and global methods. LatLRR has an extraordinary capacity for extracting
SR by integrating CSR and the morphological component texture from the image, but the ability to extract high-
analysis into a unified framework. However, since the SR- frequency information is not as good as MST.
Computational and Mathematical Methods in Medicine 3

Input VGG-19 Weight map

Conv3-64

Conv3-64

Pool1

Conv3-128

IL1 Conv3-128 W1

Pool2

Conv3-256

Conv3-256

Conv3-256

IL2 Pool3 W2

Conv3-512 Eq. (11)

Eq. (12)
Conv3-512
Activity level map
Conv3-512

Pool4 C1 C2

Soft-max Conv3-512
Eq. (10)
FC1000 Conv3-512

FC4096 Conv3-512 Feature maps

FC4096 Pool5 m 512 m 512

{𝜑1 }m = 1 {𝜑2 }m = 1

Figure 3: The procedure of low-rank part fusion.

Another type of image fusion method that is more widely CNN to generate weight maps and use the Laplacian pyramid
used is with the help of weighted maps [34, 35], and deep and Gaussian pyramid to decompose the source images and
learning-based methods are particularly suitable for generating weight maps, respectively, though the Hadamard product to
weighted maps due to their superior feature recognition capa- obtain fused layers, and the fused image can be reconstructed
bilities. The deep learning-based methods [36–42] have been through the Laplacian pyramid. Li et al. [39] applied the aver-
widely used in image fusion with the development of artificial age filter to decompose the source images, and the fused base
intelligence. These methods have a prominent ability to extract layer can be obtained by comparing the max absolute value of
feature information from the image. Therefore, it is wise to use images in four convolutional layers of the VGG-19 neural net-
deep learning-based methods to deal with high-frequency infor- work. The fused image can be obtained by superimposing the
mation after image decomposition or generate the weight map base layer and detail layer. Yin et al. [40], Tan et al. [41], and
as the basis of image region fusion. Wang et al. [36] applied a Panigrahy et al. [42] applied nonsubsampled shearlet trans-
convolutional neural network (CNN) to generate weight maps form (NSST) to decompose the source images and selected
and decompose the source images and weight maps by contrast the fused high-frequency subbands by more firing times in
pyramid and Laplacian pyramid, respectively, and make the the parameter-adaptive pulse coupled neural network
Hadamard product of each decomposition layer. Finally, the (PAPCNN), bounded measured pulse coupled neural network
fused image can be reconstructed through the contrast pyramid. (BMPCNN), and weighted parameter adaptive dual-channel
Xu et al. [37] applied LatLRR to decompose source images and pulse coupled neural network (WPADCPCNN), respectively.
processed the low-rank parts by CNN and pyramid-based As mentioned above, each method has its drawbacks and
methods, superimposed the fused low-rank part, and fused advantages. In this paper, the source images are repeatedly
saliency part to obtain the fused image. Liu et al. [38] applied decomposed through LatLRR to extract the saliency parts.
4 Computational and Mathematical Methods in Medicine

YUV space
CT

Proposed fusion
method
RGB space RGB space
Y Fused Y

SPECT U Fused image

Figure 4: Schematic of the grayscale and color image.

