Fig. 1. Symmetric windows in a direction

Fig. 2. Schematic of symmetric windows of MRDKE(xi,yi,θ*) obtained and not obtained

Fig. 3. LiDAR point cloud data and optical image for test data. (a) LiDAR point cloud for test data; (b) optical image for test data

Fig. 4. Directions of symmetrical window connection corresponding to the data points in different regions in the test LiDAR point cloud data and the magnitude of MRDKE(xi, yi, θ*) and σi(θ*). (a) Connection directions and the magnitude of MRDKE(xi, yi, θ*); (b) connection directions and the magnitude of σi(θ*)

Fig. 5. Diagram of movement of data point i and windows. (a) Symmetric windows with θ*=0 and l=2r; (b) data point i through the boundary between building and ground

Fig. 6. Change trend of MRDKE(xi, yi, θ*) and σi(θ*). (a) MRDKE(xi, yi, θ*); (b) σi(θ*)

Fig. 7. Schematic of overall point cloud corresponding to the change trend of MRDKE(xi, yi, θ*) and σi(θ*) in Fig. 6. (a) Corresponding to a red line with a width of 2(l-r) and a larger MRDKE(xi, yi, θ*) in Fig. 6(a); (b1) corresponding to a red line with a width of 2(l-r) and a smaller σi(θ*) in Fig. 6(b); (b2) Corresponding to the top of the two raised peak in Fig. 6(b)

Fig. 8. Diagrammatic sketch of data point i passing through the tree and the change trend of σi(θ#).(a) Data point i at tree; (b) data point i passing through tree; (c) change trend of σi(θ#)

Fig. 9. Diagrammatic sketch of data point i through the boundary between building and ground and the corresponding change trend to σi(θ#). (a) Symmetric windows when θ*=0 and l=r; (b) schematic of the process of data point i through the boundary between building and ground; (c) change trend of σi(θ#)

Fig. 10. Directions of symmetric window connection and magnitude of σi(θ#) corresponding to the data points in different regions when l=r

Fig. 11. Range of point sets after threshold procession. (a) T1 and T2; (b) T3

Fig. 12. Diagram of the process of filtering trees. (a) Three dimensional version of tree points; (b) two dimensional version of tree points; (c) elevation histogram; (d) two dimensional version of treated tree points

Fig. 13. Histograms and thresholds of σi(θ*) and σi(θ#). (a) Histogram of σi(θ*); (b) local magnification of Fig.13(a); (c) histogram of σi(θ#); (d) local magnification of Fig.13(c)

Fig. 14. Extraction process of the boundaries between buildings and ground. (a) Magnitude distribution of MRDKE(xi, yi, θ*); (b) vector field consisting of MRDKE(xi, yi, θ*) and θ*; (c) magnitude distribution of σi(θ*); (d) vector field consisting of σi(θ*) and θ*; (e) the extracted boundaries between buildings and ground

Fig. 15. Schematic of the extraction process of trees and the boundaries between buildings and ground. (a) Magnitude distribution of σi(θ#); (b) vector field consisting of σi(θ#) and θ#; (c) trees and buildings and the boundaries between ground

Fig. 16. Extraction process of building edges. (a) Point set O; (b) point set Of; (c) regular block partition of Of; (d) point set Ph; (e) building edges

Fig. 17. Point cloud data of residential areas and corresponding optical images. (a)-(d) Point cloud data; (a1)-(d1) optical images corresponding to the point cloud data

Fig. 18. A、B、Of、Ph for first to fourth experimental data. (a1)-(d1) A; (a2)-(d2) B; (a3)-(d3) Of; (a4)-(d4) Ph

Fig. 19. Extraction results based on proposed algorithm. (a) Data 1; (b) data 2; (c) data 3; (d) data 4

Fig. 20. Extraction results based on compared methods. (a1)-(d1) Terrasolid software; (a2)-(d2) the method in Ref. [19]; (a3)-(d3) the method in Ref. [20]

Fig. 21. Superposition results of proposed algorithm. (a) Data 1; (b) data 2; (c) data 3; (d) data 4

Fig. 22. Superposition results of compared methods. (a1)-(d1) Terrasolid software; (a2)-(d2) the method in Ref. [19]; (a3)-(d3) the method in Ref. [20]