Speckle Noise Reduction in Optical Coherence Tomography Using Two-dimensional Curvelet-based Dictionary Learning

Abstract

The process of interpretation of high-speed optical coherence tomography (OCT) images is restricted due to the large speckle noise. To address this problem, this paper proposes a new method using two-dimensional (2D) curvelet-based K-SVD algorithm for speckle noise reduction and contrast enhancement of intra-retinal layers of 2D spectral-domain OCT images. For this purpose, we take curvelet transform of the noisy image. In the next step, noisy sub-bands of different scales and rotations are separately thresholded with an adaptive data-driven thresholding method, then, each thresholded sub-band is denoised based on K-SVD dictionary learning with a variable size initial dictionary dependent on the size of curvelet coefficients’ matrix in each sub-band. We also modify each coefficient matrix to enhance intra-retinal layers, with noise suppression at the same time. We demonstrate the ability of the proposed algorithm in speckle noise reduction of 100 publically available OCT B-scans with and without non-neovascular age-related macular degeneration (AMD), and improvement of contrast-to-noise ratio from 1.27 to 5.12 and mean-to-standard deviation ratio from 3.20 to 14.41 are obtained.

Full text

86 © 2017 Journal of Medical Signals & Sensors | Published by Wolters Kluwer ‑ Medknow
Address for correspondence:
Dr. Hossein Rabbani,
Department of Advanced
Medical Technology, Isfahan
University of Medical Sciences,
Isfahan, Iran.
E-mail: h_rabbani@med.mui.
ac.ir
Website: www.jmss.mui.ac.ir
Abstract
The process of interpretation of high-speed optical coherence tomography (OCT) images is restricted
due to the large speckle noise. To address this problem, this paper proposes a new method using
two-dimensional (2D) curvelet-based K-SVD algorithm for speckle noise reduction and contrast
enhancement of intra-retinal layers of 2D spectral-domain OCT images. For this purpose, we take
curvelet transform of the noisy image. In the next step, noisy sub-bands of different scales and
rotations are separately thresholded with an adaptive data-driven thresholding method, then, each
thresholded sub-band is denoised based on K-SVD dictionary learning with a variable size initial
dictionary dependent on the size of curvelet coefcients’ matrix in each sub-band. We also modify
each coefcient matrix to enhance intra-retinal layers, with noise suppression at the same time.
We demonstrate the ability of the proposed algorithm in speckle noise reduction of 100 publically
available OCT B-scans with and without non-neovascular age-related macular degeneration (AMD),
and improvement of contrast-to-noise ratio from 1.27 to 5.12 and mean-to-standard deviation ratio
from 3.20 to 14.41 are obtained.
Keywords: Curvelet transform, dictionary learning, optical coherence tomography, speckle noise
Speckle Noise Reduction in Optical Coherence Tomography Using Two-
dimensional Curvelet-based Dictionary Learning
