# A Direct Approach for the Segmentation of Unorganized Points and Recognition of Simple Algebraic Surfaces

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1 A Direct Approach for the Segmentation of Unorganized Points and Recognition of Simple Algebraic Surfaces PhD Thesis M. Vančo University of Technology Chemnitz October 2002

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3 Ein direktes Verfahren zur Segmentierung unstrukturierter Punktdaten und Bestimmung algebraischer Oberflächenelemente Dissertation zur Erlangung des akademischen Grades Doktoringenieur (Dr.-Ing.) vorgelegt der Fakultät für Informatik der Technischen Universität Chemnitz von: geboren am: in: Dipl.-Inf.(SK) Marek Vančo 6. Juni 1974 Bratislava Gutachter: Prof. Dr. Guido Brunnett Prof. Dr. Beat D. Brüderlin Prof. Dr. Hans Hagen Chemnitz, 10. Oktober 2002

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5 Abstract In Reverse Engineering a physical object is digitally reconstructed from a set of boundary points. In the segmentation phase these points are grouped into subsets to facilitate consecutive steps as surface fitting. In this thesis we present a segmentation method with subsequent classification of simple algebraic surfaces. Our method is direct in the sense that it operates directly on the point set in contrast to other approaches that are based on a triangulation of the data set. The reconstruction process involves a fast algorithm for k-nearest neighbors search and an estimation of first and second order surface properties. The first order segmentation, that is based on normal vectors, provides an initial subdivision of the surface and detects sharp edges as well as flat or highly curved areas. One of the main features of our method is to proceed by alternating the steps of segmentation and normal vector estimation. The second order segmentation subdivides the surface according to principal curvatures and provides a sufficient foundation for the classification of simple algebraic surfaces. If the boundary of the original object contains such surfaces the segmentation is optimized based on the result of a surface fitting procedure.

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7 Zusammenfassung Im Reverse Engineering wird ein existierendes Objekt aus einer Menge von Oberflächenpunkten digital rekonstruiert. Während der Segmentierungsphase werden diese Punkte in Teilmengen zusammengefügt, um die nachfolgenden Schritte wie Flächenerkennung (surface fitting) zu vereinfachen. Wir präsentieren in dieser Arbeit eine Methode zur Segmentierung der Punkte und die anschließende Klassifikation einfacher algebraischen Flächen. Unser Verfahren ist direkt in dem Sinne, dass es direkt an den Punkten arbeitet, im Gegensatz zu anderen Verfahren, die auf einer Triangulierung der Punktmenge basieren. Der Rekonstruktionsprozess schließt einen neuen Algorithmus zur Berechnung der k-nächsten Nachbarn eines Oberflächenpunktes und Verfahren zur Schätzung der Flächeneigenschaften ersten und zweiten Grades ein. Die normalenbasierte Segmentierung (Segmentierung ersten Grades) liefert eine Aufteilung des Objektes und detekiert scharfe Kanten, sowie flache oder stark gekrümmte Gebiete des Objektes. Ein zentrales Element unserer Methode ist die Wiederholung der Schritte der Segmentierung und der Schätzung der Normalen. Erst die Iteration ermöglicht die Schätzung der Normalen in der benötigten Genauigkeit und die Generierung einer zufriedenstellender Segmentierung. Die Segmentierung zweiten Grades teilt die Oberfläche nach den Hauptkrümmungen auf und bietet eine zuverlässige Grundlage für die Klassifizierung einfacher algebraischen Flächen. Falls der Rand des Ausgangsobjektes solche Flächen enthält, wird der Segmentierungsprozess auf der Grundlage des Ergebnisses der Flächenerkennungsprozedur optimiert.

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9 Contents 1 Introduction 1 2 Background and Fundamentals of Reverse Engineering Data acquisition Multiview registration Surface reconstruction Triangulation based on 3D tetrahedrization Direct triangulation Surface segmentation Surface fitting Surface reconstruction from cross sections Definitions Prerequisites for the segmentation k-nearest neighbors computation Organizing of the data set The search algorithm Complexity Memory complexity Time comparison Approximation of the normal vectors Local Centre Triangulation (LCT) Local Delaunay Triangulation (LDT) Approximation with an analytic surface (AwAS) Comparison of the methods for normal vector estimation Consistent orientation of the normal vectors Computation of principal curvatures Segmentation Segmentation of the surface based on normal vectors Cleaning up the segmentation Iterating segmentation and normals estimation Second order Segmentation Processing principal curvatures

