Desplazamiento material

For simplicity, we must first consider a terrain surface that satisfies a Lambertian model, which means that the surface is an ideal diffusely and consistently reflecting surface. Under given solar illumination L, azimuth angle 𝜏𝜏 and zenith angle 𝜎𝜎, the intensity of hill shading image 𝐼𝐼(𝑥𝑥, 𝑦𝑦) can be expressed as [64]

𝐼𝐼(𝑥𝑥, 𝑦𝑦) = 𝐿𝐿𝑝𝑝cos𝜏𝜏sin𝜎𝜎 + 𝑞𝑞sin𝜏𝜏sin𝜎𝜎 + cos𝜎𝜎_{(𝑝𝑝2+ 𝑞𝑞2+ 1)1/2} (4.1) where

𝑝𝑝 =_∂x∂ 𝑉𝑉𝐻𝐻(𝑥𝑥, 𝑦𝑦)

𝑞𝑞 =_∂y∂ 𝑉𝑉𝐻𝐻(𝑥𝑥, 𝑦𝑦)

𝑉𝑉𝐻𝐻(𝑥𝑥, 𝑦𝑦) is the map of height or DEM.

The Surface Normal vector is determined by slope S, the gradient angle of the terrain surface, ranging from 0° to 90°, and aspect A, the direction maximum gradient (given as a compass bearing), ranging from 0° to 360°, illustrated in Figure 4.1. The slope angle is defined as

tan 𝑅𝑅 = �𝑝𝑝2_{+ 𝑞𝑞}2 (4.2)

The aspect angle is defined as

tan ܣ =݌ ݍ

(4.3)

Figure 4.1 Definition of slope and aspect

Conversely,

݌ = cos ܣ tan ܵ ݍ = sin ܣ tan ܵ

(4.4)

If we take Equation (4.4) into Equation (4.1), the relationship between the terrain surface brightness (intensity) I and surface slope and aspect is [65]

ܫ = ܮ(cos ߪ cos ܵ + sin ߪ sin ܵ cos (߬ െ ܣ)) (4.5) For 8 bits raster data, L is given as 255 but it can be set to 1 and then, Equation (4.5) is expressed in a normalised form without L and the outmost brackets. Compared to real illumination conditions in aerial and satellite images, this model ignores scattering sky light and overcast shadows relating to the geometry of topography as well as the spectral variation of different terrain surface materials. Most software packages for automatic generation of simulated hill shading images from DEMs are based on the Lambertian model of ideal diffusing surfaces, such as in ArcGIS, ERMapper and MATLAB. In our experiments, the Lambertian model has been applied to simulate hill shading images from DEM data under various solar illumination conditions in quantitative analysis of their effect on image matching. According to Equation (4.5), the intensity of a hill shading image is determined by two factors, the topography (DEM), defined by slope S and aspect A, and the illumination orientation, defined by zenith ߪ and azimuth ߬. In order to separate

0° 90° 180° 270° Slope Aspect 38

the effects of topography from that of illumination, we propose a 3D space coordinates system defined by Slope, Aspect and Intensity and denote as SAI. The SAI space facilitates our investigation of the effects of illumination on hill shading images. While every pixel in a hill shading image derived from a DEM has its own intensity, slope and aspect, pixels of the same slope and aspect share the same intensity forming a 3D spatial distribution as a curved surface, named as SAI surface. For a given illumination geometry to the DEM, the slope and aspect are the variables while the zenith and azimuth are constants. Thus, the Equation (4.5) in normalized form can be re-written as,

𝐼𝐼(𝑅𝑅, 𝐴𝐴) = 𝑚𝑚 cos 𝑅𝑅 + 𝑖𝑖 sin 𝑅𝑅 cos (𝜏𝜏 − 𝐴𝐴) (4.6) where 𝑚𝑚 = cos𝜎𝜎, 𝑖𝑖 = sin𝜎𝜎

In Equation (4.6), m, n and 𝜏𝜏 are constant coefficients, and the intensity I of each pixel is determined by the slope and aspect at the pixel’s position.

