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Abstraction of man-made shapes 2009
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Title | Abstraction of man-made shapes |
Creator |
Zhou, Qingnan |
Publisher | University of British Columbia |
Date Created | 2009-08-31T20:58:03Z |
Date Issued | 2009-08-31T20:58:03Z |
Date | 2009 |
Description | Man-made objects are ubiquitous in the real world and in virtual environments. While such objects can be very detailed, capturing every small feature, they are often identiﬁed and characterized by a small set of deﬁning curves. Compact, abstracted shape descriptions based on such curves are often visually more appealing than the original models, which can appear to be visually cluttered. We introduce a novel algorithm for abstracting three-dimensional geometric models using characteristic curves or contours as building blocks for the abstraction. Our method robustly handles models with poor connectivity, including the extreme cases of polygon soups, common in models of man-made objects taken from online repositories. In our algorithm, we use a two-step procedure that ﬁrst approximates the input model using a manifold, closed envelope surface and then extracts from it a hierarchical abstraction curve network along with suitable normal information. The constructed curve networks form a compact, yet powerful, representation for the input shapes, retaining their key shape characteristics while discarding minor details and irregularities. |
Extent | 15536443 bytes |
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Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2009-08-31T20:58:03Z |
DOI | 10.14288/1.0051201 |
Degree |
Master of Science - MSc |
Program |
Computer Science |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2009-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/12633 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0051201/source |
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Abstraction of Man-Made Shapes by Qingnan Zhou B.CS., The University of Waterloo, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Computer Science) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2009 c© Qingnan Zhou 2009 Abstract Man-made objects are ubiquitous in the real world and in virtual envi- ronments. While such objects can be very detailed, capturing every small feature, they are often identified and characterized by a small set of defining curves. Compact, abstracted shape descriptions based on such curves are often visually more appealing than the original models, which can appear to be visually cluttered. We introduce a novel algorithm for abstracting three-dimensional geometric models using characteristic curves or contours as building blocks for the abstraction. Our method robustly handles mod- els with poor connectivity, including the extreme cases of polygon soups, common in models of man-made objects taken from online repositories. In our algorithm, we use a two-step procedure that first approximates the in- put model using a manifold, closed envelope surface and then extracts from it a hierarchical abstraction curve network along with suitable normal in- formation. The constructed curve networks form a compact, yet powerful, representation for the input shapes, retaining their key shape characteristics while discarding minor details and irregularities. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . x Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Statement of Co-Authorship . . . . . . . . . . . . . . . . . . . . . xii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Man-made models . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Abstraction of man-made models . . . . . . . . . . . . . . . 2 2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Artistic and human perception of shape . . . . . . . . . . . . 5 2.2 Non-realistic rendering and modeling . . . . . . . . . . . . . 6 2.3 Shape analysis and reverse engineering . . . . . . . . . . . . 8 3 Algorithm Overview . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.2 Iterative fitting . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.1 Matching . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.2 Deformation . . . . . . . . . . . . . . . . . . . . . . . 21 4.2.3 Mesh regularization . . . . . . . . . . . . . . . . . . . 22 5 Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 iii Table of Contents 5.1 Extracting network topology . . . . . . . . . . . . . . . . . . 31 5.2 Extracting network geometry . . . . . . . . . . . . . . . . . 33 5.3 Network regularization and simplification . . . . . . . . . . . 37 6 Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 iv List of Tables 7.1 Abstraction statistics. . . . . . . . . . . . . . . . . . . . . . . 43 v List of Figures 1.1 Commonly available man-made models often have multiple connected components, self-intersections, and triangles with bad aspect ratios. . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 (Left to right) Detailed 3D model, an artist’s drawing, a crys- tal souvenir, and our abstraction of the Eiffel Tower. . . . . . 2 1.3 (Left to right) Tourist map (courtesy of Arnaud Siminski) depicting landmark buildings, Picasso’s famous abstraction of a bull (courtesy of the Museum of Modern Art), and a wooden toy (courtesy of NOVA68.com)designed by Calder all make use of distinctive attributes to generate minimalist, yet powerful, abstractions. . . . . . . . . . . . . . . . . . . . . . . 3 1.4 (Left to right) Input model, model simplified to 200 triangles, simplified envelope surface with 200 triangles. While the sim- plified envelope surface fares better, it does not preserve the core features of the Arc de Triomphe. . . . . . . . . . . . . . 4 2.1 Eiffel tower rendered using crease lines (left), suggestive con- tours (middle), and crease lines on our 3D abstraction (right). 6 2.2 Comparison of common NPR line representations. Generated using the executable provided in [14]. For suggestive contour, ridges and valleys, and apparent ridges, the outer contours are overlaid on top as black lines. . . . . . . . . . . . . . . . . 7 2.3 Man-made objects may often contain large interior features, such as the interior wall of the famous Utah teapot. Left: the original Utah teapot. Right: the same teapot cut in the middle showing its interior wall. . . . . . . . . . . . . . . . . . 10 3.1 (Left to right) Input model with 353 components, envelope surface, vector representation, and reconstructed model. . . . 12 vi List of Figures 4.1 Iterative envelope generation process: (Top left to bottom right) original mesh, initial envelope generated (grid size 9× 23× 9 and subdivided twice), iteration 1 to 9, and iteration 11. 14 4.2 There might be multiple surfaces that separates the voxel hull with the non-intersecting voxels. (a): anatomic illustration of the Arc de Triomphe model which produces two separating surfaces. The exterior separating surface (yellow) encloses the input model (red), and the interior separating surface (green) is enclosed by the input model. (b): (left to right) exterior separating surface, input model, interior separating surface. . 15 4.3 Cross section of the voxel hull (red). Notice that the input model lies between the exterior separating surface and the interior separating surface. . . . . . . . . . . . . . . . . . . . . 15 4.4 A synthetic model with hidden surfaces that produces multi- ple interior separating surfaces. . . . . . . . . . . . . . . . . . 16 4.5 Initial envelope generation. Given an input mesh (a), a grid of voxels is created to enclose it. (b) The intersecting vox- els (gold) and the outer layer (blue) is computed. (c) The exterior region propagates inwards until it reaches the voxel hull. (d) Lastly, the initial envelope is extracted as the surface separating the exterior voxels from the voxel hull. . . . . . . 17 4.6 The effect of different grid sizes on the voxel hulls and the final envelopes. . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.7 Concavity not captured in the initial envelope (i.e. w is nar- rower than the grid resolution) might be captured during the matching step. Envelope is in red, and original mesh is in black. Left: the concave feature is captured because d ≤ 0.5w, and v is mapped to the bottom of the concavity. Right: if d > 0.5w, v will map to one of the sides, thus the concavity is removed. . . . . . . . . . . . . . . . . . . . . . . . 21 4.8 The deformation step, during envelope fitting, smooths out narrow concavities and suppresses details of the input model. 22 4.9 Effect of mesh regularization. . . . . . . . . . . . . . . . . . . 23 4.10 Local mesh smoothing. . . . . . . . . . . . . . . . . . . . . . . 24 4.11 Left: in locally flat regions, all vertex normals of a triangle are approximately equal. Right: in non-flat regions such as near a crease, the vertex normals of a triangle could be quite different. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 vii List of Figures 4.12 Cross section view of mesh triangles (solid black lines) and the smooth surface (dashed black curves) they approximate. Before smoothing (left), the face normal nf is close to the vertex normals n0 and n1. After smoothing (right), the new nf might be very different from the original n0 and n1. . . . 