Original Article
Mahdad Esmaeili,
Alireza Mehri
Dehnavi,
Hossein Rabbani,
Fedra Hajizadeh1
Department of Advanced
Medical Technology, Isfahan
University of Medical
Sciences, Isfahan, 1Noor
Ophthalmology Research Center,
Noor Eye Hospital, Tehran, Iran
How to cite this article: Esmaeili M, Dehnavi AM,
Rabbani H, Hajizadeh F. Speckle noise reduction in
optical coherence tomography using two-dimensional
curvelet-based dictionary learning. J Med Sign Sence
2017;7:86-91.
This is an open access arcle distributed under the terms of the
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License, which allows others to remix, tweak, and build upon the
work non-commercially, as long as the author is credited and the
new creaons are licensed under the idencal terms.
For reprints contact: reprints@medknow.com
Introduction
Spectral domain optical coherence
tomography (SD-OCT) is a high-resolution,
noninvasive imaging technique in the
identication and assessment of internal
structures of retinal abnormalities and
to image various aspects of biological
tissues with high resolving power (5 µm
resolution in depth).[1,2] The main problem
regarding these images is in their inherent
corruption by speckle noise due to its
coherent detection nature. Traditional
digital ltering methods including median
and Lee ltering,[3] adaptive median and
Wiener ltering,[4,5] and iterative maximum
a posteriori-based algorithm[6] were
employed for reducing speckle noise. These
methods provide inadequate noise reduction
under the high speckle noise contamination,
as well as result in the meaningful loss
of faint features. In recent years, some
other approaches have been explored for
speckle noise reduction such as anisotropic
diffusion-based methods,[7-10] wavelet-
based methods,[11,12] curvelet shrinkage
technique,[13] dictionary learning-based
denoising,[14,15] and robust principal
component analysis-based method.[16]
However, the necessity of the development
of more advanced methods to provide
minimum detail loss under suppression
high speckle noise makes the speckle noise
reduction as an important part of the OCT
image processing.
Here, a novel speckle noise reduction
algorithm is developed, which is
optimized OCT image despeckling while
preserving strong edge sharpness. For this
purpose, we introduce K-SVD dictionary
learning in curvelet transform (CUT)
domain for speckle noise reduction of
two-dimensional (2D) OCT images. As
the low scale of curvelet coefficients
are more affected by the noise and to
take advantage of this sparse multiscale
directional transform, we introduce a
new scheme in dictionary learning and
take CUT of noisy image, then a nearly
optimal threshold for thresholding of
curvelet coefficients for each scale and
rotation is found based on the standard
deviation of each coefficient matrix.
Thresholded coefficients are employed,
and a curvelet-based K-SVD with
varying size dictionary dependent on the
scale and rotation of coefficient matrix is
introduced. This method does not need

