5.6 Gamma a priori models
Except for the exponential prior thea priorimodels described so far have been symmetrical. However, as mentioned earlier, the conditional distribution of the observed amplitude data is the Rayleigh distribution, which is asymmetric. In the need of a priori models that model the skewed distribution of the data a new model is presented. This model utilizes the Gamma distribution, which provides a good basis for modeling the SAR amplitude data.
The Gamma a priorimodel has the form
p(xi|xj, j∈Ni)∝ U(xi)k−1
where Γ is the Gamma function and
U(xi) =|pi−xi|+βX
c∈C
ωc|pi−xj|,
wherepi is the perturbed value.
The constant k governs the peak of the distribution and U(xi) is the energy function involving the single-site cliquexi and the eight pair-site interactions in the configuration. The constantk is estimated so that (5.4) models the actual distribution ofU(xi). The set of the eight pair-site interactions in the 2ndorder neighbourhood configuration is denoted C. The quantity wc is a weighting factor between the centre pixelxi and its neighboursxj depending on the clique is horizontal, vertical or diagonal, see Figure4.2.
As demonstrated earlier (5.3) is more convincing in terms of preserving disconti-nuities than (5.1) and (5.2), because it has its mode closer to the median than to the mean of the neighbourhood configuration. In order to utilize that advantage the energy function of (5.3) is implemented in (5.4).
The value νi represents the energy U(xi) in accordance to a predefined mean level of the local configuration. This is a crucial step in the development of the algorithm because νi governs the shape of the energy function. Hence, using β = 0.5, two versions of the Gammaa priorimodel are proposed.
5.6.1 Gamma mean a prior model
The first version we denote the Gammamean priorand here the estimate of the local energy level νi is defined as the estimated mean mof the pixels involved in the neighbourhood configuration. Letνi be given by
νi=|xi−m|+βX
whereNcis the 9 pixels involved in the configuration. As was the case with the previous a priori models (5.5) has the disadvantage of not being satisfactory in terms of reconstructing e.g. edges and discontinuities. For a more detailed description of that phenomenon refer to Section5.5.
In Figure 5.18 is illustrated to what extent (5.5) approximates U(xi) in the initial state of the restoration of the synthetic SAR data in Figure5.1 (a). In Figure5.18the observed histogram of U(xi) is shown together with the pdf of the optimized mean a priorusing the parameters α = 2, β = 0.5, n= 9 and k= 7. Although the mode of (5.5) is not coincident with the mode of observed histogram, its pdf possesses a positive skewness like the observed frequencies of U(xi).
The restoration of the synthetic one grey-level SAR data in Figure5.1(a), us-ing (5.5) and the optimal parameter settus-ing above, is illustrated in Figure5.19.
Statistics derived from the ratio between the synthetic one grey-level SAR data and the restored data are listed in Table A.1(a). Reading from the table we find z= 1.0554, S2(z) = 0.2901 and the test statistics χ2(74) is estimated to 797. Since p <0.05 H0 is rejected. The structure in the ratio image in Fig-ure5.20(a) is not evident, however, according to the histogram in Figure5.20(b) the observed frequencies in the range 0.95–1.05 and 1.05–3.5 are exceeding the theoretical distribution. The bias in the interval 1.05–3.5 is due to the same conditions accounted for in Section5.5, namely that (5.5) has its mode closer to the median than to the mean and consequentlyz >1. The mechanism causing the small peak in the interval 0.95–1.05 is also explained in Section5.5.
5.6 Gamma a priorimodels 93
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(b)
Figure 5.19: (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 mean prior and ICM with the optimized parameters α= 2,k= 7, β = 0.5 andn= 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 5.20: (a) The ratio between the homogeneous synthetic one grey-level SAR data in Figure 5.1(a) and the restored SAR data in Figure5.3 (a) using the Gamma mean prior and ICM. In (b) is the histogram of the ratio image shown together with the theoretical Rayleigh distribution.
5.6 Gamma a priorimodels 95
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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 5.21: (a) The ratio between the synthetic five grey-levels SAR data in Figure 5.1(b) and the restored data in Figure5.19(b) using the Gammamean prior and ICM. In (b) is the histogram of the ratio image shown together with the theoretical Rayleigh distribution.
Figure 5.22: The restored C-band VV-polarized EMISAR data in Figure 5.2 after 9 iterations using the Gamma mean prior and ICM with the optimized parameters α= 2,β = 0.5 andk= 7. The data are histogram equalized using the beta distribution with the parameters 3 and 2.
In Figure5.19(b) is shown the restoration of the synthetic five grey-levels SAR data in Figure 5.1 (b). With reference to Table A.1(b) the statistics of the corresponding ratio image in Figure5.21(a) are z= 1.0598,S2(z) = 0.311 and χ2(54) = 1404. Again H0 is rejected, which is in perfect accordance with the unwanted structure in Figure5.21(a). It is striking that the histograms derived from the one grey-level and five grey-levels SAR data in the Figures5.20(b) and 5.21(b) seem very similar. This implies that (5.5) is well suited for preserving structure.
