4.2 Proposed System
4.2.2 Statistical Modelling of Prediction Error Frames
The motion-compensated prediction error frames generally exhibit highly non-stationary spatial statis- tics. There may be areas with low energy where the motion-model well captures the motion and areas with high energy where the motion-model is not adequate. In the proposed method, we model each PEF of the host sequence as a locally iid non-stationary Gaussian random field which is one of the most widely used statistical models in image processing [MKR99, VDPP01]. This model is characterized by two parameters, the local mean and the local variance. According to this model, each sample of a PEF follows a Gaussian distribution with specific parameters:
∀n∈Λ, Pxk[n]∼ N
µPxk[n], σ2Px
k[n]
(4.2.10) whereµPxk[n]andσ2Px
k[n]are respectively the local mean and the local variance.
Let us now analyze how the distribution of the PEFs changes due to the addition of an MIC water- mark. As shown in Equations 4.2.6 and 4.2.8, the PEF of a sequence carrying MIC watermark is the sum of two independent random fields : the PEF of the host which is locally iid non-stationary Gaussian and that of the watermark which is iid Gaussian. By using the properties of the sum of independent Gaussian random variables one can show that the distribution the PEFs of a sequence carrying an MIC watermark are also locally iid non-stationary Gaussian, i.e.,
∀n∈Λ, Pyk[n]∼ N µPy
k[n], σ2Py
k[n]
(4.2.11) withµPy
k[n] =µPxk[n]andσ2Py
k[n] =σP2x
k[n] +σP2w
k . The local mean of the PEFs does not change after the addition of MIC watermark sincePwk is assumed to be zero-mean. Similarly in the case of a B-PEF,
∀n∈Λ, Byk[n]∼ N µBy
k[n], σB2y
k[n]
(4.2.12) whereµBy
k[n] =µBxk[n]andσB2y
k[n] =σB2x
k[n] +σB2w
k.
We have seen that the variance parameter of the locally iid Gaussian model of the PEF changes with the addition of MIC watermark. For a model-based description of this change, a suitable model for the local variances is needed. Voloshynovskiy et.al [VDPP01] suggested deriving such a model for the local variances of a host image from the estimated local variances. We propose to use the estimated local variances to derive similar models for the local variances of the host and the watermarked PEFs.
Since the PEFs are modelled as iid Gaussian within a local window, the maximum likelihood (ML) estimator for the local variance ofPyk at a pointnin is given as
ˆ σP2y
k[n] = 1 N2−1
X
m∈N(n)
Pyk[m]−µˆPy
k[n]2
(4.2.13)
4.2. PROPOSED SYSTEM 75 whereN(n)is the local neighborhood ofN ×N points centered atn, and
ˆ µPy
k[n] = 1 N2
X
m∈N(n)
Pyk[m]
is the ML estimator for local mean. Substituting the value ofPyk[n]from Equation 4.2.6 and neglect- ing the cross-terms by using the fact that Pxk[n] andPwk[n]are independent, Equation 4.2.13 can be approximated as
ˆ σ2Py
k[n] ≈ 1
N2−1 X
m∈N(n)
Pxk[m]−µˆPxk[n]2
+ 1
N2−1 X
m∈N(n)
Pwk[m]−µˆPwk[n]2
= ˆσ2Px
k[n] + ˆσP2w
k[n] . (4.2.14)
Here we assumed that
X
m∈N(n)
Pxk[m]−µˆPxk[m]
Pwk[m]−µˆPwk[m]
'0.
Thus the local variance estimated at each point of the MIC watermarked PEF is the sum of the estimated local variances of the corresponding host PEF and the watermark PEF.