The fused saliency parts can be obtained by selecting the max 2. Multiple Latent Low-Rank Representation
value. After superimposing these saliency parts, the structure
and edge information of the source images will be well pre- LatLRR is an improved version of LRR, whose principle is
served and enhance the lesion’s display. Then, the VGG-19 date X = fx1 , x2 ,⋯,xM g in space Rn can be represented by
network is used to extract features of the low-rank part, and a linear combination of vectors in an overcomplete dictio-
the weight maps can be generated to be the basis of the low- nary D ∈ Rn×m ðn < mÞ, as
rank part’s activity level. The weight maps and low-rank parts
then make the Hadamard product to obtain the fused low- X = DZ, ð1Þ
rank part. Finally, the fused image can be obtained by super-
imposing the fused saliency part and fused low-rank part. where Z = fz1 , z2 ,⋯,zM g is the coefficient matrix in space
The experimental results also show that the proposed method Rm ; it can be determined through
is significantly better than the comparison method regarding
image information retention. The main contributions of this min kZ k∗ s:t:X = DZ, ð2Þ
Z
paper are described as follows:
where k·k∗ denotes the nuclear norm. The idea of this algo-
(1) The proposed method applies the image detail rithm is similar to that of SR, in that it finds the coefficients
retention capability of LatLRR while fully extract- of an image under certain dictionary conditions. LRR using
ing the high-frequency information of an image date X itself as the dictionary, just like equation (3), that is
by iteratively decomposing the original image. It the reason LRR does not have the problems of dictionary
compensates for the deficiency of LatLRR and training or completeness.
enhances the display of the lesion by superimpos-
ing saliency parts min kZk∗ s:t:X = XZ: ð3Þ
Z
(2) The feature map of the low-rank part of the original
image is extracted using the VGG-19 network and The noise component E is added in equation (3); this is
then scaled up to match the size of the original because the original purpose of creating the low-rank algo-
image. The weight map generated in this way can rithm is to remove noise from the image. And Equation
well fit the low-rank part of the original image with (4) is the formula of LRR.
pixel information blockwise distributed
min kZk∗ + λkEk1,2 s:t:X = XZ + E, ð4Þ
The rest of this paper is organized as follows. Section 2 Z ,E
introduces the multiple LatLRR decomposition algorithm,
Section 3 introduces the fusion rules, Section 4 describes where λ > 0 is balance coefficient and kEk1,2 denotes the l1,2
the algorithmic structure of the proposed method, Section -norm of E.
5 provides a detailed discussion of the experimental results, However, there are two prerequisites for using X itself as a
and Section 6 concludes this paper. dictionary. One is that the data vector of X must be sufficiently
Computational and Mathematical Methods in Medicine 5

YUV
Color image? Yes space

No Y
I1 I2
U

V
Multiple LatLRR decomposition (N layers)

I1L I2L

I1S,N I2S,N

Max-rule VGG-19

Superpose

Superpose
IFS,N IL
IS

YUV convert
RGB
to RGB

IF Color fused
image

Figure 5: Algorithm framework.

complete. Second, the noise of X must be controlled in a small equation (7) can be obtained.
range. In many practical conditions, such requirements are
challenging to achieve. For this reason, the method of adding min kZk∗ + kLk∗ + λkEk1 s:t:X = XZ + LX + E, ð7Þ
hidden items in the dictionary is proposed in [32] Z ,L ,E

where L denotes the saliency coefficient and Z denotes the

min kZk∗ s:t:XO = ½XO , XH Z, ð5Þ
Z low-rank coefficient. Equation (7) can be solved by the aug-
mented Lagrange multiplier (ALM) [43]; the low-rank part
and saliency part of the image could be represented as XZ
where XO denotes the known image data and XH denotes
and LX accordingly. The LatLRR decomposition results are
the unknown hidden data. Since the dictionary contains hid-
shown in Figure 1. LatLRR decomposes the source image X
den data, this improved algorithm is called the latent low-
to the saliency part LX, the low-rank part XZ, and the noise
rank representation. The influence of noise is taken into
part E. LX contains the local structure information and
account; then, we rewrite equation (5) into
saliency features or can be thought of as high-frequency infor-
mation. XZ contains more global structure information and
min kZk∗ + λkEk1 s:t:XO = ½XO , XH Z + E, ð6Þ brightness information or can be thought of as low-
Z ,E
frequency information. E denotes the’superfluous’ part that
LatLRR separates.
where λ > 0 is the balance coefficient and kEk1 denotes the It is worth noting that in images with high spatial resolu-
l1 -norm of E. Simplify equation (6) by computing the skinny tion, such as CT and MRI, detailed textures may contain essen-
singular value decomposition (SVD) of ½XO , XH  [32], and tial diagnostic information. Therefore, denoising such images
6 Computational and Mathematical Methods in Medicine

Input:I1 and I2 .
Output: fused image I F
/∗ Part 1: N layer multiple LatLRR decomposition. ∗/
1 for each M = ½I1 , I2 do
2 for each i = ½1 : Ndo
3 Perform LatLRR decomposition on S to obtain fIS,i L S,i L
1 , I1 g and fI2 , I2 g.
4 end
5 end
/∗ Part 2: Fusion of saliency part. ∗/
6 for each i = ½1 : Ndo
7 perform max-rule on fIS,i S,i i
1 , I2 g to obtain IS as euqation (8).
8 end
N
9 Superimpose fIiS gi=1 to obtain the final fused saliency part IS as equation (9).
/∗ Part 3: Fusion of low-rank part.∗/
10 for each k = ½1, 2do
512
11 Input ILk to VGG-19 network to obtain fϕm k gm=1 extract from the 5th layer of the network;
m 512
12 Solve the l1 -norm of fϕk gm=1 to obtain activity level map Ck as equation (10);
13 Calculate the initial weight map W ̂ k by equation (11);
14 Enlarge W ̂ k to get the final weight map Wk as equation (12).
15 end
16 Calculate the final fused low-rank part IL as equation (13).
/∗Part 4: Reconstruction.∗/
17 Superimpose the fused saliency part IS and the fused low-rank part IL to obtain the fused image I F as equation (14).