25 4.13 Collapsing any of the 3-loop edges (red) will cause the mesh to be non-manifold. . . . . . . . . . . . . . . . . . . . . . . . . 26 4.14 The red edge is flipped if θ1 + θ2 > 180 ◦. . . . . . . . . . . . . 26 4.15 Flipping an edge (red) that fails condition 4.3 may break the crease (center). Thus, we split such edge into two halves (right). 27 4.16 Left: degenerate triangle (v0,v1,v2) was not fixed by mesh smoothing and edge flips because condition 4.3 failed. Also, since none of its edges are excessively long, edge splits did not fix it either. Center: during degenerate triangle removal step, edge (v1,v2) is first split. Right: edge (v0,v3) is then collapsed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.17 Eiffel Tower (left). Low resolution envelope (middle) is ini- tialized with grid size 9× 23× 9. The construction converged in 11 iterations. High resolution envelope (right) is initialized with grid size 27× 65 × 27. The construction converged in 7 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.18 Arc de Triomphe (left). Low resolution envelope (middle) is initialized with grid size 21 × 35 × 33. The construction converged in 12 iterations. High resolution envelope (right) is initialized with grid size 35 × 61 × 57. The construction converged in 8 iterations. . . . . . . . . . . . . . . . . . . . . 29 4.19 Empire State Building (left). Low resolution envelope (mid- dle) is initialized with grid size 7 × 9 × 7. The construction converged in 6 iterations. High resolution envelope (right) is initialized with grid size 41 × 141 × 29. The construction converged in 5 iterations. . . . . . . . . . . . . . . . . . . . . 30 4.20 Dome of the Rock (left). Low resolution envelope (middle) is initialized with grid size 37 × 23 × 37. The construction converged in 10 iterations. High resolution envelope (right) is initialized with grid size 67 × 43 × 67. The construction converged in 8 iterations. . . . . . . . . . . . . . . . . . . . . 30 viii List of Figures 5.1 Vectorization stages: (Left to right) VSA segmentation, seg- mentation after boundary improvement, smooth approxima- tion geometry, extracted regularized curve network, surface after hierarchical simplification, regularized simplified curve network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 Left: unsmoothed model. Right: the same model rendered using smoothed normals. . . . . . . . . . . . . . . . . . . . . . 34 5.3 Left: original partition. Middle: after one iteration of chart smoothing without boundary curve smoothing. Right: af- ter one iteration of chart smoothing with boundary curve smoothing. Chart boundaries are indicated using yellow lines. 35 5.4 Each chart is indicated by a unique color. Top: the tower model before chart smoothing (left), after per-triangle solve (center), and after global assembly (right). Bottom: Zoomed view of the top of the tower before chart smoothing (left), after per-triangle solve (center), and after global assembly (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5.5 Input model, processed segments, vectorized curve network, and reconstructed abstraction. Zoom panels show section of curve network, and normals along the curves. Connectivity- only or virtual edges are marked in brown. For ease of vi- sualization, normals from same curve loops are marked in identical colors. . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.1 Even for noisy input models, the regularization step allows creation of quality abstractions. We present abstractions with 2% and 5% (of bounding box diagonal length) noise added to the dome model. . . . . . . . . . . . . . . . . . . . . . . . . . 42 7.2 (Left) Fine topological features are easily combined by our envelope construction stage. However, once such features are extracted by a finer grid resolution, we have no easy method to remove them, independent of their size. (Right) Some objects, perhaps those less familiar to us, have no obvious natural abstraction. . . . . . . . . . . . . . . . . . . . . . . . 44 7.3 Result gallery showing various input models, extracted curve networks (with normals), and reconstructed abstractions. The high-resolution abstractions are rendered in yellow, while the low-resolution ones are in blue. . . . . . . . . . . . . . . . . . 45 ix Acknowledgments First of all, I would like to thank my supervisor Alla Sheffer, without whom this work would not have been possible. I would also like to thank my collaborators: Ravish Mehra, Jeremy Long, Niloy J. Mitra, and Amy Gooch. It was fun and rewarding to work with you guys. In addition, I want to thank my second reader, Robert Bridson, for his valuable feedback. I would like to give special thanks to Tiberiu Popa, Vladislav Kraevoy, Xi Chen, Benjamin T. Cecchetto, Derek Bradley, Ian South-Dickinson and Cody Robson for their help and support. I also want to thanks all my friends for their companionship, support and encouragement. Thank you Jingwen, Xi, Xiaofei, Kaida, Yilan, Wei, Mianwei, Hongbo, Lan, Bo, Kwun Kit and anyone else I have forgotten. Thank you to my father and mother for all their love, patience, understand- ing and support. Thank you to my cousin for her sharing and guidance. Lastly, I would like to thank NOVA68.com, Arnaud Siminski, and Museum of Modern Arts for granting us to use their images in our figures. Qingnan Zhou August 2009 x Dedication I would like to dedicate this work to my parents: Jian Zhou and Hua Lu; to my cousin: Jiajia Li; and to my girlfriend. xi Statement of Co-Authorship The algorithm described in this paper was developed together with Prof. Alla Sheffer, Prof. Niloy J. Mitra, Prof. Amy Gooch, Ravish Mehra, and Jeremy Long. Prof. Sheffer and Prof. Mitra supervised the project, and Ravish and I performed research and implementation. Prof. Gooch and Jeremy helped us trying out various image-based approaches in the project. The implementation is mainly built on the CML code base and the Graphite code base. xii Chapter 1 Introduction 1.1 Man-made models Engineered objects constitute a large fraction of the models populating vir- tual environments such as games, movies, or simulations. In recent years, easy access to 3D modeling tools and the rapid growth of online modeling communities have resulted in large collections of such models. Most models of man-made objects present in such collections do not satisfy the notion of “good” geometry processing models [27]. They frequently consist of nu- merous disconnected components, have self-intersections, and lack accurate information about part junctions and interconnections. From a processing point of view, such models can be considered as polygon soups (Figure 1.1). Models of man-made objects can be very detailed, capturing every hole and protrusion. However, such shapes are often characterized and identified by just a few defining features (Figure 1.2). Shape descriptions based on those features, commonly involving a handful of characteristic curves, potentially mimic the minimalist representations that we, as humans, possibly store and Figure 1.1: Commonly available man-made models often have multiple connected components, self-intersections, and triangles with bad aspect ra- tios. 1 1.2. Abstraction of man-made models use for our inference needs [38]. These compact, abstracted descriptions are visually more appealing than the detailed original ones, which may appear visually cluttered (Figure 1.2-left). Using this observation, artists frequently create recognizable images or icons of known objects by employing only a few brush strokes (see drawing in Figure 1.2). Such stylization is also common in tourist maps (Figure 1.3), where landmark buildings are depicted by just a few strokes that highlight their main features (see also [21]). Figure 1.2: (Left to right) Detailed 3D model, an artist’s drawing, a crystal souvenir, and our abstraction of the Eiffel Tower. 1.2 Abstraction of man-made models In this thesis, we introduce a novel algorithm for abstracting three-dimensional shapes. Inspired by human shape perception literature [4, 13] and artistic techniques, we use characteristic curves or contours as building blocks for the abstraction. The choice is motivated by the observation that the shape of many man-made objects is clearly delineated by contour lines and can be faithfully modeled as a union of smooth patches welded together along these junction curves. Specifically, our method extracts a sparse network of space curves and associated normals as an abstraction of input models. This compact representation makes explicit the main features of the shape, which are challenging to identify from a polygon mesh or other low-level representations. While shape abstraction can be attempted at the rendering level by devel- oping suitable NPR tools, applying it directly to the models has a number of advantages. Model-level abstraction allows consistent rendering of a shape from a variety of views and is independent of the rendering resolution or 2 1.2. Abstraction of man-made models Figure 1.3: (Left to right) Tourist map (courtesy of Arnaud Simin- ski) depicting landmark buildings, Picasso’s famous abstraction of a bull (courtesy of the Museum of Modern Art), and a wooden toy (courtesy of NOVA68.com)designed by Calder all make use of distinctive