dem construction from contour lines based on regional...

DEM Construction from Contour lines based on Regional

Optimum Control Dunjiang Song(宋敦江)

2010-7-12Institute of Policy and Management,

Chinese Academy of Sciences(CAS)

Tel:13488734079 e-Mail: [email protected]

Volume 2, p162-165

DEM Construction from Contour lines based on Regional Optimum Control

I. Background

II. HASMROC(Regional Optimum Control )Method

III. Case Study

IV. Discussion

V. References

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I Background

Digital Elevation Model (DEM) is a representation of terrain elevation as a function of geographic location.

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I Background

A contour line is a series of connected points of the same heights. Contour lines form "line group", which describes the geomorphologic characteristics, such as the terrain valley and ridge.

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I Background

Methods for transforming Contour lines to DEM

1. TIN (Triangulated Irregular Network)

2. Thin Plate Spline(TPS, implemented in software ANUDEM)

3. Contour Dilation

4. Steep Slope

5. Skeleton

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I. Background


III. Case Study

IV. Discussion

V. References

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HASM (High Accuracy Surface Modeling)

HASM (High Accuracy Surface Modeling) is a method for surface modeling, based on the theory of surface, which means that except its position in space, a surface is uniquely defined by its first fundamental coefficients and second fundamental coefficients. If we can get more precise its first fundamental coefficients and second fundamental coefficients, then we can get a more precise shape about the surface. Theoretically, through iterative method, with the boundary value and by use of iteration, we can achieve this goal.

HASM (Continued)

If a surface is graphed by a function ( )yxfz ,= or the monge patch by )),(,,(),( yxfyxyxr = , the first fundamental coefficients, E and G , and the second fundamental coefficients, L and N , have the following relationship described in terms of finite difference equations:

( ) ( )

( ) ( )

1 1 11, , 1, 1, 1, , 1 , 1 ,1 2

11 112 , ,, ,

1 1 1, 1 , , 1 1, 1, , 1 , 1 ,1 2

22 222 , ,, ,

22 2 1

22 2

n n n n n n n nn ni j i j i j i j i j i j i j i j

i j i j n ni j i j

n n n n n n n nn ni j i j i j i j i j i j i j i j

i j i j n ni j i j

f f f f f f f Lh h h E G

f f f f f f f Nh h h E G

+ + ++ − + − + −

+ + ++ − + − + −

− + − −= Γ + Γ +

+ −

− + − −= Γ + Γ +

+ 1

−

(1)

where h represents simulation step length, njif , denotes ( )yxf , value at the grid (i,j) at the nth iteration,

,( )j nii i jΓ denotes j

iiΓ value at the grid (i,j) at the nth iteration. details about )2,1,2,1( ==Γ jijii , E , G , L

and N can be referred to [1-2].

HASM (Continued)The matrix formulation of equations set (1) can be expressed as,

==

+

+

nn

nn

bFAbFA

21

2

11

1 (1)

where

1 1 1 1 1 1 1 1 1 11,1 1,2 1, 2,1 2,2 2, ,1 ,2 ,[ , , , , , , , , , , , , ]

y y x x x y

n n n n n n n n n n TM M M M M MF f f f f f f f f f+ + + + + + + + + +=

; 1A and 2A respectively represent coefficient

matrices of the first equation and the second equation in equations set (1); nb1 and

nb2 are respectively the

right-hand vectors of equations set (1) at the nth iteration.

Integrating sampling points constraint equations, the HASM simulation equations could be reformulated as, 1

21

min || ||

. .

n n

n

AF bs t CF d

+

+

−

= (1)

For a number λ sufficiently large, HASM could be transformed to unconstrained least squares approximation [15],

21 ||][][||min

db

FCA n

n

f λλ−+ (2)

or

][][]][[ 1

db

CAFCA

CA TTnTT

λλ

λλ =+ (3)

HASMROC(Regional Optimum Control ) Method

=×

−+

+

tFStsqPF

n

nn

12

1

..

||||minDEM simulation( from scattered points) equations could be reformulated as：

When more information is available, DEM simulation can be expressed as：

<<

=×

−

+

+

+

uFltFS

ts

qPF

n

n

nn

1

1

2

1

..

min

(1)Equations from

sampling

Equations from

sampled points

Equations from

unsampled points <#>

HASM

HASMROC

HASMROC vs. HASM

HASMHASMROCRed is outer bound，yellow inner bound(buffer)，green is valid simulation region (Minimum Rectangular Boundary of the Original data)

Solid point is the boundary, solid triangle is the sampling point, circle points is to be simulated.

lower and upper bounds for every DEM grids

a)

10

11

12

13

1

6

5

4

2

3

7

8

9 b)

图 1 等高线及等高线树图，a)等高线，b)等高线树

For each Grid(i,j) While(1) Find any contour C(k) which contains Grid(i,j) in contours C;

