A Novel Blind Watermarking Algorithm in Contourlet Domain
Haifeng Li, Weiwei Song, and Shuxun Wang
College of Communication and Engineering, Jilin University, R. P. of China
lhfvip_2000@https://www.doczj.com/doc/4017776618.html,
Abstract
A novel watermarking algorithm based on Contourlet transform is proposed in this paper. The watermark composed of pseudo-random sequence is embedded in the selected Contourlet transform coefficients by means of multiplicative method. The Contourlet coefficients are modeled with Generalized Gaussian Distribution with zero mean, and then watermark detection method is proposed based on maximum likelihood detection. Furthermore the decision rule is optimized via Neyman-Pearson criterion. Experimental results show that the fidelity of the watermarked image is good and robust to signal processing and small geometrical attacks.
1. Introduction
The idea of watermarking technique is to embed an indelible watermark into an original signal without perceptual artifacts. The watermark can be detected or extracted to solve ownership disputes, track piracy etc. Under the assumption that the watermarked feature coefficients follow a Gaussian distribution, the correlation-based detection method can be optimal in that it permits to minimize the error probability. However, correlation-based detector is not the optimum choice when the coefficients do not follow a Gaussian distribution. Hu [1] et al proposed a new DWT- domain decoder structure based on describing the wavelet coefficients with the Laplacian model. As nonlinear receivers have been shown to be particularly well suited for the detection of weak signal in heavy tailed noise, Briassouli and Strintzis [2] adopted the Gaussian tailed zero-memory nonlinearity as well as the local optimal Cauchy nonlinearity for detection of watermarks in DCT transformed images.
In this paper, we propose a blind watermarking algorithm using the Contourlet transform. The Contourlet transform has been developed as a true two-dimensional representation that can capture the geometrical structure in pictorial information [3]. Based on a multiresolution and multidirection image expansion using non-separable filter banks, a flexible multiresolution, local, and directional image expansion using contour segments is obtained. Within the selected scale subband, the most significant directions are selected. A watermark is embedded into these Contourlet coefficients, and it is detected using the Maximum Likelihood (ML) estimation based on modeling the Contourlet coefficients as the Generalized Gaussian Distribution (GGD) with zero mean. The determined threshold is determined based on Neyman-Pearson criterion, which maximizes the detection probability for a given false alarm probability.
2. The Contourlet transform
As a result of a separable extension from 1-D bases, wavelet in 2-D are good at isolating the discontinuities at edge points, and separable wavelets can capture only limited directional information, which is an important
Multiscale decomposition Multidirection decomposition Figure 1.The diagram of the Contourlet transform. The image is first decomposed into subbands by the Laplacian pyramid and then each detail image is analyzed by the directional filter banks.
The Contourlet transform is a novel geometrical image-based transform, recently introduced by Do and Vetterli [4], which can efficiently represent images containing contours and textures. By constructing a discrete-domain multiresolution and multidirection expansion using non-separable filter banks, we can obtain a flexible multiresolution, local, and directional image expansion using contour segments, and thus it is
named the Contourlet transform [3]. The main difference between the 2-D Gabor wavelets, the steerable pyramid and the Contourlet construction is that the previous methods do not allow for a different number of directions at each scale while achieving nearly critical sampling. Figure 1. shows a flow graph of the Contourlet transform.
The Contourlet transform is a multiscale and directional decomposition of a signal using a combination of a modified Laplacian Pyramid (LP) and a Directional Filter Bank (DFB). DFB is designed to capture high frequency components (representing directionality), and the LP part of the PDFB permits subband decomposition to avoid “leaking” of low frequencies into several directional subbands.
Let 0I be the input image. J I represents a lowpass image and j B , denotes -th bandpass images after LP stage. The -th level of the LP
decomposes the image 1,2,,j "J j j 1j I
into a coarser image J I and a detail image j B . Each bandpass image j B is further decomposed by j l -level DFB into 2bandpass directional images j
l
(),j
l j k d ,.
0,1,,21j l
k
"a b
Figure 2.The Contourlet transform of the Man image
using 2 LP levels and 8 directions at finest level. a Contourlet transform of Man; b Coefficients distributions of the second finest level.