attributes to generate minimalist, yet powerful, abstractions. zoom level. The set of extracted curves forms a minimalist representation, or an icon, of the modeled object. By maintaining the extracted curve network properties, subsequent processing can generate new models that automati- cally retain the defining characteristics of the original one [17]. Analogous to the diffusion curves for images [36], our curve network, along with suit- able normal information, presents a vectorized representation of the input models. Our abstraction method operates in two stages. First, it maps the given geometry to a voxel grid of suitable resolution, and extract a corresponding closed, manifold envelope surface, that wraps around the input model while smoothing out minor details (Figure 3.1 middle). This allows us to robustly handle non-manifold meshes and multiple component meshes, including the extreme case of polygon soups. In the second stage, the method extracts a network of curves or vectors from the envelope. After extracting the network connectivity using a mesh segmentation approach, it establishes the network geometry using a combination of regularization and approximation criteria (Figure 3.1 right). An alternative approach for filtering out insignificant shape details is sim- plification, which, operating at triangle-vertex level, aims to reduce polygon count while controlling the deviation of the simplified object from the origi- nal one. Unfortunately, in the process, characteristic shape curves are likely to be disturbed, especially under extreme simplification [19], see Figure 1.4. Additional artifacts arise when the input is a polygon soup instead of a well- formed manifold mesh. In contrast, abstraction attempts to directly extract 3 1.2. Abstraction of man-made models Figure 1.4: (Left to right) Input model, model simplified to 200 triangles, simplified envelope surface with 200 triangles. While the simplified envelope surface fares better, it does not preserve the core features of the Arc de Triomphe. the high-level structure of objects, intentionally removing insignificant visual details and potentially allowing significant topological changes. In addition to the main contribution of providing a method to perform abstraction of 3D geometric shapes, our two secondary contributions are: a novel vector-based representation of 3D geometry, which can be used for a variety of mesh editing tasks; and a simple yet robust mechanism for approximating polygon soup models by a manifold surface envelope. 4 Chapter 2 Related Work Abstraction, the process of identifying characteristic properties and extract- ing their mutual relationships and topology [16], has been studied in many fields and disciplines including art, non-realistic rendering and modeling, and human perception, for purposes such as shape analysis, generation of compact descriptors, and recognition. 2.1 Artistic and human perception of shape In the twentieth century, artists like Kandinsky, Mondrian, and Picasso pushed the boundaries of geometric abstraction in 2D representations of the surrounding world to the extremes. Alexander Calder ingeniously used wires in addition to sheet metal, wood, and bronze to create abstract 3D sculp- tures. While automatically generating abstraction levels like those created by Kandinsky and Mondrian is unrealistic, we draw motivation from the curve-based abstraction portrayed by Calder in his 3D wire sculptures. Koenderink and Doorn [26] hypothesized that humans internally represent shapes as functions that measure the visual complexity of solid shapes. Later, Nackman and Pizer [33] differentiated between representation and description of an object where an object representation contains enough information to enable an approximate reconstruction, while an object de- scription needs only to contain enough information to identify an object as a member of some object class, which is exactly what abstraction aims to do. 5 2.2. Non-realistic rendering and modeling 2.2 Non-realistic rendering and modeling A major goal of non-photorealistic rendering (NPR) is to highlight or am- plify defining object characteristics. Since low-level geometry does not pro- vide a natural prioritization of the shape features (Figure 2.1), NPR tech- niques strive to identify view-specific important features, which should be rendered or exaggerated to convey form [12]. Significant research has been devoted to identifying candidate feature lines including contours, ridge or valley lines [32], and suggestive contours [14]. Figure 2.1: Eiffel tower rendered using crease lines (left), suggestive con- tours (middle), and crease lines on our 3D abstraction (right). Contours 1 (points where the surface normals are orthogonal to the viewing direction) are the most commonly used feature lines for shape representation purposes [14, 20, 22, 26]. However, contours alone cannot fully convey even moderately complex shapes [12]. For instance, as shown in Figure 2.2(c), the cylindrical features on the front face of the pulley are not captured in the contour drawing. An important subset of contours, called outer contours or bounding contours (Figure 2.2(b)), are lines that separate the object from the background [24]. Outer contours are often combined with other NPR line representations to convey shape (Figure 2.2(d), 2.2(e), and 2.2(f)). DeCarlo et al. [14] presented a new set of lines, the suggestive contours, which contains not only traditional contours but also features that would 1Also called occluding contours [24]. In certain literatures such as [20] and [22], the term “silhouette” is used instead of “contour”. We will use the term “contour” in this thesis to avoid confusion. 6 2.2. Non-realistic rendering and modeling (a) Original model (b) Outer contours (c) Contours (d) Suggestive contours [14] (e) Ridges and valleys [35] (f) Apparent ridges [24] Figure 2.2: Comparison of common NPR line representations. Generated using the executable provided in [14]. For suggestive contour, ridges and valleys, and apparent ridges, the outer contours are overlaid on top as black lines. become contours in a nearby view as shape representations. As shown in Figure 2.2(d), suggestive contours extend the actual contours and contain lines that are almost contours. In the presence of noise, suggestive contours might be unstable and capture excess shallow surface features. Ridges and valleys are another family of lines commonly used to convey shape. Conceptually, they capture places where the surface bends sharply. Formally, they are defined as the extrema of the principle curvatures in the corresponding curvature directions [35]. In certain situations such as in the absence of concavities, they yield better results than suggestive contours [12, 14]. However, due to the view-independent nature of ridges and valleys, they often cause the line representation to exaggerate the sharpness of features. In order to compensate for this, Na et al. [32] proposed to assign strength to each ridge and valley line based on the viewing angle. Only the features 7 2.3. Shape analysis and reverse engineering with significant strengths are displayed in the final rendering. Inspired by ridges and valleys, Judd et al. [24] introduced apparent ridges, which are positions where the surface normal changes at a locally maximal rate with respect to the image space. Apparent ridges essentially provide a reweighting scheme that favors regions where the projection foreshortening is most apparent, making the result view dependent. In the extreme case, contours have weights near infinity due to the big per pixel normal difference near those regions. Figure 2.2(e) and 2.2(f) show the result of rendering the pulley model using ridges and valleys as well as apparent ridges. Although they are able to represent the shape visually from the given view point, the lines are inconsistent between viewing directions. All of the NPR algorithms mentioned above are illustrated in Figure 2.2. Notice that the lines extracted are often view-specific. While the view de- pendent nature of NPR line representations helps to convey the shapes vi- sually, it is hard to use them for 3D abstract representations due to incon- sistency across views. Moreover, input models often contain surface noise which causes excess small strokes in line representations and increase the difficulty for viewers to extract the underlying geometry. In contrast, the curve networks extracted by our abstraction method can be used to create view-independent NPR effects, which remain persistent across motion and animation and are robust against surface noises. Little work exists for stylizing or creating iconic representations of 3D mod- els. A notable exception is the research by Gal et al. [18], which creates 3D collages on top of target shapes using a database of objects as primitive building blocks. The resulting collages, though artistically powerful, are not intended for other uses. 2.3 Shape analysis and reverse engineering Motivated by procedural modeling and constructive solid geometry (CSG), researchers have long proposed to approximate a given 3D model with para- metric parts [3, 46]. Such parametric descriptions can indicate the struc- tural composition of the given model, where