- Wavelet analysis:
the purpose of this analysis is to find out the cluster pattern of laser points, i.e. smooth versions of elevation images, at different resolutions. This analysis applied a trous algorithm as described briefly below.
Let F(x) and Y(x) be scaling function and wavelet function, respectively. The scaling function is chosen to satisfy the dilation equation as follow.
where h is a discrete low pass filter associated with scaling function F. This equation shows the link between two consecutive resolutions, which are different by a factor of 2, by low pass filtering. Then, the smoothed data cj(k) at a given resolution j and at position k is the scalar product
In implementation, this smoothed data can be obtained by convolution
The difference between two consecutive resolutions is calculated as
Which is the definition of wavelet transform at resolution j. The wavelet function Y(x) is defined by
Assume the smooth operation is stopped at resolution p, the reconstruction formula of a signal is
The above wavelet transform, which is described for a single index can be extended to higher dimensional space. For example, in two-dimensional space, x in all equations above implies (x,y).
The smooth images by wavelet analysis at four consecutive resolutions are depicted in Figure 3. It is obvious to recognize the distribution of laser points, which is the buildings, in fact, in urban area. The small and low buildings gradually disappear when moving to coarser resolution. This is a clue for the multi-resolution segmentation in the following step.
- Multi-resolution segmentation:
based on the signatures in multi-resolution space, the buildings were easily detected from the elevation images. This simple idea is illustrated in Figure 4 below.
The segmentation result presented most of main buildings in the study area. There were a few low buildings missing or being distorted due to their very low signatures. The detected buildings are showed in Figure 5 both in raster and vector format. Figure 5 also depicted the missing buildings at the top-right corner of images.
The detected buildings then were masked on laser cloud points to obtain the laser points that belong to buildings. The acquired laser data was quite low density with 0.2point/m2 approximately. Therefore, after being interpolated at resolution of 1m, there existed the confusion along the edge of buildings. Furthermore, the wavelet analysis, in fact, is one kind of linear multi-resolution approach. As a result, wavelet analysis introduces the distortion of object edge. A non-linear approach should be taken into account to improve this point. Being aware, when masking with laser points, there was a careful refinement the laser points along the edge of buildings. The fusion with aerial photo or existing 2D data can be a useful aid to correct the edges of detected buildings. Figure 6 below depicted the original digital surface model on the left and the perspective 3D detected building on the right. The digital surface model illustrated the very complicated scene of the study area, especially very small and low building locates beside the tower. In addition, this area also presented the complicated roads with different levels. This kind of object will be detected and presented in future paper.
The results of segmentation by applying multi-resolution approach showed quite good result in testing area. It is necessary to emphasize the purpose of this research is to detect the buildings in urban area. The other man-made object such as roads or high ways, which are also necessary for reconstruction of 3D city, will be detected in the further development of this algorithm. In addition, this algorithm analyzed laser points in grid format to reduce the computation time.
Conclusions and Recommendations
After several introductions to apply wavelet in segmentation of airborne laser scanner data (Vu, T.T. and Tokunaga, M. 2001, Vu, T.T. and Tokunaga, M. 2002), this research is a step forward to the real detection of specific man-made objects in urban area. In conclusion, the proposed algorithm successfully detected most of buildings in a testing area. The key point is how to detect over multi-resolution that is simple tracking the signature over multi-resolution space. However, this algorithm introduced the problem at the edge of objects. It is continued in development with the integration of non-linear multi-resolution approach and the fusion of some other existing data such as aerial photo or 2D vector data. Not only buildings, all of man-made objects in urban area will be detected in the further step of this algorithm to accomplish the all requirements of 3D city model.
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