ISIJ International
Online ISSN : 1347-5460
Print ISSN : 0915-1559
ISSN-L : 0915-1559
Regular Article
A Gaussian Filter for Plate Flatness Evaluation System with 3-D Scanner
Shinichiro AoeMasaru MiyakeKazuhisa Kabeya
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2017 Volume 57 Issue 6 Pages 1054-1061

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Abstract

A LIDAR (light detection and ranging) system was applied to a plate flatness evaluation system. Plate flatness surfaces are reconstructed from many points generated by LIDAR by a smoothing spline method. We defined a smoothing spline functional with sampling measure weights. The equivalent number of parameters defined on this functional does not depend on the distributions of samples. The approximation of the equivalent number of parameters is derived when the number of samples becomes infinity. This approximation greatly reduced the calculation time needed to estimate the optimal smoothing. The smoothing spline calculation cost was so high that new algorithms (FMM: fast multi-pole method) were introduced and we developed a smoothing engine which was applied to practical problems. The engine generated clear surfaces and was robust against various dirty point clouds.

1. Introduction

Recently, the 3-D laser scanner (3-D scanner) has become a general-purpose, low-cost technology, and 3-D scanners are now generally applied in various fields. In shipyards, 3-D scanners are used to measure the flatness of steel plates. In the steel industry, 3-D scanners cannot be used for online measurement of the flatness of moving plates in a manner similar to an online shape meter, but they have high portability and are considered to be suitable for online measurement of the flatness of fixed plates.

The problem for application of 3-D scanners to online plate flatness measurement is the difficulty of data handling with the millions of point data measured by a 3-D scanner. These point data, which are projected on the plate surface, are not grid based and are distributed inhomogeneously. Therefore, the procedures for approximation to a curved surface are very complicated. Also, each point datum has measurement error, so a smoothing procedure is necessary to enhance measurement accuracy. Therefore, we applied a smoothing spline method to this problem.

The smoothing spline method is a well-known regression method which is used to estimate non-parametric curves or non-parametric surfaces from noisy samples. This method is a Gaussian filter and is applied to various fields of science and engineering, such as signal processing applications for noise filters,9,13) image processing applications for image reconstruction18) or noise filters,10) inverse problems for gravitational and magnetic fields,6) statistical processing applications for medical data8) and surface reconstructions from noisy data12,14) measured by LIDAR (light detection and ranging).

The smoothing parameter in the smoothing spline method enables calculation of a regression function with arbitrary smoothness. When the smoothing parameter is too small, the regression function is overfitted and the function takes a zigzag shape. Conversely, when the smoothing parameter is too large, the regression function becomes too smooth. In this case, the regression function loses important information. Thus, there is an optimal smoothing parameter that makes it possible to calculate an adequate regression function with a better balance between fitness and complexity. The GCV (Generalized Cross Validation) method2,3) is often used for automatic estimation of the optimal smoothing parameter. GCV includes iterative calculations of an inverse matrix of a full matrix. However, much time is required to calculate the iterations, and when the number of points is large, it is practically impossible to calculate the optimal smoothing parameter.

Similarly, much time is required to calculate the smoothing spline surface, and when the number of samples is large, it is practically impossible to calculate the smoothing spline surface. Fast methods to calculate a theoretical smoothing spline surface by using FMM (Fast Multi-pole Method) and fast methods to calculate an approximate smoothing spline surface by using a discrete model have been studied.

Beatson et al.4) proposed a fast evaluation method for spline surfaces by using FMM. This method enables a drastic reduction in calculation time. Beatson et al.5) proposed a fast method based on preconditioned GMRES (Generalized Minimal Residual) iteration to calculate spline coefficients.

In this paper, we define a smoothing spline functional with sampling measure weights.16,17) The functional greatly reduces the calculation time needed to estimate the optimal smoothing parameter. We applied our proposed method to practical problems of plate surface estimation for points measured with LIDAR (Light Detection and Ranging).