The restored C-band VV-polarized EMISAR data are presented in Figure5.22 and the data prior to the restoration are shown in Figure 5.2. The ratio of the EMISAR data to the restored data is illustrated in Figure 5.23(a), where z= 1.1250,S2(z) = 0.5375 andχ2(54) is>5×105. These values are far from the parameters for the ideal ratio image derived in Section4.3and naturallyH0
is rejected. The frequencies responsible for the bias between the observed and the theoretical distribution are located in the intervals 0–0.6, 0.95–1.05 and 2–
3.5 in Figure5.23(b). The corresponding pixels, which account for the structure, are indicated with blue, red and green colours in Figure 5.23(a). The bias is due to the same reasons given above, which caused the disturbances in the ratio
5.6 Gamma a priorimodels 97
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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 5.23: (a) The ratio between the C-band VV-polarized EMISAR data in Figure 5.2 and the restored data in Figure 5.22using the Gammamean prior and ICM and (b) a comparison of the histogram of the ratio image and the theoretical Rayleigh distribution. The red, blue and green areas in (a) indicate pixels located in the intervals 0.95–1.05, 0–0.6 and 2–3.5.
images in the Figures5.20(a) and5.21(a).
5.6.2 Gamma pixel a prior model
The second version we call the Gammapixel priorand here the estimated energy levelνi simply is defined as
νi=βX
c∈C
wc|m−xj|). (5.6)
where m=xi.
In other words the value of the centre pixel xi itself is taken to represent the mean of the neighbourhood configuration. The approximation to U(xi) using (5.6) is shown in Figure5.18. Again the synthetic one grey-level SAR data in Figure 5.1 (a) are used in the process of deriving the observed histogram of U(xi) in the initial state of the restoration. As it appears the mode and the long tail of the observedU(xi) are well represented by (5.6) using the optimized parametersα= 2, β= 0.5,n= 9 andk= 5.
These parameters from the fine tuning of (5.6) are also used in the restoration of the synthetic one grey-level SAR data in Figure5.1(a) shown in Figure5.24.
Referring to Table A.1(a) the statistics of the corresponding ratio between the synthetic one grey-level SAR data and the restored data are z = 1.0440, S2(z) = 0.2813 and χ2(76) = 573. This is far from within the range of ac-ceptable deviation and we conclude thatH0 is rejected. The rejection ofH0 is supported by the histogram in Figure5.25(b) where the observed frequencies do not match the theoretical Rayleigh distribution. Here the frequencies exceeding the Rayleigh distribution are located in the intervals 0.95–1.05 and 1.05–3.5.
This bias is due to the same condition as explained above concerning themean prior (5.5). However, in the choice of (5.6) these effects are less pronounced because the reference levelmnow isxi instead of the mean of the pixels in the neighbourhood configuration.
We now turn to the synthetic five grey-levels SAR data in Figure5.1(b), which restored equivalent is illustrated in Figure5.24(b). Referring to the information provided in TableA.1(b) the meanzof the corresponding ratio image is 1.0450,
5.6 Gamma a priorimodels 99
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Figure 5.24: (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 Figure5.1(b). The restorations are performed using the Gammapixel priorand ICM with the optimized parametersα= 2,k= 5, β= 0.5 andn= 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 5.25: (a) The ratio between the homogeneous synthetic one grey-level SAR data in Figure5.1(a) and the restored SAR data in Figure 5.24(a) using the Gamma pixel prior and ICM. In (b) is the histogram of the ratio image shown together with the theoretical Rayleigh distribution.
5.6 Gamma a priorimodels 101
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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 5.26: (a) The ratio between the synthetic five grey-levels SAR data in Figure 5.1(b) and the restored data in Figure5.24(b) using the Gammapixel prior and ICM. In (b) is the histogram of the ratio image shown together with the theoretical Rayleigh distribution.
Figure 5.27: The restored C-band VV-polarized EMISAR data in Figure 5.2 after 9 iterations using the Gamma pixel prior and ICM with the optimized parameters α= 2,β = 0.5 andk= 5. The data are histogram equalized using the beta distribution with the parameters 3 and 2.
S2(z) = 0.2970 andχ2(55) is 748. HerebyH0 is rejected and we conclude that some factor other than chance is operating for the deviation. The frequencies which account for the bias are located in the intervals 0.95–1.05 and 1.05–3.5 according to the histogram in Figure 5.26(b). This bias is due to the same conditions that caused the bias in the histogram for the one grey-level ratio image in Figure 5.25(b). The small differences between the ratio images and the histograms of the synthetic one grey-level SAR data and the five grey-levels SAR data in the Figures5.25(a)–(b) and Figures5.26(a)–(b) indicate, that edges and discontinuities are well preserved using (5.6).
Figure5.27illustrates the restoration of the C-band VV-polarized EMISAR data in Figure 5.2using (5.6). The ratio of the EMISAR data to the restored data is illustrated in Figure5.28(a) and the corresponding statistics arez= 1.1037, S2(z) = 0.5092, andχ2(54)>1.7×105. This rejectsH0, which is supported by the histogram in Figure5.28(b).
A comparison of the statistics in the TablesA.1(a)–(b) suggests that the expo-nentiala priorimodel (5.2) using ICM has the best over-all performance. This
5.6 Gamma a priorimodels 103
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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 5.28: (a) The ratio between the C-band VV-polarized EMISAR data in Figure 5.2 and the restored data in Figure 5.27using the Gamma pixel prior and ICM and (b) a comparison of the histogram of the ratio image and the theoretical Rayleigh distribution. The red, blue and green areas in (a) indicate pixels located in the intervals 0.95–1.05, 0–0.6 and 2–3.5.
is due to the energy function, which makes (5.2) superior in terms of preserving mean levels of homogeneous regions. Unfortunately, as Figure5.8(b) indicates, (5.2) is less convincing when it comes to preserving discontinuities. According to Figure 5.24 (b) and the Tables A.1(a)–(b) the Gamma pixel prior (5.6) is superior when it comes to the preservation of structure and sharp transitions but less satisfactory in terms of preserving mean levels of homogeneous regions.