激光雷达数据滤波方法
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4 splines 5 splines
How many splines can be used?
9 spline 5 spline 7 spline spline 113spline observations
How many splines can be used?
• Too few The interpolated surface is very smooth and doesn’t represent very well the observations trend in the zone with a high variability • Too many The coefficients are highly affected by a single observation where there is a low point density. Some unwanted oscillations appear. If the number of splines is higher than the maximum, there is a singularity in the least square normal matrix so the coefficient estimation is impossible.
h0(ti): observations νi: noise (i=0,N-1) M spline functions ⇒ M grid nodes
h0 ( t i ) = ∑ a k sk (
k =0
M −1
Each observation (•) can be interpreted as a linear combination of the spline functions that are related with the interval in which the observation is lying in.
Interpolation with spline functions
• Lets be ∆ = a ≡ x0< x1< x2<…. xM< xM-1 ≡ monodimensional interval [a,b]. b one decomposition of the
•
Definition: It is said spline function with degree m ≥ 1 related to the decomposition ∆ a function s(x) that satisfies:
•
In general, the splines functions are used in all the situations where the polynomial approximation within the whole interval is not satisfactory. For m=1, we have the linear splines; and for m=3, we have the cubic splines.
Bidimensional Generalization
s15
s16
s45
s46
36 spline coefficients and 37 observation points Bilinear spline DTM: The coefficients λ15, λ16, λ45, λ46 will be zero
Interpolation with spline functions
The unknowns are resolved by a least square adjustment:
⎡ h00 ⎤ ⎡ s s 0 … ⎢ ⎥ ⎢ 0 1 ⎢ h01 ⎥ ⎢ s0 s1 s2 0 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ h0k ⎥ = ⎢ 0 … 0 sk −1 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢h ⎥ ⎢ 0 … … … ⎢ 0 N −2 ⎥ ⎢ ⎢ h0N −1 ⎦ ⎣ 0 … … … ⎥ ⎣ h0
LIDAR data filtering phase
• Removal of outlier points • Detection of the building edges • Determination of the areas inside the contours • Final correction and DTM computation
The Thychonov regularization
If the data is not distributed regularly, and there are some zones with no data, the spline coefficients in these zones are zero. Thus, the interpolated surface has an unnatural trend:
Interpolation with spline functions
Linear Splines 1/2
-1
0
1
Interpolation with spline functions
2/3
Bicubic splines
-2
-1
1
2
Interpolation with spline functions
– s(x) respect each interval [xk, xk+1], k = 0,…, M-2 is a polynomial of maximum degree m; – s
( j)
dj ( x) = j s ( x) dx
is a continous function in the interval [a,b], for j = 0, …, m-1.
For: t = t0
t0 t1
3 ⎛ t − t0 ⎞ 3 ⎛ t − t1 ⎞ + a1s ( ) ⎜ h0 ( t ) = a0 s ( ) ⎜ ⎟ ⎟ ⎝ ∆t ⎠ ⎝ ∆t ⎠
Interpolation with spline functions
For t ∈ [t0 , t1 ]
t0 t1 t2
A
ˆ a = ( At A) −1 At h 0
a
The coefficient estimation, ak, is:
How many splines can be used?
Each spline mustБайду номын сангаасcover at least one observation in its domain, so the resolution must be higher than the maximum distance between the observations:
The regularization is done by minimizing the slope or the curvature of the interpolating function
Bidimensional Generalization
Bilinear spline
1
−1
ϕ12 ϕ22 ϕ11 ϕ21
3)
( t i ) +ν i ; i = 1,… , N , k = 0,… , M − 1
ak are the unknowns
⎡ 3 ⎛ t − tk sk ( t ) = s ( 3) ⎜ ⎢ ⎝ ∆t ⎣
⎞⎤ ⎟⎥ ⎠⎦
Interpolation with spline functions
3 ⎛ t − t0 ⎞ 3 ⎛ t − t1 ⎞ 3 ⎛ t − t2 ⎞ + a1s ( ) ⎜ + a2 s ( ) ⎜ h0 ( t ) = a0 s ( ) ⎜ ⎟ ⎟ ⎟ ⎝ ∆t ⎠ ⎝ ∆t ⎠ ⎝ ∆t ⎠
Interpolation with spline functions
For t ∈ [tk , tk +1 ]
A regularization must be done.
