Constraints

4.5. Matching 89 building presented in Figure 4.6a. Whereas EN L and SSI show a better result for the two higher levels of the Gaussian pyramid, SM P I is better for the Laplacian pyramid, at all levels.

Concerning the indexes showing the quality of the edge preservation, bothESI^h andESI^v show better results for the Laplacian pyramid at the two highest levels. Having a look at Figure 4.5, edges are less smoothed than using the Gaussian pyramid. Similar observations can be made for EEI calculated for the double-bounce line. However, for the layover edge, EEI is higher for the Gaussian pyramid, at all levels. For matching, it is important that the start images, i.e. G4 or P4, at low resolution, have good quality, as their result is used as input for the next levels. A matching error at the highest level would propagate down to the lowest level, i.e. the full resolution image. Both speckle ltering and edge preserving indexes indicate that the use of Laplacian pyramid outperforms the use of Gaussian pyramid for the two highest levels, therefore the Laplacian pyramid is employed in this work. The lowest level G₁ of the Gaussian pyramid represents the original, unltered image. In comparison, the lowest levelP1 of the reconstructed Laplacian pyramid represents the speckle ltered image, which explains the poorer results of edge preservation indexes of the Laplacian at the lower levels.

Starting from the highest level (P₄ in Figure 4.5), disparities are calculated using the dened matching criterion. The disparity map of one level (P_i+1) is used as input for the disparity calcu-lation of the next lower level (Pi). Therefore, the disparity map of the highest level is upsampled by a factor of 2. Before upsampling, backmatching is performed, i.e. master and slave images are switched. Only pixels presenting the same -opposite- disparity values are transmitted to the next level. At the last level, at full resolution, the disparity map is ltered in order to smooth the nal disparity map. Indeed, the disparity values between neighbor pixels are expected to be homogeneous. Therefore, for each pixel in the created disparity map, the most occurring disparity value within the pixel's direct neighborhood (3x3pixels) is retained as its nal disparity value.

m

A

s

G‘

G‘‘

R‘R‘‘

θ_m

θ_s

da d dr

A A‘

da

dr

ζ

lA‘

lA‘‘

d

hA

a b

Ground representation

Slant representation

Figure 4.7: Epipolar constraint for building layover areas: (a) ground geometry; (b) slant range geometry of master imagem

three contributions, which are aligned in range direction: considering the master image m, those points are represented asG⁰ on the ground,Aon the facade andR⁰ on the roof (Figure 4.7a). In the slant range image, they are all represented at the same position A⁰, whereby A⁰ =G⁰ =R⁰ (Figure 4.7b). Due to the dierent heading and incidence angles of the master imagemand slave images, the facade pointAis imaged inA⁰⁰ in images(Figure 4.7b). A⁰andA⁰⁰, representations of the same facade point A, are not on the same azimuth line, and correspond to two distinct ground points, G⁰ = A⁰ and G⁰⁰ = A⁰⁰ (Figure 4.7a). In the same way, the roof information contained in A⁰ and A⁰⁰ comes from two distinct roof points, represented byR⁰ and R⁰⁰ in Fig-ure 4.7a, respectively. The distance dbetween A⁰ and A⁰⁰ is the disparity to determine. It can be split into two parts: dr, due to the dierence of incidence angles of both images, andda, due to the dierence of heading angles ζ of both images. Estimatingdr and da allows reducing the search area for matches along the range and the azimuth direction, respectively. Due to layover, matching of the two facade's contributionsA⁰ andA⁰⁰involves matching dierent ground and roof contributions, which do not represent the same scatterers. However, the facade contribution is the most important contribution of layover, and the most important one for retrieving disparity at facade points and consequently estimating building height. Hence in this work, the focus is put on matching facade points. Determining the maximum dr and da arising in layover areas gives an idea about the maximal matching error induced to ground and roof points, but also allows to determine the necessary dimensions of the search area for matching. Regarding some geometric considerations in Figure 4.7, the layover length l_A⁰ at point A⁰ in slant range for the master image m is expressed as:

l_A⁰ =h_Acosθ_m (4.16)

whereby h_A is the height of facade point A, and θ_m the incidence angle of master image m. A similar equation can be written for the layover length l_A⁰⁰_s in the slave image:

l_A⁰⁰_s =h_Acosθ_s (4.17)

