Fig. 9.3 Global DEM from ASTER. Copyright METI/NASA, reprinted with permission from ERSDAC, http://www.ersdac.or.jp/GDEM/E/4.htm

GDEM covers land surfaces between 83◦N and 83◦S with estimated accuracy of 20 m at 95 % confidence for vertical data and 30 m at 95 % confidence for horizontal data. Although ASTER GDEM was found to contain significant anomalies and artifacts, METI and NASA decided to release it for public use with belief of its potential benefits outweighing its flaws and with expectation of improving it via the user community.

Research into DEM generation has also expanded to Mars by a combination of stereoscopic imagery and the Mars orbiter’s laser altimeter [58]. It can be foreseen that there will be the following improvements.

1.Further evaluation of geometric models to adaptively correct DEM errors caused by sensor platform attitude instability.

2.Development of more robust image matching algorithms for increasing automation and matching coverage.

For advanced stereo vision techniques that can be used for DEM generation from stereoscopic imagery, please refer to Chap. 2 of this book.

9.2.2 Accuracy Evaluation of DEMs

Quantitative evaluation of DEMs generated by satellite stereoscopic imagery can be conducted by comparing the reconstructed elevation values with GCPs collected by ground surveys or compared with corresponding DEMs generated by higher accuracy devices, such as LIDAR. In both cases, the measure of Root Mean Square Error

(RMSE) is used in assessing the DEM accuracy. It is defined as

where n is the number of assessed points in evaluation, and h is the height difference between the assessed DEM and GCPs or reference DEM at point i. The standard deviation of the height difference can be calculated in Eq. (9.2).

the measured data has a normal distribution, ±1σ gives 68 % level of confidence and ±2σ gives 95 % level of confidence in measurements. In the remote sensing community, the elevation accuracy is usually represented by RMSE with a level of confidence, for example, 20 m with LE95 (Linear Error with confidence of 95 %).

Possible error sources are considered in order to improve DEM accuracy. Error propagation can be tracked along the processes of DEM generation, for example, errors due to image matching and 3D reconstruction from the geometric modeling. Geometric modeling to recover elevations from stereo images is the well-known perspective projection from image coordinates to cartographic coordinates. The projection involves elementary transformations (rotations and translations), which are functions of the cameras’ interior and exterior parameters. This requires prior knowledge of the cameras, platforms, and cartographic coordinate systems. Based on such prior knowledge, a rigorous model based on collinearity conditions can be used for DEM generation. This geometric model integrates the following transformations [149, 150], where the xoy refers to the image coordinate system, rotations and translations are also referred to images, and the z-axis represents the direction perpendicular to the image common plane (usually same as the z-axis to the cartographic system).

•Rotation from the camera reference to the platform reference.

•Translation to the Earth’s center (refers to the cartographic coordinate system).

•Rotation that takes into account the platform variation over time.

•Rotation to align the z-axis with the image center on the Earth’s surface.

•Translation to the image center.

•Rotation to align the y-axis in the meridian plane.

•Rotation to have xoy (the image plane) tangential to the Earth.

•Rotation to align the x-axis in the image scan direction.

•Rotation-translation into the cartographic coordinate system.

With the geometric model and possible errors introduced by each step of the process, the integration of different distortions and their impact to the final elevation can be derived [147]. In the derivation, each of the model parameters is the combination of several correlated variables of the total viewing geometry. These include the following.

•The orientation of the image is a combination of the platform heading due to orbital inclination, the yaw of the platform, and the convergence of the meridian.

•The scale factor in the along-track direction is a combination of the velocity, the altitude and the pitch of the platform, the detection signal time of the sensor and the component of the Earth’s rotation in the along-track direction.

•The leveling angle in the across-track direction is a combination of platform roll, the viewing angle, the orientation of the sensor and the Earth’s curvature.

Mathematical models establishing geometrical relationships between the image and cartographic coordinates can be rigorous or approximate [64]. Rigorous modeling can be applied when a comprehensive understanding of the imaging geometry exists. However in many cases, it is difficult to obtain accurate interior and exterior parameters of the imaging system due to the lack of sufficient control. Therefore, approximate modeling has been developed for real-world use. Approximate models include direct linear transformation (DLT), self-calibration DLT, rational function models, and parallel projection [64]. In analyzing the accuracy potential of DEMs generated by high-resolution satellite stereoscopic imagery, Fraser [48] pointed out that a mathematical model, such as collinearity equations, needs to be modified for different settings in a rigorous model with stereo-bundle adjustments, while in the absence of the sensors’ attitude data and sensors’ orbital parameters, approximate models are recommended.