The Thychonov regularization
Applying one principle to the least squares:
(A A
t
ˆ + λ ⋅ K ) a = At h 0
( λ⋅K)
This is the factor that helps to the interpolation in the intervals where there is no data
… … … … sk
… …
… … 0 sM −3 0
… … … sM −2 sM −2
sk +1 sk + 2 0 …
… … … …
0 ⎤ ⎡ a0 ⎤ 0 ⎥ ⎢ a1 ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 ⎥ ⋅ ⎢ ak ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ sM −1 ⎥ ⎢ aM −2 ⎥ sM −1 ⎥ ⎢ aM −1 ⎥ ⎦ ⎣ ⎦
−1
1
Bidimensional Generalization
Bicubic spline
2
ϕ34 ϕ44
-2
ϕ33 ϕ43
2
-2
Bidimensional Generalization
Construction of the normal matrix
In the case the Thychonov regularization is being used, the construction of the normal matrix must take into account the correction due to the regularization matrix, K. In this matrix is used the reguralization based on the gradient or the laplacian, respectively.
System solution
Choosing the regularization coefficients is very important: • If a small value is considered in correspondence with zone where there are not too many points, the normal matrix is wrong conditioned. Then the parameter estimation is not possible. • If a high value is considered, the contribution of the regularization is very important, so a smooth interpolated surface is obtained.
The LIDAR measurement
Aim
Removal of the objects on the terrain. Interpolation of the terrain points for the continuous model development.
Spline functions with Tikhonov regularization are used.
tk tk+1
⎛ t − tk -1 ⎞ ⎛ t − tk h0 ( t ) = ak −1 s (3) ⎜ + ak s (3) ⎜ ⎟ ∆t ⎠ ⎝ ⎝ ∆t
⎞ ⎛ t − tk +1 ⎞ ⎛ t − tk + 2 ⎞ + ak +1s (3) ⎜ + ak + 2 s ( 3 ) ⎜ ⎟ ⎟ ⎟ ∆t ⎠ ∆t ⎠ ⎠ ⎝ ⎝
Laser data filtering for the object detection
Roberto Antolín and Maria Antonia Brovelli Politecnico di Milano - Polo Regionale di Como E-mail: roberto.antolin@polimi.it web page: o.polimi.it
How many splines can be used?
9 spline 5 spline 7 spline spline 113spline observations
How many splines can be used?
• Too few The interpolated surface is very smooth and doesn’t represent very well the observations trend in the zone with a high variability • Too many The coefficients are highly affected by a single observation where there is a low point density. Some unwanted oscillations appear. If the number of splines is higher than the maximum, there is a singularity in the least square normal matrix so the coefficient estimation is impossible.
h0(ti): observations νi: noise (i=0,N-1) M spline functions ⇒ M grid nodes
h0 ( t i ) = ∑ a k sk (
k =0
M −1
Each observation (•) can be interpreted as a linear combination of the spline functions that are related with the interval in which the observation is lying in.
Interpolation with spline functions
• Lets be ∆ = a ≡ x0< x1< x2<…. xM< xM-1 ≡ monodimensional interval [a,b]. b one decomposition of the
•
Definition: It is said spline function with degree m ≥ 1 related to the decomposition ∆ a function s(x) that satisfies:
•
In general, the splines functions are used in all the situations where the polynomial approximation within the whole interval is not satisfactory. For m=1, we have the linear splines; and for m=3, we have the cubic splines.
Bidimensional Generalization
s15
s16
s45
s46
36 spline coefficients and 37 observation points Bilinear spline DTM: The coefficients λ15, λ16, λ45, λ46 will be zero
Interpolation with spline functions
The unknowns are resolved by a least square adjustment:
⎡ h00 ⎤ ⎡ s s 0 … ⎢ ⎥ ⎢ 0 1 ⎢ h01 ⎥ ⎢ s0 s1 s2 0 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ h0k ⎥ = ⎢ 0 … 0 sk −1 ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢h ⎥ ⎢ 0 … … … ⎢ 0 N −2 ⎥ ⎢ ⎢ h0N −1 ⎦ ⎣ 0 … … … ⎥ ⎣ h0
LIDAR data filtering phase
• Removal of outlier points • Detection of the building edges • Determination of the areas inside the contours • Final correction and DTM computation
The Thychonov regularization
If the data is not distributed regularly, and there are some zones with no data, the spline coefficients in these zones are zero. Thus, the interpolated surface has an unnatural trend:
Interpolation with spline functions
Linear Splines 1/2
-1
0
1
Interpolation with spline functions
2/3
Bicubic splines
-2
-1
1
2
Interpolation with spline functions
– s(x) respect each interval [xk, xk+1], k = 0,…, M-2 is a polynomial of maximum degree m; – s
( j)
dj ( x) = j s ( x) dx
is a continous function in the interval [a,b], for j = 0, …, m-1.
