curvature interpolation method for surface reconstruction...

Curvature Interpolation Method for Surface

Reconstruction for Geospatial Point Cloud Data

Hwamog Kim∗, Jeffrey L. Willers†, and Seongjai Kim‡

August 23, 2017

Abstract

Surface reconstruction for scattered data is an ill-posed problem and most compu-tational algorithms become overly expensive as the number of sample points increases.This article studies an effective partial differential equation (PDE)-based algorithm,called the curvature interpolation method with iterative refinement (IR-CIM). The newmethod iteratively utilizes curvature-related information which is estimated from anintermediate surface of the nonuniform data and plays a role of driving force for thereconstruction of a reliable image surface. The IR-CIM is applied for apparent soilelectro-conductivity (ECa) data sets and digital elevation modeling for geospatial pointcloud data of overlapping strip scans acquired by light detection and ranging (LiDAR)technology. This article also introduces an effective method for the elimination of Moireeffect over strip overlaps. The resulting algorithm converges to a piecewise smooth im-age, requiring O(N) operations independently of the number of sample points, whereN is the number of grid points; outperforming inverse-distance weighting methods.

Key words. Image reconstruction, point cloud data, curvature interpolation method (CIM),

Moire effect, partial differential equation (PDE).

2000 MSC: 65M06, 68U10, 86A22, 94A08

∗Mississippi State University, Department of Mathematics and Statistics, Mississippi State, MS 39762USA. Email: [email protected].†USDA-ARS, Genetics and Precision Agriculture Research, Mississippi State, MS 39762 USA. Email:

[email protected].‡Mississippi State University, Department of Mathematics and Statistics, Mississippi State, MS 39762

USA. Email: [email protected].

1

CIM for Surface Reconstruction 2

Contents

1. Introduction 3

2. Preliminaries 4

2.1. General remarks on surface reconstruction for point cloud data . . . . . . . . 4

2.2. The curvature interpolation method (CIM) . . . . . . . . . . . . . . . . . . . 6

2.3. CIM with iterative refinement (IR-CIM) . . . . . . . . . . . . . . . . . . . . 7

3. Surface Reconstruction for Point Cloud Data 9

3.1. The intermediate surface φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2. A 9-point scheme: Evaluation of A and K . . . . . . . . . . . . . . . . . . . 11

3.3. Smoothing K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.4. Construction of a smooth correction surface w . . . . . . . . . . . . . . . . . 14

3.5. A synthetic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4. Elimination of Moire Effect in LiDAR Processing 15

4.1. Moire patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2. Correction strategy for overlapped scans . . . . . . . . . . . . . . . . . . . . 17

5. Numerical Experiments 20

5.1. Apparent soil electro-conductivity data . . . . . . . . . . . . . . . . . . . . . 20

5.2. LiDAR data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

6. Conclusions 31


1. Introduction

Various interpolation methods of nonuniformly sampled data (or, scattered data) are

widely applied to the fields of e.g. computer graphics and geosciences [18, 41]. The most cited

are polynomial interpolation such as nearest-neighbor, linear, and cubic methods [24, 38];

these methods are easy to implement, but offer only low-quality results. Inverse-distance

methods [33] are also used, although they are computationally expensive and become im-

practical as the number of samples increases. Another common interpolation model for

arbitrarily spaced data is the method of thin-plate splines, which is based on radial basis

functions [37]. However, the method is hard to be practical due to a high computational

complexity. Since the solution must be found by solving a dense algebraic system, with the

dimension being equal to the number of sample points, the problem becomes intractable even

for mid-sized images. See [10, 17] for efforts for the reduction of computational complexity

of the method. Various other methods have been proposed for the resampling of scattered

data in terms of the approximation theory; see [1, 23, 35, 36, 40] for approximations in spline

spaces and [11, 29] for those in wavelet spaces. For relevant works in computer graphics, see

[30] and references therein.

In this article, the authors are interested in a novel partial differential equation (PDE)-

based method for surface reconstruction of scattered data. The new method utilizes a

curvature-related information which is estimated from an intermediate surface of the nonuni-

form data and plays a role of driving force for the construction of a reliable image surface.

It is often the case that the constructed image surface does not contain all of the data values

due to the estimated curvature. However, the error can be corrected by a recursive applica-

tion of the curvature interpolation method having four major components: (1) measure the

misfit on the data points, (2) construct an intermediate surface over the image domain based

on the misfit values, (3) estimate the corresponding curvature-related information, and (4)

build a correction surface to update the last iterate. We will call the new algorithm the

curvature interpolation method with iterative refinement (IR-CIM) for the image reconstruc-

tion of scattered data. The IR-CIM is an application of the curvature interpolation method

(CIM) studied for image zooming [8, 22]. It has been verified that the IR-CIM shows a

minimum oscillatory behavior, and yet it results in piecewise smooth images containing all

the data values.

This article is an application of [13] for point cloud data. Here we newly consider or

introduce the following.

• Application of the IR-CIM to the surface reconstruction for point cloud data: The IR-

CIM is applied is applied to apparent soil electro-conductivity (ECa) data sets and

digital elevation modeling for geospatial point cloud data of overlapping strip scans

acquired by light detection and ranging (LiDAR) technology. In image zooming, rela-

tively accurate curvature information can be obtained utilizing the given low resolution

image; however, it is hardly the case for scatter data. In order to estimate the curvature


accurately, an intermediate surface is constructed by solving a constrained harmonic

equation. Such a harmonic surface is known to be a good approximation of the minimal

surface [3].

• Elimination of Moire patterns In LiDAR technology, a data set consists of overlap-

ping strip scans. Since each strip scan is carried out in different conditions (in flight

direction, illumination level, etc.), it is often the case that the scanner is calibrated

just before each scan. Thus the scanned data set may involve systematic misfits on

elevation values which in return introduces oscillatory patterns to the resulting sur-

face, a kind of the Moire patterns. This article introduces an effective scheme for the

elimination of Moire patterns from overlapping data sets.

• Effective 9-point difference schemes for curvature computation: The IR-CIM in each

iteration carries out an update by constructing and adding an intermediate surface

(correction surface) to the resulting image. Thus it is necessary to make all the cor-

rection surfaces smooth enough in order to build up a smooth resulting image surface.

This has motivated the authors to employ 9-point finite difference (FD) schemes, which

can construct image surfaces smoother than the standard 5-point scheme.

The article is organized as follows. In Section 2, we consider some preliminaries including

issues on surface reconstruction for point cloud data and a brief review of the IR-CIM for

image zooming. Section 3 presents the main algorithm, the IR-CIM for point cloud data, as

an iterative algorithm having four major components. Each component is discussed in detail.

