LETTERIEICE Electronics Express, Vol.11, No.17, 1–11

A novel optimization design
approach for Contourlet
directional ﬁlter banks
Songjun Zhang1, Guoan Yang2a), Zhengxing Cheng1,
Huub van de Wetering3, Chihiro Ikuta4, and Yoshifumi Nishio4
1

School of Mathematics and Statistics, Xi’an Jiaotong University,

Xi’an 710049, China
2

School of Electronics and Informations, Xi’an Jiaotong University,

Xi’an 710049, China
3

Department of Mathematics and Computer Science,

Eindhoven University of Technology, The Netherlands
4

Department of Electrical and Electronic Engineering, Tokushima University,

770–8506 Japana) gayang@mail.xjtu.edu.cn© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014

Abstract: A Contourlet transform, an expansion of a wavelet transform,
is a double ﬁlter bank structure composed of Laplacian Pyramid and directional ﬁlter banks. Several wavelet ﬁlters of preferable performance havebeen developed for wavelet transforms, e.g. CDF (Cohen, Daubechies andFeauveau) 9/7 ﬁlter. However, there is still only a limited number of wavelet
ﬁlters applicable for Contourlet transforms. Therefore, it has become anurgent issue to ﬁnd effective contourlet ﬁlters and design methods in the ﬁeldof multiscale geometric analysis. In order to design a new directional ﬁlter
bank for Contourlet transforms, this paper uses parametric modeling toobtain a novel PKVA (See-May Phoong, Chai W. Kim, P. P. Vaidyanathan,and Rashid Ansari) ﬁlter, by ﬁrst implementing Chebyshev best uniform
approximation, and then reaching the optimal solution by means of ParksMcClellan algorithm. Using Brodatz standard texture image database for test
images, and using image denoising treated with hidden Markov tree (HMT)models in the Contourlet domain, the optimal PKVA ﬁlter was obtained on
the basis of the peak signal to noise ratio (PSNR) maximum criterion withhuman visual properties considered. Experiment results show that the image
denoising performance of our ﬁlter is better than that of Po and Do’s. The
PSNR obtained from the experiment is 1.011449 higher than that of Po andDo’s in average. Therefore, Contourlet transforms using the proposed PKVA
ﬁlter as DFB can ensure that the local error in images is of a uniform
minimum value, and that good overall visual effect can be achieved.
Keywords: Contourlet, PKVA ﬁlter, directional ﬁlter banks, Chebyshev
best uniform approximation, image denoising
Classiﬁcation: Electron devices, circuits, and systems

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IEICE Electronics Express, Vol.11, No.17, 1–11

References[1] M. N. Do and M. Vetterli: in Beyond Wavelets, ed. G. V. Welland (Academic Press,
New York, NY, USA, 2003).
[2] M. N. Do and M. Vetterli: IEEE Trans. Image Process. 14 (2005) 2091. DOI:10.
1109/TIP.2005.859376[3] D. D. Y. Po and M. N. Do: IEEE Trans. Image Process. 15 (2006) 1610. DOI:10.
1109/TIP.2006.873450[4] G. Yang, H. Van de Wetering and S. Zhang: J. Signal Process. Syst. Signal Image
Video Technol. 68 (2012) 247. DOI:10.1007/s11265-011-0609-7[5] S. M. Phoong, C. W. Kim, P. P. Vaidyanathan and R. Ansari: IEEE Trans. Signal
Process. 43 (1995) 649. DOI:10.1109/78.370620[6] M. N. Do: Ph.D. dissertation, School of Computer and Communication Sciences at
EPFL-Swiss Federal Institute of Technology, Lausanne, Switzerland (December2001).
[7] G. Yang, X. Fang, M. Jing, S. Zhang and M. Hou: EURASIP J. Adv. Signal
Process. (2010). DOI:10.1155/2010/398385[8] G. Yang, H. Van De Wetering, M. Hou, C. Ikuta and Y. Liu: IEICE Trans. Inf. Syst.
E93-D (2010) 2009. DOI:10.1587/transinf.E93.D.2009[9] T. W. Parks and J. H. McClellan: IEEE Trans. Circuit Theory 19 (1972) 189.
DOI:10.1109/TCT.1972.1083419

1© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014

Introduction

Contourlet transforms, proposed by M. N. Do and M. Vetterli in 2002, form a newtool for multiscale geometric analysis, which can represent natural images moresparsely, and has less redundancy and better approximation performance comparedwith Ridgelets, Curvelets, and other multiscale geometric analysis tools [1]. Since
the support set of a Contourlet transform is a rectangle, the end result is toapproximate contours with lines [2]. Contourlet transforms have anisotropic scaling
relations similar to Curvelets, which can effectively represent the contours and edge
information of images. Therefore, a Contourlet transform is a “true” two-dimensional representation of images. However, the regularity of the basis function ofContourlet transforms is still not high enough, and time-frequency-localized image
representation is also not ideal. Furthermore, frequency aliasing and defects in boththe mathematical model and the theory of Contourlet transform exist. At present,
research of Contourlet transforms is focusing on, among others, anti-aliasing,Contourlet transforms based on wavelets, and non-subsampled Contourlet transforms. Applications of Contourlet transforms mainly include image denoising,image coding, image recognition, and image retrieval. So far, some progress has
been made in image denoising and image retrieval in Contourlet transforms basedon hidden Markov models [3].
A contourlet transform, also named pyramidal directional ﬁlter bank, is adouble ﬁlter bank structure composed of a Laplacian Pyramid (LP) and Directional
Filter Banks (DFB). Currently, there are few ﬁlters for Contourlet transforms, andthey are basically borrowed from wavelet ﬁlters which have better performance(e.g., 9/7 ﬁlter, Haar ﬁlter, PKVA ﬁlter, etc.). Although primary research results
have been achieved concerning Contourlet transforms using 9/7 ﬁlter and PKVA
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IEICE Electronics Express, Vol.11, No.17, 1–11

ﬁlters, signiﬁcant differences exist between the wavelet bases and the Contourlet
bases, and large defects appear when applying wavelet bases directly [4, 5].
Therefore, improving the design of ﬁlters for Contourlet transforms remains a core
issue in multiscale geometric analysis.
Consequently, although Contourlet transforms form an effective approach forrepresentation of singular lines, especially in contour and feature extraction,
research on the design and optimization of Contourlet ﬁlters is still needed.
Therefore, DFB design for Contourlet transforms is a challenging area of multiscale
geometric analysis, and any progress in this area will contribute to the further
deepening and improvement of multiscale geometric analysis, especially Contourlet
transform theory [7, 8]. Contourlet transforms can be applied in image processing,computer vision, and pattern recognition, for improving the efﬁciency of edge
detection, the correct recognition rate in image retrieval, and so on. The study
presented is an exploration of a design and optimization approach for DFB aimed toﬁnd optimal PKVA ﬁlters for Brodatz standard texture image database for image
denoising in Contourlet HMT models.
This paper is organized as follows: Section 2 proposes a PKVA ﬁlter design
approach based on the theory of Chebyshev best uniform approximation. In
section 3, we demonstrate the image denoising application of the proposed PKVA
using HMT models. An optimal DFB is recommended, and experiment results arepresented and discussed. Section 4 concludes the paper with a summary of the mostimportant ﬁndings and an evaluation of the study in comparison with otherresearch.
2

