In Section 5.6 two Gamma a priori models were introduced. They were im-plemented in an ICM algorithm and designed to model the skewed distribution of SAR amplitude data. As mentioned previously a core aspect of this the-sis is the development of a model that is superior in terms of preserving both discontinuities and homogeneous areas.
In that sense the Gammapixel a priorimodel (5.6) turned out to be well suited, although it fails in terms of preserving homogeneous areas. This insufficiency in terms of restoring homogeneous areas is ascribed to ICM because it is likely to get trapped in local minima. Simulated Annealing (SA), however, is a tech-nique able to escape these local minima and finally reach the global minimum, see Section 4.4. In order to improve the restoration of homogeneous regions a promising choice is therefore to implement the Gamma pixel prior in the SA algorithm.
Using (4.1) and (5.4) the Gammapixel priornow takes the form
p(xi|xj, j∈Ni)∝ U(xi)k−T1+T−1
where Γ is the Gamma function, T the temperature and k a constant. The variance is
5.7 The Gamma sampler 105
T
U(xi)×10−3
Iterations600 800 1000
400 200
0 2
1.5
1
0.5
0
Figure 5.29: The evolution of the energy functionU(xi) and the temperatureT for the Gamma pixel priorand SA during the annealing process. The cooling schedule used is logarithmic.
wherepi is the perturbed value and
νi=βX
c∈C
wc|m−xj|).
The eight pair-site interactions in the neighbourhood configuration are involved and the quantityνi again denotes the local energy ofU(xi) where
m=xi.
The set of all pair-site cliques in a second order neighbourhood configuration is C. The weighting factor wc depends on the clique c is horizontal, vertical or diagonal, see Figure 4.2.
(a)
(b)
Figure 5.30: (a) The restored homogeneous synthetic one grey-level SAR data in Figure 5.1(a) and (b) the restored synthetic five grey-levels SAR data in Figure 5.1(b). The restorations are performed using the Gamma pixel prior and SA. The Markov chain is inhomogeneous and the cooling schedule used is logarithmic. The data are stretched linearly between their mean±3 std.
5.7 The Gamma sampler 107
(a)
Ratio
Frequency
3.5 3
2.5 2
1.5 1
0.5 0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
(b)
Figure 5.31: (a) The ratio between the homogeneous synthetic one grey-level SAR data in Figure5.1 (a) and the restored data in Figure5.30(a) using the Gamma pixel priorand SA. In (b) is the histogram of the ratio image shown together with the theoretical Rayleigh distribution.
(a)
Ratio
Frequency
3.5 3
2.5 2
1.5 1
0.5 0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
(b)
Figure 5.32: (a) The ratio between the synthetic five grey-levels SAR data in Figure5.1(b) and the restored data in Figure 5.30(b) using the Gammapixel prior and SA. In (b) is the histogram of the ratio image shown together with the theoretical Rayleigh distribution.
5.7 The Gamma sampler 109
Figure 5.33: The restored C-band VV-polarized EMISAR data in Figure 5.2 using the Gamma pixel priorand SA. The cooling schedule is logarithmic with an inhomogeneous Markov chain. The data are histogram equalized using the beta distribution with the parameters 3 and 2.
A modified version of the Gamma pixel prior, which is implemented in the annealing algorithm is
p(xi|xj, j∈Ni)∝ U(xi)k/T−1 Γ(Tk)(T νi/k)k/T exp
−kU(xi) T νi
. (5.8)
It is important to emphasize, that (5.8) is not converging towards a uniform distribution when the temperature is raised. The proposed (5.8) is therefore inconsistent with (4.1) and the fundamental thermodynamic principle that the energy distribution must converge towards a uniform distribution forT → ∞. We recall that the estimated variance S2of (5.8) is given by
S2=T νi2
k . (5.9)
As mentioned in Section 4.4.4the decrement rule is of paramount importance
(a)
Ratio
Frequency
3.5 3
2.5 2
1.5 1
0.5 0
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
(b)
Figure 5.34: (a) The ratio between the C-band VV-polarized EMISAR data in Figure5.2and the restored data in Figure5.33using the Gammapixel priorand SA and (b) a comparison of the corresponding histogram and the theoretical Rayleigh distribution. The cooling schedule is logarithmic with an inhomoge-neous Markov chain.
5.7 The Gamma sampler 111
for the quality of the final result. Although the decrements ofT quickly become very small using the logarithmic cooling (4.10) our experiments show that it is more convincing in restoring singe-look SAR amplitude data than the exponen-tial schedule (4.9). The logarithmic cooling schedule (4.10) is therefore applied in the SA algorithm. The stochastic sampling scheme used is the Metropolis algorithm and here the random variable used to perturb the system is sampled from a uniform distribution. In order to keep the system close to its thermal equilibrium the random variable is sampled within a range close to the mean of the local neighbourhood configuration.