In particular, an effective 9-point finite difference scheme is introduced in order to help the IR-

CIM produce locally smooth image surfaces. In Section 4, an effective computational method

is discussed for the elimination of Moire patterns appeared over the overlapped scan strips.

Section 5 gives various examples to show effectiveness of the suggested algorithm applied to

data sets of ECa and LiDAR. We have found that the resulting algorithm results in reliable

image surfaces efficiently with little dependence on sample rates. Section 6 summarizes our

work.

2. Preliminaries

This section presents a brief review for surface reconstruction from point cloud data,

followed by the curvature interpolation method (CIM) studied for image zooming [8, 22].

2.1. General remarks on surface reconstruction for point cloud data

Point clouds are gained by scanning three-dimensional (3D) objects using various mea-

suring techniques. The point cloud represents the set of points, each of which is defined by

(x, y, z) coordinates. Point clouds are used for many purposes, including 3D computer-aided

design (CAD) modeling for manufactured parts (reverse engineering), meteorology/quality

inspection, visualization, animation, and mass customization applications. While point

clouds can be directly rendered and inspected [32], point clouds themselves are not directly


usable in most 3D applications. Thus they are usually converted to polygon mesh or triangle

mesh models, NURBS surface models, or CAD models through a process commonly referred

to as surface reconstruction. There are many techniques for surface reconstruction. Some

approaches build a network of triangles over the existing vertices of the point cloud (De-

launay triangulation, marching triangles [19], and ball-pivoting [4]), while other approaches

convert the point cloud into a volumetric distance field and reconstruct the implicit surface

through a marching cubes algorithm [20]. The Delaunay triangulation becomes inefficient

for complex objects due to the large number of possible geometries. The marching trian-

gle ball-pivoting are in general more efficient than Delaunay triangulation and may provide

similar accuracy in reconstruction.

In nonuniform sampling theory, the image reconstruction from an arbitrarily sampled

data set is a challenging problem when no constraint is imposed on their locations. The

problem is ill-posed, first of all, and numerical methods solving its optimization formulation

become overly expensive as the number of sample points increases [1, 2]. Furthermore, it is

often the case that the constructed image is not an interpolator but an approximator, i.e., the

reconstructed surface may not include the data values but approximate. The reconstruction

of approximating image surfaces might be useful for some applications particularly when the

data are erroneous. However, it is better for the reconstruction algorithm to be able to build

image surfaces which contain exact data values. It is known in the literature of interpolation

that superior kernels are interpolators, the converse is not always true though [7, 22, 24, 25].

In LiDAR data processing, for example, a frequently-used interpolation algorithm is the

inverse-distance weighting (IDW) method [33], which is defined as

u(x) =

∑k w(xk)u(xk)∑

k w(xk), w(xk) =

1

‖x− xk‖p0, (2.1)

where u(x) is the estimated value at location x, u(xk) is the value at a neighboring point

xk, w(xk) denotes the weight for xk, ‖ · ‖0 is the Euclidean norm, and p is an exponential

number (usually, greater than or equal to 2). Here the summation takes place over a vicinity

of x, which often is a rectangular window centered at x and having a radius r. The IDW

method is one of state-of-the-art interpolation algorithms particularly for scattered data.

In Figure 1, we present the reconstructed image surfaces for the marching-triangle method

and IDW of search radii r, for a synthetic example. (The summation window size is (2r +

1) × (2r + 1).) Figure 1(a) shows a set of nonuniformly sampled data, of which the image

domain contains 100× 100 pixels and 20 random data points having values ranging between

0 and 255; the background image values are set 128, for the figuring purpose. The marching-

triangle method, a triangulation-based algorithm, has resulted in an unacceptable surface

as in Figure 1(b). This example shows that triangulation-based algorithms may produce

oscillatory surfaces over regions where data points are coarse and of rapidly-varying values.

As one can see from Figures 1(c) and 1(d), the IDW method produced reasonable surfaces.

However, the search radius of the IDW must be large enough to reduce interpolation artifacts,

which in return makes the method computationally expensive. Furthermore, the resulting


(a) (b)

(c) (d)

Figure 1: Synthetic example: Surface reconstruction by the marching-triangle method and IDW:(a) a scattered data, and reconstructed image surfaces by (b) the marching triangle method, (c)the IDW with search radius r = 80 (window size =161× 161), and (d) the IDW with search radiusr = 100 (window size =201× 201). For IDW (2.1), we set p = 2.

elevation surface is not smooth near some data points.

The above example has motivated the authors to develop an effective surface reconstruc-

tion algorithm for point cloud data. The suggested algorithm to be presented in Section 3

can produce a smooth surface which contains all data points and reveals a minimum oscilla-

tory behavior. The new algorithm is an application of the so-called curvature interpolation

method (CIM) [8, 22] which is originally developed for image zooming.

2.2. The curvature interpolation method (CIM)

Image zooming is a processing task to enlarge images by applying interpolation methods.

The CIM is a PDE-based model; it begins with the selection of a curvature-related term which

is to be estimated from the low resolution (LR) image, interpolated to the high resolution

(HR) image grid, and incorporated as a driving force for the construction of HR image.

PDE-based models that employ the (mean) curvature itself as the smoothing operator (e.g.,


the total variation (TV) model [31]) are known to have a tendency to converge to a piecewise

constant image [9, 26]. Such a phenomenon is called the staircasing. Thus the curvature

would better be replaced by a more effective and convenient operator K. In [22], the authors

adopted the following gradient-weighted (GW) curvature

K(u) = −|∇u| ∇ ·( ∇u|∇u|

). (2.2)

Let Ω and Ω be the original LR image domain and its α-times magnified HR image

domain, α > 1, respectively. Let u denote the HR image, the α-times magnification of an

LR image u. Then we should have

u(x) = u(x), x = αx, (2.3)

where x = (x, y) and x = (x, y) are the coordinates of the LR and HR images, respectively.

Since ∇x = α∇x, the scaling factor between the GW curvatures on Ω and Ω is α2. That is,

|∇xu| ∇x ·( ∇xu

|∇xu|

)= α2 |∇xu| ∇x ·

( ∇xu

|∇xu|

). (2.4)

Now, let Ω0 denote the set of pixel points in Ω which can be expressed as αq, where q is a

pixel point in Ω. Then the CIM [22] can be outlined as follows.

I. Compute the GW curvature of the given LR image v0:

K = K(v0) on Ω. (2.5)

II. Interpolate K to obtain K on Ω.

III. Solve, for u on Ω, the following constrained problem

K(u) =1

α2K, u|Ω0 = v0. (2.6)

In the above algorithm, the GW curvature measured from the LR image is interpolated

and incorporated as an explicit driving force for the same GW curvature model on Ω. The

driving force would help the model construct the HR image more effectively, enforcing the

resulting image to satisfy the given curvature profile locally.