Esmaeili, et al.: Speckle noise reduction by the use of curvelet-based dictionary learning
Journal of Medical Signals & Sensors | Volume 7 | Issue 2 | April‑June 2017 87
any high signal-to-noise ratio (SNR) scans (a fraction
of repeated B scans from a unique position are captured
slowly, then these images are registered and averaged
to create a less noisy image with a sufficiently high
SNR) for dictionary learning, which is used in other
works.[14,15]
The paper is organized as follows. Section 2 provides an
introduction to 2D digital CUT (DCUT). In Section 3, we
describe the principles of conventional dictionary learning.
Our proposed method is described in Section 4, and the
results and performance evaluation are presented in Section
5. Finally, this paper is concluded in Section 6.
Two-dimensional Digital Curvelet Transform
The CUT is a high-dimensional time–frequency analysis
of images that gives a sparse representation of objects,
and it has been developed to overcome the inherent
limitations of conventional multiscale representations
such as wavelets (e.g., poor directional selectivity). The
directional selectivity of curvelets and localized spatial
property of each curvelet can be utilized to preserve the
image features along certain directions in each sub-band.
The good directional selectivity, tightness, and sparse
representation properties of this multiscale transform give
new opportunities to analyze and study large datasets in
medical image processing.[17]
This transform can be implemented by employing two
simpler, faster, and less redundant methods, i.e., the
unequally-spaced fast Fourier transform (USFFT) and
the wrapping transform.[17,18] The main difference of these
implementations is related to their choice of spatial grid
to construct the curvelet atoms in each subband. Both
algorithms have the same output, but the wrapping-based
transform has faster computational time and is easier to
implement than USFFT method.[18] The architecture of
CUT via wrapping is roughly presented in the following
form:
1. Take the 2D FFT of the image f and obtain Fourier
samples f
ˆ (n1, n2),…−n/2 ≤ n1, n2,…< n/2 (f is the
original image with the size of n × n)
2. For each scale/angle pair (j, l), form the product
d (n1, n2) = U
~
j,l (n1, n2) f
ˆ (n1, n2), here, U
~
j,l (n1, n2) is the
discrete localizing window[18]
3. Wrap this product around the origin and obtain
f
~
j,l(n1, n2) = W (U
~
j,l f
ˆ) (n1, n2). If the corresponding
periodization of the windowed data, i.e., d (n1, n2) is
dened as Wd nn dn mL nmL
mm
(,
),
12 11 22
2
1
=++
()
∈∈
∑∑
1,
j2
,j
ΖΖ
,
then at each scale j, Wd (n1, n2) is restricted to indices
(n1, n2) inside a rectangle with sides of length L1,j × L2,j
near the origin (L1,j ~ 2j and L2,j ~ 2j/2) to construct the
wrapped windowed data
4. Take the inverse 2D FFT of each f
~
j,l for collecting the
discrete coefcients, i.e., cD (j, l, k).
K-SVD Dictionary Learning For Image Denoising
Image denoising problem can be viewed as an inverse
problem. One of the most recent methods to solve an
inverse problem is a sparse decomposition over over-
complete dictionaries.[19,20] For a given set of signals
delineated by Y, suitable dictionary D can be found such
that yi ≈ Dxi, where xi is a vector which involves the
coefcients for the linear combination and yi ∈ Y. The
problem of sparse representation can thus be dened as an
optimization problem of nding D and xi, which satises:
min
,Dx
i
yD
xx
T
ii i
subjected to
−<
20
(1)
where T is a predened threshold which restraints the
sparseness of the representation and || · ||0 indicates the l0
norm which counts the number of nonzero elements of the
vector. This problem is thus involved in a selection of the
dictionary and a sparse linear combination of the atoms in
the dictionary to illustrate each desired signal. For image
denoising, the noisy image is broken up into patches and
the vectorized version of each patch is considered as a
signal. For a given image, which can be considered as a set
of signals Y, the denoising problem can be done by nding
a set of patches Z which are related by:
Y = Z + η (2)
where η is noise, which corrupts the patches.
To nd the denoised patches Z
ˆ, the following optimization
problem should be solved:[19,20]
argmin
()
,,xZD
ij
ijij
ij
YZ Dx RZ x
λµ
−+ −+
∑∑
2
22
0
ij ij Fij (3)
where λ and μ are Lagrange multipliers and Rij is dened
as the matrix which selects the ijth patch from Z, i.e.,
Zij = RijZ.
The rst term in (3) makes sure that the measured image Y
is similar to its denoised version Z and the second and third
parts are sparsity-inducing regulation terms.
For solving the above equation:
1. Initialization is done by setting
Z=Y, D = initial dictionary
2. Repeat K times
• For each patch, RijZ computes the representation
vector xij by using orthogonal matching pursuit
(OMP) algorithm.[21,22] The OMP is easy in
implementation and provides a satisfactory stable
result. The algorithm attempts to nd the best
basis vectors (atoms) iteratively such that the
representation error is reduced in each iteration
∀−
<
ij ij ij ij
ij
such thatmi
n(
)
xxRZD
xC
02
 (4)
where C is the noise gain and σ is the standard
deviation of noise