A short homogeneous Markov chain is with that sufficient to obtain stages be-tween temperature decrements which are close to equilibrium. In addition the temperature decrements quickly become very small using (4.10) as illustrated in Figure5.29. As the iterations proceed the thermal equilibrium is therefore con-sidered re-established after each decrement using a short Markov chain. In the SA algorithm presented a short Markov chain of length l = 1 is applied. This inhomogeneous Markov chain is chosen in order to prevent too much smoothing between sharp transitions and smaller objects.
Due to the Rayleigh distribution the variance of SAR amplitude data is propor-tional to the squared mean amplitude level, see (A.1). Since the variance again reflects the thermal energy of the system, different types of e.g. grassland areas may be represented by different temperatures. A common temperatureT for re-gions with different variances is therefore not appropriate. In order to overcome this problem a Multiple Temperature Annealing (MTA) schedule is introduced.
Here S2, which according to (5.9) is proportional to T, is re-estimated from the characteristics of the local neighbourhood configuration for each iteration.
Furthermore by using the logarithmic cooling schedule (4.10) the variance of the restored EMISAR data is strongly reduced as the annealing proceeds but it never reaches zero. Because of the MTA-schedule and the fact that the variance never reaches zero the variance of the restored EMISAR data is larger in regions with large mean amplitude levels than in regions with small mean amplitude levels. It should be noted, that by using (5.8) instead of (5.7)S2is proportional toT and not (k−1 +T)T. This has the implication for the restored result, that the smoothing effect of (5.8) is larger than the smoothing effect of (5.7).
Using the uninformed sampling strategy above the SA-MTA algorithm quickly converges towards the global energy minimum. This is demonstrated in Fig-ure 5.29where the thermal energyU(xi) and its fluctuations have diminished considerably after only 1000 iterations. The cooling stops when no significant improvements of the statistics of the ratio image are found. Experiments show that this stop criterion is met after 1000 iterations.
The restored result of the synthetic one grey-level SAR data in Figure 5.1(a)
using (5.8) after the fine tuning, is shown in Figure5.30(a). The corresponding tuned parameters are α= 2,k= 3.75 and T0 = 0.65. In TableA.2(a) is listed a number of statistics derived from the ratio between the synthetic one grey-level SAR data and the restored data. Reading from the TableA.2(a) we find z= 1.010,S2(z) = 0.267 and the test statisticsχ2(78) is estimated to 91. Since p >0.05 theH0hypothesis is accepted. The absent structure in the ratio image in Figure5.31(a) is therefore expected as well as the perfect match between the observed frequencies and the theoretical Rayleigh distribution in Figure5.31(b).
In Figure5.30(b) is shown the restoration of the synthetic five grey-levels SAR data in Figure5.1 (b). In the restored result the transitions between homoge-neous regions are relatively sharp, the homogehomoge-neous regions appear uniform and there is a high degree of detail preservation, which gives the first impression that the restoration is good. This is supported in the ratio image in Figure5.32(a), where there is only faint evidence of structure and in the corresponding his-togram in Figure5.32(b), where the observed frequencies are almost Rayleigh distributed. However, with reference to the statisticsz= 1.023,S2(z) = 0.310 andχ2(53) = 156 derived from the ratio image in TableA.2(b), theH0 hypoth-esis is rejected.
Figure5.33illustrates the restoration of the C-band VV-polarized EMISAR data in Figure 5.2 using (5.8). Again the transitions between homogeneous regions are relatively sharp and homogeneous areas have a smooth appearance. The statistics of the corresponding ratio image in Figure 5.34(a) are z = 1.0556, S2(z) = 0.4781 andχ2(63) is estimated to 38855. H0is hereby rejected and the bias between the observed histogram and the theoretical Rayleigh distribution is visualized in Figure5.34(b). The pixels causing this disturbance are located in the intervals 0–0.6 and 2–3.5 and indicated with blue and green colours in Figure 5.34(a). Obviously the test statistics based on ratios of the C-band VV-polarized EMISAR data are worse than statistics based on ratios of the synthetic five-grey-levels SAR data above. This is due to the high number of discontinuities in the C-band VV-polarized EMISAR data. For a description of how discontinuities affect the restorations refer to Section5.3.
A characteristic feature in the Figures5.30,5.30(b) and5.33is the clutter. This clutter is not due to artifacts, but is reflecting the original structure in the SAR data, which is preserved by the annealing algorithm. Another characteristic feature is the salt-and-pepper like appearance caused by single pixels. Some of these single pixels are frozen artifacts generated by the algorithm. This is due to the inhomogeneous Markov chain, which is too short to ensure thermal equilibrium between temperature decrements. Others of these single pixels were originally outliers that are preserved as such by the annealing. This is partly due to an inaccuracy in the optimized parameters, partly due to (5.8), which is not perfect in terms of modeling the actual shape of U(xi). Finally, a number