different abstraction levels could be obtained by direct part manipulation such as removing some of the parts while preserving others. For example, in [3], the authors described a hier- archical segmentation algorithm based on fitting primitives such as planes, spheres, and cylinders. The paper shows very promising results for decom- 8 2.3. Shape analysis and reverse engineering posing and denoising CAD models, where the input were primarily com- posed of those primitive shapes. However, due to the variety of man-made objects in general, it is hard to pick a pre-defined set of primitives as the basic building block. Thus, in general, reverse engineering structure and regularity from 3D geometry is still considered a difficult open problem [37], with algorithmic solutions unlikely to reach human performance levels in the near future. Instead, we present a method that uses low-level analysis of the models to automatically extract the main feature curves, providing a compact vector representation of the model at a desired abstraction level. Our approach bypasses the difficult reverse engineering task of detecting the global structure, while implicitly preserving the main characteristic features of the models. In the domain of 2D shapes, it is well known that features that are stable over scale changes are significant [5, 47]. Since the extrema in a signal and its derivatives often provide important information (e.g. image edges [10]), Witkin et al. [47] proposed to detect and track signal extrema at different scales and introduced scale-space representation. The scale-space concept has been widely used in computer vision and image analysis. Bengtsson et al. [5] obtained abstractions by studying contours at different scales. In 3D settings, finding features that are stable over scale changes is diffi- cult. It would require to sample the input model at different density levels, reconstruct surfaces from these sampled point clouds, and analyze features that are consistent in all reconstructions. In theory, this approach may be sound, and the resulting stable features could be used for shape abstraction. However, given that the input could be non-manifold meshes of multiple components with self-intersections, even surface reconstruction from point clouds alone could be quite challenging because the points are unlikely to lie on a manifold surface. Even if the input model is manifold, the sampling randomness might still cause the final abstraction to miss certain important details while capturing other undesirable features. Moreover, such algorithm would rely on sampling to decide the importance of features, where large but hardly visible features (e.g. the interior wall of the teapot shown in Figure 2.3) are likely to be kept due to their large surface areas even though these features have very little contribution to the user perceived shape. While the general shape analysis problem is difficult even for manifold, con- nected meshes, we show that for man-made objects, due to their inherent regularity and structure, creating effective abstraction, even from polygon soups, is possible. Our abstraction comes in the form of a curve network, 9 2.3. Shape analysis and reverse engineering Figure 2.3: Man-made objects may often contain large interior features, such as the interior wall of the famous Utah teapot. Left: the original Utah teapot. Right: the same teapot cut in the middle showing its interior wall. which can be used as input for modeling and editing systems. One of such systems is iWires [17], where the authors used a collection of 1D wires and their mutual relationship to manipulate a given shape in a structure- preserving manner. These wires were either extracted from the model based on dihedral angles or defined by a user to reflect the input structure. The curve network generated by our abstraction algorithm captures similar struc- tural information for the reconstructed surface. Therefore, the position of the curve network can be considered as iWires to allow further manipulation of the abstracted shape. Another curve based surface modeling system is FiberMesh [34], where users draw a collection of 3D control curves and a manifold surface is created to interpolates them. Although both the Fiber- Mesh control curves and our curve network can be used to define a surface, they are very different and serve distinct goals. Specifically, FiberMesh con- trol curves are designed to allow smooth surface interpolation and intuitive model manipulation. In particular, the default type of FiberMesh control curves requires the interpolating surface smoothly interpolate the curves, and there is little correspondence between these control curves and surface features. In contrast, our curve networks aims to capture those shape defin- ing features of a given model. 10 Chapter 3 Algorithm Overview Our goal is to extract a vector representation for three-dimensional shapes, targeted specifically toward abstraction of man-made objects, i.e., objects whose main features can be captured by a few smooth surfaces glued to- gether along characteristic curves. Our representation satisfies the following properties: Reconstruction: The network of curves, or vectors, combined with the normals prescribed along them, is sufficient to define the abstracted shape, encoding both the connectivity and the geometry of the reconstructed model (Chapter 6). To describe the connectivity, the network is required to be a connected B-Rep2 representation, as this significantly simplifies the recon- struction step. To adequately reconstruct the geometry we define the surface normals across the model as weighed combinations of the curve normals. Abstraction: Depending on the desired level of abstraction, our represen- tation controls which geometric features are appropriate, while smoothing out the less significant ones. Structure: Regularity and structure, which lead to simplicity of design, fabrication, and installation, are properties common to most man-made ob- jects [29]. Viewers are known to be sensitive to breakup of regular structures present in the input models when those are processed [27], and they seem to remember or identify shapes based on symmetry and regularity, ignoring the deviations [2]. Thus we expect abstractions of man-made shapes to be as regular or structured as possible. This translates to the curves being lo- cally as simple as possible, i.e., being planar, linear, circular arcs, etc., while satisfying global regularity requirements such as symmetry, parallelism, and orthogonality. Our abstraction method generates such vector representations in two steps. 2Boundary representation, where a solid shape is represented using the surface bound- aries. 11 Chapter 3. Algorithm Overview Figure 3.1: (Left to right) Input model with 353 components, envelope surface, vector representation, and reconstructed model. First, it constructs an envelope surface, a closed, manifold surface approxi- mating or “enveloping” the input model (Chapter 4). The surface provides a manifold approximation of the potentially poorly-connected data, enabling subsequent extraction of a meaningful network of curves. It also helps to approximate the input at a desired degree of abstraction, controlling the genus of the final model as well as smoothing out minor details (Figure 3.1). The second step extracts the curve network that serves as the vector rep- resentation of the input model (Chapter 5) . First it extracts the network connectivity using mesh segmentation and then establishes the network ge- ometry using a combination of regularization and approximation criteria. The geometry computation is performed using an optimization procedure that simultaneously computes the vector representation and an approxima- tion of the reconstructed surface obtained from this representation. The combined computation controls the tradeoff between the smoothness and regularity of the reconstruction, and the level of approximation of the in- put. This project is a collaboration with several other researchers. Therefore, in this thesis, I will mainly focused on the parts that I contributed. Namely, I will focus on the envelope generation process (Chapter 4) and the smooth surface approximation (Section 5.2). The other components are also present in this thesis for completeness. 12 Chapter 4 Envelope The envelope construction stage defines a tight manifold approximation of the input models, providing both a first level of abstraction and a well- defined domain for further processing. This step can be skipped if the input model is a priori described by a single manifold surface. However, in this case the abstracted model will preserve the topology of the input. While it is conceivable to have an implicit, distance function based approach followed by isosurface extraction for constructing the envelope [8], such an approach is ill-suited for our setup, as our input models often contain many self intersections and may have inconsistent triangle orientation, making distance computations problematic. Instead, we start by constructing an initial envelope that contains the input model and loosely follows its geometry. Using a local refinement process sim- ilar to iterative closest point (ICP) [7], the envelope is attracted toward the model to obtain the desired approximation quality (Figure 4.1). The itera- tive fitting process helps satisfy the potentially conflicting goals of bringing the envelope mesh close to the input model while maintaining a quality tri- angulation of the envelope. The fitting process, which provides a tradeoff between envelope resolution and input approximation, smoothes out narrow concavities, yielding a first level of abstraction. Section 4.1 provides details about initializing the envelope construction pro- cess. Section 4.2 explains the operations performed in each iteration. En- velopes of various input models are shown at the end of this chapter. 