PKVA design for DFB

Proposition 1 The DFB provides a biorthogonal or orthogonal expansion if andonly if its kernel QFB’s are biorthogonal or orthogonal, respectively.
Therefore, the design issue for the DFB essentially amounts to the design of
QFB’s with the desired properties. For synthesis of biorthogonal quincunx systems,
there are two main approaches. One uses the McClellan transformation to map 1-D
biorthogonal two channel ﬁlter banks to 2-D biorthogonal solutions [9]. The otheris based on polyphase mapping and leads to efﬁcient ﬁlter banks that use the ladderstructure where all the operations are separable. The ﬁlter obtained by the second
approach is a PKVA ﬁlter.
2.1 PKVA ﬁlter
Let fh; g; eh;egg be a biorthogonal FIR wavelet ﬁlter bank with an analysis highpass ﬁlter g, an analysis lowpass ﬁlter h, and with their polyphase representations
given byhðzÞ ¼ he ðz2 Þ þ z1 ho ðz2 Þ;© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014gðzÞ ¼ ge ðz2 Þ þ z1 go ðz2 Þ;PPP
where he ðzÞ ¼ k h2k zk , ho ðzÞ ¼ k h2kþ1 zk , ge ðzÞ ¼ k g2k zk , and go ðzÞ ¼
Pkk g2kþ1 z . Thus, the polyphase matrices of the ﬁlter bank are3

IEICE Electronics Express, Vol.11, No.17, 1–11"
PðzÞ ¼
and

"
e ¼
PðzÞ

Let

ho ðzÞ

ge ðzÞ

go ðzÞ

eho ðzÞ
he ðzÞ e##:ego ðzÞ
ge ðzÞ e"PðzÞ ¼he ðzÞ

zN1

ðzÞ

0

zN2

#
;

then the analysis ﬁlter pair is:hðzÞ ¼ z2N1 þ z1 ðz2 Þ;gðzÞ ¼ zð2N2 þ1Þ :So, hðzÞ can be designed to be a good lowpass ﬁlter, but gðzÞ is all-pass, which is
not useful for subband coding applications. We can modify gðzÞ without affectinghðzÞ by taking the polyphase matrix to be:2
3
1=2
0 " zN1 ðzÞ #5PðzÞ ¼ 4 1
 ðzÞ 1
0
zN2
2
3
2
1 N1
1
z
ðzÞ
7
6
2
2
7:
¼6
5
4 1
1
N1
N2
 z ðzÞ  ðzÞðzÞ þ z
2
2
In fact, for ðzÞ ¼ ðzÞ and N2 ¼ 2N1  1, gðzÞ is an ideal highpass ﬁlter. In thiscase, we have the polyphase matrices:
2
3
1 N
1
z
ðzÞ
6
7
2
2
PðzÞ ¼ 4
5
1 N
1 2
2Nþ1
 z ðzÞ   ðzÞ þ z
2
2 2
3
1 2
1
2Nþ1
  ðzÞ þ z ðzÞ
7
2
e ¼6
PðzÞ
4 2 1
5:
1
zN ðzÞzN
2
2
With this, we get the following expressions for the analysis and the synthesis ﬁlter
pair, respectively:
8hðzÞ ¼ ðz2N þ z1 ðz2 ÞÞ=2>>>>
< gðzÞ ¼ ðz2 ÞhðzÞ þ z4Nþ1ð1Þ>ehðzÞ ¼ gðzÞ
>
>
>
:
e
gðzÞ ¼ hðzÞIf ðzÞ has the following magnitude and phase response:© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014jðej2! Þj ¼ 1; 8 !
(ð2N þ 1Þ!
; for ! 2 ½0; =2 ;j2!
ﬀðe Þ ¼
ð2N þ 1Þ!   ; for ! 2 ½=2; ð2Þ

fh; gg is an ideal lowpass and highpass ﬁlter pair.
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IEICE Electronics Express, Vol.11, No.17, 1–11

The ideal choice of ðzÞ as in (2) requires inﬁnite complexity. Therefore, wehave to design ðzÞ to approximate the conditions in (2). However, the approximation will not change the perfect reconstruction property because the analysis
e satisfy PðzÞPðzÞ
e ¼polyphase matrix PðzÞ and the synthesis polyphase matrix PðzÞ1 3Nþ1
I, regardless of the choice of ðzÞ.
2zLetðzÞ ¼

NX

vk ðzNþk þ zNkþ1 Þ;