2.3. CIM with iterative refinement (IR-CIM)

The CIM applied for image zooming can construct an image surface on the HR grid

which satisfies the interpolated curvature of the given LR image, as the solution of (2.6).

However, due to the involved interpolation operations, the constructed surface may have

image values different from those in the corresponding LR grid points. A natural remedy for

this drawback is to update image values iteratively by utilizing the difference between the

LR image and the last updated image in the LR grid (misfit). Such an algorithm is called


the CIM with iterative refinement (IR-CIM) [8]. Utilizing the misfit, the IR-CIM in each

iteration constructs a smooth image surface, a correction image, as shown in the following.

Initialize u0 = 0, on the HR image domain ΩSelect the tolerance τ > 0For k = 1, 2, · · ·

(i) On the LR domain Ω, computepk = v0 − uk−1

(ii) Evaluate A and K for given pk:K = Apk ≈ K(pk)

(iii) Apply the bilinear method to zoom:

A→ A, K → K

(iv) On Ω, solve for wk:

Awk =1

α2K

(v) Update: uk = uk−1 + wk

(vi) If ‖wk‖ < τ , stop

(2.7)

Here u is the zoom-out of u defined on the LR grid and ‖ · ‖ denotes either the L2-norm or

the maximum norm which measures the magnitude of the k-th correction vector wk.

The above iterative algorithm, applied for image zooming, deserves the following remarks.

• The values of the final image uk are real-valued and to be rounded to become the

nearest integers, which is called the quantization. Such a quantization process allows

the image values to change by 0.5 in maximum, which in practice hardly alters the

image outlook. One may choose the tolerance τ = 0.5, or slightly larger than that.

• The algebraic system in Step (iv) of IR-CIM (2.7) can be solved efficiently by the

Richardson’s iterative method: For an initial solution wk,0, find wk,m, m = 1, 2, · · · ,given by

wk,m = wk,m−1 + r( 1

α2K − Awk,m−1

), m ≥ 1, (2.8)

for some small r > 0. In this case, the above iteration serves as the inner loop, while

the iteration updating uk (the k-loop) becomes the outer loop. The inner loop may

start from a reasonably accurate initial value, for example, the bilinear interpolation

of the misfit pk.

• The correction image wk in every outer iteration must be smooth enough not to in-

troduce ringing artifacts, because otherwise the ringing artifacts would remain in the

resulting HR image. This article introduces an effective 9-point scheme for surface

reconstruction for point cloud data; see Section 3.2 below.

We close the section with a numerical experiment which shows effectiveness of the IR-

CIM; for simplicity, we consider a simple example of signal reconstruction from an 1-D

regular data. See Figure 2, in which the domain is set Ω = [0, 200] and samples are obtained


Figure 2: Curve construction for 1D signal data. The dashed and dot-dashed curves are obtainedby applying respectively the cubic B-spline method and the IDW, while the solid curve is resultedfrom three outer iterations of the IR-CIM.

at x = 20, 60, 100, 140, and 180. There the dashed and dot-dashed curves are obtained

by applying respectively the cubic B-spline method and the IDW, while the solid curve is

resulted from the IR-CIM in three outer iterations. As one can see from the figure, the cubic

B-spline method has constructed a signal showing a large oscillation, which may introduce

a severe ringing artifact for more generally sampled data in multiple dimensions. Note that

the B-spline method gives no way to construct a signal on the regions outside the convex hull

of sample points. The IDW interpolation method performs better than the cubic B-spline

method, involving yet a certain degree of oscillations. On the other hand, the IR-CIM can

build a smooth curve, over the whole domain, which reveals a minimum oscillatory behavior.

3. Surface Reconstruction for Point Cloud Data

When the CIM is applied for image zooming as in (2.7), the curvature of the LR image

is first computed and then interpolated for an approximation of the curvature of the HR

image. Such an approximated curvature shows a reasonable accuracy so that the IR-CIM in

each iteration can produce a reliable correction image to update the image surface. In image

zooming, the given image may be viewed as an LR approximation of the target HR image.

However, the case is quite different for the surface reconstruction for scattered data, because

the data loci are nonuniform and it is hard to estimate the curvature. Thus we first have to

introduce an efficient scheme to estimate the surface and its curvature; we will construct an

intermediate surface, as the solution of the Laplace equation, from which a useful curvature

information would be obtained.

Let Ω be the image domain and Ω0 the set of data pixels where image values are initialized

as v0. Our new surface reconstruction algorithm to be presented below involves three major

steps: (1) the construction of an intermediate surface, (2) the curvature evaluation and

smoothing, and (3) the construction of a correction surface. When the reconstructed surface

does not contain all of the prescribed image values (v0), the difference can be corrected


by applying the procedure iteratively. The following outlines our reconstruction algorithm.

Details will follow.

Initialize u0 = 0, on the image domain ΩSelect the tolerance τ > 0For k = 1, 2, · · ·

(i) Compute the misfit on Ω0:rk−1 = v0 − uk−1

∣∣Ω0

(ii) If ‖rk−1‖ < τ , stop

(iii) Construct an intermediate surface φk:−∆φk = 0, x ∈ Ω \ Ω0

φk = rk−1, x ∈ Ω0

∇φk · n = 0, x ∈ ∂Ω(iv) Evaluate Ak and Kk from φk:

Kk = Akφk ≈ K(φk)

(v) Smoothen Kk to get Kk

(vi) Construct the correction surface wk:

Akwk = Kk

(vii) Update :uk = uk−1 + wk

(3.1)

Here v0 is the vector representation of the sampled data v0, rk−1 denotes the misfit defined

on the sample points Ω0, φk is an intermediate surface, a solution of an interior point value

problem of the Laplace (harmonic) equation, n denotes the unit outward normal defined on

the image boundary ∂Ω, and ‖ · ‖ is either L2 norm or the maximum norm. The equation

∇φk · n = 0 is called the no-flux boundary condition. The interior point value problem in

(3.1.iii) is incorporated for the construction of an intermediate surface, due to its simplicity.

As aforementioned, such a harmonic surface is a good approximation of the minimal surface

[3].

We will call the algorithm (3.1) the curvature interpolation method with iterative refine-

ment (IR-CIM). Hereafter, we will omit the subscript k, for a simpler presentation, whenever

no confusion is involved.

3.1. The intermediate surface φ

The construction of an intermediate surface φ is somewhat arbitrary. That is, there are

many different ways to fulfill the task. In this article, we will adopt the Laplacian smoothing

(local averaging), due to its simplicity. It is easy to implement and results in a useful

curvature information.

The diffusive PDE in (3.1.iii) can be discretized as follows.