Hk=C(k).Z; If C(k) is a leaf node Grid(i,j)’s lower bound is Hk; Grid(i,j)’s upper bound is Hk plus one contour interval; Else Look for among C(k)’s children and

find the contour C(w) which contains Grid(i,j); If contour C(w) is empty Grid(i,j)’s lower bound is Hk; Grid(i,j)’s upper bound is Hk plus one contour

interval; Else Set C(k)= C(w); End If End If

End While End For

图 2 Single Hills

图 3 Multi-hills

. P1

. P2

. P1. P2. P3

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0

0

0

0

0

0

2

2

2

2

2

-2

-2

-2

4

4

4

-4

-4

2

2

6

6

-2

-2

-6

08

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

Some results of lower and upper bounds for every DEM grids

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I. Background


III. Case Study

IV. Discussion

V. References

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III Case Study

1. Mathematical (Gaussian Synthetic ) surface

2. Real terrain surface

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Case I: Gaussian Synthetic Surface

• Z range inbetween (-6.551, 8.106)2 2 2 2 2 22 ( ( 1) ) 3 5 ( ) ( ( 1) )13*(1 ) * 10*( )* *

5 3x y x y x yxz x e x y e e− − + − − − + −= − − − − −

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Simulation results

3D Shaded relief from HASMROC simulated DEM

Derived contour lines from HASMROC and original Gauss Synthesis surface

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Resolution=0.12

•

HASMROC

TPS(Thin Plate Spline)

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Resolution=0.06

HASMROC

TPS(Thin Plate Spline)

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Case II: Manually vectorized data

江西省千烟洲生态试验站地理位置千烟洲扫描矢量化原始等高线图

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3、Results

Interpolated DEM of, a)TIN(Triangulated Irregular Network), b)HASMROC

a) b)

3、Results(2)

Overlay of the derived contour lines with original contour lines from, a)TIN(Triangulated Irregular Network),b)HASMROC

a) b)

a)

b)

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3、Results(3)

Histogram of DEM fromfrom, a)TIN(Triangulated Irregular Network),b)HASMROC

a) b)

a)

b)70 80 90 100 110 120 130 140

0

1

2

3

4

5

6

7

8 x 104

高程

频数

70 80 90 100 110 120 130 1400

0.5

1

1.5

2

2.5

3 x 104

高程

频数


I. Background


III. Case Study

IV. Discussion

V. References

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Discussion

• By solving an optimization problem, DEM can be secured, which is smooth and of high fidelity to original contours. Mathematical and real contour lines examples are given, and results from HASMROC are compared with that from TIN (Triangulated Irregular Network) and TPS (Thin Plate Spline) method, HASMROC is superior to the latters in DEM construction in two kinds of cases.

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Future work

1. Multiscale data modeling:多尺度数据综合的研究，基于等高线与地形特征的地形简化研究；

2. Quick algorithms:快速任意点的值范围(上下界)的确定方法，结合等高线树、约束TIN及射线求交法;

3. Large-scale constraint Optimization: 大规模约束优化求解方法的底层

实现，根据模拟精度（分辨率）的要求提取地形特征点、特征线与特征面，或从遥感数据或实地测量获得有用地形信息，并建立对应的约束优化方程，从而建立更加符合实际的DEM；

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I. Background


III. Case Study

IV. Discussion

V. References

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Main References1. YUE T.X., Du Z.P., SONG D.J., GONG J., A new method of surface modeling and its application

to DEM construction, Geomorphology, 2007, 91 (12), 161-172

2. Yue T.X., Song D.J., Du Z.P., Wang W., High accuracy surface modeling and its application to DEM generation [J]. International Journal of Remote Sensing, 2010, 2205 - 2226

3. Ciarlet, Philippe G. (2002). A surface is a continuous function of its two fundamental forms. C. R. Acad. Sci. Paris, Ser. I 335: 609-614.

4. Ciarlet, Philippe G., F. Larsonneur (2002). On the recovery of a surface with prescribed first and second fundamental forms. J. Math. Pures Appl. 81: 167-185.

5. Ciarlet, Philippe G. (2003). The Continuity of a surface as a function of its two fundamental forms. J. Math. Pures Appl. 82: 253–274.

6. SONG D.J., YUE T.X., Du Z.P., DEM Construction from Contour lines based on Regional Optimum Control[C], in Proceedings of the Third International Joint Conference on Computational Sciences and Optimization, CSO 2010, 162-165.

7. 宋敦江,岳天祥, 杜正平,等高线树的构建及其在建立高保真DEM中的应用, 中国图像图形学报, 2010(Accepted)

8. 宋敦江,岳天祥,杜正平, 陈传法. 简单地形特征建立DEM的HASM方法[J],武汉大学学报信息科学版, 2010(Accepted)

9. Z.L. Li, Q. Zhu, Christopher Gold. Digital terrain modeling : principles and methodology. Boca Raton, Florida New York, N.Y., Taylor & Francis,2005

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Tel:13488734079 e-Mail: [email protected]

Dunjiang Song(宋敦江)

dem construction from contour lines based on regional...

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