Image Anal Stereol 2018;37:213-223 doi:10.5566/ias.1887
Original Research Paper
PARAMETRIC BLIND IMAGE DEBLURRING WITH GRADIENT BASED
SPECTRAL KURTOSIS MAXIMIZATION
AFTAB KHANB,1 AND HUJUN YIN2
1Department of Computer Systems Engineering (DCSE), University of Engineering and Technology (UET),
25000 Peshawar, Pakistan.; 2School of Electrical and Electronic Engineering, The University of Manchester,
Manchester M13 9PL, UK.
e-mail: aftab.khan@uetpeshawar.edu.pk, hujun.yin@manchester.ac.uk
(Received January 12, 2018; revised September 9, 2018; accepted September 17, 2018)
ABSTRACT
Blind image deconvolution/deblurring (BID) is a challenging task due to lack of prior information about the
blurring process and image. Noise and ringing artefacts resulted during the restoration process further deter
fine restoration of the pristine image. These artefacts mainly arise from using a poorly estimated point spread
function (PSF) combined with an ineffective restoration filter. This paper presents a BID scheme based on the
steepest descent in kurtosis maximization. Assuming uniform blur, the PSF can be modelled by a parametric
form. The scheme tries to estimate the blur parameters by maximizing kurtosis of the deblurred image. The
scheme is devised to handle any type of blur that can be framed into a parametric form such as Gaussian,
motion and out-of-focus. Gradients for the blur parameters are computed and optimized in the direction of
increasing kurtosis value using a steepest descent scheme. The algorithms for several common blurs are
derived and the effectiveness has been corroborated through a set of experiments. Validation has also been
carried out on various real examples. It is shown that the scheme optimizes on the parameters in a close
vicinity of the true parameters. Results of both benchmark and real images are presented. Both full-reference
and non-reference image quality measures have been used in quantifying the deblurring performance. The
results show that the proposed method offers marked improvements over the existing methods.
Keywords: blind image deblurring (BID), gradient descent, image quality measures (IQMs), image restoration,
kurtosis.
INTRODUCTION
Since the first realization of blind image
deconvolution/ deblurring (BID) in early 1960s, it still
remains a challenging task to find an efficient, reliable
and most importantly, diversely applicable restoration
scheme. A scheme should not only estimate the actual
point spread function (PSF) but also restore the image.
The main challenge arises from little or no prior
information about the image or the blurring process
as well as lack of optimal restoration filters that can
eliminate the blurring effect. The two common side
effects of deblurring include noise amplification and
ringing artefacts in the deblurred image. Developing
a scheme that can handle different types of blur
especially for real images is yet to be achieved to a
satisfactory extent.
A variety of BID schemes and restoration
filters have been proposed over the years, ranging
from the spatial domain to the frequency domain,
parametric to non-parametric. Typical methods include
the Richardson-Lucy method, total variation, Wiener
filter, maximum likelihood method, minimum entropy
deconvolution, recursive inverse filter, simulated
annealing, and multi-channel blind deconvolution
(Richardson, 1972; Wiggins, 1978; Lagendijk et al.,
1988; 1990; Kundur and Hatzinakos, 1996; Banham
and Katsaggelos, 1997; Chan and Wong, 1998), as
well as their recent improvements (Chen and Cheng,
2011; Szolgay and Szirányi, 2011; Wang and Li, 2011;
Yang and Liu, 2011). These methods do provide, to
some extent, solutions to the BID problem; however,
many are not satisfactory in terms of robustness
and performance. The robustness of these schemes
becomes questionable when real-life blurred images
are to be restored. This is because unlike synthetic
deblurring, real deblurring often suffers from noise and
ringing effects. The blurred images may have been
degraded by arbitrarily shaped PSFs that are complex
and cannot be easily modelled (Gupta et al., 2010;
Harmeling et al., 2011; Lu, 2012; Whyte et al., 2011;
Yuquan et al., 2013).
Recently some BID schemes focus mainly on
arbitrary motion blur. Not only because the problem
has common occurrence in image acquisition but also
because it is challenging. This type of blur ensues from
camera shakes or movements of objects or background
in the focal range. The PSFs of such blur are usually
of arbitrary shape and are sometimes space variant.
Restoring such images requires more effort and
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complex procedures are applied to approximate the
blur kernel as well as restore the image while deterring
deblurring noise and ringing artefacts. Restoring
motion blurred images as compared to Gaussian
and out-of-focus blurred ones is less problematic in
restoration terms as for other types more ringing
artefacts and significant amount of residual blur are
often produced.
The image processing community and researchers
over the years have strived to come up with BID
schemes that can effectively handle different types
of blurs. Our motivation is to design a BID scheme
that is efficient in terms of computational costs and
effective in terms of restoration quality. The proposed
scheme uses a parametric representation for uniform
blur. This reduces the search space for PSF parameters.