Algorithm 1: Proposed image fusion algorithm.

may filter out some critical information. In this case, a reason- diagnostic information as much as possible. On the other
able approach is to superimpose the noise part and low-rank hand, Simonyan and Zisserman [44] first applied the VGG
part. Moreover, the saliency part LX contains most edge and network to extract features at different layers from images
structure information of the image so that the lesions may and obtain a splendid result. With the development of deep
mainly reflect in the saliency part. If the low-rank part of the learning, the operation efficiency and precision of the VGG
image is decomposed repeatedly, the saliency part will be fur- network have been significantly improved. As the number
ther extracted. As shown in Figure 2, a two-layer LatLRR of LatLRR decomposition layers increases, the source
decomposition structure, after completing the first layer of image’s low-rank part will contain less information. If the
LatLRR decomposition, the new object of LatLRR decomposi- recognition results of VGG-19 are extracted and processed,
tion could be obtained through X1 = XZ + E, which can be fur- the weight map with regional emphasis can be generated.
ther decomposed into saliency part LX1 , low-rank part X1 Z, Then, the fused low-rank part can be obtained by multiply-
and noise part E. The source images are denoted as I1 and I2. ing the weight map with the source image’s low-rank part.
If the number of LatLRR decomposition layers is N, there will Besides, because PET and SPECT images are in color, they
N N
be N saliency parts fIS1,i gi=1 or fIS2,i gi=1 and one low-rank part need to be converted into YUV color space before fusing
IL1 or IL2 for each source image. The display of the edge and them with the grayscale image as MRI.
structure information will be strengthened in the new image
to highlight the lesions by superimposing saliency parts. How- 3.1. Fusion of Saliency Parts. Each LatLRR decomposition of
ever, suppose the number N of LatLRR decomposition layers the two source images will produce a saliency part of each.
is blindly increased, which will reduce the efficiency of calcula- By adopting max-rule for all saliency parts of N layers of
tion. More importantly, the final fused image will display some LatLRR decomposition, N fused significant parts can be
artificial information unacceptable for medical images. In this obtained, as
paper, the optimal number of LatLRR decomposition layers will
be determined through the experiment in Section 5.1.
 i N h  N  N i
IS i=1 ðx, yÞ = max IS1,i i=1 ðx, yÞ, IS2,i i=1 ðx, yÞ , ð8Þ
3. Fusion Regulation
The saliency parts of the image include most high-frequency N
information. For multimodal medical images, the critical where fIiS gi=1 ðx, yÞ denote the position ðx, yÞ of the ith
N N
diagnostic information reflected by a single image is not layer saliency part of fused image fIiS gi=1 , so as fIS1,i gi=1 ðx,
the same. Therefore, the max-rule is applied to fuse the N
saliency parts of the image can preserve a single image’s yÞ and fIS2,i gi=1 ðx, yÞ. The final fused saliency part can be
Computational and Mathematical Methods in Medicine 7

MR-PD CT MR-T1 CT
(a) (b) (c) (d)

MR-T2 SPECT-Tl MR-T2 PET-FDG

(e) (f) (g) (h)

MR-T2 SPECT-Tc
(i) (j)

Figure 6: Five sets of multimodal medical images for the experiment. (a) MR-PD. (b) CT. (c) MR-T1. (d) CT. (e) MR-T2. (f) SPECT-Tl. (g)
MR-T2. (h) PET-FDG. (i) MR-T2. (j) SPECT-Tc.