Figure 4.1(a) shows the estimated local variance of a P-PEF from the Mobile sequence using a 3× 3 window. The local variances, estimated from the areas in the PEF corresponding to the oc-
0 2 4 6 8 10 12
x 104 0
5000 10000 15000
array index
local variance
(a) (b)
Figure 4.1: (a) Local variances of a P-PEF from the Mobile sequence and (b) the plot of the sorted array of local variances. Darker areas in (a) indicate regions where the estimated variance is high valued.
cluded/uncovered regions and object boundaries in the current frame, are generally very high com- pared to those estimated from other areas in the PEF. This is evident from the darker areas in the local variance plot, shown in Figure 4.1(a). The local variance estimates, rearranged to a 1-D array and then sorted in the ascending order, are plotted in Figure 4.1(b). It is clear from this plot that the local
4.2. PROPOSED SYSTEM 76 variance estimated from the above mentioned areas, though a few in numbers, are of extremely high values. These extreme-valued estimates, resulting from the failure of motion-compensated prediction, are considered as outliers. This outliers are discarded from the analysis by setting an upper-threshold τ on the local variance estimate.
Motivated by the works [MKRM99, VDPP01] we now derive a statistical model for the ML esti- matorsσˆP2x
k. It has been suggested that the histogram of the local variance estimates of the high-pass wavelet bands of an image can be well approximated by using the exponential, Weibull, Rice or Gamma distributions. We propose that such approximation is valid for the local variance estimates of the host PEFs. From the above mentioned distributions, we choose the more general Gamma distribution to approximate the histogram ofσˆP2x
k.
The probability density function(PDF) of the two-parameter Gamma distribution [RS00] is given by
f(x) = ( b−a
Γ(a)xa−1exp[−xb], x >0
0, x≤0 (4.2.15)
wherea >0is the shape parameter,b >0is the scale parameter and Γ(a) =
Z ∞
0
ta−1exp[−t]dt (4.2.16)
is the Gamma function. The shape parameter allows the gamma distribution to take on a wide range of shapes. The scale parameter is a measure of the spread of the distribution. Many distributions like the exponential, Weibull and the chi-squared (χ2) distributions are the special cases of the Gamma distribution.
The normalized histograms of σˆP2x
k, estimated using a 3 × 3 sliding window and their Gamma approximations for different sequences, are shown in Figure 4.2(a-c). The Gamma approximation parameters are obtained by the maximum likelihood estimation and the statistical tool box of MATLAB.
These plots suggest that the Gamma distribution is indeed a reasonable approximation to the distribution of the ML estimatorσˆP2x
t. In other words, the estimated local varianceσˆ2Px
t[n]at each point can be seen as the realization of a Gamma random variable.
Deriving a statistical model for the estimator σˆ2Pw
k is rather straightforward. Since Pwk is an iid Gaussian random field, we can show that theσˆ2Pw
k will follow aχ2 distribution withN2−1degrees of freedom [RS00]. The PDF of the two-parameterχ2distribution is given by
f(x) =
(2σ2)−n2
Γ(n2) xn2−1exp[−2σx2], x >0
0, x≤0
(4.2.17)
4.2. PROPOSED SYSTEM 77 wheren ∈Z+is the degree of freedom andσis the scale parameter. On comparing Equations (4.2.15) and (4.2.17), we can see that theχ2distribution is a special case of the Gamma distribution witha = n2 andb = 2σ2.
The local variance estimatorσˆP2y
k
can be now seen as the sum of gamma random variablesσˆP2x
k and ˆ
σP2w
k. Further, it can be assumed that the random variablesσˆP2x
k andσˆP2w
k are independent, since they are the functions of independent random variablesPxk[n]and Pwk[n]respectively (Equation (4.2.14)).
The sum of the two independent Gamma variables is Gamma distributed only if the scale parameters of the distributions are equal. Nevertheless, the sum of two independent Gamma random variables with unequal scale parameters can still be reasonably approximated as a Gamma random variable [Mat82].
Using this approximation, we model the local variance estimators of each watermarked PEF as Gamma distributed. The normalized histograms of the local variances of the PEFs for MIC watermarked se- quences and their Gamma approximations are shown in Figure 4.2 (d-f). In summary,
1. The estimated local variances of each host PEF is Gamma distributed.
2. With the addition of the MIC watermark, the estimated local variance of the PEFs are still Gamma distributed, but with different parameters.
This change in the distribution parameters with the addition of MIC watermark is exploited in the proposed oracle.