2. Smoothing Spline Regression

The smoothing 3rd order spline functional is defined as   

Π= i=1 m 1 2 ( z M ( i ) -f( x M ( i ) ) ) 2 +γ 0 L 1 2 ( d 2 f d x 2 ) 2 dx (1)
where m is the number of samples, L is the length of the sampling region, x M ( i ) ( i=1,,m ) is the position of the ith sample such that 0 x M ( 1 ) << x M ( m ) L , z M ( i ) ( i=1,,m ) is the value of the ith sample, f is the regression curve and γ is the smoothing parameter. Applying the variational principle to the functional in Eq. (1),   
δΠ= 0 L { γ d 4 f d x 4 + i=1 m ( f- z M ( i ) ) Dirac( x- x M ( i ) ) }δfdx             +γ d 2 f d x 2 δ( df dx ) | x=0 x=L -γ d 3 f d x 3 δf | x=0 x=L =0 (2)
where, Dirac(x) is Dirac’s delta function. From Eq. (2), the differential equation of the smoothing 3rd order spline is given by   
γ d 4 f d x 4 + i=1 m ( f- z M ( i ) ) Dirac( x- x M ( i ) ) =0 (3)
and the boundary conditions are given by   
d 2 f( 0 ) d x 2 = d 2 f( L ) d x 2 = d 3 f( 0 ) d x 3 = d 3 f( L ) d x 3 =0. (4)
Solving the differential Eq. (3) with the boundary conditions(4), the regression curve is given by   
f( x ) = c 1 + c 2 x+ i=1 m d M ( i ) 12 | x- x M ( i ) | 3 (5)
and the constraints are given by   
i=1 m d M ( i ) = i=1 m x M ( i ) d M ( i ) =0 (6)
where c1, c2, and d M ( i ) ( i=1,,m ) are the unknown parameters.

Substituting Eq. (5) into Eq. (3),   

f( x M ( i ) ) +γ d M ( i ) = z M ( i ) ( i=1,,m ) . (7)
Equations (6) and (7) engender the following linear system of equations:   
[ [ A M ]+γ[ I m×m ] [ P M ] [ P M ] T [ O 2×2 ] ]{ { d M } { c } }={ { z M } { O 2 } } (8)
where [Im×m] is the m-by-m identity matrix, [O2×2] is the 2-by-2 zero matrix, {O2} is the 2-by-1 zero vector,   
[ A M ] T =[ [ a M ( x M ( 1 ) ) ] T [ a M ( x M ( m ) ) ] T ], [ a M ( x ) ]=[ 1 12 | x- x M ( 1 ) | 3 1 12 | x- x M ( m ) | 3 ], (9)
  
[ P M ] T =[ [ p( x M ( 1 ) ) ] T [ p( x M ( m ) ) ] T ], [ p( x ) ]=[ 1 x ], (10)
  
{ d M } T =[ d M ( 1 ) d M ( m ) ], (11)
  
{ c } T =[ c 1 c 2 ], (12)
and   
{ z M } T =[ z M ( 1 ) z M ( m ) ].
Substituting Eqs. (9), (10), (11), (12) into Eq. (5), the regression curve can be expressed in matrix form as   
f( x ) =[ a M ( x ) ]{ d M }+[ p( x ) ]{ c }. (13)

3. Generalized Cross Validation and Information Criterion

GCV is one of the most popular methods to obtain the required optimal smoothing parameter. GCV is defined such that the optimal smoothing parameter is the one which minimizes the evaluation function   

V( γ ) = i=1 m ( z M ( i ) -f( x M ( i ) ) ) 2 / ( 1- 1 m trace( [ H ] ) ) 2 (14)
where [H] is the so-called Hat matrix, defined as   
{ f( { x M } ) }=[ H ]{ z M } (15)
where   
{ f( { x M } ) } T =[ f( x M ( 1 ) ) f( x M ( m ) ) ].

Substituting into Eq. (15) the regression curve of Eq. (13), in which the unknown parameters {c} and {dM} from Eq. (8) are substituted, the Hat matrix of the smoothing 3rd order spline can be written as   

[ H ]=[ [ A M ] [ P M ] ] [ [ A M ]+γ[ I m×m ] [ P M ] [ P M ] T [ O 2×2 ] ] -1 [ [ I m×m ] [ O 2×m ] ].