512
calculated by maps, where k ∈ f1, 2g. The l1 -norm of fϕmk gm=1 ðx, yÞ could
be the activity level measure of the low-rank part. So, the
N activity level map Ck can be calculated by
IS = 〠 IiS : ð9Þ
 
i=1  512 
Ck ðx, yÞ = fϕm g
k m=1 ð x, y Þ : ð10Þ
1
3.2. Fusion of Low-Rank Parts. VGG-19 is a convolutional
neural network that is 19 layers deep, including 16 convolu- ̂ k can be obtained by
Then, the initial weight map W
tional layers and 3 fully connected layers. Its structure is
shown in Figure 3. Each convolutional layer is denoted as ̂ k ðx, yÞ = Ck ðx, yÞ
W : ð11Þ
conv‘size of the filter’-‘number of such filters,’ and max- C1 ðx, yÞ + C2 ðx, yÞ
pooling layers have 2 × 2 filter with the stride of 2. The infor-
mation of the low-rank part of the image after multiple
As feature maps are only ð0:5Þ5 , the size of the source
LatLRR decompositions is relatively fuzzy and presents a ̂ k , which is generated by
regional-like distribution. According to this feature, the fea- image, so the initial weight map W
5
ture maps extracted from the fifth convolution layer of feature maps, is ð0:5Þ the size of the source image too. For
VGG-19 can match the low-rank part of the image’s infor- matching the size of the source image, W ̂ k need the upsam-
mation distribution state after amplification. pling procedure as
512 m 512
For low-rank parts IL1 and IL2 , fϕm 1 gm=1 and fϕ2 gm=1
denote the feature maps extracted from the fifth convolu- ̂ k ðx + p, y + qÞ
Wk ðx, yÞ = W p, q ∈ f1, 2, ⋯ ,15g: ð12Þ
tional layer of VGG-19. As shown in Figure 3, the 5th con-
volutional layer is conv3-512, so there are 512 feature The fused low-rank part IL can be calculated by
maps of each low-rank part. Moreover, because of max-
pooling layers, these feature maps are only ð0:5Þ5 the size IL = W1 ∘ IL1 + W2 ∘ IL2 , ð13Þ
512
of the source image. According to [39], let fϕm k gm=1 ðx, yÞ
denote the ðx, yÞ position of the kth low-rank part’s feature where ∘ denotes the Hadamard product.
8 Computational and Mathematical Methods in Medicine

0.8000 0.7000

0.7000 0.6500

Value of QG
Value of QTE 0.6000 0.6000

0.5000 0.5500

0.4000 0.5000

0.3000 0.4500
1 2 3 4 1 2 3 4

LatLRR decomposition layer LatLRR decomposition layer

(a) (b)
0.7350 0.6500
0.7100 0.6000

Value of QCB
Value of Qc

0.6850 0.5500
0.6600 0.5000
0.6350 0.4500
0.6100 0.4000
1 2 3 4 1 2 3 4

LatLRR decomposition layer LatLRR decomposition layer

(c) (d)

Figure 7: Parametric experimental results. (a) QTE of five sets of images. (b) QG of five sets of images. (c) QC of five sets of images. (d) QCB of
five sets of images.

1 Layer 2 Layers 3 Layers 4 Layers

(a) (b) (c) (d)

1 Layer 2 Layers 3 Layers 4 Layers

(e) (f) (g) (h)

1 Layer 2 Layers 3 Layers 4 Layers

(i) (j) (k) (l)

Figure 8: The fusion result of each LatLRR decomposition level. (a) 1 layer. (b) 2 layers. (c) 3 layers. (d) 4 layers. (e) 1 layer. (f) 2 layers. (g) 3
layers. (h) 4 layers. (i) 1 layer. (j) 2 layers. (k) 3 layers. (l) 4 layers.

3.3. YUV Color Space. For color images such as SPECT and converted to YUV space and decomposed into one lumi-
PET, Yin et al. [40] proposed a YUV space to solve color and nance component, ‘Y’ and two chrominance components,
grayscale images’ fusion problems. The color image is first ‘U’ and ‘V.’ Then, the ‘Y’ component of the color image
Computational and Mathematical Methods in Medicine 9

MR-PD CT CSR ASR

(a) (b) (c) (d)

CSMCA CNN BMPCNN MDLatLRR

(e) (f) (g) (h)

VGG-19 NSCT_SR LP_SR Proposed

method
(i) (j) (k) (l)

Figure 9: Comparison of the fusion results in the first set images. (a) MR-PD. (b) CT. (c) CSR. (d) ASR. (e) CSMCA. (f) CNN. (g)
BMPCNN. (h) MDLatLRR. (i) VGG-19. (j) NSCT_SR. (k) LP_SR. (l) Proposed method.