To improve the imaging geometry, researchers have paid special attention to the B/H ratio in acquiring satellite stereo pairs. A systematic investigation was conducted by Hasegawa et al. [67]. In their research, the impact of the B/H ratio to DEM accuracy was analyzed and the conclusion was made that B/H ratios ranging from 0.5 to 0.9 give better results for automatic DEM generation from stereo pairs. Li et al. designed an accurate model of the intersection angle and B/H ratio for a spaceborne three linear array camera system [100]. It was indicated that the B/H ratio was directly related to the DEM accuracy. A favourable imaging geometry can be achieved by a B/H ratio of 0.8 or more [48]. With SPOT-5, the viewing angle can be adjusted to tune the across-track B/H ratio between 0.6 and 1.1 and the along-track B/H ratio to around 0.8 [150].

From an application point of view, errors of terrain representation (ETR) are also taken into account since these may propagate through GIS operations and affect the quality of final products which use DEMs [31]. When interpolation is needed, the way to represent the terrain surface contributes to DEM accuracy. Chen and Yue [31] developed a promising method of surface modeling based on the theorem of surface. In their work, a terrain surface was defined by the first and second fundamental coefficients with information of the surface geometric properties and its deviation from the tangent plane at the point under consideration. It was demonstrated in their work that a good criterion for DEM accuracy evaluation should have included not only errors generated in 3D reconstruction from stereoscopic geometry but also ETR at a global level. When using a DEM in an application product, ETR should be counted as an input error.

9.2.3 An Example of DEM Generation from SPOT-5 Imagery

In this section, the main steps in DEM generation from a satellite stereoscopic image pair are outlined. The example presented in this section is from [149]. The study site is the area around Quebec City, QC, Canada (47◦N, 71◦30 W). The information of the SPOT-5 stereo images in panchromatic mode is listed in Table 9.1.

A perspective projection model is established based on the geometric positions of the satellite, the camera, and the cartographic coordinate. This model links the 3D cartographic coordinate to the image coordinates, and the mathematical expression is given by Eqs. (9.3) and (9.4) [147, 148]:

where

Parameters involved in Eqs. (9.3)–(9.6) are explained as follows.

His the altitude of the point corrected for Earth curvature;

ais mainly a function of the rotation of the Earth;

αis the instantaneous field-of-view;

Fig. 9.4 Left: SPOT-5 image captured on 5 May 2003. Right: DEM generated from the stereo pair. A: melting snow; B: frozen lakes; C: the St. Lawrence River with significant melting ice; D: down-hill ski stations with snow. Figure courtesy of [149]

The ground control points (GCPs) with known (x, y, z) coordinates and corresponding (u, v) image coordinates are employed for the bundle adjustment to obtain parameters in the mathematical model. The processing steps of DEM generation from SPOT-5 stereo images (see Fig. 9.4(left)) are as follows.

1.Acquisition and pre-processing of the remote sensed data (images and metadata showing configuration of image acquisition) to determine an approximate value for each parameter of the 3D projection model.

2.Collection of GCPs with their 3D cartographic coordinates and 2D image coordinates. GCPs covered the total surface with points at the lowest and highest elevation to avoid extrapolations, both in x, y and elevation.

3.Computation of the 3D projection model, initialized with the approximate parameter values and refined by an iterative least-squares bundle adjustment with the GCPs.

4.Extraction of matching points from the two stereo images by using a multi-scale normalized cross-correlation method with computation of the maximum of the correlation coefficient.

5.Computation of (x, y, z) cartographic coordinates from the matching points in a regular grid spacing using the adjusted projection model (from step 3).

The full DEM (60 km × 60 km with a 5 m grid spacing) is extracted as shown in Fig. 9.4(right). It reproduces the terrain features, such as the St. Lawrence River and a large island in the middle. The black areas correspond to mismatched areas due to radiometric differences between the multi-date images. In this case, they are a result of snow in the mountains and on frozen lakes. The quantitative evaluation is conducted by comparison of the DEM generated from the SPOT-5 stereo images to a LIDAR acquired DEM with the accuracy of 0.15 m in elevation. Accuracies of 6.5 m (LE68) and 10 m (LE90) were achieved, corresponding to an image matching error of ±1 pixel.