An important property of SAI space is that the shape of the SAI surface is determined only by illumination conditions and is independent of the DEM. According to Equation (4.6), the intensity I is a trigonometric function of variables S and A, and the shape of SAI surface is uniquely determined by the coefficients m and n which are calculated from the zenith and azimuth angles. This means that under the same solar zenith and azimuth angles, the shape of the surface is invariant or unique for the whole range of slopes and aspects while for a DEM that covers only partial ranges of slopes and aspects, the theoretical SAI surface is partially populated but the shape of the SAI surface is unaltered. This property is clearly illustrated in Figure 4.2. From two different DEMs (Figure 4.2 (a)-(b)), hill shading images (Figure 4.2 (c)-(d)) under the same illumination condition of 𝜎𝜎=60°, 𝜏𝜏=315° are generated. Figure 4.2(e) shows the SAI space in which the blue surface represents the theoretical SAI surface for the given zenith and azimuth angles while the red and green dots plot the SAI surfaces of images Figure 4.2(c)-(d)

respectively; both the red and the green surfaces conform exactly to the blue surface in different parts of it.

Figure 4.2 Hill shading images produced from two DEMs (a) and (b) under the same illumination condition (࣌=60°, ࣎=315°). (c)-(d): Hill shading image from (a) and (b) respectively. (e): The 3D view of

SAI surface in which, the red dots are from (c) and the green from (d), and they both cover parts of the

theoretical surface under the given illumination shown as the blue surface.

On the other hand, under different illumination conditions, the shape of SAI surfaces will be altered even when the DEM remains the same. As shown in Figure 4.3, (a) and (c) are generated from the same DEM, but under different illumination conditions: ߪ=60°, ߬=315° for (a); and ߪ=45°, ߬=125° for (c). The corresponding

SAI surfaces presented in Figure 4.3(b) and Figure 4.3(d) are of different shapes as

a result of the differing illumination conditions. Therefore, judging from 3D shapes of the SAI surfaces, it is possible to group the images under the same illumination condition.

(e)

(a) (b) (c) (d)

Another key property of SAI space is that the shape of a SAI 3D surface is invariant to image translation and rotation, because such image transformation is independent of the relationship between image intensity and illumination conditions. As shown in Figure 4.4, a hill shading image under the illumination condition: ߪ = 60°_{, ߬ = 315}° _{is presented in Figure 4.4(a) and its SAI surface in} Figure 4.4(b). Figure 4.4(a) was then shifted 10 pixels to the right and downwards as shown in Figure 4.4(c) together with its SAI surface in Figure 4.4(d). The shape of SAI surface in Figure 4.4(d) remains unchanged compared to Figure 4.4(b). Furthermore, the initial image Figure 4.4(a) was rotated 90° clockwise, and the resultant image Figure 4.4 (e) and the corresponding SAI surface Figure 4.4(f) again show that the shape of SAI surface is unchanged. In summary, by introducing

SAI space, we are able to focus on the effect of variable illumination conditions on

hill shading images regardless of the DEM coverage, linear translation and rotation.

Figure 4.3 Hill shading images under different illumination conditions. (a)-(b): ࣌=60° and ࣎=315°, (c)- (d): ࣌=45° and ࣎=125° 7UDQVIRUP LQWR6$, 7UDQVIRUP LQWR6$, (a) (c) (d) (b) 41

Figure 4.4 The effects of image shift and rotation on SAI space. (a) Original hill shading image, (b) SAI surface of (a), (c) 10 pixel shifted right and downwards from (a), (d) SAI surface of (c), (e) 90° degree rotation from (a) and (f) SAI surface of (e).

The shape change of the SAI surface indicates that changes in illumination geometry can be differentiated by the SAI surface shapes. The SAI coordinates system will be used to investigate the illumination invariant property of the PC algorithm in the next section.