can be fused with the grayscale image by the proposed 5. Experiment

method. The final fused image can be obtained through
transforming the fused component ‘Y’ and other two chro- Five sets of registered multimodal medical images collected
minance components ‘U’ and ‘V’ from YUV space to the from The Whole Brain Atlas [45] are used to verify the effec-
RGB space, as shown in Figure 4. tiveness of the proposed method, as shown in Figure 6.
Figures 6(a) and 6(b) are the first set of images from a 55-
3.4. Reconstruction. Superimpose the fused low-rank part IL year-old patient with multiple embolic infarctions;
and fused saliency part IS to reconstruct fused image I F as Figures 6(c) and 6(d) are the second set of images from a
31-year-old man with cerebral toxoplasmosis; Figures 6(e)
and 6(f) are the third set of images from a 51-year-old
I F = IS + IL : ð14Þ patient with anaplastic astrocytoma; Figures 6(g) and 6(h)
are the fourth set of images from a 70-year-old patient with
mild Alzheimer’s disease; Figures 6(i) and 6(j) are the fifth
4. Structure of Algorithm set of images from a 36-year-old patient with infectious dis-
ease due to HIV positive.
Source images I1 and I2 have been registered, the algorithm All the experiments in this paper are conducted on a PC
framework in this paper as showed in Figure 5. equipped Intel(R) Xeon(R) CPU E3-1231 v3 (3.40 GHz) and
The main steps of the proposed method are summarized 16 GB RAM. The software environment is MATLAB R2019b
as Algorithm 1. installed on Win 10 64-bit operating system.
10 Computational and Mathematical Methods in Medicine

MR-T1 CT CSR ASR

(a) (b) (c) (d)

CSMCA CNN BMPCNN MDLatLRR

(e) (f) (g) (h)

VGG-19 NSCT_SR LP_SR Proposed

method
(i) (j) (k) (l)

Figure 10: Comparison of the fusion results in the first set images. (a) MR-T1. (b) CT. (c) CSR. (d) ASR. (e) CSMCA. (f) CNN. (g)
BMPCNN. (h) MDLatLRR. (i) VGG-19. (j) NSCT_SR. (k) LP_SR. (l) Proposed method.

5.1. Parametric Experiment. In order to determine the image. The larger the QC is, the better the structure of the
decomposition layers of LatLRR in this paper, five sets of source images is preserved. The calculation process of QCB
images in Figure 6 were fused by the proposed method. is complex and consists of five steps: contrast sensitivity fil-
The results are objectively evaluated by four fusion image tering, local contrast computation, contrast preservation cal-
evaluation indexes: fusion metric-based on Tsallis entropy culation, saliency map generation, and global quality map
(QTE ) [46], gradient-based fusion performance (QG ) [47], computation. QCB takes the mean value of the global quality
image structural similarity metric (QC ) [48], and human map. The larger the QCB value is, the richer the contrast
perception inspired fusion metric (QCB ) [49]. QTE is a diver- information of the fused image is.
gence measure of the degree of dependence between two dis- The test decomposition layers of LatLRR are set from 1
crete random variables, and it calculates information from to 4. Parameter experimental results of indexes of four sets
the source images is transferred to the fused image. There- of images are shown in Figures 7(a)–7(d). It can be observed
fore, the larger the QTE value, the better the fusion effect. that as the number of LatLRR decomposition layers
QG uses the Sobel edge operator to calculate the intensity increases, not all indexes show a uniform trend. The value
and direction information of the edges in the source image of QTE increased with the increase of decomposition layers,
and the fused image. The larger the QG value is, the richer while QCB are optimal in the case of one-level decomposi-
the edge information of the fused image is. QC is used to tion. As for QG and QC , the changing trend is related to
measure the preservation degree of structure of the fused the image set. It is reasonable because the more decomposi-
image, so it calculates how much of the salient information tion layers, the image background information contained in
in each source image has been transferred into the fused the low-rank part will be fuzzier and more contained in the
Computational and Mathematical Methods in Medicine 11

MR-T2 SPECT-T1 CSR ASR

(a) (b) (c) (d)

CSMCA CNN BMPCNN MDLatLRR

(e) (f) (g) (h)

VGG-19 NSCT_SR LP_SR Proposed

method
(i) (j) (k) (l)

Figure 11: Comparison of the fusion results in the second set images. (a) MR-T2. (b) SPECT-T1. (c) CSR. (d) ASR. (e) CSMCA. (f) CNN.
(g) BMPCNN. (h) MDLatLRR. (i) VGG-19. (j) NSCT_SR. (k) LP_SR. (l) Proposed method.