4.1 Initialization The goal of the initialization step is to construct a surface that approximates the shape of the given model. We embed the input model in a regular grid and define the voxel hull of the model as the set of all grid voxels that 13 4.1. Initialization Figure 4.1: Iterative envelope generation process: (Top left to bottom right) original mesh, initial envelope generated (grid size 9× 23× 9 and subdivided twice), iteration 1 to 9, and iteration 11. intersect any of the model triangles. Often, the voxel hull is formed by a closed layer of grid cells which encompass a space made of non-intersecting voxels in the middle. As shown in Figure 4.2, such voxel hull may have multiple disconnected surfaces that separate the intersecting voxels from the non-intersecting ones. We use the term interior separating surface to denote the surface that separates regions surrounded by the voxel hull from the hull, and the term exterior separating surface for the surface between the 14 4.1. Initialization (a) Anatomic view (b) Exterior, original, and interior surfaces Figure 4.2: There might be multiple surfaces that sep- arates the voxel hull with the non-intersecting voxels. (a): anatomic illustration of the Arc de Triomphe model which produces two separating sur- faces. The exterior separat- ing surface (yellow) encloses the input model (red), and the interior separating surface (green) is enclosed by the in- put model. (b): (left to right) exterior separating surface, in- put model, interior separating surface. ExteriorInterior Exterior Separating Surface Interior Separating Surface Original Mesh Figure 4.3: Cross section of the voxel hull (red). Notice that the input model lies be- tween the exterior separating surface and the interior sepa- rating surface. voxel hull and the region outside it (Figure 4.3). We do not use interior separating surface in our algorithm because there could be several interior separating surfaces due to hidden surfaces in the input model (Figure 4.4). In contrast, the exterior separating surface is unique for each component of the voxel hull. We use the exterior separating surface as the initial envelope because it loosely approximates the visible structure of the input model. The initial envelope construction consists of three steps: 15 4.1. Initialization (a) Input model Hidden surfaces (b) Cross section (c) Exterior separating sur- face (d) Interior separating surfaces Figure 4.4: A synthetic model with hidden surfaces that produces multiple interior separating surfaces. 1. Computing the Voxel Hull: The goal of the computation is to determine all voxels intersecting the input model. The brute force way to achieve this is to perform intersection tests of all voxels with every edge and every triangle. The running time is O (|Vh| · (|E| + |F |)), where |Vh|, |E|, |F | are the number of voxels, the number of edges, and the number facets respectively. Such method is very inefficient especially when there are many non-intersecting voxels (Figure 4.5(c)). Instead, for each triangle, we perform intersection test only on voxels within its bounding box, and for each edge, we find intersecting voxels by ray tracing from one endpoint to the other using the algorithm described in [1]. The running time for this step is O(m1|F |+m2|E|), where m1 is the maximum number of voxels in the bounding box of a facet, and m2 is the maximum number of voxels that intersect an edge. In all models used in this thesis, both m1 and m2 can be bounded by a constant number c, which makes the running time O(c|F |+ c|E|) ∈ O(|E|). Figure 4.5(b) shows the voxel hull of the tower model. 16 4.1. Initialization 2. Computing the Exterior Voxels: The next step is to find all non- intersecting voxels that are exterior. The outer most layer of voxels in the grid is exterior by construction, shown as blue boxes in Fig- ure 4.5(b). Since any non-intersecting voxels adjacent to an exterior voxel are also exterior, we expand the exterior region inwards until it reaches the voxel hull. This operation is O(|Vh|). Figure 4.5(c) shows the grid with exterior voxels colored blue. 3. Surface Extraction: The final step is to extract the surface that separates the exterior voxels from the voxel hull. This operation is O(|Vh|). Figure 4.5(d) gives an example of such extracted surface. The resulting surface is the initial envelope. Overall, the initialization step has running time O(|E|+ |Vh|). Once the envelope is constructed we refine the initial coarse mesh using one or two iterations of regular subdivision to enable better subsequent approximation. (a) Input mesh (b) Intersecting vox- els and exterior vox- els (c) Exterior region propagation (d) Initial envelope Figure 4.5: Initial envelope generation. Given an input mesh (a), a grid of voxels is created to enclose it. (b) The intersecting voxels (gold) and the outer layer (blue) is computed. (c) The exterior region propagates inwards until it reaches the voxel hull. (d) Lastly, the initial envelope is extracted as the surface separating the exterior voxels from the voxel hull. The voxel grid resolution directly determines the topology of the resulting 17 4.2. Iterative fitting abstraction. Therefore, we allow users to choose a suitable resolution for each given model. Figure 4.6 shows the effect of different voxel grid resolu- tion for the Eiffel Tower model. In addition, the voxel grid resolution plays an important role in determining the abstraction level of the final envelope. As shown in Figure 4.7, narrow concavities that are not captured in the initial envelope will remain ab- stracted in the final envelope as well. See Section 4.2 for detailed discussion. Besides its effects on the final topology and the abstraction of narrow fea- tures, the initial envelope only needs to follow the input model geometry loosely. For instance, both the the center and right voxel hulls of Figure 4.6 capture the cavity in the center of the tower but produce different cavity shapes. Despite the distinct cavity shapes in these two initial envelopes, the corresponding final envelopes are very similar. 4.2 Iterative fitting Once we have an initial envelope that loosely follows the input geometry, the iterative fitting process will progressively tighten the gap between the envelope and the input model. Similar to ICP [7], where a model is itera- tively rotated and translated to align with the given shape, we iteratively deform the envelope towards the input model while preserving the overall shape and mesh quality. Each iteration consists of the following steps. • Matching, which maps the vertices of the envelope to the closest posi- tions on the input model; • deformation, which deforms the envelope to better approximate the input based on the computed matches; • and mesh regularization, which maintains the quality of the envelope mesh as it deforms. The process terminates when the envelope stabilizes or a maximum number of iterations is reached. We now describe the steps in detail. 4.2.1 Matching Each vertex of the envelope is matched to the closest position on the input model. The straight forward matching method is to check the distance 18 4.2. Iterative fitting (a) Left to right: grid size (9× 23× 9), (17× 41× 17), and (27× 65× 27). (b) The corresponding envelopes generated with two level of subdivision (left) and one level of subdivision (center, right). Figure 4.6: The effect of different grid sizes on the voxel hulls and the final envelopes. from an envelope vertex to every facet of the original mesh and find the shortest distance. This will take O(|Ve| · |F |), where Ve is the set of vertices of the envelope, and F is the set of facets in the original mesh. To accelerate this, we observe that the input model is entirely contained within the voxel hull (Figure 4.3). Since the initial envelope is the exterior separating surface of the voxel hull, any envelope vertex is at most dcell away from the input mesh, where dcell is the diagonal length of a voxel. In addition, each iteration of the fitting step pulls the envelope towards the input model, so the distance between each individual envelope vertex and the input mesh is likely to decrease as well. Therefore, when searching for the 19 4.2. Iterative fitting closest match of a target vertex, we eliminate the checks for facets known to be further than 2dcell away. We use space subdivision to prune these unnecessary distance computations. The bounding box of the envelope is subdivided into a regular grid with the same grid size as in the initialization step (each cell’s diagonal length is dcell). For each envelope vertex, the grid