It is known that the GCV evaluation function of Eq. (14) is associated with the information criterion. In particular, AIC1) is given by   

AIC=-2LL+2k (16)
where LL is the maximum log likelihood defined as   
LL=- m 2 log( i=1 m ( z M ( i ) -f( x M ( i ) ) ) 2 ) ,
and k is the number of parameters or the degree of freedom for a regression model. AIC is one of the most popular and simplest information criteria; it is nearly equivalent to the GCV evaluation function when m is large. Equation (14) can be transformed to the information criterion   
I C GCV =mlog( V( γ ) ) =mlog( i=1 m ( z M ( i ) -f( x M ( i ) ) ) 2 ) -mlog( ( 1- 1 m trace( [ H ] ) ) 2 ) =-2LL-mlog( ( 1- 1 m trace( [ H ] ) ) 2 ) (17)
by using the increasing function. In the limit of infinite samples, Eq. (17) becomes   
AIC=-2LL+2trace( [ H ] ) . (18)
Comparing Eqs. (16) with (18), trace([H]) replaces the number of parameters for the smoothing spline. Consequently, ENOP (Equivalent Number of Parameters) is defined as   
k GCV =trace( [ H ] ) . (19)

As the number of samples becomes large, the time required to calculate ENOP becomes prohibitively long, because ENOP includes computing the m+2-by-m+2 inverse matrix. Moreover, the inverse matrix must be computed every time the smoothing parameter is changed. De Nicolao et al.7) invented a fast approximation method for ENOP. This is a useful approximation, but it cannot be applied to a surface or a higher dimensional model. Since the approximation of ENOP7) includes the sampling length, it is influenced by the sampling length. As ENOP is comparable to the complexity of a regression model, ENOP should not depend on the distributions of samples. Thus, we must consider the smoothing spline without depending on the distributions of samples.

4. Smoothing Spline Regression with Sampling Measure Weights

The smoothing lth order spline functional with weights is defined as   

Π= i=1 m 1 2 ω M ( i ) ( z M ( i ) -f( x M ( i ) ) ) 2 +γ 0 L 1 2 ( d ( l+1 ) /2 f d x ( l+1 ) /2 ) 2 dx (20)
where ω M ( i ) ( i=1,,m ) is the weight of the ith sample and is the order of spline such that l=1,3,5,···. We consider the weights are to be defined as   
0 L g( x ) dx i=1 m ω M ( i ) g( x M ( i ) ) , i=1 m ω M ( i ) =L,      and       ω M ( i ) 0 (21)
where i=1,···,m and g(x) is an arbitrary function. Equation (21) is the numerical integral formula. If g(x) is the twisted interpolation line, the weights become   
ω M ( i ) = x M ( i+1 ) - x M ( i-1 ) 2 ( i=1,,m )
where x M ( 0 ) = x M ( 1 ) and x M ( m+1 ) = x M ( m ) .

Applying the variational principle to the functional of Eq. (20),   

δΠ= 0 L { ( -1 ) ( l+1 ) /2 γ d l+1 f d x l+1 + i=1 m ω M ( i ) ( f- z M ( i ) ) Dirac( x- x M ( i ) ) }δfdx             + j=1 l+1 2 ( -1 ) j-1 γ d ( l+1 ) /2 +j-1 f d x ( l+1 ) /2 +j-1 δ( d ( l+1 ) /2 -j f d x ( l+1 ) /2 -j ) | x=0 x=L =0. (22)

From Eq. (22), the differential equation of the smoothing lth order spline with weights is given by   

( -1 ) ( l+1 ) /2 γ d l+1 f d x l+1 + i=1 m ω M ( i ) ( f- z M ( i ) ) Dirac( x- x M ( i ) ) =0 (23)
and the boundary conditions are given by   
d ( l+1 ) /2 +j-1 f( 0 ) d x ( l+1 ) /2 +j-1 = d ( l+1 ) /2 +j-1 f( L ) d x ( l+1 ) /2 +j-1 =0( j=1,, ( l+1 ) /2 ) . (24)
Solving the differential Eq. (23) with the boundary conditions (24), the regression curve is given by   
f( x ) = j=1 ( l+1 ) /2 c j x j-1 + i=1 m d M ( i ) 2×( l ) ! | x- x M ( i ) | l (25)
and the constraints are given by   
i=1 m ( x M ( i ) ) j-1 d M ( i ) =0( j=1,, ( l+1 ) /2 ) . (26)