The parameters are optimized by a steepest descent
scheme on a restoration measure. The method offers
a unified framework and can estimate any parametric
blur type.
METHODS
In this following sub-sections, the various
concepts used in implementing our proposed scheme
are discussed. It includes the introduction and
modelling of parametric blurs, the principle of kurtosis
maximization and the gradient descent scheme based
on this principle.
PARAMETRIC BLUR MODELS
Parametric models help simplify the search for the
blur function, while a restoration measure is used to
aid the optimization to a desired result. In the quest
for a simple deblurring scheme, the proposed method
relies on higher order statistics based non-Gaussianity
measure to lead in the direction of increased quality
of the deblurred image. Previously the authors have
proposed schemes using non-Gaussianity measures in
either spatial or frequency domain, namely spatial
and spectral kurtosis (Yin and Hussain, 2008; Khan
and Yin, 2011). In addition, re-blurring based image
quality measure was also investigated but its use was
limited as it is highly sensitive to restoration noise and
artefacts.
Let x(m,n) represent the original image without any
form of degradation, h(m,n) be the PSF, and y(m,n) the
output of the image acquisition system. For a system
of stationary impulse response across the entire image
(i.e. a spatially invariant PSF), the convolution is given
by (Oppenheim et al., 1968; Sondhi, 1972),
y(m,n) = h(m,n)∗ f (m,n)+η(m,n) , (1)
where ∗ denotes the convolution operator, η(m,n)
represents additive noise; and m and n are spatial
coordinates. The frequency domain representation of
Eq. (1), with spectral coordinates i and j, is given as
Y (i, j) = H(i, j)F(i, j)+∏(i, j) . (2)
Parametric blur models are discusses as follows.
The blur PSFs studied here are spatially invariant.
Modelling, restoration and estimation of the original
image from invariant PSFs is less cumbersome as
compared to spatially variant ones. The classes of
blur that can be easily modelled include atmospheric
turbulence, motion and out-of-focus. The PSF for
atmospheric turbulence blur for long exposures
can be approximated by Gaussian blur and the
degraded image regarded as a Gaussian blurred image
(Lagendijk and Biemond, 2009). The 2-D Gaussian
PSF is given by
h(m,n) =
1
2πσ2
exp
(
−m
2 +n2
2σ2
)
, (3)
where σ is the width of the blurring kernel.
Motion blurs commonly result from relative
motions between the scene and the recording device.
They can be in the form of translation, sudden change
in scale, rotation or a combination of these. The motion
blur dealt with here is the case of linear translation.
Denoting the length of motion by L and the angle by
ϕ , the motion blur PSF is given as (Lagendijk and
Biemond, 2009),
h(m,n;L,ϕ) =

1/L if
√
m2 +n2 ≤ L/2
and m/n =− tanϕ ,
0 elsewhere.
(4)
When a camera captures a 2-D image of a 3-D
scene, certain points in the scene may not be focused.
For a camera with a circular aperture, the PSF of a
point looks like a circular disk with a specific radius.
The spatially continuous out-of-focus blur of radius R
can be expressed as (Lagendijk and Biemond, 2009),
h(m,n;R) =
{
(CπR2)−1 if
√
m2 +n2 ≤ R2,
0 elsewhere,
(5)
where C is a constant and is chosen so that energy
conservation law is satisfied.
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Image Anal Stereol 2018;37:213-223
KURTOSIS MAXIMIZATION
If statistical independence can be assumed for
the unknown source signals, then according to the
Central Limit Theorem (CLT) the output signals of a
linear system are more Gaussian. Hence a general BID
scheme can be designed to estimate the independent
signals that are maximally non- Gaussian. A measure
of non-Gaussianity such as kurtosis or negentropy can
be employed to achieve this goal. The normalized
kurtosis of an image y is defined as
K(y) =
E{y4}
(E{y2})2
−3 . (6)
Using a suitable optimization scheme, the kurtosis
can be optimized in the direction of maximum
non-Gaussianity for the deblurred image. However,
optimizing the kurtosis function is not a trivial task
and resulting learning algorithm can be difficult to
implement or may induce high computational costs.
The proposed scheme utilizes the Wiener filter to
estimate the deblurred image for a candidate PSF.
Kurtosis calculated for the spatial domain, referred to
as spatial kurtosis, is computationally costly because
its calculation on the deblurred image needs to be
transformed from the frequency domain back to the
spatial domain in each iteration. By calculating the
non-Gaussianity of the spectrum images, this overhead
is drastically reduced. For this purpose, the kurtosis
of an image is calculated using the spectral kurtosis
(Khan and Yin, 2011) defined for the frequency
domain image, Y, as
K(Y ) =
E{Y 4}
(E{Y 2})2
−3 . (7)
Even so, the previous work (Yin and Hussain,
2008) had used a genetic algorithm to search for
optimal blur parameters. The process is often time
consuming and there is no guarantee that an optimal
solution will be reached. Based on the statistical nature
of the image, i.e. super-Gaussian or sub-Gaussian,
spatial kurtosis is either minimized or maximized at
the correct blur parameters. However, in the case
of spectral kurtosis, where magnitude spectrum is
used,the maximum is always reached for the deblurred
image with the correct parameters. A new gradient
based BID scheme that utilizes parametric blur models
and spectral kurtosis as the non-Gaussianity measure
is proposed in this research work.