cell that contains this vertex is calculated using its coordinates. To find the closest match, we only check the facets intersecting the same grid cell as the target vertex plus the neighboring grid cells, which contain all facets within the 2dcell distance range. Thus, the running time becomes O(|Ve| ·m), where m is the maximum number of facets intersecting a grid cell. Although, in the worst case, this algorithm is still O(|Ve| · |F |), each grid cell on average only intersect a constant number of facets. Therefore, the running time is roughly linear in |Ve| for most of the input models. Similar to ICP setup, we discard outlier matches. We use two simple but effective heuristics to validate each match. First, matched positions too far from the envelope are likely wrong. This check is implicitly enforced by the space subdivision algorithm described above, where matches with distance greater than 2dcell are not considered. Second, since the initial envelope is strictly outside the input model, we force every envelope vertex to map inward. More specifically, for each envelope vertex, we enforce its matched point to be in the opposite direction with respect to its normal. This requirement works well for the earlier iterations. However, in latter iterations, the envelope might intersect the input mesh and the mappings could be in the direction of the normals. Therefore, we only apply this validation when the envelope is sufficiently far from the input mesh (average distance is greater than 0.01 times the diagonal of the bounding box). The matching step plays a key role in removing narrow concavities. A concavity that is not captured by the initial voxel hull could be captured during the matching step if an envelope vertex is mapped to the bottom of the concavity. There are two factors that affect this possibility: the depth to width ratio of the concavity, and the position of the envelope vertex. For concavities with depth more than half of its width, it is impossible for an envelope vertex outside of the concavity to map to the bottom of the concavity because the boundary of the concavity will always be closer to the vertex than the bottom (Figure 4.7 right). For other cases, shallow concavity could be captured if there is an envelope vertex outside of the concavity that is closer to the concavity bottom than to the boundaries (Figure 4.7 left). Thus, such shallow concavities are more likely to be captured by dense envelopes than sparse envelopes. 20 4.2. Iterative fitting w d v w d v Figure 4.7: Concavity not captured in the initial envelope (i.e. w is nar- rower than the grid resolution) might be captured during the matching step. Envelope is in red, and original mesh is in black. Left: the concave feature is captured because d ≤ 0.5w, and v is mapped to the bottom of the concavity. Right: if d > 0.5w, v will map to one of the sides, thus the concavity is removed. 4.2.2 Deformation Despite filtering out outliers, the target positions provided by the correspon- dences established in the matching step can still sometimes be inconsistent, leading to self-intersections and other mesh degeneracies if strictly enforced. To avoid such artifacts we introduce a deformation formulation that pro- vides a tradeoff between enforcing the matches and maintaining the shape and position of the current envelope. Specifically we optimize min vi ∑ i c1Tl + c2To + c3Tn (4.1) where Tl = ∣∣∣∣∣∣ ∣∣∣∣∣∣ vi − 1 |(i, j)| ∑ e=(i,j) vj − li ∣∣∣∣∣∣ ∣∣∣∣∣∣ 2 , To = ‖vi − v ′ i‖ 2, Tn = ‖vi −wi‖ 2, vi are the new positions of the mesh vertices, v ′ i are the current vertex positions, li are the Laplacian vectors [42] given by (v ′ i− ∑ e=(i,j) v ′ j/|(i, j)|) computed on the current envelope, and wi are the positions proposed by the matching. The term Tl is the objective function of many state of the art deformation algorithms [42], where it is often used to preserve the local details of the 21 4.2. Iterative fitting Figure 4.8: The deformation step, during envelope fitting, smooths out narrow concavities and suppresses details of the input model. mesh. To and Tn pull the envelope towards its original position and its mapped position respectively. The weights determine the balance between the original shape and the target mesh shape as well as the speed of the convergence. We used c1 = 1, c2 = 0.5, c3 = 2 for all the models shown in this thesis. The shape preservation terms Tl and To combined with the regularization step further help smooth over narrow concavities in the input model where the size of the narrow features is smaller than the initial grid resolution (see Figure 4.8). This aids the abstraction process by removing potentially deep, but poorly visible, features. The resulting linear system is solved using a sparse linear solver [45]. We choose to use Cholesky factorization to speed up the solving process. The following option is passed to the TAUCS linear solver. "taucs.factor.LLT=true","taucs.factor.ordering=metis",NULL 4.2.3 Mesh regularization As the envelope deforms, the quality of the mesh triangles can increasingly deteriorate as the initial connectivity may not accurately reflect the fitted geometry. Such mismatching connectivity can negatively affect the qual- ity of the approximation (Figure 4.9). Our iterative fitting addresses this concern by performing mesh regularization after every iteration. The basic 22 4.2. Iterative fitting Figure 4.9: Zooms of the mesh envelope, for the Dome of the Rock model, after four iterations of fitting without (left) and with (right) regularization. regularization step is based on [44] and involves a combination of local mesh smoothing, edge collapses, edge flips, edge splits, and degenerate triangle removal operations aimed at maintaining reasonable mesh quality. Mesh Smoothing. As the envelope deforms, multiple envelope vertices often maps to the same or close by positions on the input mesh, which may produce poorly shaped or tiny triangles. The smoothing step improves the triangle quality by sliding vertices along the surface. Specifically, for each vertex, we compute the average of its neighbors and use the projection of this average onto the tangent plane as the new position. As shown in Figure 4.10, the new position vi of vertex i is vi = v ′ i + t where t = (p′i − v ′ i)− ( (p′i − v ′ i) · n ) n, v′i is the position before smoothing, p ′ i = ∑ e=(i,j) v ′ j/|(i, j)| is the average of neighbors of vertex i, and ni is the vertex normal. Note that all vertex normals are computed as the area weighted average of the facets normals adjacent to them. Since each vertex stays on its tangent plane, the shrinkage problem associated with standard mesh smoothing is largely alleviated. To avoid smoothing out sharp details on the envelope, the following two metrics [44] are used to determine the flatness of the local neighborhood around a triangle and the fidelity of the resulting mesh: Edist = max i={0,1,2} arccos(ni · ni+1) < θdist (4.2) Esmth = max i={0,1,2} arccos(nf · ni) < θsmth (4.3) 23 4.2. Iterative fitting n v′i vi p′i Tangent planeTangent plane t Figure 4.10: Mesh smoothing: each vertex is moved to the projection of p′i onto the tangent plane, where p′i is the average of neighbors. where ni are the vertex normal before smoothing, and nf is the face normal after smoothing. All indices are modulo 3. A vertex is smoothed if and only if both Edist and Esmth satisfy the corresponding thresholds. The term Edist measures how flat the surface is locally before smoothing, and it is defined over the local neighborhood of a triangle (i.e. all triangles sharing a vertex with the given triangle). Figure 4.11 shows two triangles along with their local neighborhoods. In relatively flat neighborhoods (left), all vertices of a triangle have similar normals, and hence Edist is small. On the other hand, in a non-flat neighborhood (right), Edist is big due to large normal difference between the triangle vertices. We set the threshold θdist to 45◦ for all examples shown in this thesis. In addition to the smoothness criterion, the term Esmth serves as a heuristic to preserve shape. Consider each envelope triangle as a linear local approx- imation of a smooth surface, where the vertex normal is the surface normal at the corresponding vertex, and each triangle approximately coincides with the local tangent plane. Notice that the triangle normal is an interpolation of the three vertex normals. Thus, Esmth requires the resulting triangle normal to remain close to the vertex normals in order to prevent deviation from the original shape. As shown in Figure 4.12, whenever the resulting triangle normal differs from any of the vertex normals by a large amount, θsmth, the smoothing operation is not carried out. For all examples shown in this thesis, θsmth is set to 30 ◦. Edge Collapses. Although the smoothing step greatly improves the trian- gle shapes, tiny triangles often remain tiny. To eliminate these, we collapse edges shorter than a given tolerance. Each collapse operation removes an edge by moving one of its end points (call it s) to the other end point (call 24 4.2. Iterative fitting v0 v1 v2 n0 n1 n2 v0 v1 v2 n0 n1 n2 Figure 4.11: Left: in locally flat regions, all vertex normals of a triangle are approximately equal. Right: in non-flat regions such