saliency part. The max-rule selects the saliency part of the as possible, and artifacts should be strictly controlled. There-
fused image, so the background information of the source fore, the decomposition layer of LatLRR in this paper is set
images may also be strengthened in the fused image, as two.
enhancing the appearance of the lesion. If the greater the
amount of information, the larger the QTE value will be. 5.2. Contrast Experiment. Nine typical multimodal medical
However, strengthening image background information will image fusion methods are selected to compare with the pro-
weaken the boundary observation and reduce the image posed method: four sorts of SR-based methods, CSR [22],
contrast, undoubtedly leading to a lower QCB . Medical image ASR [21], CSMCA [23], and MDLatLRR [33]; three sorts
fusion aims to show the information of lesions in the fused of deep learning-based algorithm, CNN [36], VGG-19 [39],
image, so the QTE is more impotent than the other three and BMPCNN [41]; and two sorts of MST integrated SR
indexes. It can be seen from Figure 7(a) that QTE of two methods are nonsubsampled contourlet transform (NSCT)
decomposition layers is significantly improved than that of combined with SR (NSCT_SR) [24] and Laplacian pyramid
one decomposition layer. Still, the more decomposition (LP) combined with SR (LP_SR) [24]. The MST decomposi-
layers could not considerably improve the QTE . tion level is set to 4.
Besides, as Figure 8 shows, it can be seen that with the Figure 9 shows the fusion results of the first set images,
increase of the number of LatLRR decomposition layers, Figures 9(a) and 9(b) are the source images. It can be seen
the artifact around the object will be aggravated in several from Figure 9(l) of the proposed method, both the brain tis-
sets of images. In order to improve the image fusion quality, sue texture in the green box and the lesion edge in the red
the lesion in the fused image should be highlighted as much box are the clearest from other fusion methods. Besides,
12 Computational and Mathematical Methods in Medicine

MR-T2 PET-FDG CSR ASR

(a) (b) (c) (d)

CSMCA CNN BMPCNN MDLatLRR

(e) (f) (g) (h)

VGG-19 NSCT_SR LP_SR Proposed

method
(i) (j) (k) (l)

Figure 12: Comparison of the fusion results in the third set images. (a) MR-T2. (b) PET-FDG. (c) CSR. (d) ASR. (e) CSMCA. (f) CNN. (g)
BMPCNN. (h) MDLatLRR. (i) VGG-19. (j) NSCT_SR. (k) LP_SR. (l) Proposed method.

the pixel consistency of bone in the proposed method is the serious color distortion, as can be seen in Figures 12(c),
best. Figure 10 shows the fusion results of the second set 12(d), and 12(j), the color rendering of the proposed method
images; Figures 10(a) and 10(b) are the source images. As is closest to the source image. Figure 13 shows the fusion
can be seen in Figure 10(l), the pixel consistency of the skel- results of the fourth set images, Figures 13(a) and 13(b) are
etal structure of the fused images is the best, and the widen- the source images. Compared with Figure 13(b), the graphic
ing of the brain tissue sulcus in the green box, as well as the structure of metabolic abnormalities (red box) in other
calcified lesions and edema in the red box, are also clearly fusion methods has been deformed to a certain extent. The
visible. Figure 11 shows the fusion results of the second set structure is kept intact in the proposed method, and the
images; Figures 11(a) and 11(b) are the source images. In chromatic aberration is most consistent with the source
Figure 11(l) of the proposed method, the boundary of the image.
metabolic abnormality in the red box is the clearest, and All the fusion images are evaluated by four indexes intro-
the chromatic aberration is most consistent with duced in Section 5.1. As shown in Tables 1–5, in all sets of
Figure 11(b). The texture information in the green box is images, the proposed method leads in QTE , especially in
also clearly visible. Figure 12 shows the fusion results of the first set of images, the lead is more than 30 percent. It
the third set images; Figures 12(a) and 12(b) are the source means that the proposed method is superior to other
images. In Figure 12(l) of the proposed method, the bound- methods in the information conversion of source images.
ary of the metabolic abnormality in the red box is the most Moreover, for the first set and the fourth set of images, the
distinct, and the texture in the green box also holds the best. proposed method also leads in the QC . And it indicates that
Besides, in this set of images, some fusion methods appear the proposed method can retain the structure of the source
Computational and Mathematical Methods in Medicine 13