IMAGE QUALITY MEASURES
The proposed scheme has been tested on a
variety of images, synthetic and real blurred, degraded
with various functions. A number of image quality
measures were employed to evaluate the quality
of deblurred images. The full reference Image
Quality Assessment (IQA) measures include the Mean
Structural SIMilarity (MSSIM) index (Wang et al.,
2004; Wang and Li, 2011; Brunet et al., 2012) and
the Universal Quality Index (UQI) (Wang and Bovik,
2002). The blind/no-reference based IQA measures
are the Blind/Reference-less Image Spatial QUality
Evaluator (BRISQUE) (Mittal et al., 2011; 2012b) and
the Natural Image Quality Evaluator (NIQE) (Mittal
et al., 2012a; 2013). These quality measures are briefly
described below.
The SSIM is an objective image quality metric
that measures the structural similarity between two
images by comparing local patterns of pixel intensities
normalized for luminance and contrast. Given two
images, f and g, the SSIM is a function of luminance
l( f ,g), contrast c( f ,g) and structural similarity s( f ,g).
SSIM( f ,g) = t(l( f ,g),c( f ,g),s( f ,g)) , (8)
l( f ,g) =
2µ f µg +C1
µ2f +µ2g +C1
, (9)
c( f ,g) =
2σ f σg +C2
σ2f +σ2g +C2
, (10)
s( f ,g) =
σ f g +C3
σ f σg +C3
, (11)
where µ and σ are the mean and the standard
deviation of the image, respectively; C1, C2 and C3 are
constants; and t represents the SSIM transfer function.
SSIM is computed at each pixel and generally its mean
value is used, so named Mean SSIM (MSSIM). SSIM
has been used for denoising and classification (Gao
et al., 2011; Rehman and Wang, 2011).
The UQI analyzes the loss of correlation,
luminance and contrast distortion among the two
images as the base for quality perception. The UQI for
two images, f and g, is defined as (Wang and Bovik,
2002),
Q( f ,g) =
4σ f gµ f µg
(σ2f +σ2g )(µ
2
f +µ2g )
, (12)
Q( f ,g) =
σ f g
σ f σg
.
2µ f µg
µ2f +µ2g
.
2σ f σg
σ2f +σ2g
. (13)
The first term is the correlation coefficient of
the two images whereas the second and third terms
are the measures of mean luminance and structural
similarities.
The BRISQUE measures distortion based on
spatial domain statistics. It is also a non-reference
IQA model. Rather than computing distortion specific
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features such as ringing, blur or blocking, it uses scene
statistics of locally normalized luminance coefficients
to quantify possible losses of naturalness in the image
due to the presence of distortions, thereby leading to
a holistic measure of quality. The BRISQUE uses a
mapping from feature space to quality scores with
a regression module that yields a measure of image
quality. The features comprise statistic measures of a
generalized Gaussian distribution fitting of the Mean
Subtracted Contrast Normalized (MSCN) coefficients.
The MSCN for the distorted image g is obtained by
subtracting the local mean signal value µg and then
dividing by the local contrast function σg such that
ĝ(m,n) =
g(m,n)−µg(m,n)
σg(m,n)+C
, (14)
where
µg(m,n) =
U
∑
k=−U
V
∑
l=−V
wk,lgk,l(m,n) , (15)
σg(m,n) =
√√√√ U∑
k=−U
V
∑
l=−V
wk,l(gk,l(m,n)−µg(m,n))2 ,
(16)
where C is a constant, and w= {wk,l|k =−U, ...,U, l =
−V, ...,V} is a 2-D circular-symmetric Gaussian
weighting function and U = V = 3 was used for
calculating the measure.
The non-reference NIQE is a blind image quality
analyzer that only makes use of measurable deviations
from statistical regularities observed in natural images,
without being trained on human-rated distorted images
and without any exposure to distorted images. The
same computation for image quality as in the
BRISQUE is used with the exception that NIQE uses
natural scene statistics features from a staple of natural
images, while the BRISQUE is trained on features
obtained from both natural and distorted images as
well as human judgment of quality of the images.
IMAGE DATASET
Images from the Desktop Nexus image wallpaper
database (Nexus, 2010) were used in the experiments
in the synthetic blur case. Real blurred natural images,
captured by either ourselves or others, were used in
deblurring real blind images in the experiments.
