to what extent discontinuities interfere with the statistics of the ratio images.
This is done by applying the optimal parameter settings, derived from the ho-mogeneous synthetic one grey-level SAR data, on the synthetic five grey-levels SAR data in Figure 5.1 (b). Here β is a factor that governs the correlation between neighbouring pixels. The weighting factor wc depends on the orienta-tion of the clique c, see Figure4.2. The GRF is said to be isotropic if wc = 1 [46]. Finally, in order to evaluate the model performances on real SAR data, the optimal parameter settings are applied on the C-band VV polarized EMISAR data in Figure5.2.
A description of the synthetic SAR data and tabled statistics derived from the ratio images is given in AppendixA.
5.3 Gaussian a priori model
The anisotropic Gauss-Markov Random Field (GMRF) model
p(xi|xj, j∈Ni) = 1
√2πσ2exp{− 1
2σ2U(xi)}, (5.1)
where
U(xi) ={xi−µ−βX
c∈C
ωc(xj−µ)}2,
and
µ= 1 Nc
Nc
X
i=1
xi,
whereNc is the 9 pixels involved in the configuration.
The model (5.1) is a simple choice in terms of describing continuous phenomena in a 2ndorder neighbourhood system. The model (5.1), here involving the eight pair-site interactions, is known as a conditional autoregressive (CAR) model.
The energy function isU(xi) andσ2 is the variance of the conditional distribu-tion given by the neighbours ofxi. The mean value isµandC denotes in this
Figure 5.2: The C-band VV-polarized EMISAR amplitude data covering Lade-gaards Enge 3 June 1997. The data are histogram equalized using the beta distribution with the parameters 3 and 2.
context the set of the pair-site cliques [6]. The quantityβ is a factor that con-trols the dependence betweenxi and its neighbours. The weights of the clique potentials as they appear in Figure4.2areωc.
The distribution (5.1) is symmetrical while the amplitude data to be used are Rayleigh distributed. Because of the positive skewness the mode of the Rayleigh distribution is not at the mean. Unfortunately this skewness has the effect that (5.1) performs badly in terms of preserving mean levels in amplitude data. In other words the restored mean levels will be smaller than the true mean levels.
This again implies that the mean value z of the ratio between the SAR data and the restored data becomes larger than 1.
According to the concept of the Boltzmann distribution (4.1) and the prin-ciples of thermodynamics, particles attempt to arrange themselves towards a configuration that has the lowest energy. Consequently we are searching for a ˆ
xi= arg maxxip(xi|xj, j∈Ni) that reflects that the neighbourhood configura-tion is in a minimum state of energy, that is e.g. to say the sum of absolute dif-ferences between the centre pixelxiand its neighbours has reached a minimum.
However, this is not always the case with the Gaussian model where (5.1) has the disadvantage of sometimes choosing estimates ofµin the opposite direction
5.3 Gaussian a priori model 71
(a)
(b)
Figure 5.3: (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 Gauss prior and ICM with the optimized parameters α = 2, β = 3, n = 9. The data are stretched linearly between their mean±3 std.
(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 5.4: (a) The ratio between the homogeneous synthetic one grey-level SAR data in Figure 5.1 (a) and the restored SAR data in Figure 5.3 (a) using the Gaussian prior and the ICM algorithm and (b) a comparison of the histogram of the ratio image and the theoretical Rayleigh distribution.
5.3 Gaussian a priori model 73
(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 5.5: (a) The ratio between the synthetic five grey-levels SAR data in Figure 5.1(b) and the restored data in Figure 5.3 (b) using the Gauss prior and ICM. In (b) is the histogram of the ratio image shown together with the theoretical Rayleigh distribution.
Figure 5.6: Restored C-band VV-polarized EMISAR data in Figure5.2after 9 iterations using a Gaussian prior and ICM withα= 2, andβ = 3. The data are histogram equalized using the beta distribution with the parameters 3 and 2.
of what is to be expected given the neighbours of pixel xi. This occurs when the mean of the neighbourhood is larger than xi and is a consequence of the fact that it is not the sum of the absolute values of the differences between xi
and its neighbours that enter the energy function in (5.1).