4φp,q − φp,q−1 − φp−1,q − φp+1,q

−φp,q+1 = 0, xp,q 6∈ Ω0,(3.2)


Figure 3: The standard coordinates x = (x, y) and the 45-rotated coordinates x′ = (x′, y′) super-posed on the vicinity of the pixel point (p, q).

where φp,q denotes the value of φ at xp,q. Utilizing

φp,q = rk−1,p,q, xp,q ∈ Ω0, (3.3)

and applying the no-flux condition along the boundary points, the associated algebraic sys-

tem for the Laplacian smoothing can be formulated as

Bφ = gk−1, (3.4)

where the source vector gk−1 has nontrivial values on only the rows corresponding to sample

points. It is easy to see that the matrix B is nonsingular.

The solution of (3.4) is the most expensive component in the IR-CIM. We have imple-

mented various algebraic solvers for (3.4) including the successive over relaxation (SOR)

method, the strongly implicit procedure (SIP) [34], and the alternating direction implicit

(ADI) method [15, 16]; the SOR turns out to be simple to implement and efficient enough.

3.2. A 9-point scheme: Evaluation of A and K

The IR-CIM in each iteration constructs a correction surface, incorporating the misfit

which measures the difference between the scattered data and the last updated image. Thus

it is necessary to make all the correction surfaces smooth enough in order to build up a

smooth resulting image surface. This has motivated the authors to employ the following

9-point FD schemes for both the curvature evaluation and the construction of correction

surfaces.

Let x′ = (x′, y′) be the coordinates which are 45-rotated counterclockwise from the

standard coordinates x = (x, y). See Figure 3. Then, as an alternative to (2.2), we may

consider

K(φ) = −|∇xφ|1( φx

|∇xφ|

)x− |∇xφ|2

( φy

|∇xφ|

)y

−|∇x′φ|1( φx′

|∇x′φ|

)x′− |∇x′φ|2

( φy′

|∇x′φ|

)y′,

(3.5)


where |∇xφ|1 and |∇xφ|2 are the same as |∇xφ|, and |∇x′φ|1 and |∇x′φ|2 are the same as

|∇x′φ|; however, terms of different subscripts will be approximated in different ways.

Note that the curvature in (3.5) evaluates twice the conventional gradient-weighted curva-

ture in (2.2). This does not require any modification in the IR-CIM (2.7), because the matrix

A will be correspondingly defined as in (2.7.iv) and utilized for the surface reconstruction in

(2.7.vi).

Let N, S, E, and W denote respectively the north, south, east, and west directions. For the

(p, q)-th pixel of the intermediate image φ, we first compute the second-order FD approxima-

tions of |∇φ| at xp−1/2,q (W ), xp+1/2,q (E), xp,q−1/2 (S), and xp,q+1/2 (N) and approximations

of |∇x′φ| at xp−1/2,q−1/2 (SW ), xp+1/2,q+1/2 (NE), xp+1/2,q−1/2 (SE), and xp−1/2,q+1/2 (NW ):

dpq,W = [(φp,q − φp−1,q)2 + (φp−1,q+1 + φp,q+1

−φp−1,q−1 − φp,q−1)2/16 + ε2]1/2,

dpq,E = dp+1,q,W ,

dpq,S = [(φp,q − φp,q−1)2 + (φp+1,q + φp+1,q−1

−φp−1,q − φp−1,q−1)2/16 + ε2]1/2,

dpq,N = dp,q+1,S,

dpq,SW =[(φp,q − φp−1,q−1)2

2+

(φp−1,q − φp,q−1)2

2+ ε2

]1/2

,

dpq,NE = dp+1,q+1,SW ,

dpq,SE =[(φp+1,q − φp,q−1)2

2+

(φp,q − φp+1,q−1)2

2+ ε2

]1/2

,

dpq,NW = dp−1,q+1,SE,

(3.6)

where ε > 0 is the regularization parameter introduced to prevent the denominators from

approaching zero. (We will set ε = 0.01 for all examples presented in this article.) Then,

the directional curvature terms at xpq can be approximated as( φx

|∇φ|

)x(xpq) ≈ 1

dpq,Wφp−1,q −

( 1

dpq,W+

1

dpq,E

)φpq +

1

dpq,Eφp+1,q,( φy

|∇φ|

)y(xpq) ≈ 1

dpq,Sφp,q−1 −

( 1

dpq,S+

1

dpq,N

)φpq +

1

dpq,Nφp,q+1,( φx′

|∇x′φ|

)x′

(xpq) ≈1

2

[ 1

dpq,SWφp−1,q−1 −

( 1

dpq,SW+

1

dpq,NE

)φpq

+1

dpq,NE

φp+1,q+1

],( φy′

|∇x′φ|

)y′

(xpq) ≈1

2

[ 1

dpq,SEφp+1,q−1 −

( 1

dpq,SE+

1

dpq,NW

)φpq

+1

dpq,NW

φp−1,q+1

].

(3.7)


Now, we discretize the gradient magnitudes as follows.

|∇φ|1(xpq) ≈[1

2

( 1

dpq,W+

1

dpq,E

)]−1

,

|∇φ|2(xpq) ≈[1

2

( 1

dpq,S+

1

dpq,N

)]−1

,

|∇x′φ|1(xpq) ≈[1

2

( 1

dpq,SW+

1

dpq,NE

)]−1

,

|∇x′φ|2(xpq) ≈[1

2

( 1

dpq,SE+

1

dpq,NW

)]−1

,

(3.8)

where the right-hand sides are harmonic averages of FD approximations of the gradient

magnitudes in x-, y-, x′-, and y′-coordinate directions, respectively. Then, it follows from

(3.5), (3.7), and (3.8) that

K(φ)(xpq) ≈ 6φpq − apq,W φp−1,q − apq,E φp+1,q

−apq,S φp,q−1 − apq,N φp,q+1

−apq,SW φp−1,q−1 − apq,NE φp+1,q+1

−apq,SE φp+1,q−1 − apq,NW φp−1,q+1,

(3.9)

where

apq,W =2 dpq,E

dpq,W + dpq,E, apq,E =

2 dpq,Wdpq,W + dpq,E

,

apq,S =2 dpq,N

dpq,S + dpq,N, apq,N =

2 dpq,Sdpq,S + dpq,N

,

apq,SW =dpq,NE

dpq,SW + dpq,NE, apq,NE =

dpq,SWdpq,SW + dpq,NE

,

apq,SE =dpq,NW

dpq,SE + dpq,NW, apq,NW =

dpq,SEdpq,SE + dpq,NW

.

Here it is easy to see that apq,W+apq,E+apq,S+apq,N = 4 and apq,SW+apq,NE+apq,SE+apq,NW =

2.