Fig. 1. Test images used for deblurring of synthetic
blurred images.
A new gradient based BID scheme that utilizes
parametric blur models and spectral kurtosis as the
non-Gaussianity measure is proposed next.
PROPOSED GRADIENT BASED
DEBLURRING SCHEME
The derivation of the gradient based BID scheme
starts with differentiation of the cost function with
respect to the parameter of the blurring PSF model. A
Kurtosis based cost function for the deblurred image is
employed. The parameter is updated iteratively using
the steepest descent,
λp+1 = λp +α∇
∣∣∣Kλp(X̂)∣∣∣ , (17)
under the constraint,∣∣∣Kλp+1(X̂)∣∣∣> ∣∣∣Kλp(X̂)∣∣∣ , (18)
where λ is a blur PSF parameter, such as kernel width,
length or radius in case of Gaussian, motion or out-
of-focus blur respectively; p is the iteration number;
Kλp(X̂) is the spectral kurtosis of the deblurred image
for the current value of λ . ∇|Kλp(X̂)| represents the
gradient matrix for the magnitude of the complex
kurtosis variable and α is the step size. Since
|Kλ (X̂)|= Kλ (X̂)sign(Kλ (X̂)) , (19)
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Image Anal Stereol 2018;37:213-223
Eq. (8) can be rewritten as
λp+1 = λp +α∇Kλp(X̂) sign(Kλp(X̂)) . (20)
For α , a fixed scalar value can be used, 0 < α ≤
1. For faster convergence, more efficient computation
schemes can also be used such as the Newton method
or conjugate gradient. As X̂ is the deblurred image
resulting from deconvolution of the blurred image with
the Wiener restoration filter, which is given by
Ĥ∗
|H|2 +δ
, (21)
where δ is the noise to signal ratio. For simplicity of
future derivations and considering the magnitude of the
complex data, we rewrite the equation as
Ĥ
Ĥ
Ĥ∗
|Ĥ|2 +δ
=
(
ĤĤ∗
|Ĥ|2Ĥ + Ĥδ
)
=
Ĥ2
Ĥ3 + Ĥδ
. (22)
The kurtosis of the deblurred image X̂ is given by
K(X̂) =
µX̂(4)
µX̂(2)
2 −3 , (23)
where µX̂(k) is the k
th central moment. Let µX̂(0)
represent the expected value or mean of X̂ .
µX̂(0) =
1
M−Q1
1
N−Q2
M
∑
i=Q1+1
N
∑
j=Q2+1
X̂(i, j) . (24)
Q1 and Q2 are regions of uniform convolution, while M
and N are image rows and columns respectively. The
gradient matrix is derived as follows:
∇Kλ (X̂) =
∂
∂λ
K(X̂) =
∂
∂λ
(
µX̂(4)
µX̂(2)
2
)
. (25)
Expanding the cumulants, we get
∇Kλ (X̂) =
∂
∂λ
[(
1
M−Q1
1
N−Q2
·
M,N
∑
i=Q1+1, j=Q2+1
[X̂(i, j)−µX̂(0)]
4
)/
(
1
M−Q1
1
N−Q2
·
M,N
∑
i=Q1+1, j=Q2+1
[X̂(i, j)−µX̂(0)]
2
)2]
.
(26)
∇Kλ (X̂) =
4
(
1
µX̂(2)
2
1
M−Q1
1
N−Q2
·
M
∑
i=Q1+1
N
∑
j=Q2+1
η [X̂(i, j)−µX̂(0)]
3
)
−
4
(
µX̂(4)
µX̂(2)
3
1
M−Q1
1
N−Q2
·
M
∑
i=Q1+1
N
∑
j=Q2+1
η [X̂(i, j)−µX̂(0)]
)
,
(27)
where
η =
∂
∂λ
(X̂−µX̂(0)) =
∂
∂λ
(X̂)− ∂
∂λ
(µX̂(0)) . (28)
∂ (µX̂(0))/∂λ is equivalent to the mean value of
∂ X̂/∂λ and
∂
∂λ
X̂ =
∂
∂λ
(
G
(
Ĥ2
Ĥ3 + Ĥδ
))
= G
∂
∂λ
(
Ĥ2
Ĥ3 + Ĥδ
)
.
(29)
Through further manipulations, we get
∂
∂λ
X̂ =
G
(
2Ĥ
Ĥ3 + Ĥδ
+ Ĥ2
(
−(3Ĥ2 +δ )
(Ĥ3 + Ĥδ )2
))
∂
∂λ
Ĥ .
(30)
The term ∂ X̂/∂λ needs to be calculated when
a specific type of blur is used. As an example, the
derivation for the Gaussian blur PSF is as follows.
