Determination of the Bayes' Parameters

un-derwent changes between the dierent acquisition dates, specic parameters change. Especially, the automatic adaptation of the parallelogram extent can be observed. For example, for build-ing B12, the detected parallelogram lengths and heights dier from date to date, lettbuild-ing expect changes. Indeed, depending on the acquisition date and the conguration, either the extracted parallelogram lengths or the heights or both dier from the original building parameters. The case of this building is however more complex, as reconstruction occurs in a short time span after demolition (cf. Table 6.4). For building B7, already demolished in October 2012, the standard deviation of the disparity values within the extracted parallelograms is always higher as for the other not demolished buildings. Finally, building B11 is still standing in October 2012 but de-molished in January 2013. Even if the extracted parallelograms have the same original length, the height parameter undergoes a higher change, and the standard deviations become higher from January 2013. A probabilistic analysis of the changes for zone B is given in Section 6.4.

6.4. Results of Change Detection 143

Table 6.18: Building parameters of zone B at different dates

B3 B5 B7 B11 B12

Reference l(m) 90 90 50 70 90

h(m) 32 33 16 34 34

October 2012

36^◦/22^◦

l0_radarg 89 89 49 68 91

lh_radarg 89 89 49 68 82

h_radarg 28 29 34 28 27

σ0_radarg 0.0065 0.0114 0.0257 0.0132 0.0131 σh_radarg 0.0152 0.0164 0.0275 0.0174 0.0136

47^◦/36^◦

l0_radarg 91 91 49 68 68

lh_radarg 91 91 49 68 68

h_radarg 31 31 25 26 32

σ0_radarg 0.0025 0.0025 0.0092 0.0039 0.0049 σh_radarg 0.0050 0.0051 0.0099 0.0066 0.0079

January 2013

36^◦/22^◦

l0_radarg 89 91 49 68 91

lh_radarg 89 91 23 68 65

hradarg 31 31 29 21 27

σ0_radarg 0.0082 0.0084 0.0243 0.0209 0.0133 σ_hradarg 0.0201 0.0159 0.0419 0.0208 0.0204

47^◦/36^◦

l0_radarg 89 91 49 70 93

lh_radarg 89 91 23 70 93

hradarg 32 31 24 14 24

σ0_radarg 0.0022 0.0038 0.0087 0.0054 0.0036 σ_hradarg 0.0062 0.0056 0.0179 0.0065 0.0049

October 2014

36^◦/22^◦

l0_radarg 91 91 49 70 26

l_hradarg 91 91 49 70 26

hradarg 43 26 32 19 22

σ0_radarg 0.0079 0.0093 0.0304 0.0263 0.0402 σh_radarg 0.0196 0.0166 0.0220 0.0150 0.0510

47^◦/36^◦

l0_radarg 91 91 49 70 91

l_hradarg 91 91 49 70 91

hradarg 32 34 18 10 28

σ0_radarg 0.0009 0.0042 0.0067 0.0054 0.0039 σh_radarg 0.0066 0.0061 0.0062 0.0044 0.0045

20% shorter than the original building lengthlGIS, the probability for this building to be categorized into class (2) increases.

• M H: this is the equivalent ofM Lfor the height, chosen to be here also20%of the original building height.

• M R: this parameter allows the dierentiation of class (3) from classes (1) and (2). Here also, a dierence of 20% between l0radarg and lh_radarg would be considered as a building change corresponding to class (3). M Ris greater than 1, specifying thatl_hradarg should be shorter thanl_0radarg.

• M_{ST D0}: this parameter is the mean standard deviation of the disparity values withinP₀. In the ideal case, all disparity values within P0 are equal to0and their standard deviation is 0. However, depending on the conguration and also on the building surroundings that inuence the disparity calculation, the standard deviation of the disparity values is not 0 in reality. This parameter is determined by averaging the standard deviation within P₀ for all buildings of zone A, for each conguration. The parameter set M_{ST D0} =

[0.0102,0.0044,0.0026,0.0053] pixel is obtained for the congurations 36^◦/22^◦, 47^◦/36^◦, 52^◦/42^◦ and 42^◦/29^◦, respectively. For larger intersection angles, the standard deviation is higher, showing that the matching of homologous points is more ambiguous due to the radiometric dierence.

