a two-dimensional wavelet frame spanned by two one-
dimensional wavelets can just describe the location of sin-
gular points in the image. In the support region, the wavelet
basis function only has horizontal, vertical and diagonal
direction, and its shape is isotropic square. In fact, the wavelet
frame is optimal for approximating data with point-wise
singularities only and cannot equally well handle distributed
singularities such as singularities along curves. Therefore, the
wavelet frame, lacking of direction and anisotropy, is hard to
sparsely represent high dimensional singular characteristics
like edges and textures [11] .
In order to overcome the classical wavelet frame defects,
scholars propose a variety of multi-scale geometric analysis
methods, in light of the characteristics of visual cortex
receiving outside scene information. Notable methods include
the curvelet [12] and the contourlet [13]. The curvelet is the
first and so far the only construction providing an essentially
optimal approximation property for 2D piecewise smooth
functions with discontinuities along C
2
curves. However, the
curvelet is not generated from the action of a finite family of
operators on a single function, as is the case with wavelets.
This means its construction is not associated with a multi-
resolution analysis. This and other issues make the discrete
implementation of the curvelet very challenging, as is evident
by the fact that two different implementations of it have been
suggested by the originators. In an attempt to provide a better
discrete implementation of the curvelet, the contourlet
representation has been recently introduced. This is a discrete
time-domain construction, which is designed to achieve
essentially the same frequency tiling as the curvelet repre-
sentation. But the directional sub-bands of contourlet exists
spectrum aliasing. This leads to high similarity between the
high frequency sub-band coefficients. However, both of them
do not exhibit the same advantages the wavelet has, namely a
unified treatment of the continuum and digital situation, a
simple structure such as certain operators applied to very few
generating functions, and a theory for compactly supported
systems to guarantee high spatial localization.
The shearlet frame was described in 2005 by Labate,
Lim, Kutnyiok and Weiss with the goal of constructing sys-
tems of basis-functions nicely suited for representing aniso-
tropic features (e.g. curvilinear singularities ) [14, 19, 20].
Many scholars put forward image enhancement algorithms
based on the shearlet frame for different types of images, such
as medical, cultural relic, remote sensing, infrared and ultra-
sound images. The common idea is decomposing the original
image into low frequency coefficients and high frequency
sub-band coefficients of various scales and directions in the
shearlet domain, and then enlarging or reducing these two
components according to the aim of enhancement. These
methods enlarge high frequency coefficients include hard
thresholding [15] and soft thresholding [16]. Meanwhile,
there is fuzzy contrast enhancement to deal with low fre-
quency coefficients [17].
The above-described enhancement methods using the
shearlet frame notably enhance the weak edges and textures
while suppress noise amplification. However, these methods
are not very effective in keeping the geometric structures of
the image, and there may appear over-enhancement or details
loss by some unexpected and uncertain enhancement. Inves-
tigating this, when those algorithms process coefficients in the
shearlet domain, they just change the values of coefficients
without considering the local structural information of coef-
ficients. The advantage of the shearlet frame, in particular, is
to provide a unique ability to control the geometric infor-
mation associated with multidimensional data. Thus, the
shearlet appears to be particularly promising as a tool,
enhancing the component of the 2D data associated with the
weak edges. We linearly amplify the high frequency coeffi-
cients based on their structure information and correct the low
frequency coefficients to solve any non-uniform illumination
problem the infrared image may have.
This paper is organized as follows. Section 2 provides a
brief introduction to the general theory and definition of the
shearlet and analyzes the advantages of the shearlet on opti-
mal representation anisotropic features in detail. Section 3
describes how the proposed algorithm enhances coefficients
of high frequency based on their structure information and
improves the contrast of low frequency coefficients. Results
are presented in section 4, which explains the performance of
the presented method. Conclusions are drawn in the final
section 5.
2. Shearlet frame
When designing representation systems of functions, it is
sometimes advantageous or unavoidable to go beyond the
setting of orthonormal bases and consider redundant systems.
Frame theory is nowadays used when redundant, yet stable
expansions are required. The notion of a frame, was originally
introduced by Duffin and Schaeffer, and guarantees stability
while allowing non-unique decompositions [18]. A sequence
{Φ
k
}
käZ
in H is called a frame for H, if there exist constants
0<AB<∞ for all f ä H such that
å
áFñ
Î
Af f Bf,.1
kZ
k
222
‖‖ ∣ ∣ ‖‖ ()
The frame constants A and B are the lower and upper
frame bound, respectively. If A and B can be chosen with
A=B, then the frame is called A-tight, and if A=B=1is
possible, then {Φ
k
}
käZ
is a Parseval frame. There also exist
the following signal nonlinear approximation f
M
for f:
å
=áFñF
Î
ff,. 2
M
kZ
kk
()
The choice of basis function for the frame directly affects
the performance of frame. Taking advantage of the classical
theory of affine systems, Guo and Labate constructed a
shearlet transform, a stable, efficient and optimal multi-
dimensional sparse representation [14, 19, 20]. In dimension
n=2, the affine systems with composite dilations are the
2
J. Opt. 18 (2016) 085706 ZFanet al