For Gaussian blur, the Optical Transfer Function
(OTF) for the parametric model is given by
Ĥ(i, j) =
1
2πσ2f
exp
(
− i
2 + j2
2σ2f
)
, (31)
∂
∂λ
Ĥ(i, j) =
∂
∂σ
Ĥ(i, j) =
∂
∂σ
(
1
2πσ2
exp
(
− i
2 + j2
2σ2
))
=
−2
2πσ3
e
(
− i
2+ j2
2σ2
)
+
1
2πσ2
∂
∂σ
(
e
(
− i
2+ j2
2σ2
))
= Ĥ(i, j)
(
−2
σ
+
i2 + j2
σ3
)
.
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The derivations of the gradient matrix for the
motion and out-of-focus blur can be found in
Appendix A.
The blur parameters of the PSF model are
optimized in the direction of increasing value of the
absolute kurtosis till the gradient becomes flat or the
difference of kurtosis between successive iterations is
smaller than a specified tolerance.
RESULTS
The proposed scheme has been tested on a
variety of images, synthetic and real blurred, degraded
with various functions. Image quality measures were
employed to evaluate the quality of deblurred images.
These included the IQA, MSSIM, UQI, BRISQUE and
NIQE quality measures. The results for the proposed
scheme are presented below.
DEBLURRING SYNTHETIC BLURRED
IMAGES
Fig. 1 depicts the test images from Desktop Nexus
image database. Fig. 2 shows the restorations from
their Gaussian-blurred counterparts using the proposed
scheme.
Fig. 2. Gaussian blurred images and their respective
estimates using the proposed BID scheme.
The estimated PSF parameters and comparisons of
different quality measures for the deblurring are given
in Table 1.
Fig. 3. Out-of-focus blurred images and their
respective estimates using the proposed BID scheme.
Fig. 3 illustrates the out-of-focus blurred images
and their deblurred counterparts by the proposed
scheme. Table 2 summarizes the results of the
deblurring along with a comparison by different image
quality measures. The deblurring examples for motion
blur are shown in Fig. 4, with results summarized in
Table 3.
DEBLURRING REAL BLURRED IMAGES
The proposed deblurring scheme has also been
employed to restore real images with motion blur, a
typical problem facing photographers. The deblurred
images have been compared with the estimated images
using the scheme by Shan et al. (2008) and that of Cho
and Lee (2009). These two recently proposed schemes
have received a great deal of attention due to their
advantages.
Fig. 4. Motion blurred images and their respective
estimates using the proposed BID scheme.
In the first case, an image was captured by a low
quality camera. The blurred image, LABEL image,
depicted in Fig. 5 (a), shows an approximate vertical
motion blur with the small digits at the bottom
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Image Anal Stereol 2018;37:213-223
of the image becoming unreadable. The image also
contains noise and uneven illumination due to poor
lighting one of the most difficult cases in deblurring.
Fig. 5(b) shows the deblurred image estimated using
the proposed scheme with the estimated blur kernel
shown in Fig. 5(e). Using the proposed scheme, a
uniform motion PSF of length 21 pixels and angle
eight seven degree was estimated. Fig. 5(c) shows
the deblurred image and Fig. 5(f) the estimated PSF
using the scheme of Shan et al. (Shan et al., 2008).
The image in Fig. 5(c) seems to have recovered
well; however, the digits are still unreadable since the
PSF estimated using this scheme does not completely
follow a uniform motion blur.
Fig.5(d) shows the deblurred image with the Cho
et al scheme (Cho and Lee, 2009) with the estimated
PSF shown in Fig. 5(g). The image appears smoother
and the blurring seems to be reasonably diminished.
However, the small text in the figure is not fully
recovered. Even the relatively larger text in the lower
right portion of the image is not clear. This is due
to the error in approximation of the blurring PSF
especially the angle of the motion blur. In Fig. 5(b),
the digits in the deblurred image are much clear and
easily readable. An edge-taper function was used to
reduce the ringing effect caused by the discrete Fourier
Transform.
Fig. 5. (a) Real motion blurred LABEL image.
Deblurred images: (b) using the proposed scheme, (c)
using Shan et al.s scheme, (d) using Cho et al.s scheme.
Estimated PSFs: (e) proposed scheme, (f) Shan et als
scheme, (g) Cho et al.s scheme.
Fig. 6 and 7 show the deblurring of motion blurred
images affected by object movement ”MATLAB
BOOK” image and camera handshake ”BUILDINGS”
image. The motion blur was almost linear at a certain
angle. Again recovery was achieved to a good extent.
Another motion blurred image taken by an
ordinary digital camera and its deblurred versions are
shown in Fig. 8. Fig. 8 (b) and (e) shows the deblurred
image and the estimated PSF using the proposed
scheme. Fig. 8 (c) and (f) show the deblurred image
and estimated PSF using the Shan et al scheme. Fig. 8
(d) and (g) show the deblurred image and estimated
PSF using the Cho et al. BID scheme. Fig. 8(d) is
not recovered well due to the poor estimation of the
blurring PSF. The proposed scheme is clearly better
and the main and sub title text of the book become
more readable.