ð3Þ

k¼1

P
where the coefﬁcients vk satisfy Nk¼1 vk ¼ 12 . Thus, we get a linear phase FIR
biorthogonal ﬁlter bank.
We know that the higher regularity of wavelets, the less number of the wavelet
coefﬁcients in the wavelet domain, while the regularity of a wavelet is changed
with vanishing moments of wavelet. For biorthogonal wavelets, the vanishing
moments of wavelet function Ψ and e depend on the number of zeros of hð!Þ andehð!Þ at ! ¼ ; that is, the number of zeros of hðzÞ and ehðzÞ when z ¼ 1,erespectively. From (1), we can see that hðzÞ and hðzÞ have the same number ofzeros at the same condition. Note thatPhðzÞ ¼ ðz2N þ z1 ðz2 ÞÞ=2 ¼ z2N 1 þ Nk¼1 vk ðz2k1 þ z2kþ1 Þ 2;
P
Let Fh ðzÞ ¼ 1 þ Nk¼1 vk ðz2k1 þ z2kþ1 Þ, then the number of zeros of hðzÞ atz ¼ 1 is equivalent to the number of zeros of Fh ðzÞ at z ¼ 1. Assuming hðzÞhas zeros of order p at z ¼ 1, we have the following:FhðiÞ ðzÞjz¼1 ¼ 0;

1  i  p  1:

Note that, if i is odd, the above equation always holds, and if i is even, we have thefollowing equation:
NX

vk ð2k  1Þ2l ¼ 0;

1  l  r  1;

ð4Þ

k¼1

where p ¼ 2r  1.
Theorem 2 If the vanishing moments of biorthogonal wavelets Ψ and e are pand e
p, respectively, then the length of their support is at least p þ e
p  1.
Therefore, the maximum number of zeros of hðzÞ is 2N  1, and when p ¼
2N  1, that is r ¼ N, we get the following equation:
1
10
1 0
0
x02

x0N
x01
v1
1=2
C
CB
C B
B
x22

x2N CB v2 C B 0 C
B x21
C
CB
C B
B
ð5Þ
CB . C ¼ B . C
B
..
..
..
..
CB .. C B .. C
B
.
.
.
.
A
A@
A @
@
x2ðN1Þ
1© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014x2ðN1Þ2

x2ðN1ÞN

vN

0where xk ¼ 2k  1, k ¼ 1; 2; . . . ; N. As the matrix is nonsingular, the solution for
fvk : k ¼ 1; 2; . . . ; Ng will always exist. Thus, we can get a pair of maximally ﬂat
FIR biorthogonal wavelet ﬁlters.
Next, we will map the 1-D biorthogonal ﬁlter banks into the 2-D quincunx ﬁlter
1banks with perfect reconstruction. Let ðzÞ ¼ ðz0 Þðz1 Þ with z1 ¼ z1
0 z1 , then the5

IEICE Electronics Express, Vol.11, No.17, 1–11

polyphase matrix of the 2-D quincunx ﬁlter banks can be written as:
2
3
1
1
N
ðz0 z1 Þðz0 Þðz1 Þ
6
7
2
2
Pðz0 ; z1 Þ ¼ 4
5;1
1
 ðz0 z1 ÞN ðz0 Þðz1 Þ  2 ðz0 Þ2 ðz1 Þ þ ðz0 z1 Þ2Nþ1
2
3
22
1 2
1
2
  ðz0 Þ ðz1 Þ þ ðz0 z1 Þ2Nþ1  ðz0 Þðz1 Þ
7
2
e 0 ; z1 Þ ¼ 6
Pðz
5:
4 2 1
1
N
N
ðz0 z1 Þ ðz0 Þðz1 Þ
ðz0 z1 Þ
22
e ¼ 1 ðz0 z1 Þ3Nþ1 I,
And the analysis and synthesis polyphase matrices satisfy PðzÞPðzÞ
2
so the perfect reconstruction property is preserved. From these polyphase matrices,
we can get the analysis and synthesis quincunx ﬁlter banks as follows.
81>hðz0 ; z1 Þ ¼ ðz2N
þ z1
>
0
0 ðz0 z1 Þðz0 z1 ÞÞ=2>>>
< gðz0 ; z1 Þ ¼ ðz0 z1 Þðz0 z1 Þhðz0 ; z1 Þ þ z4Nþ1
1
0
ð6Þ
>
e
>
hðz0 ; z1 Þ ¼ gðz0 ; z1 Þ
>
>
>
:
e
gðz0 ; z1 Þ ¼ hðz0 ; z1 Þ
If the 1-D transfer function ðzÞ approximates the conditions in (2) well, hðz0 ; z1 Þand gðz0 ; z1 Þ are diamond and diamond-complement ﬁlters, respectively.
2.2 Chebyshev best uniform approximation
For a continuous function fðxÞ on the interval ½a; b, there will be only one^
polynomial pðxÞin the set Pn of all n polynomials, which satisﬁes the following
equation:^  fðxÞj ¼ minf max jpðxÞ  fðxÞjg;max jpðxÞ