10 10 CONTENTS 5 Postprocessing and Surface Fitting Plane fitting Sphere fitting Obtaining the sphere parameterization from the segmentation Cylinder fitting Obtaining the cylinder parameterization from the segmentation Cone fitting Obtaining the cone parameterization from the segmentation Extension procedure Postprocessing procedure Postprocessing thresholds Results 85 7 Conclusion 89

11 List of Figures 2.1 2D and 3D object reconstruction Touch probe scanner Laser scanner Optical Scanner Multiview scanning Merged views of an scanned real object Reverse engineering of an object Triangulation results Embryo consisting of cross sectional contours - one skin (Courtesy of Anatomic Institute in Göttingen) Reconstructed embryo consisting of three skins and dead cells (Courtesy of Anatomic Institute in Göttingen) Interval histogram and median determining Median subdivision in 2D Hash table with point lists Binary tree after median subdivision nearest neighbors searching in 2D Local Centre Triangulation Local Delaunay Triangulation Comparison of a quadratic and quintic approximation Approximation of points with a quadratic curve in the original and rotated coordinate system Approximation with a cubic curve of a surface with sampling flaws Normal vector orientations problems Neighborhood depending orientation propagation Correct and estimated normal vectors on a sharp edge Segmentation using two angles An example for segment creation First order segmentation for artificially generated object consisting of 22,211 points. 41 segments result containing 21,112 points The remainder segment of the object on the figure

12 12 LIST OF FIGURES 4.6 First step normal estimation (upper image) and the normal vectors after first recomputing (lower image) Subdivision of a surface of revolution into strips during first order segmentation Curvatures smoothing problem between smooth transition between a cylinder and a sphere First order a) and second order b) segmentation of a curved box First order a) and second order b) segmentation of a mechanic part First order a) and second order b) segmentation of a rotational part Second order segmentation of a cone a) and the cone after postprocessing b) Parameterization of a sphere Parameterization of a cylinder Parameterization of a cylinder Computing of a point of the cone s axis Computing the cone apex C Approximate intersection of two lines L 1 L Computing N ϕ ϑ A distance of Q to the cone mantle Cylindrical or spherical segment? Planar segments detection Points stealing Segmentation and recognition problems with a small plane Segmentation and recognition problems on a C 1 transition Gaussian Image of one cylindrical, two conical and one spherical segment Reconstruction of a noisy cone The first order segmentation: 76 regular segments containing 89,753 points (of 90,974) The second order segmentation: 25 regular segments containing 76,343 points (of 90,974) The first postprocessing pass: 18 regular segments containing 90,963 points (of 90,974) The first order segmentation: 77 regular segments containing 88,877 points (of 90,974) The second order segmentation: 171 regular segments containing 46,793 points (of 90,974) The first postprocessing pass: 37 regular segments containing 87,814 points (of 90,974) The second postprocessing pass: 18 regular segments containing 90,898 points (of 90,974) Results Results The object from the fig. 6.1 f), if the sphere detection was switched off. 88

13 List of Tables 3.1 Searching times for median-hashing search algorithms Normal estimation comparison for a box Normal estimation comparison for a cone Normal estimation comparison for a torus Normal vector orientation times Normal recomputing for a box Automatic segmentation procedure with recomputing of the normal vectors after each segmentation Segmentation and postprocessing (including the classification) time consumption of some objects. All times are in seconds Postprocessing thresholds

14 Chapter 1 Introduction The computers affected the human life in almost all areas. Especially in the industry they became an essential part of the manufacturing process - they are used e.g. for path planning in numerical controlled machining, quality controlling, CAD/CAM (Computer Aided Design / Computer Aided Manufacturing), visualization etc. The need for digital models used in the manufacturing process caused a rapid development of reverse engineering. There are numerous real objects, for which the computer representation is demanded. The reason is obvious: the computer representation is compact, easy to transfer without loss of information (thanks to Internet). Together with an appropriate tool the digitized model can easily be visualized, is simple to modify (e.g. CAD representation can be very easy modified) etc. To get a computer representation of an object there two quite different ways: To model the object in a modeler program (e.g. AutoCAD, Pro Engineer, Catia, Surfacer etc. for creating a CAD model or 3D Studio Max, SoftImage, Maya, TrueSpace etc. for general modeling). To scan the object with a 3D scanner and to reconstruct (if possible) the surface with a surface reconstruction program. The first choice (to model the object in a modeler program) is the direct way (socalled forward method) to obtain a computer representation. The greatest disadvantage is, that for a complex object this method can be extremely time consuming and require that the human is very familiar with the modeling program. The disadvantage of the second choice (the reverse method) is that a 3D scanner is not (at present) a very common computer equipment and the high-end accurate 3D scanners are rather expensive. The advantage is a slight user interaction (the aim of surface reconstruction tools is a full automatic program with no user interaction). For complex objects the reconstruction tools are mathematically very difficult and the authors believe that some user assistance will always be necessary. In this thesis we present an approach for automatic surface segmentation and recognition of sampled objects, consisting of unorganized boundary points. Our aim is the 1