4.5. Matching 91

w

l

w

l

da d dr

Figure 4.8: Dimensions of template (wm, lm) and search (ws, ls) windows for matching

However, the slave imagesis reprojected in the slant geometry of imagemduring coregistration, inducing a scaling eect on l_A⁰⁰_s described in (Goel & Adam 2012). The coregistered layover length l_A⁰⁰ at pointA⁰⁰ for the slave image corresponds therefore to:

lA⁰⁰= sinθm

sinθs

lA⁰⁰s (4.18)

Additionally considering the dierence of heading angle ζ between both images, the necessary widthda of the search window in azimuth direction for pointA is expressed as:

da=l_A⁰⁰sinζ = sinθm

sinθ_shAcosθssinζ (4.19) In this work, the building height h, and therefore facade points' heights h_A are searched for.

However, average building heights in a specic scene are assumed to be known. Depending of the considered area, land use maps are available and permit to evaluate a maximum building height for the area. For example, in nancial districts, high-rise buildings are frequent, whereby in residential districts, low-rise buildings are predominant. Besides, prior information about the building heights are available from the InSAR processing (cf. Section 3.6). Fixing h_A to h_max, which is the expected building height known from InSAR, yields the determination ofdamaxfrom Equation(4.19), i.e. to the maximum expected disparity in azimuth direction. da_max is therefore the epipolar constraint for matching in layover areas: the search window in the slave image has the dimension damax in azimuth direction. For sake of the implementation, a search window of dimension w_s = 2da_max+ 1 in azimuth dimension is dened in order to obtain symmetrical windows, centered on the point of interest, but the area of search for the maximum value of the matching criterion is then restrained to the correct half, as represented in Figure 4.8 in blue. In Section 6.3.2, a table depicts the values of damax depending on the available acquisitions and corresponding dierence of heading angles ζ, taking into account the expected maximum height value in the scene of interest.

Besides damax in azimuth direction, the maximum expected disparity between both images at layover location in range direction, dr_max, is determined. Regarding Figure 4.7, dr is expressed as:

dr= q

l²_A00−da²−l_A⁰ = q

l_A²00(1−sin²ζ)−l_A⁰ (4.20) ReplacinghAbyhmax in Equations(4.16)and (4.17), the maximum expected disparity value in range directiondr_max is dened. Those values are also given in Section 6.3.2 for dierent cong-urations. drmax allows to determine the necessary search area for matching in range direction.

Considering Figure 4.8, the second dimensionls of the search area has to follow the rule:

dr≤ l_s 2 −l_m

2 , i.e. l_smin = 2dr_max+l_m (4.21)

lsminis the minimum length of the search window so thatdrmaxcan be retrieved, using a template window in the master image of size l_m. Knowing the incidence angles of both master and slave, it is straightforward to deduce the direction of the disparity within the layover. Considering the symmetry of the search window with respect to the point of interest, the search for the correct match can also be restrained to half of the search window, represented in green in Figure 4.8.

Additionally to the previous half dened from the azimuth direction, this leads to a privileged quarter (upper left side in Figure 4.8) where layover matches can be found. Practically, a match is retained in this area if its criterion value is at least 95% of the highest criterion value of the whole search window. This allows to privilege matching of layover areas without completely neglecting the building surroundings, that could lead to displacements in the other direction.

As a conclusion, in areas aected by important layover, a consideration of the epipolar con-straint is mandatory for setting precisely the parameters for matching, which ensure to spare computation time and avoid wrong matching. Another constraint mentioned in Section 2.4.2 is not fullled in layover areas: the assumption of continuous disparity. At the border of layover areas, at near range, the disparity values are very high, corresponding to the building height.

The neighboring pixels, situated at nearer range, belong however to the ground, and show smaller disparities. Therefore, disparity jumps occur around layover areas, yielding discontinuous dis-parities.