3. The statistical of the Contourlet coefficients The statistical model of the Contourlet coefficients is the foundation of designing the optimal watermark decoder. Figure 2. plots the Contourlet transform of the Man image using 2 LP levels and 8 directions at finest level, and the histograms of the finest subband. These
distributions exhibit a sharp peak at zero amplitude and
heavy tails to both sides of the peak. Kurtosis of the signal, defined for a zero-mean random variable x as
42(){}3({})Kurt x E x E x 2 (1) where {}E denotes the expectation operation. For a Gaussian random variable, kurtosis is 3. The kurtosis of the second-level subband are respectively 10.2953, 6.5605, 8.8150 and 9.4238, which are much larger than 3. So the subband marginal distributions of the Contourlet coefficients are highly non-Gaussian.
We adopt the zero-mean GGD to describe the Contourlet subband coefficients, and the probability distribution function is expressed as following
()exp()c
X f x A x E (2)
where
2(1)
c A c E
*
,E
dt ,
, ().10
()t z z e t f
* 30z !V is the standard
deviation, and the positive real number c is the shape parameter.
The shape parameter c and the standard deviation V need estimating to design the effective watermark detector. [5] concludes that the ML estimator is significantly superior for heavy-tailed distribution. Hence, the Maximum Likelihood is used to estimate the distribution parameters.
After a series of calculations, we can obtain the estimated parameter V
11
?(L c
i i c x L V
|) (3) where denotes the sample number. The shape parameter is the solution of the following transcendental equation
L c ??1
1??log log()?(1)10??L
L c c
i i
i i i c
i c x x x c L c c x \ ||| (4) where ()\ is the digamma function. The solution of
can be obtained using the Newton-Raphson iterative procedure [5].
c
?4. Watermarking Algorithm
4.1. Watermark embedding
Based on investigation the property of the Contourlet transform, we propose a content-dependent watermark embedding algorithm. As an embedding criterion, we resort to embed the watermark into the most energetic edges of the image. Cox et al [6]
claimed that the watermark should be embedded into the perceptually significant features of the images to protect. We avoid embedding a watermark into the coarse subband to reduce the obviously perceptual alternations.
Specially, for the whole original image 0I the Contourlet transform is performed. One lowpass
subband J I
is obtained, and bandpass directional images (),j
l j k
d ,,, wher
e represents the -th level o
f LP decomposition, k
represents the k -th bandpass directional image decomposed by a 0,1,,21j l
k "1,2,,j "J j j j l -level DFB. The energy distribution of these subimages is known to be an important characteristic for digital image processing. The energy is computed as follows:
2
()
,,111(,)j M N l j k j k m n E d MN ||m n (5)
where M and denote the width and length of the subimage N ()
,j l j k d . A larger value of ,j k E implies that
this sub-band contains more energy and should be treated as a significant sub-band in comparison with other sub-bands. We embed the watermark signal into this significant sub-band with largest energy to improve the robustness. The selected sub-band coefficients are collected in the vectors 12{,,,}L v s s s ".Assume that the watermark information
12{,,,}L W w w w " composes of a pseudo-random number sequence whose values are determinations of a random variable having a Gaussian distribution of zero mean and unit variance. Content adaptive watermark embedding is designed to insert watermarks into the selected Contourlet coefficients as follows: W i i i i s s s w D (6) where W i s
a the watermarked Contourlet coefficient, denotes an element of a watermark signal, and i w D
is the watermark strength. D controls the imperceptibility and the robustness of the watermark. The other Contourlet coefficients remain.
Then the watermarked Contourlet coefficients are inserted back in the same location where they have been taken from. The inverse Contourlet transform is performed in succession and the watermarked image W I is obtained.
4.2. Watermark detection
Generally speaking, the watermark detection can be
formulated as the binary hypothesis test
0H :i i y x , (7) 1,2,,i "L 1H :i i i y x x w i D (8) 1,2,,i "L where 0H indicates the absence of any watermark, 1H the presence of the watermark. 12(,,,)T L X x x x "is the selected Contourlet coefficients embedding the watermark, is one observation of the possible watermarked Contourlet coefficients, and
a watermark generated by the key 12(,,,)T L Y y y y "12(,,,)T L W w w w "K .
Given the received watermarked signals, maximum-likelihood (ML) detection can be performed to determine the presence of the embedded watermark. The likelihood ratio function is
()l Y 10(|,)
()(|)
Y Y f y H K l Y f y H
(9)
where 1(|,)Y f y H K represents the probability
distribution function of the random vector Y conditioned by a given K and the event 1H .