15 2 CHAPTER 1. INTRODUCTION development of an automatic (or a semi-automatic) tool, which is able to decompose the object into coherent point clusters based on estimated surface properties. These clusters are hereafter approximated by surfaces and the surface parameters are obtained. Using an existing tool (e.g. ACIS library) boundary curves of the recognized surfaces or intersection curves between two surfaces can be approximated, which together with the provided surface decomposition fully describe the object to be reconstructed. In the beginning the user should provide starting parameters for the whole reconstruction process. No further user interaction is desired in the future work, but at the moment the user should supervise the reconstruction process and, if necessary, slightly modify the initial parameters and repeat the failed reconstruction step. We based the whole segmentation and reconstruction on a general data structure, which can efficiently be computed even for large data sets: the neighborhood graph. The description of our new method for fast finding of k-nearest neighbors to a given point is shown in the chapter 3. The method is based on a median subdivision of the space containing the points (creating so-called buckets) and the searching in each bucket is accelerated by a hashing strategy with an appropriate distribution function. Further we introduce three new methods for computation of the surface normal vectors from the neighborhood graph and discuss their stability and efficiency. A very fast modification of the well known orientation propagation algorithm for consistent normal vectors orientation is presented. At the end of the chapter 3 the method for second order surface properties estimation is described. In the chapter 4 we present our first and second order surface segmentation. In the first step that is based on normal vectors propagation, sharp or nearly sharp edges (i.e. edges with a very high variation of normal vectors) in the object are detected. The detection allows us to improve our initial normal vectors estimation by modifying the neighborhood of the points close to the sharp edges. In the second order segmentation principal curvatures are used to subdivide the data set. Here, tangent continuous but curvature discontinuous edges are detected. The segmentation procedure is able to find and separate parts of simple algebraic surfaces (planes, cylinders, cone, spheres). To keep the user interaction and the number of repetitions as small as possible we provide to the user an estimation of the segmentation parameters, which are crucial for a proper second order segmentation. The method how to estimate these parameters from the first order segmentation is given at the end of the chapter 4. The task of the second order segmentation is to subdivide the object according to the specified criteria into consistent parts. These parts do not have to correspond to the fundamental geometric surfaces (as planes, spheres, etc.), which form the object. It even happens, that if the object contains some sampling flaws or noise, one simple algebraic surface can be subdivided into many segments. An important part of this thesis is the postprocessing (chapter 5) of the segmentation, which provides an improvement of the segmentation based on the surface recognition. Its task is to find the segments that lie on the same simple algebraic surface (as planes, cones, cylinders or spheres), fit them to the simple surface and extend the recognized segments. The postprocessing consists of the steps of

16 3 segment classification: an assumption about the geometric surface type of a segment is made and validated by creating an appropriate parameterization and by fitting an algebraic surface to the segment. Finally the surface parameters are obtained. segment extension: neighboring points are added to a segment, if they fulfill the surface equation within a prescribed tolerance. Similarly, as in the previous chapter, at the end of the chapter 5 we introduce our procedure for automatic postprocessing thresholds estimation, in order to reduce the necessary user assistance. Our reconstruction method is very fast (the whole segmentation is done within a few seconds for a data set of 100 thousand points), which allows the user, if necessary, to fine tune the reconstruction results. In the chapter 6 results obtained by our method are presented for various objects.

17 4 CHAPTER 1. INTRODUCTION

18 Chapter 2 Background and Fundamentals of Reverse Engineering Reverse engineering and surface reconstruction are young fields of the computer graphics. They got very popular in early 80 s with rapid development of the object scanning devices. There is a vast amount of objects (objects in real life, parts of machines, toys etc.), for which the computer representation is not known (like hand modeled prototypes) or not available. The aim of the surface reconstruction is to transform such an object into digital representation, which can be used for further visualization, processing, manufacturing etc. The methods of 3D reconstruction and the further processing of the point clouds have already been introduced in almost all fields of human life such as medicine, architecture, machine building, die making industry, design, toy industry as well as clothing. The entertainment industry (film making industry), scientific visualization as well as virtual reality worlds can also be mentioned as fields demanding 3D digitized objects. The analogy between 2D and 3D object reconstruction is illustrated in the fig. 2.1: for 2D reconstruction the raw data are paper documents (text or photos), which are digitized with an usual optical scanner, processed (e.g. OCR: transformation of an image into a text), optionally modified and prepared for printing or plotting. Analogously, a 3D object is scanned by a scanning device (for details see section 2.1), reconstructed (a computer representation is created, which fully covers the initial object and is easy to handle, modify, test, process etc.) and can optionally be manufactured. The whole process can be subdivided into following steps: Data acquisition - object scanning. Multiview registration - aligning the scanned views. Surface reconstruction - creating a computer model (CAD, polygonal approximation etc.). Manufacturing - [optional] 5