Substituting Eq. (25) into Eq. (23),   

f( x M ( i ) ) + ( -1 ) ( l+1 ) /2 γ ω M ( i ) d M ( i ) = z M ( i ) ( i=1,,m ) . (27)
Equations (26) and (27) engender the linear equation system   
[ [ A M ]+ ( -1 ) ( l+1 ) /2 γ [ Ω M ] -1 [ P M ] [ P M ] T [ O ( l+1 ) /2 × ( l+1 ) /2 ] ]{ { d M } { c } }={ { z M } { O ( l+1 ) /2 } } (28)
where   
[ Ω M ]=[ ω M ( 1 ) 0 0 ω M ( m ) ], [ a M ( x ) ]= 1 2×( l ) ! [ | x- x M ( 1 ) | l | x- x M ( m ) | l ], (29)
  
[ p( x ) ]=[ 1 x x ( l+1 ) /2 -1 ], (30)
and   
{ c } T =[ c 1 c ( l+1 ) /2 ]. (31)

The unknown parameters {c} and {dM} can be obtained by solving Eq. (28). A regression curve in matrix form, Eq. (13), may be obtained by substituting Eqs. (29), (30), (11) and (31) into Eq. (25). The regression curve f(x) can then be obtained by substituting the unknown parameters into Eq. (13).

The optimal smoothing parameter for the smoothing spline with weights is defined as the smoothing parameter obtained by minimizing the GCV evaluation function   

V( γ ) = i=1 m ω M ( i ) ( z M ( i ) -f( x M ( i ) ) ) 2 / ( 1- k GCV m ) 2 . (32)
The Hat matrix of the smoothing spline with weights becomes   
[ H ]=[ [ A M ] [ P M ] ] × [ [ A M ]+ ( -1 ) ( l+1 ) /2 γ [ Ω M ] -1 [ P M ] [ P M ] T [ O ( l+1 ) /2 × ( l+1 ) /2 ] ] -1 [ [ I m×m ] [ O ( l+1 ) /2 ×m ] ]. (33)

5. Approximation of Equivalent Number of Parameters

In the limit of infinite samples, the differential equation of the smoothing spline with sampling measure weights, Eq. (23), becomes   

( -1 ) ( l+1 ) /2 γ d l+1 f d x l+1 +f- z M =0 (34)
where zM is the sampling value function. Applying the Laplace transformation to Eq. (34),   
f( s ) =h( s ) z M ( s ) (35)
where s is the Laplace variable and h(s) is the transfer function given by   
h( s ) = 1 ( -1 ) ( l+1 ) /2 γ s l+1 +1 . (36)
Comparing Eqs. (35) and (15), it can be understood that the transfer function h(s) plays the same role as the Hat matrix. Applying the Fourier transformation to Eq. (36), the frequency response function h(ω) is given by   
h( ω ) = 1 ( -1 ) ( l+1 ) /2 γ ( jω ) l+1 +1 = 1 γ ω l+1 +1 (37)
where j is an imaginary unit and ω is the angular frequency. The frequency response function h(ω) shows that the smoothing splines act as a low pass filter and a Gaussian filter with phase compensation.

We define the degree of freedom k for trigonometric functions as   

k=L/ L ω (38)
where Lω is the half wavelength given by   
L ω =π/ω . (39)
Substituting Eq. (39) into Eq. (38),   
k= Lω /π . (40)
Substituting Eq. (40) into Eq. (37), the response function h(k) for the degree of freedom has the form   
h( k ) = ( γ ( π L ) l+1 k l+1 +1 ) -1 .

From the definition of ENOP (Eq. (19)), ENOP is the summation of eigenvalues of the Hat matrix; then, ENOP for an infinite number of samples is given by   

