New approach to multispectral image fusion based on a weighted merge

International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN
INTERNATIONAL JOURNAL OF ELECTRONICS AND
0976 – 6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 1, January- February (2013), © IAEME
COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
ISSN 0976 – 6464(Print)
ISSN 0976 – 6472(Online)
Volume 4, Issue 1, January- February (2013), pp. 25-34
IJECET
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NEW APPROACH TO MULTISPECTRAL IMAGE FUSION
BASED ON A WEIGHTED MERGE

Benayad NSIRI1, Salma NAGID1, Najlae IDRISSI2
1
(LIAD Faculty of Science Ain Chock, University Hassan II Ain Chock,
Casablanca 20100, Morocco, b.nsiri@fsac.ac.ma)
2
(TIT-Team Faculty of Sciences and Techniques, University Sultan Moulay Slimane,
Beni-Mellal,B.P 523 Mguilla, Morocco, n.idrissi@usms.ma)

ABSTRACT

Multispectral image fusion seeks to combine information from different images to
obtain more relevant information than can derive from a single one. A wide variety of
approaches addressing fusion at pixel level has been developed, but they suffer from several
disadvantages, (1) the number of bands merging is limited, (2) color distortion, (3)Spectral
content of small objects often lost in the fused images. The paper presents a new approach of
image fusion based on a weighted merge of multispectral bands, each band is modeled by
two or three Gaussian distributions, the mixture parameters (weights, mean vectors, and
covariance matrices) are estimated by the Expectation Maximization algorithm (EM) which
maximizes the log-likelihood criterion, the weighted coefficients of each band are extracted
from the degree of similarity between this one and the other bands, it’s calculated by a cost
function based on the distance between the parameters of the Gaussian distribution of each
band. In our work we use an Extended Malhanobis distance. This cost function allows us to
reduce data redundancy and given greater weight to the complementary data. We applied this
approach to MRI and satellite images by using respectively a weighted average and weighted
color composite method, the given results are promising.

Keywords : Pixel-based Image Fusion, Expectation Maximization, Multispectral Merge,
Mixture of Gaussian distributions, Image Similarity.

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I. INTRODUCTION

The goal of image fusion[1][2][3] is to combine information from two or more images
of a scene into a single composite image that is more informative and is more suitable for
visual perception or further image-processing tasks. However, the information provided by
different images may be complementary and redundant. The fusion at pixel level is
traditionally handled by the Multiscale Decomposition based methods, the principal
component analysis (PCA) methods, the color composite transform method.
Multiscale transforms combine the multiscale decomposition of the source images[18], this
approach, constructs a composite representation of multiscale transform on the source images
using some sort of fusion rule, and then construct the fused image by applying the inverse
multiscale transform. Pyramid transforms and wavelet transforms [29][30] are the most
commonly used multiscale decomposition fusion methods.
A pyramid transforms fusion consists of a number of images at different scales which
together represent the original image; the Laplacian Pyramid is an example of a pyramid
transform. Each level of the Laplacian Pyramid is constructed from its lower level using
blurs, size reduction, and interpolation and differencing in this order [18]. Alternative
pyramid transforms are contrast pyramid which preserves local luminance contrast in the
source images [19], and finally a gradient pyramid applies the gradient operator on each level
of the Gaussian pyramid representation [20].
Discrete Wavelet transforms are a type of multi-resolution function approximation
that allow for the hierarchical decomposition of a an image [21][22][18]. The wavelet
transforms W are first calculated for two input images ‫ܫ‬ଵ ሺ݅, ݆ሻ and ‫ܫ‬ଶ ሺ݅, ݆ሻ, then the results are
combined using the ߔ fusion rules. Finally, the inverse wavelet transform ܹ ିଵ is computed
and the image fusion ‫ܫ‬ሺ݅, ݆ሻ is re-constructed. The wavelet transform has several advantages
over other pyramid-based transforms: It provides a more compact presentation, separate
spatial orientation in different bands, and decorrelates interesting attributes in the original
image.
PCA (Principal Component Analysis)[26] is a general statistical technique which
transforms multivariate data with correlated variables into multivariate data with
uncorrelated ones. These new variables are obtained as a linear combination of the original
variables. The PCA have been used to fuse the images by two ways researchers. The first
approach assigns the first principal component (PC) band to one of the RGB bands and the
second component to another RGB band in a color composite technique while the second
method separates the first and the second PCs to intensity and hue band in an IHS image
[23][24].
The color composite method [25], assigns in order the first, the second and the third
band to the R, B and G channel. It will work well if we merge three images but problems
occur beyond this number. To overcome these problems we developed during earlier work a
new method termed a color composite Composed CCC (4) [9] that is an extension of the
color composite method for merging four bands. The principle is to give each band a certain
coefficient αi during the merge. Suppose that we have four images ሺ‫ܫ‬ଵ , ‫ܫ‬ଶ , ‫ܫ‬ଷ , ‫ܫ‬ସ ሻ to merge, the
merger will be done as follows:

ܴ ߙଵ . ‫ܫ‬ଵ ൅ ߙଶ . ‫ܫ‬ଶ
൥‫ ܩ‬൩ ൌ ൥ߙଷ . ‫ܫ‬ଶ ൅ ߙସ . ‫ܫ‬ଷ ൩ …..……………………... (1)
‫ܤ‬ ߙହ . ‫ܫ‬ଷ ൅ ߙ଺ . ‫ܫ‬ସ
26


We can see from (1) that we use a weighted fusion by a set of constant coefficients αi
, where :

{αi /i = {1..6}} = {0.75, 0.25, 0.50, 0.50, 0.25, and 0.75} ……………….... (2)

This method has given promising results in view of the amount of information retrieved, but
it remains limited by the number of bands to fuse (increasing number of bands will cause a
color distortion) and the management of complementary and of redundant data. Despite the
advantages of the pixel level methods they still suffer from several disadvantages, (1) the
number of bands merging is limited, (2) color distortion, (3) Spectral content of small objects
often lost in the fused images. In this paper we develop a new approach that addresses both
the problem of band numbers and the weight assigned to each band according to the
information it contains and its reliability with other bands. The idea is to give a weight to the
images during the merger process to handle the redundancy and complementarily of data.
Two ways are addressed to compare our approach with the PCA: visual evaluation and
quantitative evaluations-based on RMSE and UIQI indexes quality. The results are
promising.
The remainder of this paper is organized as follows. In Section 2, we explain our fusion
method in detail, including how to select the similarity characteristics of source images,
obtain the weight of each image, and fuse images. Section 3, provides the simulation
scenarios and evaluates the results. Finally, conclusions are drawn in Section 4.

II. THE WEIGHTED MERGING OF BANDS

1. The image modeling by mixture of Gaussian distributions
A Gaussian mixture model is a weighted sum of k component Gaussian densities
given by the equation

݂൫‫⁄ݔ‬Θ୮ ൯ ൌ ∑௣ ߙ௞ ݂௞ ሺ‫⁄ݔ‬Θ୩ ሻ ൌ ∑௣ ߙ௞ ݂௞ ൫‫⁄ ݔ‬µ୩ , Σ௞ ൯ …………….. (3)
௞ୀଵ ௞ୀଵ

Where p is the numbers of components in the mixture, (α k ≥ 0) are the mixing proportions of
components satisfyingΣ௣ ߙ௞ ൌ 1, and each component density ݂௞ ൫‫⁄ݔ‬µ୩ , Σ௞ ൯ is a Gaussian
௞ୀଵ
probability density function given by
ଵ
݂௞ ൫‫⁄ ݔ‬µ୩ , Σ௞ ൯ ൌ ሺଶగሻ೙/మ |Σ exp ሺെ1⁄2ሻሺx െ µ୩ ሻ் Σିଵ ൫‫ ݔ‬െ µ୩ ൯ሻ ………….. (4)
௞
ೖ|
భ/మ

Where ݊ is the dimensionality of the vector x, µ୩ is the mean vector and Σ௞ is the covariance
matrix assumed to be positive definite. We suppose Θ୮ the collection of all the parameters in
the mixture Θ୮ ൌ ሺߠଵ , … , ߠ௣ , ߙଵ , … ߙ௣ ሻ the log-likelihood function for the Gaussian mixture
mo Del of a set of ܰ i.i.d. samples, ܺ ൌ ሼ‫ݔ‬௜ ሽே is
௜ୀଵ

log ሺ݂ሺܺ⁄Θ୮ ሻሻ ൌ ݈‫݃݋‬ሺΠ୒ ሺ‫ݔ‬௜ ⁄Θ୮ ሻሻ ൌ ∑ே ݈‫݃݋‬ሺ∑ே ߙ௞ ݂ሺ‫ݔ‬௜ ⁄ߠ௞ ሻሻ …………….
୧ୀଵ ௜ୀଵ ௜ୀଵ
(5)

27


Then we maximize (5) to get a Maximum Likelihood (ML) and estimate of Θ୩ via the EM
algorithm as follow:

ߙ௞ ൌ 1⁄݊ ∑ே ݂ ሺ݇ ⁄‫ݔ‬௜ ሻ
௜ୀଵ ………………………………….. (6)