The Gaussian prior in the choice of (5.1) compensates by choosing an estimate ˆ
xi< µ, which is inconsistent with the principles of thermodynamics. This bias is significant forβ <0.2 and smalln. For largeβ andnthe effect is negligible.
Another disadvantage of (5.1) is that too much weight is put upon pixels that diverge. That is specific features in a neighbourhood configuration such as e.g.
edges are not easily preserved. Or the opposite case where an outlier severely can change the local characteristics. It therefore follows that discontinuities in the restored data will have a smooth appearance in the choice of (5.1).
An unfavorable side effect of the previously mentioned problem concerning di-verging pixels in a neighbourhood configuration is that mean values of areas close to discontinuities will be affected too. That is to say the areas with small am-plitudes will increase their mean value whereas areas with large amam-plitudes will decrease their mean value. This side effect, which is amplified by the iterating nature of the ICM-algorithm, gets more pronounced as the iterations proceed
5.3 Gaussian a priori model 75
(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 5.7: (a) Ratio of the C-band VV-polarized EMISAR data in Figure5.2 to the restored data in Figure 5.6 using the Gaussian prior and ICM and (b) a comparison of the observed and the theoretical ratio distributions. The blue and green areas in (a) correspond to the pixels in the range 0–0.6 and 1.8–3.5 in (b) that exceed the theoretical curve.
and for n → ∞ all discontinuities eventually will be smeared out leaving a single grey-level image. On the other hand the property of (5.1) of smearing out discontinuities makes (5.1) a good a priorimodel in terms of reproducing homogeneous areas.
Figure5.3(a) shows the optimal restoration of the synthetic one grey-level SAR data in Figure 5.1(a) usingα= 2,β = 3 and n= 9. The characteristic para-meters from the restoration are listed in TableA.1(a). The mean valuez of the ratio between the synthetic one grey-level test data and the restored data is 1.108 and its variance S2(z) = 0.331. As we see z >1 which is in perfect agreement with the above mentioned inconsistency between (5.1) and the Rayleigh distri-bution. The chi-square statistics χ2(75) is 1216. Since we are interested in a significance level of 0.05 we may reject the null hypothesisH0. In other words there is a significant difference between the expected and the observed results which is evident examining the histogram in Figure5.4(b). It is therefore safe to say that (5.1) fails in terms of preserving the level of homogeneous regions.
In Figure 5.3 (b) is presented the restoration of the synthetic five grey-levels SAR data shown in Figure5.1 (b). With reference to TableA.1(b)z= 1.091, S2(z) = 0.406 and the chi-square statisticsχ2(77) is 5496 which means that the H0hypothesis is rejected. This supposition is also supported by the ratio image in Figure5.5(a), which does not look homogeneous, and the corresponding his-togram in Figure5.5(b). Obviously the observed histogram and the theoretical curve are not coincident as one would expect given the restoration was perfect.
The frequencies causing the observed histogram to exceed the theoretical curve are in the range 0.3–0.4 and 1.6–3.5.
After the restoration and as expected the image appears blurred with a smooth transition from areas with small amplitudes to areas with large amplitudes.
The previously mentioned tendency of (5.1) to pick low estimates ofµ has not proven to have any significant effect on the restored data in Figure 5.3 (b).
All discontinuities in Figure 5.1 (b) are badly preserved. Areas with small amplitudes close to areas with larger amplitudes in Figure 5.1 (b) correspond to frequencies concentrated in the range 0.3–0.4 in Figure5.5(b). These outlier frequencies are caused by the former mentioned tendency that areas with small amplitudes close to a discontinuity will increase their mean values during the iterations and therefore the ratio will be smaller than 1. Subsequently pixels or areas with large amplitudes in the immediate neighbourhood of areas with smaller amplitudes will decrease their mean values by the same amount but the ratio will now be larger than 1. These pixels are reflected in frequencies in the range 1.6–3.5 in Figure5.5(b).
Finally, the optimal setting from the synthetic one grey-level SAR data is ap-plied upon the EMISAR data illustrated in Figure 5.2. Figure 5.6 shows the