k A ( γ ) = 0 h( k ) dk = 0 ( γ ( π L ) l+1 k l+1 +1 ) -1 dk . (41)
Thus, kA becomes the approximation of ENOP. Applying the change of the variable   
κ= π L γ 1 l+1 k
to the integral of Eq. (41), the approximation of ENOP, kA, becomes   
k A ( γ ) = L π γ - 1 l+1 0 1 κ l+1 +1 dκ . (42)
Applying the change of the variable   
1-ξ= 1 κ l+1 +1
to the integral of Eq. (42), kA becomes   
k A ( γ ) = L π γ - 1 l+1 × 1 l+1 0 1 ξ 1 l+1 -1 ( 1-ξ ) l l+1 -1 dξ = L π γ - 1 l+1 × 1 l+1 B( 1 l+1 , l l+1 ) = L π( l+1 ) B( 1 l+1 , l l+1 ) γ - 1 l+1
where B is Euler’s Beta function. This approximation of ENOP is a very simple expression and does not depend on the distributions of samples. Values of the Beta function have closed-form expressions. For example, if the degree of spline is 3, the Beta function becomes B(1/4,3/4)= 2 π and the approximation of ENOP becomes   
k A ( γ ) = 2 L 4 γ - 1 4 . (43)

The above-mentioned derivation of the approximation of ENOP (43) is formal. The equivalent equation to Eq. (42) can be found in Craven et al.2) and Golub et al.3) The details of the theories are described in these references.

Figure 1 shows the approximation of ENOP obtained from Eq. (43) and the exact ENOP, kGCV, obtained from Eq. (19) when the degree of spline is 3, the number of samples is 101 and the length of the sampling region is 1.0. Bias is added to the approximation of ENOP to be compared with the exact ENOP. The value of bias is (l+1)/4 experimentally, but the bias is not important for GCV or information criteria. The approximation of ENOP, kA, agrees well with the exact ENOP, kGCV, in the ENOP range from 0 to half the number of samples; however, the correlation is not good for ENOP ranging from half to the total number of samples. The typical practice is that ENOP should not be used when it is larger than half the number of samples. Therefore, if the optimal ENOP decided by GCV or information criteria is more than half the number of samples, the number of samples should be increased.

Fig. 1.

Equivalent number of parameters.

6. Smoothing Thin Plate Spline Regression with Sampling Measure Weights

The smoothing thin plate spline (TPS) functional with weights is defined as   

Π= i=1 m 1 2 ω M ( i ) ( z M ( i ) -f( x M ( i ) , y M ( i ) ) ) 2 +γ Ω 1 2 ( 2 f ) 2 dΩ , (44)
where =[ / x / y ] , Ω is the area of the sampling region, and ( x M ( i ) , y M ( i ) )    ( i=1,,m ) is the position of the ith sample. Extending the one-dimensional setting of Chapter 4 to the case of plates, we consider the weights are to be defined as   
0 Ω g( x,y ) dΩ i=1 m ω M ( i ) g( x M ( i ) , y M ( i ) ) , i=1 m ω M ( i ) =Ω,   and    ω M ( i ) 0 (45)
where i=1,…,m and g(x,y) is an arbitrary function. Equation (45) is the numerical integral formula. For example, the weights are obtained by the numerical cubature method.11,15) In general, however, the weights calculated by numerical cubature11,15) are not all positive. If some negative weights are found, the sampling points corresponding to these negative weights must be removed.

Applying the variational principle to the functional of Eq. (44),   

δΠ= Ω { γ 4 f+ i=1 m ω M ( i ) ( f( x M ( i ) , y M ( i ) ) - z M ( i ) ) Dirac( x- x M ( i ) ,y- y M ( i ) ) }δfdΩ -γ Γ ( ( 2 f ) μ ) δfdΓ +γ Γ 2 f( δ( f ) μ ) dΓ =0 (46)
where Γ is the boundary of the region Ω and μ is the normal unit vector of the boundary Γ. From Eq. (46), the partial differential equation of the smoothing TPS is given by   
γ 4 f+ i=1 m ω M ( i ) ( f( x M ( i ) , y M ( i ) ) - z M ( i ) ) Dirac( x- x M ( i ) ,y- y M ( i ) ) =0 (47)
and the boundary conditions are given by

2 f=( 2 f ) μ=0 on Γ.

In general, a closed-form solution of the partial differential Eq. (47) is not possible, but if the region Ω is infinity, the solution of the partial differential Eq. (47) becomes very simple and the regression surface is given by   

f( x,y ) = c 1 + c 2 x+ c 3 y+ i=1 m d M ( i ) 8π r M ( i ) ( x,y ) 2 log( r M ( i ) ( x,y ) ) (48)
and the constraints are given by   
i=1 m d M ( i ) = i=1 m x M ( i ) d M ( i ) = i=1 m y M ( i ) d M ( i ) =0
where   
r M ( i ) ( x,y ) = ( x- x M ( i ) ) 2 + ( y- y M ( i ) ) 2       ( i=1,,m ) .

