# Co-registration of surfaces by 3D least squares matching.

Abstract: A method for the automatic co-registration of 3D surfaces is presented. The method utilizes the mathematical model of Least Squares 2D image matching and extends it for solving the 3D surface matching problem. The transformation parameters of the search surfaces are estimated with respect to a template surface. The solution is achieved when the sum of the squares of the 3D spatial (Euclidean) distances between the surfaces are minimized. The parameter estimation is achieved using the Generalized Gauss-Markov model. Execution level implementation details are given. Apart from the co-registration of the point clouds generated from spaceborne, airborne and terrestrial sensors and techniques, the proposed method is also useful for change detection, 3D comparison, and quality assessment tasks. Experiments using terrain data examples show the capabilities of the method.

## Summary (3 min read)

### Introduction

- With the availability of the various sensors and automated methods, the production of large numbers of point clouds is no longer particularly notable.
- In terrestrial laser scanning practice, special targets provided by the vendors, e.g., Zoller Fröhlich, Leica, and Riegl, are mostly used for the co-registration of point clouds.
- Such a strategy has several deficiencies with respect to fieldwork time, personnel and equipment costs, and accuracy.
- In a recent study, Sternberg et al. (2004) reported that registration and geodetic measurements comprise 10 to 20 percent of the total project time.
- In another study, a collapsed 1,000-car parking garage was documented in order to assess the damage and structural soundness of the building.

### 34980 Sile, Istanbul, Turkey, and formerly with the Institute of Geodesy and Photogrammetry, ETH Zurich,

- Targets are essentially required in projects where the absolute orientation to an object coordinate system is needed.
- The co-registration is crucially needed wherever spatially related data sets can be described as surfaces and has to be transformed to each other.
- This work, called 3D Least Squares surface matching (LS3D), is another straightforward extension of the 2D Least Squares image matching and has the same underlying ideas and concepts.
- The execution aspects and the implementation details are extensively elaborated.

### Least Squares 3D Surface Matching

- The Basic Estimation Model Two 3D surfaces are subject to a co-registration procedure.
- This extension is especially useful to avoid such over-parameterization problems and for the flexible selection of the appropriate degree of freedom (DOF).
- For every template surface element, the correspondence operator seeks a minimum Euclidean distance location on the search surface.
- The central point is discarded according to the number of distances that are greater than a given distance threshold.
- The mesh boundaries represent the model borders, and in addition the data holes inside the model.

### Initialization: Defining the Box Size

- For the sake of simplicity, they are given the same (M) here.
- Step 2. Scan L1 and fill B. For any point ai, the box indices are as follows: (17) where stands for the truncation operator, and DX, DY, and DZ are dimensions of any box along the x y z axes, respectively.
- In L2, the first point of the (u,v,w)th box is indexed by I while the address of the subsequent points is controlled using B whose value is incremented each time a new point enters the box.

### Access Procedure

- Step 1. Using Equation 17, compute the indices ui, vi, and wi of the box that contains point ai.
- In L2, I indexes the first point, while the number of points in the box is given by the following formula: (19) The access procedure requires O(q) operations, where q is the average number of points in the box.
- One of the main advantages of the boxing structure is a faster and easier access mechanism than the tree search-based methods provide.
- In the LS3D surface matching case, the search surface, for which the boxing structure is established, is transformed to a new state by the current set of transformation parameters.
- The access procedure is the same, except the following formula is used for the calculation of indices: (20) Where ‘ ’ stands for a vector dot product.

### Simultaneous Matching of Surface Geometry and Intensity

- When the object surface lacks sufficient geometric information, i.e., homogeneity or isotropicity of curvatures, the basic algorithm will either fail or will find a side minimum.
- Available attribute information, e.g., intensity, color, temperature, etc., is used to form quasi-surfaces in addition to the actual ones.
- The matching is performed by simultaneous use of surface geometry and attribute information under a combined estimation model (Akca, 2007a).

### Experimental Results

- The algorithm was implemented as a stand-alone MS Windows™ application with a graphical user interface.
- The software package was developed with the C/C programming language.
- Thus, angle was excluded from the system, and the second version of the computation was run in 5-DOF mode.
- (a) SRTM C-Band DEM with data holes, (b) registration of a local DEM onto the SRTM C-Band DEM by use of the LS3D matching, and (c) filled data holes.

### Image Data

- The image data consisted of 28 DMC images with a ground sampling distance (GSD) of 22 cm arranged in four parallel flight strips in the E/W direction, each of seven images.
- The forward and side overlap of the DMC images were 60 percent and 75 percent, respectively.
- The automated DSM generation was performed using the SAT-PP software 314 March 2010 PHOTOGRAMMETRIC ENGINEER ING & REMOTE SENS ING TABLE 3.