Esmaeili, et al.: Speckle noise reduction by the use of curvelet-based dictionary learning
88 Journal of Medical Signals & Sensors | Volume 7 | Issue 2 | April‑June 2017
• Once this sparse coding stage is done, the algorithm
proceeds to update the atoms l = 1, 2,…, k of the
dictionary one by one to reduce the error term. For
this purpose the set of patches that use this atom
Dl = ([i, j]|xij (l) ≠ 0) are calculated and l-th atom
from dictionary is deselected, then the coding error
matrix (El) of these signals is calculated whose
columns are:
eRXdxm
ml
ij
l
ij ij mij
=−
≠
∑()
(5)
• Minimize this error matrix with rank-1
approximately from SVD that El = UΔVT. Replace
coefcient values of atom Dl with entries of V1Δ1
and updated dictionary column to be D=
l
U
U
1
12
.
3. Set
T1 T
ij ij ij ij
ˆ
( )( )
ij ij
Z I R R Y R Dx

−
=++
∑∑
(6)
Proposed Denoising Method
Our curvelet-based approach consists of rst taking the
2D forward DCUT of noisy image to produce the curvelet
coefcients, then for each sub-band in the transform
domain, the coefcients’ matrix is independently denoised
based on K-SVD dictionary learning with the initial
dictionary of discrete cosine transform (DCT), in which its
size is specic for each scale.
In the proposed method for efcient representation of
different structures in image, we select initial dictionary
to be variable in size (depends on the size of curvelet
coefcients’ matrix in each sub-band) instead of traditional
xed form. By increasing the scale of curvelet coefcients’
matrix (or reducing in resolution), the block size (indicates
the size of the blocks to operate on) is also increased,
while in high resolutions, the block size is reduced which
results in better representation of particular structure in
image.
The proposed method for image denoising is as follows.
Forward Digital Curvelet Transform
Take the 2D CUT of the data to produce the curvelet
coefcients C (j, l) (j is the scale and l is the orientation).
According to our image size (512 × 1000), each image is
decomposed into six scales (it is recommended to take the
number of scales to be equal or less than the default value,
⌈(log2
[min(M, N)] – 3)⌉, here M, N is the image size and
x denotes the smallest integer being greater than or equal
to x) then each scale is further partitioned into a number
of orientations. The number of orientations is l=1, n, 2n,
2n, 4n, 4n… from ner to coarser scales, where n is the
number of orientation at the second scale.
Initial denoising
For each scale, apply hard thresholding based on the
standard deviation of each scale. The hard threshold Tj,l to
each curvelet coefcient is selected such that:
CjlCj
lT
Cjl
(,)([,])
(,)
=≤



0ifabs
else
j,l (7)
The threshold Tj,l is selected based on the standard deviation
of selected coefcient matrix (C) in that scale and rotation
(Tj,l = 0.5 standard deviation [C]).
Initial dictionary selection and K-SVD dictionary
learning denoising
For each 2D-coefcient matrix in each scale and rotation,
the varying size initial dictionary for each scale is chosen
by employing DCT on each sub-band.
We let the block size in dictionary learning to be dependent
on the size of each coefcient matrix, so the dictionary size
also varies with block size. After nding the appropriate 2D
initial dictionary, D, for each sub-band, the noisy curvelet
coefcient matrices of noisy image in the same scale
and rotation are despeckled based on K-SVD dictionary
learning as described in Section 3.
According to the size of curvelet coefcient matrix C, the
dictionary size and block size are set empirically to be:
Blocksize=



.min (,)5 mn (8)
Dictionary size = Block size.^3 (9)
where m, n are respectively the number of rows and
columns of coefcient matrix C and [x] indicates the largest
integer smaller than or equal to x.
Contrast enhancement
Since the CUT is successful in dealing with edge
discontinuities, it is a good candidate for edge
enhancement. Hence, to enhance the contrast of intra-
retinal layer boundaries, denoised curvelet coefcients
can be modied to enhance edges in a B-scan image,[23,24]
before taking 2D inverse discrete CUT (2D-IDCUT). For
OCT images, a function kc (Cj,l) denes empirically that
is similar to function dened by Starck for gray and color
image enhancement,[25] which modies the values of the
curvelet coefcients as follows:
Kx
xxN
xs NxN
sx
Nx
c
if abs
if abs
if abs
()
.()
.(
)(
)
() ()
=
<
<<
<


14
83
3
1
2








(10)
In this equation, N = 0.1M, where M is the maximum
curvelet coefcient of the relative band, and s1 and s2 are
dened as follows:

Esmaeili, et al.: Speckle noise reduction by the use of curvelet-based dictionary learning
Journal of Medical Signals & Sensors | Volume 7 | Issue 2 | April‑June 2017 89
sxN
N
M
N
Nx
N
1
22
=−



+−()
()
.
(11)
sM
x
2
2
=




.
(12)
Converting to image domain
Then, we reconstruct the enhanced image from the
denoised and modied curvelet coefcients by applying
IDCUT. The outline of the whole denoising process is
shown in Figure 1.
Results
We tested our algorithm on 100 selected 2D OCT B-scans
of size 512 × 1000 from publically available datasets[15,26]
that were acquired using SD-OCT, Bioptigen imaging
systems, with and without non-neovascular AMD. For
the better representation of image details in low scale,
high-frequency components, the block size is selected
to be dependent on the scale of coefcient matrix. On
the other hand, for low scales, the coefcient matrix is
small in size and the size of this matrix will be increased
in high scale, low-frequency components of the image.
Figure 2 demonstrates the samples of the variable size
initial dictionaries in curvelet domain used for K-SVD-
based denoising of each curvelet sub-band.
Figure 3 shows the reconstructed OCT images from
curvelet-based K-SVD enhancement method.
For K-SVD denoising in each scale and rotation of curvelet
coefcients, 1000 patches with equal distance between
the samples in each dimension are selected. To obtain a
compromise between having enough iterations to obtain a
good result and having a correct processing time, we set
K empirically to be 15 for our dataset. According to Eqs.
3 and 4, we also set C = 1.15 and λ = 30/σ, where σ is
selected to be 25 for our dataset.
To compare the performance of different denoising
algorithms quantitatively, we compute the averaged mean
SNR[27] and contrast-to-noise ratio[28] obtained from ten
regions of interest (ROIs) from B-scan OCT images
[similar to the foreground ellipse boxes in Figure 4].
The algorithm that has been implemented in MATLAB
requires around 2 min of computation time for denoising
each 512 × 1000 B-scan on an Intel (R) Core i7 CPU with
4 GB of RAM. The drawback of the proposed method[15] is
its time complexity so that it takes more than 31 min for
denoising each B-scan on the same Intel (R) Core i7 system.
Table 1 compares the quantitative performance measure
values of our method with those from the available
well-known denoising approaches[15] such as: Tikonov,[6]
K-SVD[14] and multiscale sparsity-based tomographic
denoising approach[15] (the reported results[15] are on
17 images of this dataset). To show the ability of proposed
method in edge preserving, the average of the edge
preservation (EP)[29] measure over the selected ROIs is
obtained, that is, 0.83 ± 0.01. This EP measure ranges
between 0 and 1, having smaller values when the edges
inside the ROI are more blurred.
Figure 5 also shows the visual performance of our proposed
method in comparison with some traditional state-of-theart
denoising methodssuch as: Bernardes method[10] Tikonov,[6]
and multiscale sparsity-based tomographic denoising
algorithm.[15]
To show the ability of our proposed method
(DCUT + K-SVD), we have demonstrated the reconstructed
image with thresholded curvelet coefcients and the ability
of K-SVD in image domain (block size = 8, dictionary
size = 256) for noise suppression in Figure 6. Table 2 also
compares the quantitative performance measure values of
our method with thresholded curvelet coefcients (without
dictionary learning) and the K-SVD-based denoising
(without CUT) method.
Figure 1: The outline of the proposed method
Figure 2: Samples of the trained two-dimensional initial dictionaries in
K-SVD-based denoising of each curvelet coefcient matrix. The block size
in (a) is 3 × 3 and in (b) is 4 × 4
b
a