as near a crease, the vertex normals of a triangle could be quite different. v0v0 v1v1 n0n0 n1n1 nfnf Tangent plane Figure 4.12: Cross section view of mesh triangles (solid black lines) and the smooth surface (dashed black curves) they approximate. Before smoothing (left), the face normal nf is close to the vertex normals n0 and n1. After smoothing (right), the new nf might be very different from the original n0 and n1. it t). s and t are chosen to minimize the shape difference introduced by this operation. Specifically, we require ∑ e=(v,s) 180◦ −A(v, s) <= ∑ e=(v,t) 180◦ −A(v, t) where A(a,b) is the dihedral angle across the edge (a,b). As usual [44], we avoid producing non-manifold meshes. Specifically, as shown in Figure 4.13, edges that are part of a 3-loop are not collapsed regardless of their length. Otherwise, the collapse will cause non-manifold mesh where an edge could connect more than two triangles. We used one 25 4.2. Iterative fitting vo va vb vc Figure 4.13: Collapsing any of the 3-loop edges (red) will cause the mesh to be non- manifold. vava vbvb vcvc vdvdθ1 θ2 Figure 4.14: The red edge is flipped if θ1 + θ2 > 180 ◦. fifth of the average edge length as the tolerance for edge collapse operation. Edge Flips. Mesh smoothing improves the geometry of the envelope while the connectivity is left unchanged. Bad connectivity, such as high or very low valence vertices, can negatively affect the mesh quality. We therefore use edge flipping to improve mesh connectivity. In 2D triangulation, edge flips are used to achieve the Delaunay property [6]: the circumcircle must contain no other mesh vertices except on its border. In 3D settings, edge flips are also the basic operation used for generating high quality meshes such as Delaunay mesh [15]. In our remeshing algorithm, as shown in Figure 4.14, edges with sum of opposite angles bigger than 180◦ are flipped. Once again, we use formula (4.3) to check if the result preserves the original shape. In Figure 4.14, the algorithm will compute Esmth for triangle (va,vb,vd) and triangle (vb,vc,vd). If either result exceeds the threshold, the flipping operation will not be carried out. As for smoothing, θsmth is set to 30 ◦. As before, to avoid introducing non-manifold regions, we check that the flipped edge does not exist before performing the flipping operation. For instance, in the case shown in Figure 4.14, we check that there should not be an edge connecting vb and vd before the flipping. Edge splits. Sometimes, an edge become excessively long, but it cannot be flipped because condition 4.3 fails. Such case typically happen on the creases of the mesh, where the flipping operation will interrupt the crease 26 4.2. Iterative fitting Figure 4.15: Flipping an edge (red) that fails condition 4.3 may break the crease (cen- ter). Thus, we split such edge into two halves (right). v0v0 v1v1v1 v2v2 v2 v3v3 Figure 4.16: Left: degenerate trian- gle (v0,v1,v2) was not fixed by mesh smoothing and edge flips because con- dition 4.3 failed. Also, since none of its edges are excessively long, edge splits did not fix it either. Cen- ter: during degenerate triangle re- moval step, edge (v1,v2) is first split. Right: edge (v0,v3) is then collapsed. (Figure 4.15). We split such edge into two halves. We use 1.5 times the average edge length as the threshold. Degenerate triangle removal. Even with all the operations described above, the resulting mesh may still contain some very skinny but obtuse triangles (Figure 4.16), where each edge is longer than the edge collapse threshold and shorter than the edge split threshold. These triangles are typically located near the creases where the smoothness requirement (for- mula 4.3) fails so that mesh smoothing and edge flip operations cannot be applied. Therefore, as the last step of the regularization, we remove these degenerate triangles. To be conservative, we define every triangle with the largest angle greater than 120◦ as a degenerated triangle. In such case, we first split the longest edge of the triangle even if its length does not exceeds the threshold (Figure 4.16 center). The split introduces four new edges, and a regular edge collapse operation is performed on these newly formed edges (Figure 4.16 right). Notice that the edge collapse operation is designed to minimize shape changes, thus the algorithm always collapses the vertex not on the crease toward the vertex on the crease. For instance, in Figure 4.16, v0 is collapsed to v3, thus the crease is preserved. Other operations. In addition to remeshing for improving the mesh qual- ity, in later iterations of the envelope contraction we use heuristic remeshing 27 4.2. Iterative fitting operations to improve the approximation quality. Specifically, we flip edges if the mid-point of the flipped edge is closer to the input model than that of the current edge. If the flip creates poorly shaped triangles, we split the flipped edge at the midpoint. The order of flipping is determined by the ap- proximation improvement amount, i.e., the edge for which the improvement is largest gets flipped first. Results. The resulting envelope surfaces are manifold triangle meshes with fair mesh quality that approximate both the topology and the geometry of the input model up to a desired resolution. The envelope generation step fills up small through-holes, smooths out narrow concavities, and removes self-intersections. Figure 4.17, 4.18, 4.19, and 4.20 demonstrate envelope results for four different models, each with two distinct resolution. 28 4.2. Iterative fitting Figure 4.17: Eiffel Tower (left). Low resolution envelope (middle) is ini- tialized with grid size 9×23×9. The construction converged in 11 iterations. High resolution envelope (right) is initialized with grid size 27×65×27. The construction converged in 7 iterations. Figure 4.18: Arc de Triomphe (left). Low resolution envelope (middle) is initialized with grid size 21 × 35 × 33. The construction converged in 12 iterations. High resolution envelope (right) is initialized with grid size 35× 61× 57. The construction converged in 8 iterations. 29 4.2. Iterative fitting Figure 4.19: Empire State Building (left). Low resolution envelope (mid- dle) is initialized with grid size 7 × 9 × 7. The construction converged in 6 iterations. High resolution envelope (right) is initialized with grid size 41× 141 × 29. The construction converged in 5 iterations. Figure 4.20: Dome of the Rock (left). Low resolution envelope (middle) is initialized with grid size 37 × 23 × 37. The construction converged in 10 iterations. High resolution envelope (right) is initialized with grid size 67× 43× 67. The construction converged in 8 iterations. 30 Chapter 5 Segmentation Having generated an envelope surface we now proceed to extract the net- work of curves and associated normals sufficient to reconstruct an abstracted replica of the envelope and hence the input shape. To avoid storing unnec- essary geometric information, we distinguish between two types of curves in our network: regular curves that have a geometric definition (positions and normals) along the curve, and virtual or connectivity-only curves that have no such information and which are used solely to define the connectivity of the network. The network is extracted in two steps. We first extract the network con- nectivity using a mesh segmentation mechanism. We then finalize the curve geometry using smoothing and regularization enforcing spatial, relational, and metric constraints (Figure 5.1). 5.1 Extracting network topology We recast the network extraction problem as one of mesh segmentation, with the curve network defined as the set of chart boundaries. Explicitly aiming at a segmentation that satisfies the reconstruction and approximation criteria in Chapter 3 is likely to be rather time consuming, since even evaluating a chart quality would be computationally expensive. Instead we opt for a simpler, conservative approach, requiring charts to be relatively planar. Clearly any chart that satisfies a near-planarity requirement would satisfy the other two. Though alternative criteria such as chart developability [25] are likely to work as well, the advantage of the planarity criterion is the simplicity and speed of the resulting method. While this approach may initially result in an over-segmented network, the number of charts is later reduced when the network is hierarchically simplified (Section 5.3). We employ the Variational Segmentation Algorithm (VSA) [11] to obtain 31 5.1. Extracting network topology Figure 5.1: Vectorization stages: (Left to right) VSA segmentation, seg- mentation after boundary improvement, smooth approximation geometry, extracted regularized curve network, surface after hierarchical simplification, regularized simplified curve network. the near-planar segmentation, using the approximation error to control the number of charts. We start VSA with an initial number of charts (typically just one) and measure the total approximation error. If the error is above the user-prescribed threshold we add another chart, using the triangle with the maximal error as the seed and repeat the process. The segmentation is improved using a standard post-processing step [25] of straightening bound- aries and merging small charts with their