Let A denote the coefficient matrix of K. Then, the GW curvature estimate K corre-

sponding to the 9-point stencil can be expressed algebraically as

K = Aφ, (3.10)

where φ is the vector representation of φ. It follows from (3.5) and (3.9) that the nonzero

entries of A corresponding to xpq, in the stencil formulation, are

[A]pq =

−apq,NW −apq,N −apq,NE

−apq,W 6 −apq,E−apq,SW −apq,S −apq,SE

. (3.11)

Thus the sum of off-diagonal elements in a row of A is clearly −6 and therefore the Jacobi

matrix of A defines a weighted averaging operator.


Note that the intermediate surface φ satisfies ∆φ = 0 except at the sample points. Thus

the GW curvature vector K in (3.10) may have values large in modulus at sample points

and small elsewhere.

We will set the zero curvature, K = 0, along the image boundary for simplicity. For the

evaluation of the coefficient matrix A at boundary points, the model should incorporate a

boundary condition, which may affect the final outcome near the boundary. In this article,

we will employ the Dirichlet boundary condition for the first pixel point (0, 0) and the no-

flux boundary condition for the other boundary points. The Dirichlet boundary condition

(assigned at a point) makes the first row of the algebraic system strictly diagonally dominant,

which in turn allows the coefficient matrix A to be nonsingular [39]. Thus (2.7.vi) can define

the correction surface that satisfies the Dirichlet condition w0,0 = φ0,0.

3.3. Smoothing K

We adopt a convenient weighted 9-point averaging scheme for the smoothing operation:

Let K(0) = K and find K(`), ` = 1, 2, · · · , defined by

K∗p,q =K

(`−1)p−1,q + cwK

(`−1)p,q +K

(`−1)p+1,q

cw + 2,

K(`)p,q =

K∗p,q−1 + cwK∗p,q +K∗p,q+1

cw + 2,

(3.12)

where cw is a weight parameter for the central pixel, cw ≥ 1. When cw = 1, the above

algorithm becomes the simple box scheme

K(`)p,q =

1

9

( q+1∑j=q−1

p+1∑i=p−1

K(`−1)i,j

). (3.13)

The iteration stops either after a certain number of iterations (`∗) or when the sign of values

of K(`) changes from that of K(0) by a prescribed per cent.

It has been observed from various numerical examples that the smoothed curvature

K(= K(`∗)) must be smooth enough to result in smooth correction surfaces. However,

an oversmoothed curvature may slow down the convergence speed of the IR-CIM iteration.

When the curvature is smoothed reasonably, the IR-CIM has converged in 3-6 outer itera-

tions for all examples we have tested.

3.4. Construction of a smooth correction surface w

In each iteration of the IR-CIM (3.1), the final nontrivial task is to construct a smooth

correction surface w by solving the linear system in (3.1.vi). Note that the curvature smooth-

ing operator introduces tangible changes to the curvature mainly in the vicinity of sample

points and therefore the solution w can be obtained correspondingly, incorporating changes

of φ there. Thus, when the initial value of w is set φ, local relaxation methods such as the

SOR method must converge in a few iterations.


Figure 4: Synthetic example: Surface reconstruction by the IR-CIM, in four iterations.

3.5. A synthetic example

In the IR-CIM, the most expensive component is the computation of intermediate surface

φ discussed in Section 3.1. On the other hand, the convergence of the IR-CIM outer iteration

depends on the curvature smoothing presented in Section 3.3. More effective strategies are

yet to be developed for the curvature smoothing.

We will close the section with a numerical result performed by the IR-CIM. Figure 4

presents a smooth surface which is obtained by four iterations of the IR-CIM, starting from

the synthetic data in Figure 1(a). Unlike the IDW which has resulted in nonsmooth surfaces

as in Figure 1(d), the IR-CIM is able to produce a smooth surface everywhere including

data points. The IR-CIM (for the four iterations) took 0.237 seconds on a 2.6 GHz laptop

computer, while the IDW (with the search radius r = 100) finished the interpolation in 0.450

seconds. For this example of a small number of sample points, the IR-CIM as an iterative

algorithm is twice more efficient than the IDW, a direct evaluation method. As the number

of sample points increases, the IDW would take proportionally more computation time, while

the cost of the IR-CIM remains the same.

4. Elimination of Moire Effect in LiDAR Processing

For the last decade or so, the LiDAR technology has grown in popularity, meeting the

need and replacing conventional surveying techniques which are time-consuming and labor-

intensive [12, 21, 27, 28]. LiDAR data are acquired in a form of point cloud; helicopter

or fixed-wing LiDAR systems scan the surface below the aircraft, collecting reflected light

signals in a scan rate of 50,000 to 100,000 pulses per second, achieving high accuracies

up to 50mm. Due to their high resolution and rich information content, LiDAR data are

utilized for a wide range of applications with different requirements in terms of resolution,

accuracy, and surface representation. Applications include digital elevation model (DEM)

topography, flood risk mapping, watershed analysis, coastal erosion analysis, landslides, tree


(a) (b) (c)

Figure 5: LiDAR data acquisition: (a) a schematic illustration of data collection, (b) the aircrafttrajectory for a field survey over Mississippi farms near Mississippi State University, May 12, 2011,and (c) one of the LiDAR scan coverages.

canopy analysis, transmission line mapping, and urban applications [6]. LiDAR is an active

remote sensing technique, analogous to radar, but using laser light.

See Figure 5, which depicts a schematic illustration of LiDAR data acquisition, along

with the aircraft trajectory and the LiDAR scan coverage for a field survey over Mississippi

farms near Mississippi State University (MSU), May 12, 2011. LiDAR instruments built on

an aircraft measure the round-trip time for a pulse of laser energy to travel between the

sensor and a target. This incident pulse of energy (usually with a near-infrared wavelength

for vegetation studies) reflects off of canopy (branches, leaves) and ground surfaces and back

to the instrument where it is collected by a telescope. The travel time of the pulse, from

initiation until it returns to the sensor, provides a distance or range from the instrument to

the object. The acquired information is then transformed, with the aid of a minimum of

four GPS satellites, to obtain a 3D position fix. The individual data points are collected to

form a set of point cloud data. As one can see from Figure 5(c), the data are collected with

the scan strips overlapped, in order to densely cover the scan area by cloud points.

4.1. Moire patterns

Digital elevation models (DEMs) derived from such LiDAR techniques are growing in

popularity as a tool for use in soil survey, in particular. This form of remotely sensed

elevation data can serve multiple purposes, not the least of which is the visual interpretation

of landforms and soil parent materials.

However, as one can see from Figure 6(a), the data may involve missing gaps; for which it

is necessary to either perform extra rounds of data acquisition or incorporate a very effective

interpolation algorithm for the reconstruction of reliable elevation surfaces. Figure 6(b)

depicts a elevation surface processed by the IDW built in ArcMap, using another subset

of the LiDAR data. (ArcMap is one of state-of-the-art geospatial processing programs.)