MR-T2 SPECT-Tc CSR ASR

(a) (b) (c) (d)

CSMCA CNN BMPCNN MDLatLRR

(e) (f) (g) (h)

VGG-19 NSCT_SR LP_SR Proposed

method
(i) (j) (k) (l)

Figure 13: Comparison of the fusion results in the fourth set images. (a) MR-T2. (b) SPECT-Tc. (c) CSR. (d) ASR. (e) CSMCA. (f) CNN. (g)
BMPCNN. (h) MDLatLRR. (i) VGG-19. (j) NSCT_SR. (k) LP_SR. (l) Proposed method.

Table 1: Objective evaluation of fusion methods in the first set of Table 2: Objective evaluation of fusion methods in the second set
images. of images.

Fusion method QTE QG QC QCB t/s Fusion method QTE QG QC QCB t/s
Proposed method 0.4359 0.5145 0.6436 0.5036 158.7860 Proposed method 0.4495 0.5647 0.6995 0.6061 50.6440
CSR 0.3096 0.6608 0.5621 0.7137 135.2650 CSR 0.3911 0.6363 0.7000 0.7454 33.1950
ASR 0.2976 0.6700 0.6121 0.6622 210.1220 ASR 0.3595 0.7122 0.7532 0.7328 64.9900
CSMCA 0.2945 0.5830 0.5587 0.6103 372.3040 CSMCA 0.3491 0.6315 0.7090 0.6974 78.8380
CNN 0.3247 0.6267 0.5560 0.5916 32.7860 CNN 0.4131 0.4757 0.6533 0.5848 13.8630
BMPCNN 0.3058 0.5554 0.5828 0.5943 67.0890 BMPCNN 0.3726 0.5354 0.6918 0.6210 15.7120
MDLatLRR 0.2882 0.6527 0.6077 0.6784 69.0270 MDLatLRR 0.3657 0.6516 0.7108 0.7357 21.1340
VGG-19 0.2835 0.5104 0.5569 0.6273 7.4480 VGG-19 0.4211 0.4030 0.6457 0.3627 5.1570
NSCT_SR 0.2953 0.6243 0.5895 0.6353 7.4890 NSCT_SR 0.3725 0.5514 0.6886 0.6389 2.2760
LP_SR 0.2948 0.6560 0.5734 0.6588 0.3380 LP_SR 0.3582 0.6025 0.6741 0.6521 0.2210

images better than other methods. On the other hand, the two indicators. As mentioned in Section 5.1, QG and QCB
proposed method has no advantage or even a considerable are mainly used to measure the degree to which the fused
gap over the best-performing method in terms of the other image retains the edge information of the source images.
14 Computational and Mathematical Methods in Medicine

Table 3: Objective evaluation of fusion methods in the third set of of CSR show undeniable advantages in the contrast experiment
images. of third set images. QTE measures the degree to which the fusion
image retains the information of the source images, and the dis-
Fusion method QTE QG QC QCB t/s tortion of image structure or color will lower this index. Image
Proposed method 0.7297 0.6602 0.6249 0.4645 233.0850 artifacts and structural distortions may mislead medical profes-
CSR 0.5025 0.8111 0.7920 0.7010 130.2410 sionals, so it is not advisable to blindly pursue high image struc-
ASR 0.4632 0.7901 0.7755 0.6611 188.6130 ture indexes. The proposed method has obvious advantages in
CSMCA 0.4576 0.7319 0.6790 0.5979 390.8500 the QTE index, and the degree of color distortion and artifact
CNN 0.7183 0.7822 0.7530 0.4966 38.6490
in the fused images is minimal. It is crucial for medical images.
Because artifacts in brain images can sometimes look very sim-
BMPCNN 0.3832 0.7900 0.8470 0.6505 75.5030
ilar to lesions, false lesions in images can directly affect the judg-
MDLatLRR 0.4303 0.7755 0.7298 0.6416 73.5810 ment of medical professionals. In addition, SPECT and PET
VGG-19 0.4828 0.7026 0.6891 0.6179 7.8900 show metabolic abnormalities through chromatic aberration.
NSCT_SR 0.4659 0.8054 0.8200 0.6757 9.2580 If significant color distortions appear in the image, it will cause
LP_SR 0.5044 0.7994 0.8120 0.5991 0.5030 deviation from the real. Based on the above analysis, the pro-
posed method is effective.