Fig. 6. (a) Real motion blurred MATLAB BOOK image.
Deblurred images in (b) using the proposed scheme,
(c) using Shan et al. scheme, (d) using Cho et al.
scheme; and their respective estimated PSFs in (e), (f)
and (g).
Fig. 7. (a) Real motion blurred BUILDING image.
Deblurred images: (b) using the proposed scheme, (c)
using Shan et al. scheme, (d) using Cho et al. scheme;
and their respective Estimated PSFs: (e), (f) and (g).
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Fig. 8. (a) Real motion blurred DIP BOOK image.
Deblurred images: (b) using the proposed scheme, (c)
using Shan et al. scheme, (d) using Cho et al. scheme;
and their respective Estimated PSFs: (e), (f) and (g).
Table 4 shows the BRISQUE and NIQE measures
for the deblurred images in Figs. 5-8. It illustrates
the quantitative measures for the deblurred images
estimated by these schemes. The BRISQUE and NIQE
values for the original, blurred and deblurred images
are shown as references and discussion on deblurring
qualities. Ideally, the lower values such non-reference
measures the higher qualities of the deblurred images.
Though the deblurred images in Fig. 5(d), Fig. 6(d)
and Fig. 8(d) by Cho et al.’s scheme did not recover
well as visually, the BRISQUE and NIQE values seem
to depict the deblurred images of high quality. This is
due to the over smoothing of the deblurred images by
Cho et al.’s scheme in order to recover from the ringing
artefacts. In some cases, the original (blurred) images
have lower measures than their deblurred version. The
tests also show that BRISQUE and NIQE IQMs may
not be consistent with blind IQMs and more effort
is still required to evaluate their consistency as a
performance measure.
DISCUSSION
In this research work, a new technique for the
restoration of blurred images under the influence of
parametric blurs is presented. The scheme estimates
the blurring parameters by utilising a gradient based
kurtosis maximization method for the deblurred image.
The proposed method can handle various forms of
parametric blurs including Gaussian, motion and
out-of-focus (Lagendijk and Biemond, 2009). The
proposed method operates on single image and is
simple, efficient and easy to implement. Gradients
for the blur parameters were computed and optimized
in the direction of increasing kurtosis value using a
steepest descent scheme. The gradients were derived
for several common blurs and the proposed method
was tested on artificial and real-life blurs. These two
image datasets validated the research work. The results
demonstrate the effectiveness of our proposed model.
Firstly, the algorithm was tested on the Desktop
Nexus image dataset (Nexus, 2010). This included the
generation and recovery of the parametric blurs from
their model equations. Uniform blurring was assumed.
Afterwards, the images were presented to the steepest
descent scheme which estimated the parameters of
the blurred. In order to do so, the steepest descent
scheme took feedback from the increasing value of
kurtosis. this would guide the proposed model in the
right direction. Kurtosis is a higher order statistic
which in conjunction with the Central Limit Theorem’s
principle, aids in the recovery of blurred images (Yin
and Hussain, 2008).
The robustness of the deblurring schemes become
questionable when real-life blurred images need to be
restored. Once our proposed scheme was validated
for artificial blurring, it was tested on real images
captured under the influence of the parametric motion
blur. Restoring such images requires more effort as
simultaneously approximating the blurring kernel and
restoring the image while deterring deblurring noise
and ringing artefacts is complex.
Among the various restoration filters, the Wiener
filter was opted over the Richardson-Lucy and Total
Variation (TV) filter for its ability to recover the image
in a single pass and with high deblurring quality. A
number of image quality measures were employed to
evaluate the quality of deblurred images. This included
both full-reference and non-reference image quality
measures in quantifying the deblurring performance.
Full-reference IQA measures included MSSIM and
UQI while the blind/non-reference IQA measures
included BRISQUE and NIQE. The proposed scheme
is highly efficient as it is independent of a pristine
reference image and is able to deblur the images
effectively. Hence, the proposed method can be used
in deblurring applications where the reference pristine
image is not available.
The deblurred images have been compared with
the schemes of Shan et al. (2008) and Cho and Lee
(2009) and observed that it performs better than the
benchmark restoration schemes. It was observed that
for both datasets the proposed scheme works better as
evident from the full reference and non-reference IQA
measures.
CONCLUSIONS
A blind deblurring is one of the most challenging
problem in image restoration. A scheme based on
220
Image Anal Stereol 2018;37:213-223
Table 1. PSF parameter estimation for Gaussian blurred images and quantitative comparison of SSIM, UQI,
BRISQUE and NIQE quality measures.