In the figure, white regions are related to trees and buildings and the ground elevation

decreases in the SE direction. The parallel features running in the SSW-NNE direction


(a) (b)

Figure 6: LiDAR soil survey over Mississippi farms near MSU: (a) a missing gap in the point clouddata and (b) the elevation surface processed by the IDW built in ArcMap.

are processing artifacts involved during the surface reconstruction of the data which have

been acquired through scan lines in the same direction with multiple overlapped scan strips.

Thus the parallel features are kinds of Moire interference patterns. Contours and topographic

parameters (gradients, curvatures) derived from such elevation surfaces must include a noisy

pattern; for most applications, they have been further processed (filtered, smoothed), often

manually, to make them suitable [28].

4.2. Correction strategy for overlapped scans

As aforementioned, it is a common practice in LiDAR data acquisition that the data

are collected with the scan strips overlapped. However, due to various technical reasons,

data sets obtained from different scans covering the same overlapped area may have misfits

in elevation values. When these data sets are used without an appropriate correction, the

resulting surface may involve serious artifacts including oscillatory patterns.

The elevation misfits can be eliminated using local elevation averages that are obtained

from each of the data sets and measured over the overlapped scan areas. However, these

elevation averages computed from data sets may not represent the real average height of the

same area, unless the data loci are distributed relatively uniformly. In this subsection, we

suggest an effective method that can eliminate misfits incorporated due to scan overlaps.

For a simple presentation, we first assume that the scan coverages overlap maximum

twice as shown in Figure 7. In the figure, Si (i = 1, 2, 3) denote the scan strips, the dash-dot

lines (in blue) indicate the center of the scan strips, and Ci are points on the center lines of

scan strips. The check points P12 and P23 represent centroids of overlapped areas between

the scans S1 and S2 and between S2 and S3, respectively. The data correction begins with

partitioning the overlapped areas and setting check points. Then the average elevation values

on each of overlapped scan strips are computed in a vicinity of each check point, using the


Figure 7: A schematic illustration of LiDAR scan coverages. The dash-dot lines (in blue) indicatethe center of the scan strips and the red bullets in the overlapped areas are check points.

Figure 8: A schematic illustration for data correction through local elevation averages.

data values at pixels that are scanned and assigned from the both sides of the adjacent scan

strips. If the average elevations obtained from the two different scan strips are different at

the check points, the data on the two adjacent strips can be adjusted linearly so that the

resulting data sets have the same average elevations at the check points.

In Figure 8, the average elevation at P12 computed from S1, E12, is larger than that from

S2, E21. Thus the data elevation values in S1 and S2 can be corrected by adjusting the data

values on half-strips linearly. For example, assuming that−−−→C1C2 is parallel to the y-axis, the

elevation values over the overlapped region can be adjusted as follows.

(x, y, z) ∈ S1 7→ (x, y, z′), z′ = z +(E21 − E12)/2

p12,y − c1,y

(y − c1,y),

(x, y, z) ∈ S2 7→ (x, y, z′′), z′′ = z +(E21 − E12)/2

c2,y − p12,y

(c2,y − y),

(4.1)

where c1,y, c2,y, p12,y are the second(y)-coordinates of C1, C2, P12, respectively. Note that

when y = p12,y,

z′ = z + (E21 − E12)/2, z′′ = z + (E21 − E12)/2;

both the corrected data strips have the same magnitude at the check point.

The algorithm for the elimination of Moire patterns should also incorporate the following

concerns.


• Array of check points: The misfits, the differences of the average elevations on over-

lapped areas, may differ depending on the data utilized. For example, in the case of

Figures 7 and 8, we may try to introduce a line of check points in each of overlapped

areas, aligned parallel to the x direction. Then, for each check point, the average ele-

vation can be computed using data values in a neighborhood of the check point. This

concern introduces an array of check points over the whole scan coverage. The Moire

effects can be reduced effectively by applying piecewise bilinear functions, as explained

earlier in this subsection.

• Multiple overlaps: The scan coverages may overlap more than twice for some regions;

in practice, it is often the case that some regions are covered three times. In this

case, the array of check points should include more rows in the cross direction of the

scan flight (the y direction), in order to represent the misfits more appropriately. For

example, let the data overlap m times in the scan strip Sk. Then the misfit correction

function (MCF) for the data in Sk can be defined to be a polynomial of degree at most

m− 1 in the y direction. The MCF is still piecewise linear in the x direction.

• Accuracy of MCFs: It is often the case that data loci are distributed with a largely

varying density; the data may involve missing gaps. Thus, local elevation averages

obtained from raw data may not accurately represent the real elevation averages and

therefore the MCFs can be erroneous. In order to overcome the difficulty, we may

consider the following strategy.

(i) Construct the surfaces for each of the scan strips.

(ii) Compute the local elevation averages using the elevation values from the recon-

structed surfaces.

(iii) Construct MCFs for each of overlapped strips and eliminate Moire patterns.

(iv) Blend the corrected surfaces for a single surface covering the whole scan area.

(v) Improve the blended surface by applying the IR-CIM.

The final step is not computationally expensive, because the blended surface would be

a good approximation of the final result.

For overlapping multi-strip point cloud data sets having large missing gaps, here we sum-

marize our surface reconstruction algorithm which effectively suppresses artifacts of Moire

patterns.


Figure 9: Data acquisition for scaled electrical conductivity.

1. Construct surfaces Uk, for each scan strip Sk

2. Misfit correction

Allocate an array of check points PijCompute local averages of Uk near each of Pij

Construct the MCFs Qk for all k

Apply Qk to Uk to eliminate Moire patterns,

for each k

3. Blend QkUk for a single surface Q4. Enhance Q, using the IR-CIM

(4.2)

5. Numerical Experiments

This section gives numerical examples to show effectiveness of the suggested algorithms.

For examples presented in this section, the smoothing operation (3.12) with cw = 2 is stopped

when the sign of values of K(m) differs from that of K(0) by 5% and the outer iteration of

the IR-CIM (3.1) is stopped when ‖r‖2 < τ = 0.1. For the purpose of comparison, we have

also implemented the IDW method (2.1); we will set p = 2 for examples to be presented in

this section.

5.1. Apparent soil electro-conductivity data

The new algorithm, IR-CIM, has been applied for surface reconstruction for data sets

of apparent soil electro-conductivity (ECa) acquired using Veris R© 3100 carts. These data

represent and are influenced by many conditions such as cation exchange capacity, soil tex-

ture, soil pH, soil moisture content, and temperature [5]. See [14] for details of Veris carts

and their applications. The ECa data used in this research was obtained in Castro County,

TX, about two months prior to late spring corn planting, during 2013 (Courtesy of Olan

Moore, High Plains Consulting, Inc., Springlake, TX). The field was traversed as shown by

the point patterns in Figure 9, where data points counted 5,143 and covered a circular region

of a diameter approximately a kilometer.