The regression surface expressed in matrix form is   

f( x,y ) =[ a M ( x,y ) ]{ d M }+[ p( x,y ) ]{ c }
where   
[ a M ( x,y ) ]= 1 8π [ r M ( 1 ) 2 log( r M ( 1 ) ) r M ( m ) 2 log( r M ( m ) ) ],
  
[ p( x,y ) ]=[ 1   x   y ] , and    { c } T =[ c 1     c 2     c 3 ].
The unknown parameters {c} and {dM} are obtained from Eq. (28), into which l=3 is substituted. The smoothing parameter is obtained from Eq. (32) and the Hat matrix is obtained from Eq. (33).

In the limit of infinite samples, the partial differential equation of the smoothing TPS with sampling measure weights, Eq. (47), becomes   

γ 4 f+f- z M =0. (49)
Applying the Laplace transformation to Eq. (49), the transfer function is given by   
h( s 1 , s 2 ) = 1 γ ( s 1 2 + s 2 2 ) 2 +1 (50)
where s1 and s2 are the Laplace variables. Applying the Fourier transformation to Eq. (50), the frequency response function is given by   
h( ω 1 , ω 2 ) = 1 γ ( ( j ω 1 ) 2 + ( j ω 2 ) 2 ) 2 +1 = 1 γ ( ω 1 2 + ω 2 2 ) 2 +1 (51)
where ω1 and ω2 are the angular frequencies. Substituting the relational expression between the angular frequency and the degree of freedom, Eq. (40), into the frequency response function (51), if the region Ω is rectangular (L1×L2),   
h( k 1 , k 2 ) = ( γ ( ( π L 1 k 1 ) 2 + ( π L 2 k 2 ) 2 ) 2 +1 ) -1
where k1 and k2 are the degree of freedom. The ENOP definition (Eq. (41)) on the one-dimensional space for the degree of freedom is extended to the ENOP definition on the two-dimensional space. From the ENOP definition on the two-dimensional space, the approximation of ENOP for the smoothing TPS is given by   
k A ( γ ) = 0 0 h( k 1 , k 2 ) d k 1 d k 2 = 0 0 ( γ ( ( π L 1 k 1 ) 2 + ( π L 2 k 2 ) 2 ) 2 +1 ) -1 d k 1 d k 2 . (52)
Applying the changes of the variables   
r= ( π L 1 k 1 ) 2 + ( π L 2 k 2 ) 2 and tanθ= L 1 k 2 L 2 k 1
to the integral of Eq. (52), the approximation of ENOP becomes   
k A ( γ ) = L 1 L 2 2 π 2 0 π 2 dθ 0 1 γ r 2 +1 dr = L 1 L 2 4π × 1 2 B( 1 2 , 1 2 ) γ - 1 2 = L 1 L 2 8 γ - 1 2 . (53)
The approximation of ENOP, Eq. (53), is only applied to a rectangular region. The approximation of ENOP generalized for application to an arbitrary region is given by   
k A ( γ ) = Ω 8 γ - 1 2 . (54)
The approximation of ENOP of the smoothing nth order poly-harmonic spline whose functional is defined as   
Π= i=1 m 1 2 ω M ( i ) ( z M ( i ) -f( x M ( i ) , y M ( i ) ) ) 2 +γ Ω 1 2 ( n+1 f ) 2 dΩ
is given by   
k A ( γ ) = Ω 4π( n+1 ) B( 1 n+1 , n n+1 ) γ - 1 n+1

Figure 2 shows the approximation of ENOP obtained from Eq. (54) and the exact ENOP, kGCV, obtained from Eq. (19) when the smoothing TPS is used, the number of samples is 21×21=441, and the sampling region is [0,1]×[0,1]. Bias is added to the approximation of ENOP to enable comparison with the exact ENOP. The approximation of ENOP, kA, shows good agreement with the exact ENOP, kGCV, in the ENOP range from 0 to half the number of samples.

Fig. 2.

Equivalent number of parameters.