• MST Dh: this is equivalent toMST D0 for parallelogramPh. The set of parametersMST Dh = [0.0158,0.0048,0.0046,0.0078] pixel is obtained for the congurations 36^◦/22^◦, 47^◦/36^◦, 52^◦/42^◦ and 42^◦/29^◦, respectively. It is observable thatM_{ST Dh} is higher thanM_{ST D0}, the high intensity of the double-bounce line yieldingP0 permitting a more reliable matching.

• M_{ST D}: this is equivalent of the two previous parameters for class (4). The building is not present anymore, thus there is no double-bounce line. The disparity values around the original double-bounce line are not homogeneous anymore. Besides, the extracted P₀ does not need to correspond to the original building position. The standard deviation of the disparity values withinP0 is therefore higher. In this work,MST D is set to0.0286pixel and 0.0098 pixel, for the congurations 36^◦/22^◦ and 47^◦/36^◦, respectively. They correspond to the mean values of σ0radarg of building B7 above the three acquisition dates for both congurations (cf. Table 6.18). Indeed, B7 is the only building that is demolished from the beginning of the analysis. This parameter could be enhanced if more training data were available.

Those parameters correspond to the expected means of the normal distributions for each vari-able of the conditional probability. The parameters l_InSAR and h_InSAR vary for each building depending on its original dimensions, but the other parameters are xed for the dened building classes. The seven remaining parameters correspond to the standard deviations of the normal distributions for each variable. They are also dened in this work based on the results obtained for the training zone A.

• σ_l0

radarg: this parameter represents the deviation of the estimated building length compared to the original length. It corresponds to the precision with which the building lengthl_0radarg is retrieved by radargrammetry. In this work, it is set to be the standard deviation between the original GIS length, considered as desired value, and the obtained building lengths, for a specic conguration. It was calculated for each conguration over all building lengths, leading each time to σl0radarg = 2 m.

• σ_hradarg: this parameter is equivalent toσ_l0

radarg for the height estimate and is determined in the same way, using all buildings of zone A. This leads to σh_radarg = [4.5,5.5,5.9,9.1]m for the congurations 36^◦/22^◦,47^◦/36^◦,52^◦/42^◦ and42^◦/29^◦, respectively.

• SR: considering only l0radarg and lh_radarg from zone A, SR should be equal to 0, as it is the standard deviation of the ratio of both parameters. Yet in this work, it is set to 0.25, in order to give the possibility that the building does not show a change even if l_0radarg and l_hradarg dier of a small amount, for example due to disturbing objects in one layover corner.

6.4. Results of Change Detection 145

• SST D0: this parameter represents the standard deviation of the standard deviation values within parallelogram P₀. It is calculated over all buildings of zone A, for each cong-uration. The obtained values are SST D0 = [0.0014,0.0003,0.0004,0.0012] pixel for the congurations36^◦/22^◦,47^◦/36^◦,52^◦/42^◦ and 42^◦/29^◦, respectively. It is observable, as for M_{ST D0}, that the conguration presenting the largest intersection angle shows the highest standard deviation values.

• S_{ST Dh}: it is the equivalent ofS_{ST D0} for parallelogramP_h, yieldingS_{ST Dh}= [0.0018,0.0009, 0.0005,0.0006]pixel for the congurations36^◦/22^◦,47^◦/36^◦,52^◦/42^◦ and42^◦/29^◦, respec-tively.

• ∆Land∆SR: these two parameters are used for class (4), in order to give an imprecision on the parametersl_0radarg andl_hradarg. Indeed, for class (4), no parallelogram is recognizable in the disparity map but the developed algorithm extracts in any case two parallelograms from the data. Their dimension and length ratio cannot be predicted with high precision, and can have a high discrepancy to the original values. These parameters are set empirically to ∆L= 2 m and∆SR= 0.5.

In this work, the building orientationαis considered as exact, as coming from GIS data, and has not been evaluated. Furthermore, the accuracy of the GIS data has not been taken into account for two reasons. First, the given planimetric accuracy of the GIS data (2.5 m) corresponds to the absolute accuracy of the building corner position in the RGF93 reference frame with the associated Lambert projection. In this work, the length is considered, which represents a relative measurement between two corners. Second, the height specication is also given with an absolute precision of 2.5m. However, for change detection, it is important to know with which accuracy the developed algorithm can estimate the relative building height. This is why the results of the training zone A are used for setting the uncertainties of the parameters of the conditional likelihoods.