Table 4: Objective evaluation of fusion methods in the fourth set of 6. Conclusion

images.
In this paper, a multimodal medical image fusion method
Fusion method QTE QG QC QCB t/s based on multiple latent low-rank representation is pro-
Proposed method 0.6267 0.5968 0.6416 0.4832 163.4530 posed. Experimental results show that the proposed method
CSR 0.5041 0.8026 0.6380 0.7370 149.6120 has advantages in preserving edge features and texture
ASR 0.4431 0.7291 0.6231 0.6573 192.5610 details and leads to objective evaluation indexes compared
CSMCA 0.4383 0.6821 0.6355 0.6141 344.2190 with other fusion algorithms. The proposed method can
CNN 0.4197 0.6749 0.5585 0.6446 33.4460
enhance the observer’s ability to identify the lesions and
contribute to practical applications such as diagnosis, treat-
BMPCNN 0.3437 0.5930 0.6328 0.5226 71.7970
ment planning, and surgical navigation. On the other hand,
MDLatLRR 0.4270 0.6752 0.6056 0.6567 61.6160 the proposed method also has some drawbacks. The high-
VGG-19 0.4730 0.6605 0.5722 0.6353 7.6620 light of lesions is based on multilayer LatLRR decomposi-
NSCT_SR 0.4529 0.6827 0.6386 0.6001 7.5950 tion, but with the increase of the number of decomposition
LP_SR 0.4457 0.6734 0.5938 0.5764 0.3240 layers, artifacts will become more evident. Besides, the more
layers decomposed by LatLRR, the less information of the
low-rank part of the source image, and the greater the error
Table 5: Objective evaluation of fusion methods in the fifth set of of VGG-19 network recognition. The next stage should
images. focus on eliminating artifacts as much as possible in the case
of multilayer LatLRR decomposition and improve the fusion
Fusion method QTE QG QC QCB t/s quality of low-rank parts of source images.
Proposed method 0.5260 0.6594 0.6234 0.6005 46.2110
CSR 0.5175 0.8783 0.8793 0.8374 22.4600 Data Availability
ASR 0.4639 0.7902 0.7986 0.6573 53.9060
CSMCA 0.4808 0.7908 0.8237 0.7346 76.9750 The data supporting the study are obtained from K. A. John-
son and J. A. Becker, The Whole Brain Atlas, 2021. URL:
CNN 0.5186 0.7668 0.8170 0.4144 11.8690
http://www.med.harvard.edu/aanlib/, (accessed 12 May
BMPCNN 0.3632 0.7289 0.8196 0.7099 14.8240
2021).
MDLatLRR 0.4591 0.7905 0.8180 0.7464 17.9870
VGG-19 0.4947 0.6974 0.6851 0.6797 5.1760 Conflicts of Interest
NSCT_SR 0.4962 0.7658 0.7943 0.9358 2.2660
LP_SR 0.5083 0.7843 0.8435 0.6496 0.2280 The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
QG , in particular, is directly affected by the gradient of the
fused image. As shown in Figure 12(c), the color of the CSR Acknowledgments
fusion image is seriously distorted, and the white area in the
middle of the image has apparent artifacts, which is unaccept- This work was partially supported by grants from the Chi-
able for brain images. However, the color distortion and arti- nese National Natural Science Foundation (Grant No.
facts bring to the fused image that the ‘border’ does not 22178036), Chongqing Nature Science Foundation for Fun-
initially exist and will undoubtedly increase the image struc- damental Science and Frontier Technologies (Grant No.
ture’s gradient value and complexity. That is why QG and QCB cstc2018jcyjAX0483), Science and Technology Research
Computational and Mathematical Methods in Medicine 15

Program of Chongqing Education Commission of China [14] S. Kollem, K. R. Reddy, and D. S. Rao, “Improved partial differ-
(Grant Nos. KJQN201900821 and KJQN202000803), Inno- ential equation-based total variation approach to non-
vative Research Group of Universities in Chongqing (Grant subsampled contourlet transform for medical image denois-
No. CXQT21024), and Graduate Innovation Project of ing,” Multimedia Tools and Applications, vol. 80, no. 2,
Chongqing Technology and Business University (Grant pp. 2663–2689, 2021.
No. yjscxx2021-112-45). [15] J.-Y. Li and C.-Z. Zhang, “Blind watermarking scheme based
on Schur decomposition and non-subsampled contourlet
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