19 6CHAPTER 2. BACKGROUND AND FUNDAMENTALS OF REVERSE ENGINEERING 2D: Paper document / Photo 2d scanner Optical Character Recognition Digital document / Image Digital document / Image Digital Image Processing Printer Plotter Paper document / Photo 3D: Real object 3d scanner 3D Model Surface reconstruction 3D Model Rapid protyping CAM Real object Figure 2.1: 2D and 3D object reconstruction 2.1 Data acquisition The acquisition of the geometry of 3D models is a longstanding difficult problem in computer graphics. In the past, forward methods such as CSG, B-rep, NURBS, and sweeping have been successful in modeling regular shapes such as mechanical and architectural parts. These methods allow a user to directly and interactively define the shape of an object. On the other hand, natural objects and scenes often have irregular shapes and complex surface structures. It is very difficult to use forward methods to interactively define the details of the complex shapes and surfaces. Further, the desire to reduce the dependence on human input in making realistic images of complex scenes has, over the past ten years, resulted in an increased role for measurements of the real world in the computer graphics pipeline. With the availability of vision equipment and the maturity of computer vision technology, there is now a trend to reverse engineer the acquisition process by recovering the geometric and topological information from measurements of real scenes and objects. The scanning process can be divided into two categories: Passive scanning (sensing).

20 2.1. DATA ACQUISITION 7 Active scanning (sensing). A few years ago computer vision scientists were mainly interested in passive sensing techniques that were supposed to reflect the way the human eye is working, i.e. no energy is emitted for the purpose of sensing, it is only received. Such passive techniques include stereo vision and monocular techniques like shape-from-contour or shape-from-shading. The problem is that recovering 3D information from a single 2D image is an ill posed problem. Thus, monocular techniques have to add a priori information such as surface smoothness to recover 3D data - a process known as regularization. Regularization is most often context dependent and is difficult to apply outside of the lab. Stereo vision techniques use two or more sensors, and thus two or more 2D images to recover 3D information. To interpret disparity between images, the matching problem must be solved, which has been formulated as an ill-posed problem too in a general context and which anyway is a task difficult to automate. To overcome the above problems of passive sensing, active sensing techniques have been developed, i.e. properly formatted light (or any other form of energy) is emitted and then received once it has interacted with the object to digitize. Typically, light is emitted in the direction of an object, reflected on its surface and recovered by the sensor and the distance to the surface is calculated. Obviously, volume digitization techniques like computerized tomography (CT), magnetic resonance imaging (MRI) or ultrasound imaging also fall in this category. Scanners, based on the type of acquiring the information from a 3D object can be categorized into these groups: Contact scanners: Touch probe scanners Non contact scanners: Laser scanners Optical scanners using structured light A touch probe scanner (sometimes called coordinate measuring machines - CMM), see fig 2.2 physically touches the object to be scanned with a probe. The probe is usually mounted on a bench which must have enough degrees of freedom to move around the part to digitize. The probe is either moved manually or the bench has motors and digital command control. Most CMMs translate along three orthogonal axes. Manual articulated arms have recently appeared at the market and offer an easier to use, although less precise, alternative. CMMs are very precise and efficient, but their biggest disadvantage is the speed - they work very slow. They digitize one sample at a time, and for each sample the probe has to carefully approach the surface, contact it, then reach for the next sample. Although path planning software tools for CMMs do exist - which make the task considerably easier - there were no tools to visualize and process the large sets of samples - clouds of points - involved in the digitization of complex surfaces.