7. Plate Surface Estimation by Using LIDAR and Smoothing TPS with Sampling Measure Weights

Figure 3 shows the point cloud of samples measured with a LIDAR on a plate surface. The plate length is 5.475 m and the plate width is 2.143 m. The x-axis is the rolling direction, and the values of the x-axis are ten times the actual ones. The y-axis is the plate width direction. The distance in the longitudinal direction x is decoupled because the wavelength in the longitudinal direction x is shorter than that in the width direction y and to show both wavelengths in the same dimension. The distribution of samples is not homogeneous. The number of samples is 25691. The values of the samples include measurement error of 2 mm. This error value is taken from the specification of the 3-D laser scanner, which is a Photon 120 manufactured by FARO Corporation. The samples (Fig. 3) are interpolated into a DEM (digital elevation model) in Fig. 4 with a mesh of triangles. Although the DEM of the plate surface has a zigzag shape, the actual plate surface was not zigzagged, as it was a rolled plate. The zigzag shape derives from LIDAR measurement error. The zigzag shape is considered to be eliminated by the smoothing TPS with sampling measure weights.

Fig. 3.

Plate surface point cloud.

Fig. 4.

Plate surface DEM.

As the number of samples becomes larger (experimentally more than 3000), the time required to solve the system of Eq. (8) or Eq. (28) becomes prohibitively long. In this case, in order to evaluate the value of Eq. (48), we applied a fast calculation method based on FMM proposed by Beatson et al.4) and solved the system of Eq. (28) with GMRES. We referred to the preconditioning proposed by Beatson et al.5) for GMRES. In the preconditioning, the approximated cardinal functions which were proposed by Beatson et al.5) were used as reference.

In this case, we use BIC (Baysian information criterion) defined as   

BI C A =mlog( i=1 m ω M ( i ) ( z M ( i ) -f( x M ( i ) , y M ( i ) ) ) 2 ) -logm×mlog( ( 1- k A m ) 2 )
to decide the optimal smoothing parameter. Practically speaking, application of the GVC method is not possible with large-scale samples. Figure 5 shows the search result of the optimal smoothing parameter. The horizontal axis shows the smoothing parameter. The vertical axis shows BIC. The points show the search process and the circle at the lower right shows the optimal search result. The value of the optimal smoothing parameter is 5.09×10−3 and the value of ENOP is 199.9. Figure 6 shows the estimated plate surface for the optimal smoothing parameter. Good agreement between the plate surface shown in Fig. 6 and the result of manual measurement was confirmed.
Fig. 5.

Results of Bayesian information criterion.

Fig. 6.

Result of optimal regression for plate surface.

The plate surface was estimated on a 2.66 GHz Intel(R) Core(TM)2 Quad CPU with 3.25 GB RAM and a Microsoft Visual C++ 2005 on a Windows XP x86. It took 9.8 sec to calculate the plate surface in Fig. 6 and 226 sec to calculate the optimal smoothing parameter.

The distribution of the wavelength on a plate surface does not depend on the size of plates if the plates are produced under the same processing conditions. Let the representative half wavelength on a plate surface be Lω. From Eqs. (38) and (43), the smoothing parameter for the smoothing TPS is given by   

γ= 1 64 L ω 4 . (55)
The smoothing parameter depends on only the half wavelength, Lω. Equation (55) means that it is not necessary to calculate the optimal smoothing parameter whenever the plate to be measured changes. Figure 7 shows the surface regressions when the same smoothing parameter is used.
Fig. 7.

Results of regressions for plate surfaces.

8. Conclusions

In this paper, we proposed a method for reconstructing the surface shape of steel plates from a large number of point data obtained by a 3-D scanner. The following conclusions were drawn from the theoretical, numerical, and experimental investigations carried out in this work:

(1) A smoothing spline method with sampling measure weights was defined theoretically.

(2) In the limit of infinite samples, the transfer function and the frequency response function for the smoothing spline system with sampling measure weights were solved theoretically.

(3) The approximation of ENOP was derived from the frequency response function theoretically. We confirmed that the approximation agreed with the theoretical ENOP.

(4) The information criteria including the approximation of ENOP enabled calculation of the optimal smoothing parameter.

(5) We applied the proposed method to the problem of actual large-scale samples measured by LIDAR. The results clarified the fact that engineering applications of the method are possible.

References
 
© 2017 by The Iron and Steel Institute of Japan
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