### DSM Comparison and Analysis

- Evaluation is done based on the height differences.
- The LS3D surface matching method was used to avoid both these shortcomings.
- This is actually the difference between the two DSMs after removing the reference frame differences.
- In planimetry, this bias is due to the different orientations of the images and the lidar, and is significant only in the Y (N/S) direction.
- The study is a cooperative project between the IGP and the Department of Landscape Inventories of the Swiss Federal Research Institute WSL.

### 75 percent forward and a 30 percent lateral overlap.

- All images were digitized with a Vexcel UltraScan® scanner with a 15 micron pixel size, which results in a GSD of 15 cm and 8.25 cm for the 1997 and 2002 images, respectively.
- The 1997 film images had severe scratches on the emulsion side, causing artifacts in the digitized images and DSM errors in the automated DSM generation .
- The national lidar data of the Swiss Federal Office of Topography was acquired in 2001 when leaves were off the trees.
- The first pulse point cloud was interpolated to a regular grid with 2.5 m grid spacing, called 2001_DSM.

### Co-registration and Change Detection

- They cannot consider the surface modeling errors.
- The estimation model is the Generalized Gauss-Markov model.
- The capability to match surfaces of different quality and resolution is another positive aspect of the proposed method.
- This is especially useful for quality assessment and change detection tasks as discussed in the Experimental Results Section.

### Acknowledgments

- I would like to express my gratitude to my advisor, Professor Armin Gruen, for his invaluable advice and support on the work presented here.
- The author is financially supported by an ETHZ internal research grant, which is gratefully acknowledged.

### Conclusions

- The basic estimation model is a generalization of the Least Squares matching concept.
- The current implementation uses a 3D similarity transformation model for the geometric relationship.
- The unknown transformation parameters are treated as observables with proper weights, so that sub-versions of the 7-parameter model can be run, i.e., rigid body (6-DOF), tilt and translation (5-DOF), translation (3-DOF), horizontal shift (2-DOF), and depth (1-DOF).

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##### Citations

225 citations

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### Cites methods from "Co-registration of surfaces by 3D l..."

...3D similarity transformation model [96] Small plateau ALS, images...

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### Cites methods from "Co-registration of surfaces by 3D l..."

...Automatic co-registration of the DEMs on the reference DEM was performed in LS3D (Least Squares 3D Matching; Akca, 2010)....

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### Cites methods from "Co-registration of surfaces by 3D l..."

...Consequently, the LiDAR DTM acquired in 2007 was taken as the reference surface for all the other processed DTMs. Automatic co-registration of the DTMs on the reference DTM was conducted with LS3D (Least Squares 3D Matching software; Akca, 2010)....

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...The DTM difference calculations in LS3D yielded a direct estimate of the location and amount of topographic change over the four time periods (Figure 6)....

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...Automatic co-registration of the DTMs on the reference DTM was conducted with LS3D (Least Squares 3D Matching software; Akca, 2010)....

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...The transformation of individual DTMs to a common reference system using LS3D reduced the offsets between DTMs that were introduced by slightly different GPS positions....

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##### References

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### "Co-registration of surfaces by 3D l..." refers methods in this paper

...A comparison study between the LS3D and the Iterative Closest Point (ICP; introduced by Besl and McKay (1992), Chen and Medioni (1992), and Zhang (1994) was carried out....

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2,013 citations

### "Co-registration of surfaces by 3D l..." refers methods in this paper

...A comparison study between the LS3D and the Iterative Closest Point (ICP; introduced by Besl and McKay (1992), Chen and Medioni (1992), and Zhang (1994) was carried out....

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### "Co-registration of surfaces by 3D l..." refers background in this paper

...More details on this issue can be found in Gruen (1985), Maas (2002), Gruen and Akca (2005), and Kraus et al. (2006)....

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...Least Squares matching is a mathematical concept, which was originally developed for automatic point transfer on stereo or multiple images (Ackermann, 1984; Pertl, 1984; Gruen, 1985)....

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...The theoretical precisions of the transformation parameters are optimistic, mainly due to the stochastic properties of the search surfaces that have not been considered as such in the estimation model, as is typically done in Least Squares matching (Gruen, 1985)....

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "Co-registration of surfaces by 3d least squares matching" ?

The further conceptual extensions are given as: the Least Squares matching of 3D curves, matching of 3D curves or 3D sparse points ( e. g., ground control points ) with a 3D surface, and a general framework, which can perform the multiple surface matching, the combined surface geometry and intensity matching, and georeferencing tasks simultaneously ( Akca, 2007b ).