According as the depiction in 3.1, the Contourlet coefficient is modeled as a realization from a GGD, and assuming the Contourlet coefficients obey the i.i.d., the distributions under 1H and 0H are respectively
11(|,)exp{ln(1)}2(1)1c
L
i
Y i i i y c f y H K w c w E E D D * (10)
01(|)exp{}2(1)L
c
Y i i c f y H y c E E * (11)
The log-likelihood ratio is expressed as 110
ln ()(ln(1))
ln 1c
L c i i i i i H y l Y y w w H E E D D ! |K (12) The threshold K
is determined based on NP criterion, which maximizes the detection probability for a given false alarm probability. The false alarm probability is given by
fa P 0ln (ln ()ln |)()fa P P l Y H f r dr K
K f ! 3
(13)
where ()f r represents the distribution of ln under ()l Y 0H . The threshold K will be solved when a is given. The resulting test guarantees that the power of the test, i.e. the probability of detection, will be maximized for a predetermined false alarm [7].
fa P
5. Experimental results
To evaluate the performance of the proposed
watermarking algorithm, the 256 Man image
with 8 bits/pixel resolution was used for watermarking.
The Man image is transformed by Contourlet using ‘9-7’ Pyramid filter and ‘pkva’ directional filter to obtain
a three-level decomposition, and 2(256u i 0,2,3i )
directional decomposition at each level. Peak Signal to
Noise Ratio (PSNR) is used measure of fidelity. Figure
3a shows the original Man image and Figure 3b shows
the watermarked Man image with
.48.3506 dB PSNR
a b
Figure 3.a Original Man image b Watermarked Man image Watermarking detection is blind, which is done by estimating the standard deviation V and shape parameter possibly from the distorted watermarked image.
c Table 1.Expeirmental results against different attacks
DETECTION
ATTACKS PSNR
CORRELATIONPROPOSED
JPEG compression 80 33.22h OK JPEG compression 40 28.46h OK Pepper & Salt noise
0.01
24.54h
OK Speckle noise 0.01 29.33h OK Gamma correction 0.6 16.60h OK Histogram equation 12.12h OK Gaussian filtering 33u 25.05h
OK Scaling 0.6 26.66h OK Scaling 1.3 33.73h OK Rotation 0.1 37.25h OK Rotation 0.2 30.38h OK Rotation 10 12.63h h Rotation 20 11.23h h Ratio x 2.0 y 0.8 30.51h OK Ratio x 0.6 y 0.8 27.41h OK Cropped 6.25%
removed 23.13h
OK Cropped 14.06%
removed
18.44h
OK Printing & Scanning 13.90h
OK The robustness of the watermark has been tested against different kind of attacks, including PEG compression, scaling, cropping and et al. Some test results are listed in Table 1. The robustness to small rotation, coupled with the robustness to scaling and additive noise, allows watermark recovery even after printing and scanning by using poor quality devices. It is evident that rotation by large angles lowers the algorithm performances because there is no more a
correspondence between the watermark embedded
subband of the original image and the rotated image.
No results are reported here about robustness to large
rotation, since it ultimately depends on the
synchronization mechanism and is not a characteristic
of the watermark embedding and retrieval algorithm [7].
6. Conclusions
A novel image watermarking scheme in the Contourlet domain is presented in this paper. A blind detection method is presented using ML estimation based on model the Contourlet coefficients as Generalizeed Gaussian Distribution. Experimental results demonstrate the good performance of the proposed algorithm which is imperceptibility and robust.
7. References
[1] Yongjian Hu, Sam Kwong, and Y. K. Chan, “The design
and application of DWT-domain optimum decoders,” LNCS 2613, 2003, pp. 22-30.
[2] A. Briassouli, and M. G. Strintzis, “Locally optimum nonlinearities for DCT watermark detection,”IEEE Trans. on Image Processing , 2004, vol. 13, no 12, pp. 1604-1617.[3] M. N. Do, and M. Vetterli, “The Contourlet transform: an efficient directional multiresolution image representation,” IEEE Trans. on Image Processing , 2005, vol.14, no. 12, pp. 2091-2106.
[4] Do, M. N., and Vetterli, M., “Contourlets: a directional multiresolution image representation,” in Proc. ICIP , vol. 1, 2002,pp. 357-360.