axb

axb

where pðxÞ 2 Pn .
Pjk!
, we may assume that hðzÞ is
For the linear phase FIR ﬁlter hðzÞ ¼ N1
k¼0 hk e
positive symmetric, and N is odd, then:hðzÞ ¼ zðN1Þ=2 hg ðzÞ; z ¼ ej! ;P
where hg ðzÞ ¼ hg ðej! Þ ¼ M
n¼0 aðnÞ cosðn!Þ, M ¼ ðN  1Þ=2, and aðnÞ is deﬁned asfollows:(; n¼0
hðN1Þ=2:aðnÞ ¼
2hðN1Þ=2n ; n ¼ 1; 2; . . . ; ðN  1Þ=2
Assume hnew ðej! Þ is the desired function:(
1 ; 0  !  !p;
hnew ðej! Þ ¼
0 ; !s  !  © IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014

ð7Þ

where !p and !s are the passband and stopband frequencies, respectively. Let
Bp ¼ ½0; !p  and Bs ¼ ½!s ;  be the frequencies of the passband and the stopband,S
respectively. Let F ¼ Bp Bs , then by deﬁnition we know that F  ½0; .
We deﬁne the weight function as follows:(
1=k ; 0  !  !p ; k ¼ 1 =2:
Wðej! Þ ¼
1
; !s  !  

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IEICE Electronics Express, Vol.11, No.17, 1–11

Consequently, the error function is as follows:

hXM
i
j!
aðnÞ
cosðn!Þ
hðeÞ:
Eðej! Þ ¼ Wðej! Þ½hg ðej! Þ  hnew ðej! Þ ¼ Wðej! Þ
newn¼0

Theorem 3 (Alternation Theorem) If pð!Þ is a linear combination of M cosinePfunctions (i.e. pð!Þ ¼ M1n¼0 aðnÞ cosðn!Þ), then a necessary and sufﬁcient condition for pð!Þ to be the unique best-weighted Chebyshev approximation to acontinuous function hnew ðej! Þ on F is that the weighted error function Eðej! Þexhibits at least M þ 2 extremal frequencies in F. That is, points f!i g such that!0 < !1 < !2 <    < !Mþ1 , Eðej!i Þ ¼ Eðej!iþ1 Þ, i ¼ 0; 1; 2;    ; M, andjEðej!i Þj ¼ max!2F jEðej! Þj.
According to the above theorem, we have:
"
#
M
X
j!k
j!k
aðnÞ cosðn!k Þ ¼ ð1Þk ;ð8ÞWðe Þ hnew ðe Þ 
n¼0

where k ¼ 0; 1;    ; M þ 1, and  ¼ max!2F jEðej! Þj.
With equation (8), we can get only one solution of aðnÞ, n ¼ 0; 1;    M and ρ,
so that we get the desired ﬁlter. However, since directly solving the equation (8) is
difﬁcult, we can solve the above equation by Remez algorithm, that is, the ParksMcClellan algorithm to obtain the desired ﬁlter.
3