(a) (b)

Figure 10: Reconstructed surfaces in 1000× 1000 pixels obtained by (a) the IDW method of radiusr = 147 and (b) the IR-CIM in three iterations.

Figure 10 depicts reconstructed surfaces obtained by (a) the IDW method and (b) the

IR-CIM. For the IDW method, we set the radius of rectangular window r = 147; when

the radius is smaller than 147, the result shows unreliable values at the center of image

due to the hole in the data set. When the radius is set larger than 147, the reconstruction

cost increases without improving the image quality. As one can see from the figure, the

main features obtained from the two methods agree well to each other, with the IR-CIM

being more reasonable. It must be noticed that the resulting image of the IDW method

contains apparent circular patterns which come from the geometry of acquisition points and

are algorithmic artifacts, while the IR-CIM can reduce the artifacts effectively as shown

in Figure 10(b). The IDW elapsed 54.50 seconds, while the IR-CIM converged in three

iterations taking 4.38 seconds, for the images in one million pixels, on a computer having an

Intel Xeon processor of 3.60 GHz.

5.2. LiDAR data processing

In order to demonstrate effectiveness of the IR-CIM and the suggested Moire-pattern

correction method, the new algorithm has also been applied to a LiDAR data set. The

12 strips of LiDAR point cloud data as shown in Figure 5(c), which are acquired over the

Mississippi North Farm near Mississippi State University, are utilized for the modeling. The

data set includes approximately 37 million points counted including multiple arrivals.

In this study, the Moire pattern is reduced by applying the formulas in (4.1) which use

the elevation averages measured locally from the data; missing gaps are not too large so that

they are treated in the same way. Figure 11 depicts the final elevation image covering a


Figure 11: Final reconstructed surface covering a region of 3Km×2Km square with a one-meterresolution.


Table 1: Elapsed times for the surface reconstruction over a 400m×400m square region, for variousspatial resolutions.

Resolution Image size IR-CIM Iterations CPU (sec) Sample rate (%)1.00 m 400× 400 4 0.80 64.650.50 m 800× 800 4 3.45 22.970.25 m 1600× 1600 4 13.89 6.69

region of 3Km×2Km square with a one-meter resolution, resulting in total 6 million pixels.

The IR-CIM with the Moire pattern correction algorithm converged in 5 iterations, taking

31.90 seconds. The center location of this image corresponds to the latitude and longitude,

approximately 33.4726 N and 88.7706 W, respectively.

In examples below in this article, portions of the LiDAR data set with various resolutions

are utilized to investigate the effectiveness of the IR-CIM and the Moire pattern correction

algorithm. The locations will be indexed based on Figure 11 where the bottom-left cor-

ner corresponds to the coordinate point (0, 0) and the top-right corner corresponds to the

coordinate point (3000, 2000).

In order to check effectiveness of the Moire pattern correction algorithm, we selected

two 26 × 26 square regions in Figure 11 having coordinate values of (2400:2525, 1025:1175)

and (1850:1975, 725:850). Here a:b means “from a to b.” Figure 12 depicts image surfaces

constructed by the IR-CIM covering the square regions: (a)-(b) without the Moire pattern

correction algorithm and (c)-(d) with it. The images are in the “colorcube” color map of

quarter-meter spatial resolution and 10cm resolution in elevation. As one can see from figure,

the surface reconstruction for overlapping multi-strip LiDAR data sets may result in serious

oscillatory patterns unless the Moire patterns are effectively corrected. It is clear to see that

contours and topographic parameters (gradients, curvatures) derived from subfigures (a) and

(b) must include noisy patterns.

We have selected various regions to investigate effectiveness of the Moire pattern correc-

tion algorithm and compared image qualities. The algorithm can reduce Moire patterns very

satisfactorily for most regions, with the case in Figures 12(b) and 12(d) being one of least

effective cases.

In order to further investigate effectiveness of the suggested Moire pattern correction

algorithm, we compare the elevation values in the resulting images by selecting a vertical

segment in the mid of Figure 11 that covers two adjacent scan strips. Figure 13 depicts the

result of our algorithm where the dotted (blue) and dashed (red) curves indicate elevation

values obtained from separate scan strips, while the solid curve (in black) represents the

corrected elevation values over the overlapped region of the two scan strips. As shown in

Section 4.2, the elevation values are corrected using a locally linear correction function; as a

result, the elevation values are blended well to produce continuous and reliable image surface.

Table 1 shows elapsed times for the surface reconstruction over a 400m×400m square re-

gion, for various spatial resolutions; the region corresponds to coordinate values (1700:2100, 750:1150)


(a) (b)

(c) (d)

Figure 12: Moire effect and its correction: (a), (b) Surfaces without correction in Moire effect, (c),(d) Corrected surfaces using the Moire-effect correction algorithm.


Figure 13: Correction of elevation values, shown on a vertical line segment. The solid curve (inblack) represents the corrected elevation values over an overlapped region of two scan strips.

Table 2: Elapsed times for the surface reconstruction over a 100m×100m square region, for variousspatial resolutions.

Resolution Image size IR-CIM Iterations CPU (sec) Sample rate (%)1.00 m 100× 100 4 0.04 75.660.50 m 200× 200 3 0.14 27.370.25 m 400× 400 4 0.73 7.43

in Figure 11. “Sample rate” indicates the percentage of pixels of which values are assigned

directly from the data, while CPU is the elapsed time measured in second (sec). It should

be noticed that the CPU is proportional to the number of pixels. The IR-CIM converged in

four iterations for all three resolutions, taking approximately 5 seconds per million pixels.

Figure 14 depicts (a) the raw data values and (b)–(d) reconstructed surfaces over the same

region considered in Table 1. It is not easy to recognize cons and pros of higher resolutions

in the surface reconstruction, particularly when large images are displayed in small images

as in the figure; Figure 14(d) is in 1600×1600 and reduced to fit in the right bottom portion

of the figure.

In order to see more details, we will consider another square region much smaller than

that of Figure 14, in the following.

Table 2 shows elapsed times for the surface reconstruction over a 100m×100m square

region which corresponds to the coordinate values of (2770:2780, 1150:1250) in Figure 11.

For this example, the IR-CIM converged in three or four outer iterations, taking again

approximately 5 seconds per million pixels.

Figure 15 contains resulting images in three different resolutions over the same region

treated in Table 2. The images look sharper as the spatial resolution increases. Particularly

for the 0.25m resolution, the sample rate becomes 7.43%. Most of traditional surface re-

construction algorithms show the following tendency: (1) resulting poor-quality images for

low sample rates and (2) becoming expensive for higher sample rates. However, the IR-CIM

results in a reliable image satisfactorily and efficiently, with little dependence on sample


(a) (b)

(c) (d)

Figure 14: Constructed images from a 400m×400m square region containing (a) Moire effect cor-rected LiDAR data and IR-CIM images with (b) one-meter scale, (c) half-meter scale and (d)quarter-meter scale.