[5] M. N. Do, and M. Vetterli, “Wavelet-based texture retrieval using generalized gaussian density and Kullback–Leibler distance,” IEEE Transactions on Image Processing ,2002, vol.11, no. 2, pp.146-158.
[6] I. J. Cox, J. Kilian, F. T. Leighton, and T. Shamoon, “Secure spread spectrum watermarking for multimedia,” IEEE Trans. on Image Processing , 1997, vol. 6, pp. 1673-1687.
[7] M. Barni, F. Bartlini, A. D. Rosa, and A. Piva, “A new decoder for optimum recovery of nonadditive watermarks,” IEEE Trans. on Image Processing , 2001, vol. 10, no. 5, pp. 755-766.
附件2、《各系列专业职称资格分类一览表》 国家各系列（专业）职称资格分类一览表 各系列（专业）分类 高、中、初级专业技术资格名称 正高级 副高级 中级 初级 （助理级） 初级 （员级） 工 程 系 列 建设专业 研 究 员 级 高 级 工 程 师 高 级 工 程 师 工 程 师 助 理 工 程 师 技 术 员 机械专业 纺织专业 轻工专业 冶金专业 石油化工专业 交通水路运输 专业 交通公路运输 专业 质量技术监督 专业 水利专业 水产专业 林业专业 环境保护专业 广播电影电视 工程专业 电子信息专业 煤炭专业 地质矿产专业 水文(工程、环境)地质专业 探矿专业 物化探与遥感 专业 地质实验测试 (选矿)专业 测绘专业 采矿专业 土地专业 岩土工程专业
工艺美术专业研究员级高 级工艺美术 师 高级工艺美 术师 工艺美术 师 助理工艺 美术师 工艺美术 员 农业系列 农技专业 农业技术推 广研究员 高级农艺师农艺师 助理农艺 师 技术员畜牧(兽医) 专 业 农业技术推 广研究员 高级畜牧 (兽医)师 畜牧(兽 医)师 助理畜牧 (兽医)师 技术员农业经济专业高级农经师 农经师 (考试) 助理农经 师(考试) 农经员 (考试) 财经系列 国际商务专业 高级国际商 务师 国际商务 师(执业资 格) 助理国际商 务师 (从业资格) 外销员 (从业资 格) 经济专业 研究员级高 级经济师 高级经济师 经济师 (考试) 助理经济 师(考试) 经济员 (考试) 会计专业 研究员级高 级会计师 高级会计师 (考评结合) 会计师 (考试) 助理会计 师(考试) 会计员 (考试) 统计专业高级统计师 统计师 (考试) 助理统计 师(考试) 统计员 (考试) 审计专业 高级审计师 (考评结合) 审计师 (考试) 助理审计 师(考试) 审计员 (考试) 思想政治 工作专业 研究员级高 级政工师 高级政工师政工师 助理政工 师 教师系列高校教师(思想 政治教育专职 教师) 教授副教授讲师助教 高校教管研究 专业 研究员副研究员 助理研究 员 研究实习 员 高校实验专业高级实验师实验师 助理实验 师 实验员中专校教师高级讲师讲师助理讲师教员中专实验专业高级实验师实验师 助理实验 师 实验员技工学校教师 教授级高级 讲师 高级讲师讲师助理讲师教员技校实习指导 教师 高级实习指 导教师 一级实习 指导教师 二级实习 指导教师 三级实习 指导教师党校教师 (市级以上) 教授副教授讲师助教 党校教师(县 级) 高级讲师讲师助理讲师