Esmaeili, et al.: Speckle noise reduction by the use of curvelet-based dictionary learning
90 Journal of Medical Signals & Sensors | Volume 7 | Issue 2 | April‑June 2017
Conclusion
Speckle noise in OCT images causes difculty in the
actual recognition of morphological characteristics which
can be viewed and quantied using OCT tomograms, such
as the thickness of intra-retinal layers and the shape of
structural features (e.g., drusens, macular holes, macular
edema, and nerve ber atrophy and cysts, which can be
used as markers in clinical investigation and diagnostics of
retinal diseases). Hence, to suppress noise while preserving
and enhancing the edges and to consider the geometric
properties of structures and exploit the regularity of edges,
we introduced a new curvelet-based K-SVD despeckling
and contrast enhancement method for OCT datasets. We
discussed the application of dictionary learning along
with CUT for denoising of SD-OCT of normal and
AMD retinal images. Our proposed method also does
not need any high-SNR scans or any repeated scans (or
averaged versions of scans) for dictionary learning (since
in some cases, there is no access to the averaged frames).
Moreover, since the proposed method decomposes the
image into lower dimension sub-components, we achieved
a signicant reduction of computational time by reducing
the size of initial dictionary to be dependent with the
size of each scale. The EP value also shows that the
proposed method can preserve edges very well while
removing speckle noise. As OCT is a medical imaging
technique that captures three-dimensional (3D) images
from within optical scattering media, it seems that the
direct analyzing of 3D images with 3D sparse transforms
and also considering the 3D geometrical nature of the
data outperform analyzing 2D slice-by-slice, which is our
ongoing research to extend this work to 3D domain.
Financial support and sponsorship
Nil.
Conicts of interest
There are no conicts of interest.
Figure 4: Selected background and foreground regions of interest for
evaluation. Bigger ellipse outside the retinal region is used as background
region of interest and other circles represent foreground regions of interest
Figure 6: Visual comparison of our proposed method with thresholded
digital curvelet transform coefficients and K-SVD (a) initial images,
(b) reconstructed image with thresholded digital curvelet transform
coefficients, (c) obtained images by using K-SVD on image domain
(d) proposed method
d
c
b
a
Figure 3: The implementation of proposed method (a and c) initial images
and (b and d) obtained images by proposed method
d
c
b
a
Figure 5: Visual performance for spectral domain optical coherence
tomography retinal image using Bernardes, Tikhonov, multiscale sparsity-
based tomographic denoising, and the proposed method. (a) Original noisy
image (b) denoising results using the Bernardes method. (c) Denoising
results using the Tikhonov method. (d) Denoising results using the
multiscale sparsity-based tomographic denoising. (e) Proposed method
d
c
b
a
e
Table 1: Mean and standard deviation of the mean signal-to-noise ratio and contrast-to-noise ratio results for 17
spectral domain optical coherence tomography retinal images using the Tikonov, K-SVD, multiscale sparsity-based
tomographic denoising, and proposed methods
Original Tikhonov[6] K-SVD[14] MSBTD[15] Proposed method
Mean (CNR) 1.27 3.26 4.11 4.76 5.12
STD (CNR) 0.43 0.22 1.23 1.54 1.81
Mean (MSR) 3.20 7.64 11.22 14.76 14.41
STD (MSR) 0.46 0.63 2.77 4.75 4.12
MSR – Mean signal-to-noise ratio; CNR – Contrast-to-noise ratio; MSBTD – Multiscale sparsity-based tomographic denoising

Esmaeili, et al.: Speckle noise reduction by the use of curvelet-based dictionary learning
Journal of Medical Signals & Sensors | Volume 7 | Issue 2 | April‑June 2017 91
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Table 2: Mean signal-to-noise ratio and contrast-to-noise
ratio results of our proposed method in comparison with
reconstructed image from thresholded digital curvelet
transform coefcients and K‑SVD‑based denoising
method in an image of Figure 6
Original Thresholded
DCUT
K-SVD Proposed
method
CNR 1.17 3.53 4.23 5.03
MSR 3.35 9.21 11.03 14.12
DCUT – Digital curvelet transform; MSR – Mean signal to noise
ratio; CNR – Contrast- to-noise ratio