∑ಿ ௫ೖ ௙ሺ௞⁄௫೔ ሻ
ߤ௞ ൌ ೔సభ
∑ಿ ௙ሺ௞⁄௫೔ ሻ
……………………………………… (7)
೔సభ

ሺ∑ಿ ௙ሺ௞⁄௫೔ ሻሺ௫೔ିఓ೔ ሻሺ௫೔ ିఓ೔ ሻ೅ ሻ
Σ௞ ൌ ೔సభ
∑ಿ ௙ሺ௞⁄௫೔ ሻ
.....…………………….. (8)
೔సభ

Where ݂௞ ሺ݇ ⁄x୧ ሻ ൌ ߙ௞ fሺx୧ ⁄θ୩ ሻ/ ∑୧ୀଵ ߙ௞ ݂ሺx୧ ⁄θ୩ ሻ are the posterior probabilities
୮

2. Distance measures between Gaussian Mixtures Models
The Extended Mahalanobis distance metric is an extension of a distance measure between
two distributions (in our case a Gaussian distribution).
The Extended Mahalanobis distance is based on the statistical distribution of data and not on
data directly. We consider two Gaussian distributions ܰଵ ሺߤଵ , Σଵ ሻ and ܰଶ ሺߤଶ , Σଶ ሻ, the measure
between the two distributions is defined as follows:

‫ܦ‬ሺܰଵ , ܰଶ ሻ ൌ ඥሺߤଵ െ ߤଵ ሻ் ሺΣଵ ൅ Σଶ ሻିଵ ሺߤଵ െ ߤଶ ሻ ………………….. (9)

However, this measure creates a singularity for singular covariance matrices. In practical
problems it often appears in learning such models mixture. The acquired covariance matrix is
not always conditioned and their inversion creates a problem. In our implementation, we
replace the inverse of singular covariance matrix by its pseudo inverse. Singular value
decomposition is used for the calculation of the pseudo inverse. Round of errors can lead to a
singular value not being exactly zero even if it should b e. Tolerance parameter places a
threshold when comparing singular values with zero and improves the numerical stability of
the method with singular or near-singular matrices.

3. Approach and Conception of the Proposed Method
The computation of co eﬃcients fusion is based on the degrees of similarity between
images to b e merged and the quantity of additional information provided with each one.

a. Extraction of parameters and the cost function
Each image is modelled by a mixture of two Gaussian distributions. This modeling
consists in estimating the parameters of the mixture (weight, mean vectors, and covariance
matrix).
We calculate the distance of Malhanobis Dij between each two models.

28


்
‫ܦ‬൫ܰ௜ , ܰ௝ ൯ ൌ ට൫ߤ௜ െ ߤ௝ ൯ ሺΣ௜ ൅ Σ௝ ሻିଵ ൫ߤ௜ െ ߤ௝ ൯ ……………………….. (10)

ܰ௜ ሺߤ௜ , Σ௜ ሻ : The Gaussian distribution of image݅.
We calculate the weighted coefficients ݀௜ of each image by the cost function as follows:

݀௜ ൌ ൛max൫‫ܦ‬௜௝ ൯ /ሺ݅ ് ݆ሻൟ ………………………….. (11)

Where ݀௜ is the weighted coefficient of the fusion attributed to the image ݅ each coefficient
then attributes to its image, and used in the fused rule.
We normalize the d i distances, we got:

ௗ೔
ߙ௜ ൌ ∑೙
೔సభ ௗ೔
…………………………………….. (12)

ߙ௜ : The normalized weighted coefficient of the ݅ image.
In the following we call ߙ௜ the weighted coefficients and we use it on the fusion rules.

b. Application to some fusion rules
We consider ሼ‫ܫ‬ଵ , … , ‫ܫ‬௡ ሽ the set of images to fuse, and ሺߙଵ , … ߙ௡ ሻ the set of weighted
coefficients of the images to fuse.

• Weighted averaging

‫ ܫ‬ൌ ∑௡ ߙ௜ ‫ܫ‬௜ ………………………………….. (13)
௜ୀଵ

• Weighted color composite

ܴ ൌ ∑௞ ߙ௜ ‫ܫ‬௜ ൅ ߦଵ ‫ܫ‬௞ାଵ ………………………………….. (14)
௜ୀଵ
Where ∑௞ ߙ௜
௜ୀଵ ൅ ߦଵ ൌ 1

‫ ܩ‬ൌ ሺߙ௞ାଵ െ ߦଵ ሻ‫ܫ‬௞ାଵ ൅ ∑௞ା௠ ߙ௜ ‫ܫ‬௜ ൅ ߦଶ ‫ܫ‬௞ା௠ାଵ ……………. (15)
௜ୀ௞ାଶ

Where ∑௞ା௠ ߙ௜ ൅ ߦଶ ൅ ሺߙ௞ାଵ െ ߦଵ ሻ ൌ 1
௜ୀ௞ାଶ

‫ ܤ‬ൌ ∑௡
௜ୀ௞ା௠ାଶ ߙ௜ ‫ܫ‬௜ ൅ ሺߙ௞ା௠ାଵ െ ߦଶ ሻ‫ܫ‬௞ା௠ାଵ …………… (16)