专业技术职称分类 系 列 高 级中 级 初 级正高级副高级 高等学校 教师 教授副教授讲师助理讲师 中等专业 学校教师 高级讲师讲师助理讲师、教员 技工学校教师 高级讲师讲师助理讲师、教员高级实习指导教师 一级实习指导教 师 二级实习指导教 师、三级实习指 导教师 中学教师中学高级教师中学一级教师中学二级教师、中学三级教师 小学（幼儿园）教 师 小学高级教师 小学一级教师、 小学二级教师、 小学三级教师幼儿园高级教师 幼儿园一级教 师、幼儿园二级 教师、幼儿园三 级教师 自然科学研究人员研究员 (Z) 副研究员 (Z) 助理研究员(Z)研究实习员(Z) 社会科学研究人员研究员 (S) 副研究员 (S) 助理研究员(S)研究实习员(S) 工程技术人员教授级高 级工程师 高级工程 师 工程师 助理工程师、技 术员 实验技术人员教授级高 级实验师 高级实验 师 实验师 助理实验师、实 验员 教授级高 级农艺师 高级农艺 师 农艺师 助理农艺师、农 业技术员
农业技术人员教授级高 级兽医师 高级兽医 师 兽医师 助理兽医师、兽 医技术员 教授级高 级畜牧师 高级畜牧 师 畜牧师 助理畜牧师、畜 牧技术员 卫生技术 人员主任医师 副主任 医师 主治（主管） 医师 医师、医士主任药师 副主任 药师 主管药师药师、药士主任护师 副主任 护师 主管护师护师、护士主任技师 副主任 技师 主管技师技师、技士 经济专业人员教授级高 级经济师 高级经济 师 经济师 助理经济师、经 济员 会计专业人员教授级高 级会计师 高级会计 师 会计师 助理会计师、会 计员 审计专业人员教授级高 级审计师 高级审计 师 审计师 助理审计师、审 计员 统计专业人员教授级高 级统计师 高级统计 师 统计师 助理统计师、统 计员 新闻专业人员高级记者主任记者记者助理记者高级编辑主任编辑编辑(X)助理编辑(X) 出版专业人员编审副编审编辑(C)助理编辑(C) 技术编辑 助理技术编辑、 技术设计员 一级校对 二级校对、三级 校对 图书资料专业人员研究馆员 (T) 副研究馆 员(T) 馆员(T) 助理馆员、管理 员(T) 文物博物专业人员研究馆员 (W) 副研究馆 员(W) 馆员(W) 助理馆员、管理 员(W)
我国专业技术职称系列级别名称 序号系列 级别名称 高级 中级 初级 正高级副高级助理级员级 1 高级教师教授副教授讲师助教 2 自然科学研究研究员副研究员助理研究员研究实习员 3 社会科学研究研究员副研究员助理研究员研究实习员 4 卫生技术主任医师 主任药师 主任护师 主任技师 副主任医师 副主任药师 副主任护师 副主任技师 主治医师 主管药师 主管护师 主管技师 医师 药师 护师 技师 医士 药士 护士 技士 5 农业技术研究员高级农艺师 高级畜牧师 高级兽医师 农艺师 畜牧师 兽医师 助理农艺师 助理畜牧师 助理兽医师 技术员 6 工程技术高级工程师 (正高级) 高级工程师工程师助理工程师技术员 7 经济高级经济师经济师助理经济师经济员 8 会计 审计 高级会计师 高级审计师 会计师 审计师 助理会计师 助理审计师 会计员 审计员 9 统计高级统计师统计师助理统计师统计员 10 中专教师高级讲师讲师助理讲师教员 11 技校教师 高级讲师 高级实习指导教师 讲师 一级实习指导教 师 助理讲师 二级实习指导教 师 教员 三级实习指导教 师 12 中学教师中学高级教师中学一级教师中学二级教师中学三级教师 13 小学教师小学高级教师小学一级教师小学二级教师小学三级教师 14 档案研究馆员副研究馆员馆员助理馆员管理员 15 文物博物 群众文化 研究馆员副研究馆员馆员助理馆员管理员 16 图书资料研究馆员副研究馆员馆员助理馆员管理员 17 翻译译审副译审翻译助理翻译 18 律师一级律师二级律师三级律师四级律师律师助理 19 公证员一级公证员二级公证员三级公证员四级公证员公证员助理 20 新闻高级记者 高级编辑 主任记者 主任编辑 记者 编辑 助理记者 助理编辑 21 播音播音指导主任播音员一级播音员二级播音员三级播音员 22 出版编审副编审 编辑 技术编辑 一级校对 助理编辑 技术助理编辑 二级校对 技术设计员 三级校对 23 体育教练国家级教练高级教练一级教练二级教练三级教练 24 船舶 高级船长 高级轮机长 高级电机员 高级机务员 船长 大副 大管轮 电机员等 二副 二管轮 二级电机员等 三副 三管轮等 25 艺术一级演员等 二级演员 主任舞台技师 三级演员 舞台技师等 四级演员 舞台技术员等 26 工艺美术高级工艺美术师工艺美术师助理工艺美术师工艺美术员 27 试验高级试验师试验师助理试验师试验员 28 海关高级关务监督关务监督助理关务监督关务员 29 飞行一级飞行员二级飞行员三级飞行员四级飞行员