MSSIM UQI BRISQUE NIQE
Image Original Sigma Estimated Sigma Blurred Proposed Scheme Blurred Proposed Scheme Blurred Proposed Scheme Blurred Proposed Scheme
Img-01 1.63 1.85 0.85 0.90 0.99 0.99 56.69 18.28 7.28 8.40
Img-02 4.98 4.76 0.73 0.81 0.95 0.97 77.15 70.69 10.84 7.97
Img-03 5.13 5.26 0.62 0.68 0.94 0.96 79.32 73.12 11.85 9.05
Img-04 3.88 4.05 0.78 0.86 0.97 0.98 72.87 55.30 10.64 7.65
Img-05 0.71 0.68 0.88 0.95 0.99 1.00 25.45 19.06 3.80 3.88
Img-06 0.94 1.21 0.97 0.88 1.00 1.00 52.71 38.15 7.00 7.21
Img-07 2.07 2.18 0.52 0.74 0.94 0.97 61.24 40.69 7.40 7.76
Img-08 2.63 2.50 0.69 0.80 0.95 0.97 67.06 34.43 9.10 6.99
Img-09 3.45 3.32 0.67 0.76 0.97 0.98 67.54 54.20 10.28 6.86
Img-10 2.44 2.24 0.75 0.86 0.97 0.98 64.76 36.55 8.56 6.45
Img-11 1.64 1.82 0.75 0.88 0.98 0.99 54.33 26.65 6.56 6.59
Img-12 0.98 0.99 0.93 0.94 0.99 1.00 47.88 11.99 5.48 3.85
Img-13 1.38 1.53 0.89 0.96 0.99 0.99 45.66 24.88 6.66 5.94
Img-14 3.13 3.26 0.62 0.80 0.93 0.97 72.01 49.99 9.79 6.85
Img-15 4.18 4.13 0.79 0.81 0.97 0.98 80.98 58.73 10.61 8.11
Img-16 5.47 5.39 0.45 0.55 0.93 0.95 82.44 70.65 10.77 9.02
Table 2. PSF parameter estimation for out-of-focus blurred images and quantitative comparison of SSIM, UQI,
BRISQUE and NIQE quality measures.
MSSIM UQI BRISQUE NIQE
Image Original Radius Estimated Radius Blurred Proposed Scheme Blurred Proposed Scheme Blurred Proposed Scheme Blurred Proposed Scheme
Img-01 7 6.84 0.7 0.81 0.96 0.98 69.52 27.22 10.24 4.32
Img-02 8.85 9.2 0.7 0.76 0.94 0.97 76.69 44.17 10.42 5.34
Img-03 10 9.4 0.59 0.7 0.93 0.95 77.68 39.09 11.99 4.97
Img-04 11.5 11.47 0.69 0.84 0.93 0.97 76.05 32.21 12.35 5.55
Img-05 1 0.74 0.95 0.97 1 1 22.08 12.82 3.29 2.7
Img-06 2.5 2.46 0.97 0.93 1 1 60.49 14.96 8.15 5.54
Img-07 4 3.91 0.48 0.83 0.93 0.98 57.52 10.78 6.92 3.77
Img-08 5.5 4.72 0.64 0.59 0.94 0.94 68.54 29.01 9.56 5.19
Img-09 13 12.58 0.63 0.68 0.96 0.97 67.17 39.76 10.84 4.61
Img-10 14.5 14.14 0.63 0.61 0.93 0.95 73.93 45.1 12.12 5.31
Img-11 16 15.27 0.51 0.58 0.93 0.95 76.35 43.96 11.36 5.2
Img-12 17.5 18.74 0.73 0.79 0.96 0.98 77.13 37.51 11.19 4.75
Img-13 19 18.82 0.4 0.7 0.86 0.95 80.4 45.59 11.93 5.96
Img-14 20.5 20.03 0.42 0.65 0.82 0.93 83.63 46.97 11.43 5.83
Img-15 22 21.58 0.68 0.64 0.93 0.93 86.56 56.11 12.76 7.12
Img-16 23.5 22.89 0.38 0.47 0.89 0.91 84.8 57.47 11.57 7.85
Table 3. PSF parameter estimation for motion blurred images and quantitative comparison of SSIM and UQI
quality measures.