For: t = t0
t0 t1
3 ⎛ t − t0 ⎞ 3 ⎛ t − t1 ⎞ + a1s ( ) ⎜ h0 ( t ) = a0 s ( ) ⎜ ⎟ ⎟ ⎝ ∆t ⎠ ⎝ ∆t ⎠
Interpolation with spline functions
For t ∈ [t0 , t1 ]
t0 t1 t2
A
ˆ a = ( At A) −1 At h 0
a
The coefficient estimation, ak, is:
How many splines can be used?
Each spline mustБайду номын сангаасcover at least one observation in its domain, so the resolution must be higher than the maximum distance between the observations:
The regularization is done by minimizing the slope or the curvature of the interpolating function
Bidimensional Generalization
Bilinear spline
1
−1
ϕ12 ϕ22 ϕ11 ϕ21
3)
( t i ) +ν i ; i = 1,… , N , k = 0,… , M − 1
ak are the unknowns
⎡ 3 ⎛ t − tk sk ( t ) = s ( 3) ⎜ ⎢ ⎝ ∆t ⎣
⎞⎤ ⎟⎥ ⎠⎦
Interpolation with spline functions
3 ⎛ t − t0 ⎞ 3 ⎛ t − t1 ⎞ 3 ⎛ t − t2 ⎞ + a1s ( ) ⎜ + a2 s ( ) ⎜ h0 ( t ) = a0 s ( ) ⎜ ⎟ ⎟ ⎟ ⎝ ∆t ⎠ ⎝ ∆t ⎠ ⎝ ∆t ⎠
Interpolation with spline functions
For t ∈ [tk , tk +1 ]
A regularization must be done.
The Thychonov regularization
Applying one principle to the least squares:
(A A
t
ˆ + λ ⋅ K ) a = At h 0
( λ⋅K)
This is the factor that helps to the interpolation in the intervals where there is no data
… … … … sk
… …
… … 0 sM −3 0
… … … sM −2 sM −2
sk +1 sk + 2 0 …
… … … …
0 ⎤ ⎡ a0 ⎤ 0 ⎥ ⎢ a1 ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ 0 ⎥ ⋅ ⎢ ak ⎥ ⎥ ⎢ ⎥ ⎥ ⎢ ⎥ sM −1 ⎥ ⎢ aM −2 ⎥ sM −1 ⎥ ⎢ aM −1 ⎥ ⎦ ⎣ ⎦
−1
1
Bidimensional Generalization
Bicubic spline
2
ϕ34 ϕ44
-2
ϕ33 ϕ43
2
-2
Bidimensional Generalization
Construction of the normal matrix
In the case the Thychonov regularization is being used, the construction of the normal matrix must take into account the correction due to the regularization matrix, K. In this matrix is used the reguralization based on the gradient or the laplacian, respectively.
System solution
Choosing the regularization coefficients is very important: • If a small value is considered in correspondence with zone where there are not too many points, the normal matrix is wrong conditioned. Then the parameter estimation is not possible. • If a high value is considered, the contribution of the regularization is very important, so a smooth interpolated surface is obtained.
The LIDAR measurement
Aim
Removal of the objects on the terrain. Interpolation of the terrain points for the continuous model development.
Spline functions with Tikhonov regularization are used.
tk tk+1
⎛ t − tk -1 ⎞ ⎛ t − tk h0 ( t ) = ak −1 s (3) ⎜ + ak s (3) ⎜ ⎟ ∆t ⎠ ⎝ ⎝ ∆t
⎞ ⎛ t − tk +1 ⎞ ⎛ t − tk + 2 ⎞ + ak +1s (3) ⎜ + ak + 2 s ( 3 ) ⎜ ⎟ ⎟ ⎟ ∆t ⎠ ∆t ⎠ ⎠ ⎝ ⎝
Laser data filtering for the object detection
Roberto Antolín and Maria Antonia Brovelli Politecnico di Milano - Polo Regionale di Como E-mail: roberto.antolin@polimi.it web page: o.polimi.it