各系列专业技术职称一览表 序号系列 专业技术职务 高级 中 级 初级正高 级 副高 级 助理 级 员 级 1高等学校教师教授副教授讲师助教 2中等专业学校教 师 高级讲师讲师助理讲师教员 3中小学（幼儿 园）教师 中学高级教师 中学一级教 师 中学二级教师中学三级教师小学中的中学高级教师 小学高级教 师 小学一级 教师 小学二级 教师 小学三级 教师 幼儿园高级 教师 幼儿园一级 教师 幼儿园二级 教师 幼儿园三级 教师 4技工学校教师 高级讲师讲师助理讲师教员 高级实习指导教师 一级实 习指导教师 二级实习指导教 师 三级实习指导 教师 5自然科学研究 人员研究员 副研究 员 助理研 究员 研究实习员 6社会科学研究 人员研究员 副研究 员 助理研 究员 研究实习员 7实验人员高级实验师实验师助理实验师实验员 8工程技术人员 教授级 高级工 程师 高级工 程师 工程师助理工程师技术员高级建筑师建筑师助理建筑师技术员高级城市规划师 城市规 划师 助理城市规划师技术员 9经济专业人员 高级经济师经济师助理经济师经济员高级农业经济师 农业经 济师 助理农业经济师农业经济员 1 0卫生技术人员主任医 师 副主任 医师 主治医 师 医师医士主任药 师 副主任 药师 主管药 师 药师药士主任护 师 副主任 护师 主管护 师 护师护士主任技 师 副主任 技师 主管技 师 技师技士 附 件4
2 3律师专业人员 一级律 师 二级律 师 三级律 师 四级律师律师助理 2 4公证专业人员 一级公 证员 二级公 证员 三级公 证员 四级公证员公证员助理 2 5群众文化系统 研究馆 员 副研究 馆员 馆员助理馆员管理员 2 6 职工教育系统高级讲师讲师助理讲师教员 2 7党校系统 教授副教授讲师助理讲师教员 高级讲师讲师助理讲师教员 2 8档案系列 研究馆 员 副研究 馆员 馆员助理馆员管理员 2 9文学创作系列 文学创作 一级 文学创作 二级 文学创作 三级 文学创作四级
专业技术职称等级表 系列 高级 中级初级正高级副高级 高等学校 教师 教授副教授讲师助理讲师 中等专业 学校教师 高级讲师讲师助理讲师、教员 技工学校教师 高级讲师讲师助理讲师、教员 高级实习指导教师 一级实习 指导教师 二级实习指导教师、三级 实习指导教师 中学教师中学高级教师 中学一级 教师中学二级教师、中学三级 教师 小学（幼儿园）教师小学高级 教师 小学一级教师、小学二级 教师、小学三级教师幼儿园高 级教师 幼儿园一级教师、幼儿园 二级教师、幼儿园三级教 师 自然科学 研究人员研究员(Z) 副研究员 (Z) 助理研究 员(Z) 研究实习员(Z) 社会科学 研究人员研究员(S) 副研究员 (S) 助理研究 员(S) 研究实习员(S) 工程技术人员教授级高 级工程师 高级工程 师 工程师助理工程师、技术员 实验技术人员教授级高 级实验师 高级实验 师 实验师助理实验师、实验员
农业技术人员教授级高 级农艺师 高级农艺 师 农艺师 助理农艺师、农业技术员 教授级高 级兽医师 高级兽医 师 兽医师 助理兽医师、兽医技术员 教授级高 级畜牧师 高级畜牧 师 畜牧师 助理畜牧师、畜牧技术员 卫生技术人员主任医师副主任医 师 主治（主 管）医师 医师、医士主任药师副主任药 师 主管药师药师、药士主任护师副主任护 师 主管护师护师、护士主任技师副主任技 师 主管技师技师、技士 经济专业人员教授级高 级经济师 高级经济 师 经济师助理经济师、经济员 会计专业人员教授级高 级会计师 高级会计 师 会计师助理会计师、会计员 审计专业人员教授级高 级审计师 高级审计 师 审计师助理审计师、审计员 统计专业人员教授级高 级统计师 高级统计 师 统计师助理统计师、统计员