Image Original Length Estimated Length Blurred Qi Shan Proposed Scheme Blurred Qi Shan Proposed Scheme Blurred Qi Shan Proposed Scheme Blurred Qi Shan Proposed Scheme
Img-01 7 8 0.92 0.867 0.944 0.99 0.984 0.992 34.759 34.759 24.062 6.364 5.663 5.989
Img-02 8 8 0.875 0.894 0.926 0.985 0.986 0.992 64.961 64.961 36.537 7.059 5.146 3.913
Img-03 9 9 0.757 0.81 0.893 0.975 0.977 0.987 56.472 56.472 42.72 6.477 5.257 4.221
Img-04 10 10 0.859 0.818 0.926 0.982 0.979 0.993 65.95 65.95 38.149 8.986 5.663 4.379
Img-05 11 12 0.74 0.73 0.828 0.983 0.974 0.987 52.707 52.707 44.811 8.393 5.916 4.311
Img-06 12 12 0.924 0.89 0.921 0.991 0.99 0.996 67.186 67.186 21.621 9.369 6.35 4.467
Img-07 13 13 0.536 0.616 0.81 0.928 0.926 0.972 39.975 39.975 24.533 7.172 7.687 5.626
Img-08 14 13 0.775 0.86 0.858 0.965 0.977 0.978 39.233 39.233 35.9 7.278 8.694 6.288
Img-09 15 14 0.725 0.76 0.828 0.972 0.969 0.98 27.367 27.367 26.921 7.151 5.484 4.77
Img-10 16 15 0.726 0.794 0.826 0.958 0.966 0.974 26.157 26.157 24.412 7.265 4.84 4.844
Img-11 17 17 0.572 0.53 0.804 0.942 0.942 0.978 13.188 13.188 53.311 7.936 6.072 3.844
Img-12 18 19 0.778 0.769 0.827 0.975 0.974 0.987 27.514 27.514 26.164 9.074 5.335 3.992
Img-13 19 19 0.513 0.718 0.806 0.919 0.962 0.976 26.608 26.608 46.489 9.244 6.058 4.449
Img-14 20 21 0.551 0.673 0.771 0.894 0.945 0.967 25.011 25.011 45.833 8.099 5.232 4.312
Img-15 21 21 0.755 0.679 0.587 0.969 0.956 0.962 58.33 58.33 55.449 8.397 6.492 5.462
Img-16 22 24 0.504 0.551 0.674 0.934 0.932 0.963 49.092 49.092 45.294 6.07 5.92 5.615
Table 4. BRISQUE and NIQE quality measures of the deblurred images.
Image BRISQUE NIQE
Original Shan Cho Proposed Original Shan Cho Proposed
LABEL 56.08 52.00 48.811 54.07 7.11 8.1 7.984 7.68
DIP-BOOK 29.77 40.21 33.302 45.68 5.78 6.31 5.078 6.51
MATLAB-BOOK 5.48 28.84 24.490 48.55 5.76 4.07 8.277 6.31
BUILDINGS 39.85 41.17 46.976 39.58 7.01 6.41 6.117 6.85
221
KHAN A ET AL: Paramteric blind image deblurring
spectral kurtosis is proposed to estimate uniform blur
for a parametric form of PSF. A steepest descent
optimization scheme is employed to optimize the
blur parameter in the direction of maximum absolute
kurtosis of the deblurred image. The proposed method
operates on single image and is simple, efficient
and easy to implement. The method is applicable to
various types of blurs such as Gaussian, motion and
out-of-focus. Experiments on both artificial and real
blurred images have shown the capability and marked
improvement of the scheme over the existing methods,
in terms of both visual perception and a range of
quality measures.
APPENDICES
A. MOTION BLUR
Without loss of generality, the linear motion
modelled is the result of translation in the horizontal
direction. In order to simplify the search for
blur parameters with the steepest descent scheme,
one can rotate the image until the blur can be
assumed horizontal. This simplifies the case from two
parameters, length (L) and angle (ϕ) of blur to only the
length that needs to be optimized. The OTF for motion
blur according to Biemond et al. (1990) is given by
Ĥ(i, j) =
1
L+1
e−kiπL
sin(π(L+1)i))
sin(πi)
, ∀ j . (32)
Differentiating the OTF with respect to parameter L of
the OTF:
∂
∂λ
Ĥ(i, j) =
∂
∂L
Ĥ(i, j)
=
∂
∂L
(
1
L+1
e−kiπL
sin(π(L+1)i))
sin(πi)
)
=
−1
(L+1)
Ĥ(i, j)− kπiĤ(i, j)
+(πi)cot(π(L+1)i)) Ĥ(i, j)
= Ĥ(i, j)
(
− 1
L+1
− kπi
+πicot(π(L+1)i)
)
. (33)
B. OUT-OF-FOCUS BLUR
The OTF for out-of-focus blur (Biemond et al.,
1990) is given by,
Ĥ(i, j) =
J1(Rp)
Rp
, p2 = i2 + j2 , (34)
where J1 is the first order Bessel function. The Bessel
function can be approximated by an exponential for
sufficiently large data (Olver et al., 2010):
Ĥ(i, j) =
eRp√
2πRp
. (35)
∂
∂R
Ĥ(i, j) =
∂
∂R
(
eRp√
2πRp
)
= eRp
∂
∂R
(
1√
2πRp
)
+
(
1√
2πRp
)
∂
∂R
(
eRp
)
= (2π p)eRp
(−1/2)√
2πRp
+
(
eRp√
2πRp
)
(Rp)
=
eRp√
2πRp
(−π p+Rp)
= Ĥ(i, j)(−π p+Rp) . (36)
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