21 8CHAPTER 2. BACKGROUND AND FUNDAMENTALS OF REVERSE ENGINEERING Figure 2.2: Touch probe scanner Figure 2.3: Laser scanner

22 2.1. DATA ACQUISITION 9 Figure 2.4: Optical Scanner The expansion of computer-aided mechanical design and computed-aided manufacturing (CAD/CAM) has brought new needs. Digitizing a few samples may be enough to measure simple geometric surfaces like perfect cylinders, but modeling complex surfaces means digitizing thousands, sometimes millions of samples. Among the active non-contact devices for 3D measurements, the laser scanner is probably the most versatile one. Its main advantages are great setup simplicity, small size and high resolution. There are a variety of ways of using lasers to measure distance. The precise 3D shape or profile of solid objects can be determined using laser scanning techniques. Common approaches include: Time of flight: it measures the time the light travels to the scanned object. It is probably the most difficult, especially for short to medium distances where high resolution is desired. The simplistic method of just measuring the round trip from the reflection of a pulsed laser beam requires extraordinarily precise timing as the range decreases and the desired resolution increases. Chirped pulse lidar/radar: the laser transmitter sends out an optical signal which include a subcarrier with a time-varying (chirped) frequency. If the frequency changes linearly, the difference between the outgoing and detected signal frequencies f is a constant (over the duration of the chirp) and the distance is then: f c Distance 2chirp rate Where c is the velocity of light.

23 10CHAPTER 2. BACKGROUND AND FUNDAMENTALS OF REVERSE ENGINEERING Triangulation: a light source illuminates an object and an image of this light spot is then formed, by means of a lens, on the surface of a linear light sensitive sensor. By measuring the location of the light spot image the distance of the object from the instrument can be determined. Common optical range finders can measure the distance using the difference in angles of the line-of-sight from an LED to the scene and back to a near photodetector. As the distance is reduced, the angle increases. This is coupled to the lens focusing mechanism. The use of a well collimated laser would increase both the maximum useful distance and resolution of such a system. Interferometry: for very precise measurement of small changes in position, the use of the wave nature of coherent laser light cannot be surpassed. Resolution scaled down to a few nanometers is possible with relatively simple equipment. Nowadays, there are two main approaches to the rapid acquisition of the models of natural objects and scenes: structured light and multiple views. The first approach projects a structured light pattern, usually a stripe of laser beam, onto the surface of an object and uses a camera to capture the surface contour as revealed by the reflected laser light. The contours along different surface patches are then combined to compute the 3D geometry of the object. Such laser scanning products are currently available in two versions: hand-held and automated. The hand-held version requires the user to manually move the laser stripe to various parts of the object s surface. It, therefore, takes a lot of manual work to capture all the surface details of an object. The automated version uses a motorized system to move the laser in a manner similar to a 2D image scanner. 3D laser scanners typically produce accurate results but are expensive, especially the automated version and the color scanners. The second approach uses one or more cameras to capture multiple views of an object and uses the multiple views to recover 3D coordinates of the feature points on the object s surface. The single-camera system either moves the camera or the object so as to capture various views of the object. The multiple-camera system typically employs static cameras to simultaneously capture multiple views of the object. However, it would require many cameras to adequately capture various facets of a complex object, and would therefore be more complex and expensive than the single-camera system. Example of a laser scanning device: Laser beam and laser plane triangulation sensors Cyberware 3D color digitizers are very often introduced in the movie industry. The sensor is mounted on proprietary benches and translates/rotates along digitized surfaces (e.g. actors) to perform rectangular/cylindrical scans. The sensor projects a fine line of laser light across the surface. The shape of this line is captured and processed to recover the depth of every sample. As the sensor moves along, the same line is illuminated with white light and a color video camera captures the apparent color - texture.

24 2.2. MULTIVIEW REGISTRATION Multiview registration The goal of registration is to transform sets of surface measurements into a common coordinate system. To capture a complete object surfaces multiple 2.5D range images from different viewpoints are required as illustrated in the fig Figure 2.5: Multiview scanning Besl and McKay [10] introduced the Iterative Closest Point (ICP) algorithm to register two sets of points on a free-form surface. ICP is a general purpose, representation independent method for the accurate and computationally efficient registration of 3D shapes including free-form curves and surfaces. Extensions of this algorithm are now widely used for registration of multiple sets of surface data. The original algorithm registers two point sets provided one is a subset of the other and that the transform between the sets is approximately known. The original ICP algorithm operates as follows given point set P and surface Q where P is a subset of Q: Nearest point search: For each point p in P find the closest point q on Q. Compute registration: Evaluate the rigid transform T that minimizes the sum of squared distances between pairs of closest points p q. Transform: Apply the rigid transform T to all points in set P. Iterate: Repeat step 1 to 3 until convergence. This approach will converge to the nearest local minimum of the sum of squared distances between closest points. A good initial estimate of the transformation between point sets is required to ensure convergence to the correct registration. Incorrect registration may occur if the error in the initial transformation is too large (typically

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