The sum of coefficients ߙ௜ attributed to each band must b e equal to 1, when∑௞ ߙ௜ ൏ 1, we
௜ୀଵ
can add a constant ߦ௧ to have∑௞ ߙ௜ ൅ ߦ௧ଵ ൌ 1.
௜ୀଵ

We use a ߦଵ quantity of the information from the image ݇ ൅ 1 in the band ܴ and ሺߙ௞ାଵ െ ߦଵ ሻ
of its information in the band‫.ܩ‬

29


III. EXPERIMENTAL RESULTS

Evaluation results can be achieved in two ways visual and quantitative.
1. Visual evaluation

a. Brain MRI images
Figure 1 show four types of Brain MRI images (T1, PD, T 2, MRGad) used in the
fusion process. The corresponding visual result of image fusion based on the weighted
average method compared to PCA is shown in figure 2. Compared to PCA, our approach
reconstructs clearly different brain structures than the original down to the smallest details.
the black spot in the right of the MRGad image is found in the merged one whereas the PCA,
some structures are less clear and/or confused.

b. Satellite images
Figure 3 shows five satellites images used in the fusion process. The corresponding
visual result of image fusion based on the weighted color composite method compared to
PCA is shown in figure 4. Compared to PCA, the color composite effect of our approach is
evident in the fused image while in the PCA method, there is no color effect.

Figure 1: The MRI images of Meningioma tumour: T1; PD; T2; MRGad

Figure2: Result of fused image from Brain MRI images. (Left) our approach, (right) PCA

30


Figure3: Examples of satellite images

Figure 4: Results of fused images from satellite examples. (Left) our approach, (right) PCA

2. Quantitative evaluation
To evaluate the proposed approach, we retain two quantitative measures widely used
in the literature to assess the quality of reconstructed images by fusion method [28]:
UIQI (Universal Image Quality Index)[27] : it measures how much of the salient
information contained in original image. The range of this measure varies from -1 to +1
where high value of UIQI significates better fusion. If A and B are respectively the original
and fused image and ߤ௔, ߤ௕ ,ߪ௔ , ߪ௕ are the mean and standard deviation of A and B, the
corresponding UIQI is defined as:
ఙ ଶఓೌ ఓ ଶఙೌ ఙ
‫ݍ‬௜ ൌ ఙ ೌ್ ఓమ ାఓ್ ఙమ ାఙ್ ……………………………… (17)
ఙ మ మ
ೌ ್ ೌ ್ ೌ ್
Where ߪ is covariance and ߤ is mean.

31


RMSE (Root Mean Squared Error): it calculates the difference of standard deviation
and the mean between the original and the fused image. Smaller value corresponds to better
fusion method.
∑ಾ ∑ಿ ሾிሺ௜,௝ሻି஺ሺ௜,௝ሻሿమ
ߜൌට ೔సభ ೕసభ
ேൈெ
…………………………... (18)

Where ‫ܣ‬ሺ݅, ݆ሻ is original image and ‫ ܨ‬ሺ݅, ݆ሻ is fusion image.
Table 1 reports the results of RMSE and UIQI applied on the Brain MRI fused image.
For both RMSE and UIQI our approach works much better than PCA. It reduces the RMS
error around 30% relative to PCA while the UIQI value retains more than 75% of the salient
information for 1st, 2nd and 3rd band.

Quality index Method Band 1 Band 2 Band 3 Band 4 Average
RMSE PCA 32.6284 33.0610 33.2983 35.6705 33.6645
Our 22.0949 22.5831 23.1733 26.4562 23.5769
UIQI approch PCA 0.3201 0.3041 0.3047 0.4899 0.3547
Our 0.6185 0.5345 0.5853 0.6486 0.5967
approch
Table 1: The comparison between the indexes quality values of our approach and PCA
Method relative to Brain MRI image

Bands Band R Band G Band B Average PCA
Band 1 0.7630 0.8711 0.7349 0.7897 0.7696
Band 2 0.8711 0.7308 0.8040 0.8020 0.7675
Band 3 0.7349 0.7736 0.7198 0.7428 0.7372
Band 4 0.6031 0.6441 0.6166 0.6213 0.6141
Band 5 0.7901 0.7583 0.7445 0.7643 0.7586
Table 2: The comparison of the UIQI index quality value of our approach and PCA method.

For the satellite images, the results are reported in table2. For most satellite bands the
weighted color composite approach is competitive to PCA.

IV. CONCLUSION

In this work, we propose a new method for multispectral image fusion based on the
weighted merge to overcome the problem of limited number of merged bands for the other
multispectral image fusion. The quality of fused image by our proposed method is much
better than obtained with PCA approach for both Brain MRI and satellite images.

V. ACKNOWLEDGEMENTS

This article is dedicated to Miss Salma NAGID who had realized first version before she
passed away.

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