contrast of x, and
s
xy
measures the tendency of x and y to
vary together.
The SSIM index is superior over MSE, SNR, PSNR, and
there are three key factors leading its success. First, the
SSIM considers image degradations as perceived changes
in structural information variation, instead of estimating
perceived errors. Second, the SSIM is a top-down approach,
mimicking the hypothesized functionality of the overall
HVS. Third, the problems of natural image co mplexity and
decorrelation are avoided to some extent because the SSIM
does not attempt to predict image quality by accumulating
the errors associated with psychophysically understood
simple patterns.
However, the SSIM has some weakness. It is difficult to
distinguish the distortion level between a blurred image
and a noisy image. For example, a very blurred image will
has a higher SSIM value than a noisy image with low noise
level (see Fig. 1). What is wrong with SSIM? We find that
the structure measure in SSIM only computes the related
coefficient of pixel value between the reference image and
the distorted image using space domain statistics, so it
can not reflect image edge and texture structural informa-
tion very well.
To overcome this drawback, one solution is to provide
more accurate measure for the structure distortion. For
example, Chen et al. [18] developed a gradient-based
structural similarity. The primary reason of performance
improvement in GSSIM is that it pays more attention to
the edges and details of images, which represents the
most important structural information of images.
2.2. Improved structure comparison measure
An important aspect of the HVS perception is its sen-
sitivity to image structure. In general, structural magnitude
and structural orientation are two important attributes of
image geometrical structure. Furthermore, philosophy
experiments show that HVS is more sensitive to structural
orientation. Hence, the orientation field of the geometrical
structure is more important in the structure similarity
measure.
It is well known that the structure tensor can capture
the image structure information very well. Especially,
it could be used to estimation the structural orientation.
In this section, the structural orientation is utilized to
measure the structure similarity.
The structure tensor of an image is formulated as a
symmetric and positive semi-definite matrix:
J
r
ð
r
u
s
Þ¼G
r
ð
r
u
s
r
u
s
Þ¼
J
11
J
12
J
12
J
22
!
, ð3Þ
where convolving u with a Gaussian kernel G(0,
s
2
) makes
orientation estimation robust against noise at scales
smaller than
s
. Convolving the tensor
r
u
s
r
u
s
with a
Gaussian kernel G(0,
r
2
) makes orientation estimation
more accurate, especially at corners. In this paper, we
set the standard deviation
s
¼0.5,
r
¼0.5. Moreover, the
corresponding eigenvectors
$
and
n
as follows,
w ¼
2J
12
J
22
J
11
þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðJ
22
J
11
Þ
2
þ4J
2
12
q
0
@
1
A
,
$
¼
w
:w:
,
n
¼
$
?
,
ð4Þ
where the eigenvector
$
points in the dominant orienta-
tion with the largest contrast, and the eigenvector
n
points
in the structure orientation with the smallest contrast. They
are shown in Fig. 2.
Different from the SSIM index, we utilize the structure
tensor to estimate the structure orientation similarity
between reference image x and distorted image y. First,
we should compute their eigenvectors:
n
x
¼(
n
x1
,
n
x2
)and
n
y
¼(
n
y1
,
n
y2
). Then the corresponding geometrical structure
orientation angle
y
x
,
y
y
and their cosine-squared value
Fig. 1. Comparing the SSIM value of different distorted type images. (a) noisy image, SSIM¼0.4332 (b) blurred image, SSIM¼0.4678.
Fig. 2. Structure tensor.
X. Fei et al. / Signal Processing: Image Communication 27 (2012) 772–783774