Results and analysis

The PKVA ﬁlter family designed in this study was applied in image denoisingthrough hidden Markov tree models in the Contourlet domain, with Brodatz
standard texture image database as the test objects. By analyzing the experimentresults, we obtained two optimal PKVA ﬁlters for texture image based on the peaksignal to noise ratio (PSNR) as shown in Table I.
Table I. The coefﬁcient vk in equation (3) for optimal PKVA ﬁlters.
ControlParameter
0.48
0.47

© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014

Coefﬁcient vk in equation (3)0.635940
0.635291−0.212241−0.209614

0.125984
0.122377−0.090210−0.085136

0.069231
0.062993−0.206109−0.136772

In the experiment, we added random Gaussian noise for all images in Brodatz
standard texture image database with the adding noise parameters α (2 =2552 ) is thevariance of the Gaussian noise normalized to the image range) being either 10, 20,
30, 40, or 50. And image denoising was implemented. It is found from the resultsthat for all adding noise parameters, the number of images with maximum PSNR
was larger than that of others when the control parameter γ (!p ¼  is the
passband edge in equation (7)) of the PKVA ﬁlter is either 0.48 or 0.47. The
results are shown in Fig. 1, 2, and 3. The experiment also shows that the PSNR
clearly reduced when the control parameter was outside ½0:3; 0:5.
Objective comparison of experimental statistics of maximum PSNR withdifferent adding noise parameters for test images in the Brodatz standard texture
image database are shown in Table II.
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IEICE Electronics Express, Vol.11, No.17, 1–11

Fig. 1.

The number of times PSNR is maximum with different control
parameters.

Fig. 2.

The number of times PSNR is maximum with different control
parameters when adding noise parameter is 10.

Table II.

Comparison between experiment results of PKVA ﬁlter
designed in this paper and performances of Po and Do’s
PKVA ﬁlter.

Adding Total
NoiseamountParameter of images© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014

10
20
30
40
50
All

111
111
111
111
111
555

Po and
Do

0.47

Mean
differences

0.48

Mean
differences

24.976405
25.114868
25.912179
26.801758
27.570614
26.081925

26.270900
26.209089
26.705407
27.517241
27.944214
26.943073+1.294495+1.094221+0.793228+0.715483+0.373600+0.861148

26.575702
26.442935
26.907501
27.555005
27.92816627.093374+1.599297+1.328067+0.995322+0.753247+0.357552+1.011449

8

IEICE Electronics Express, Vol.11, No.17, 1–11

Fig. 3.

The number of times PSNR is maximum with different control
parameters when adding noise parameter is 40.

Fig. 4.

The PSNR value of the test image D72 with different control
parameters when adding noise parameter is 20.

Subjective comparison of denoising performances between our new PKVA
ﬁlter and Po and Do’s ﬁlter for test images D62 and D72 in the Brodatz standard
texture image database with adding noise parameter 20 and 20 respectively, areshown in Fig. 5, 6, 7, 8, and 9.

© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014

Fig. 5. Original two images from left to right: D62, D72

9

IEICE Electronics Express, Vol.11, No.17, 1–11

© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 2014

Fig. 6.

The two images from left to right: D62 and D72 with adding
noise parameter 20 respectively.

Fig. 7.

Denoised images with Po and Do’s ﬁlter from left to right: D62,
D72

Fig. 8.

Denoised images with the new PKVA ﬁlter when control
parameter is 0.47 from left to right: D62, D72

Fig. 9.

Denoised images with the new PKVA ﬁlter when control
parameter is 0.48 from left to right: D62, D72

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IEICE Electronics Express, Vol.11, No.17, 1–11

4

Conclusions

This paper proposed a new PKVA ﬁlter design approach for the contourlet transform based on Chebyshev best uniform approximation method. It proved to beeffective, and PKVA 12 ﬁlter for Brodatz standard texture image denoising in thecontourlet HMT models was improved by 1.599297 dB. Meanwhile, the subjective comparison was as good as the experimental results of Po and Do in thereference [3].

© IEICE 2014
DOI: 10.1587/elex.11.20140556
Received June 10, 2014
Accepted July 22, 2014
Publicized August 13, 2014
Copyedited September 10, 201411