(a) (b) (c)

Figure 15: Reconstructed surfaces over a 100m×100m square region with (a) one-meter scale, (b)half-meter scale and (c) quarter-meter scale.

Table 3: Computation times for various rectangular regions having the same centroid at (2000, 1200)and the same quarter-meter spatial resolution.

Image size Iterations CPU (seconds) Sample rate (%)400× 400 4 0.7344 7.05800× 800 4 3.0468 7.44

1600× 1600 4 12.0781 7.16

rates.

In order to visualize the progress of the iteration in the IR-CIM, we consider the same

area in Figure 15 with the quarter-meter resolution; the resulting image has 400×400 pixels.

Figure 16 depicts the progress of the IR-CIM starting from the Moire effect-corrected LiDAR

data and intermediate surfaces in 1–3 iterations. As shown in the IR-CIM (3.1), each outer

iteration is a non-constrained reconstruction; however the first iterate in Figure 16(b) reveals

main features of the resulting surface in Figure 15(c). The third iterate in Figure 16(d) is

almost identical to the resulting image, showing all features successfully.

Figure 17 contains line plots over a horizontal line segment chosen from the 400 × 400

image in Figure 15(c), covering coordinate values (150:250, 200). Image values of the first

iterate, the second iterate, and the final image over the line segment are depicted respectively

in the dotted curve (in red), the dashed curve (in blue), and the solid curve (in black). The

bold symbol ’X’ indicates values of the original LiDAR data on the line segment, while the

symbol ’O’ shows LiDAR data values on neighboring horizontal line segments. These neigh-

boring LiDAR data values must affect the surface reconstruction and thus are reflected in

the resulting image. Note that our suggested algorithm (IR-CIM) produces smooth surfaces

through the curvature smoothing in (3.1.v). Thus, when data values are rapidly oscillatory,

the resulting image would assign average values, possibly not exactly the same as the data

values. The IR-CIM can produce a resulting image whose values are nearer to the data

values, by reducing the number of smoothing iterations in (3.12).


(a) (b)

(c) (d)

Figure 16: Progress of the IR-CIM on the square region of Figure 15: (a) Moire effect-correctedLiDAR data and intermediate surfaces in (b) 1 iteration, (c) 2 iterations, and (d) 3 iterations.


Figure 17: Line graph illustrating a single horizontal line data from the region in Figure 15(c).

(a) (b) (c)

Figure 18: Constructed images of quarter-meter resolution for rectangular regions with centroid at(2000, 1200): in (a) 400× 400 pixels, (b) 800× 800 pixels, and (c) 1600× 1600 pixels.

It can be observed from Tables 1 and 2 that the sample rate increases correspondingly as

the spatial resolution is refined. Such a change in sample rates may affect the computation

time of the IR-CIM. Thus in order to remain the sample rate the same, we choose different

(increasing) sizes of rectangular regions, all of which have the same quarter-meter spatial

resolution. Table 3 shows computation times for three rectangular regions with the same

centroid at (2000, 1200), which is linear with respect to the number of pixels (the image

size), taking approximately 4.72 seconds per million pixels. We have observed from various

examples including this one that the computation time of the IR-CIM is (1) linear with

respect to the number of pixels, (2) of little dependence on sample rates, and (3) taking

about 5 seconds per million pixels on a computer having an Intel Xeon processor of 3.60

GHz.

For visual illustrations, Figure 18 depicts the three regions treated in Table 3.


(a) (b)

(c) (d)

Figure 19: Images of a 150m×150m region covering coordinate values of (1110:1260, 1200:1350):(a) raw data, and images in quarter-meter resolution constructed by the SOR method with (b) therelative tolerance of 10−2, (c) the relative tolerance of 10−3, and (d) the relative tolerance of 10−5.


Although LiDAR data sets are acquired satisfactorily without any mechanical errors, the

data may still involve missing parts over regions of water such as ponds, creeks and rivers.

Figure 19 depicts a square region covering coordinate values of (1110:1260, 1200:1350). This

region contains a small pond and thus contains an area of coarse data points, as shown in

Figure 19(a). For the construction of the intermediate surface φk (3.1.iii) and the correction

surface (3.1.vi), such missing gaps make the iterative solver (the SOR method) computa-

tionally expensive, requiring a high number of iterations to converge. The figure depicts

images in quarter-meter resolution constructed by the SOR method with (b) the relative

tolerance of 10−2, (c) the relative tolerance of 10−3, and (d) the relative tolerance of 10−5.

The SOR method has converged respectively in 54, 167, 424 iterations for the construction

of the intermediate solution for the first outer iteration.

When the SOR method is stopped with the the relative tolerance of 10−2, the IR-CIM

takes 1.578 seconds total and the resulting image shows a big hole as shown in Figure 19(b).

For the relative tolerance of 10−3, the IR-CIM takes 1.812 seconds total to reconstruct the

image with a small dark hole as in Figure 19(c). When the SOR method is stopped with the

the relative tolerance of 10−5, the IR-CIM takes 2.328 seconds total and the resulting image

shows no hole as shown in Figure 19(d).

For all the examples presented in this article, the initial value is assigned as u0 = 0, which

may result in a high computational cost especially for the solution of the first intermediate

surface φ1. The suggest method (IR-CIM) is yet efficient; however, in order to further reduce

the computational cost, we may have to consider more effective methods for an accurate

initialization of the solution, particularly when big holes appear in the data.

6. Conclusions

This article has studied an innovative image reconstruction algorithm called the curva-

ture interpolation method with iterative refinement (IR-CIM), which is a PDE-based model

and constructs piecewise smooth image surfaces for arbitrarily spaced data, incorporating a

generalized curvature information as a driving force. Each iteration of the IR-CIM consists

of four steps: the computation of the misfit, the construction of an intermediate surface, an

estimation of the curvature, and the construction of a smooth correction surface to update

the last iterate. The iteration converges as the misfit diminishes. An effective 9-point finite

difference scheme is introduced in order to help the algorithm produce locally smooth sur-

faces. Also, locally bilinear misfit correction functions (MCFs) are introduced for an effective

elimination of Moire interference patterns from the overlapped scan strips of LiDAR data

sets. Numerical examples have been provided to show effectiveness of the IR-CIM and the

MCFs for image reconstruction of arbitrarily spaced data. It has been numerically verified

that the resulting algorithm converges to reliable images satisfactorily, outperforming the

inverse-distance weighting method, one of popular surface construction algorithms.


Acknowledgment

S. Kim’s work is supported in part by NSF grant DMS-1228337.

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