neighbors, while bounding the per-chart error (we allow up to 10% increase in per-chart error for straight- ening and 20% for merging). In the last step, to create a connected network, we split charts with multiple boundary loops into simple ones using a bottom-up triangle clustering. The new boundaries are defined as connectivity-only as they are unnecessary from a geometry point of view. 32 5.2. Extracting network geometry 5.2 Extracting network geometry We expect the reconstructed surface to consist of smooth charts bounded by the prescribed curve network. Hence, our goal is to define positions and normals along the curves, that enable such a reconstruction. We achieve this using an optimization that simultaneously edits the chart boundaries and the chart geometry such that the optimized charts reflect the reconstructed geometry. The optimization balances the following aspects: • Surface Smoothness: The normals across each surface chart should change smoothly. This ensures that the reconstruction using the nor- mals along the chart boundaries will approximate the optimized charts. • Approximation: The optimized and hence the reconstructed surface should remain close to the original one. • Curve Smoothness: The boundary curves should be smooth. This helps to regularize the curve network and thus the subsequent recon- struction result. Normals being a highly non-linear function of the vertex positions, an opti- mization functional aiming to simultaneously satisfy the above goals would be challenging to minimize. Instead we use a solution that decouples the normals and the positions, splitting the solution into three steps: normal solve, per-triangle vertex positioning, and, finally, global assembly. Normal Solve: We first compute new triangle normals which provide a tradeoff between smoothness and normal-level approximation of the envelope surface, min {ni} ∑ i ‖ni − 1 |N(i)| ∑ j∈N(i) nj‖ 2 + ω1 ∑ i ‖ni − n ′ i‖ 2 (5.1) where i indexes mesh triangles, N(i) denotes the set of triangles adjacent to i that belong to the same chart, and ni and n ′ i are the new and current trian- gle normals, respectively. We set ω1 to 0.25 for interior chart triangles, and to 0.1 for triangles adjacent to chart boundaries. We use a smaller weight for boundary triangles since their envelope normals often deviate from those of the input models and hence are less critical to preserve (Figure 5.1 top, center). Figure 5.2 compares the input envelope with the normal smoothed envelope, where the effect of normal smoothing is visualized through ren- dering each facet using the smoothed normal. Notice the new normals yield a much smoother shading. 33 5.2. Extracting network geometry Figure 5.2: Left: unsmoothed model. Right: the same model rendered using smoothed normals. Per-Triangle Solve: Next we compute the vertex positions that gener- ate the desired normals while staying close to the original positions. The computation is performed on a per-triangle basis. • For interior triangles we solve min {vk} ∑ k ‖vk − v ′ k‖ 2 + ω2(nivk + di) 2, where k indexes the three triangle vertices and di is the unknown distance component of the triangle’s plane equation (normal form). The weight ω2 is set to 1000, effectively ensuring that the new positions vk satisfy the desired normal ni. • For boundary triangles, we incorporate an additional boundary curve smoothness requirement. For every boundary edge eb = −−−→ v1bv 2 b of the triangle, we add an additional component to the minimization: min {vk} ∑ k ‖vk − v ′ k‖ 2 + ω2(nivk + di) 2 + ω3 ∑ b ‖eb − sb‖ 2, where sb is the smoothed boundary edge. More specifically, sb has the same magnitude as eb but its direction aligns with the local average boundary direction. Such average boundary direction is obtained by 34 5.2. Extracting network geometry tracing boundary edges to each side of eb, treating each edge as a vec- tor whose direction is consistent with eb, and compute their weighted average. The tracing stops if either the midpoint of a boundary edge is further than r away from the midpoint of eb or an intersection is reached where three or more charts meet at that point. The weight of an edge vector e is G(0,0.5r)(d), where G(0,0.5r) is a Gaussian distribu- tion with mean 0 and standard deviation 0.5r, and d is the distance from the midpoint of e to the midpoint of eb along the chart bound- aries. In this thesis, we chose r to be 1.5× the average edge length of the mesh, and ω3 is set to 2 in all our examples. Figure 5.3 shows the effect of boundary smoothing. Figure 5.3: Left: original partition. Middle: after one iteration of chart smoothing without boundary curve smoothing. Right: after one iteration of chart smoothing with boundary curve smoothing. Chart boundaries are indicated using yellow lines. As shown in Figure 5.4 (top and bottom middle images), the original mesh is disconnected due to per-triangle solve. Notice that when the original surface is smooth (along the sides of the tower), triangles approximately stay at their original position. When the surface is rough (near the tip of the tower), large gaps are created due to large normal changes. Global Assembly: Finally, to obtain a connected mesh we reconcile the different per-triangle positions computed for each vertex. At this stage the shape and orientation of each individual triangle can be viewed as optimal, hence we aim for new vertex positions such that the per-triangle transforma- 35 5.2. Extracting network geometry Figure 5.4: Each chart is indicated by a unique color. Top: the tower model before chart smoothing (left), after per-triangle solve (center), and af- ter global assembly (right). Bottom: Zoomed view of the top of the tower be- fore chart smoothing (left), after per-triangle solve (center), and after global assembly (right). tion gradient is close to identity. We express this requirement using a similar formulation to [43]. For each triangle we define Vk = [v3−v0,v3−v1,v3−v2] where v0,v1,v2 are the three vertices of the triangle computed in the previ- ous step and v3 = v0 + ni. We define Wk similarly but using the unknown shared vertex positions w. We optimize over w using min {wj} ∑ i ||WiV ′−1 i − I|| 2 F + ω4 ∑ j ‖wj − v ′ j‖ 2 36 5.3. Network regularization and simplification where F denotes the Frobenius norm, j indexes mesh vertices, and ω4 set to 10. We iterate the steps until the positions and normals converge, typi- cally in three to four iterations. After each iteration we perform topological cleanup, reapplying the segmentation post-processing step (Section 5.1). Finally, we associate per-chart normals with curve vertices, using the aver- age normal of the adjacent triangles around the vertex in each chart (see Figures 5.1 and 5.5). 5.3 Network regularization and simplification The obtained curve network satisfies the abstraction and reconstruction cri- teria (Chapter 3). We now further regularize the network to capture struc- ture, and then simplify it by removing appropriate curves, producing a se- quence of abstractions. Regularization. The vector representation we obtain at this stage is suf- ficient to faithfully reconstruct an approximation of the input. However, our goal is to create abstractions for man-made objects, highlighting regularity and structure, while ignoring minor deviations or inconsistencies. Hence, we regularize individual curves and loops of the network, and enhance their mutual relations. Our local regularization replaces near-regular geometric features by perfectly regular ones, in an effort to retain the essence of shapes while ignoring small variations. The process considers both positions and normals across the network. First near-planar curves are made exactly planar using a least squares plane fit. Curves, once optimized, are kept fixed during the later iterations, as side constraints for least squares optimization. Subsequently, near-linear curves are converted to line segments. Similarly, curves that fit well to circular arc segments are regularized to perfect circular arcs. Such tests are incremental, where once a curve or a sequence of curves fit into one of these categories, we incrementally check if consecutive curves sharing the same boundary loop with the current sequence also fit the same plane, line, or arc. We use a similar incremental approach to detect sequences of curves with nearly identical normal values along a shared face loop, indicating a locally near planar face region and make the normals identical. In man-made or engineered objects, symmetry and regularity play a dom- inant role due to aesthetic considerations, as well as convenience in design 37 5.3. Network regularization and simplification and manufacturing. A good abstraction should capture such relations, mak- ing them explicit. In the global regularization step we enforce mutual rela- tions between curve pairs that local processing may have missed. First, planar loop pairs that are nearly parallel (or orthogonal) are made exactly so. Such an approach clearly depends on the order in which the loops are processed. As a (heuristic) solution, we process the loop pairs in a greedy fashion (i.e., the loop-pair, which is closest to being parallel, is processed first), taking small steps towards making the loops parallel (or orthogonal), and iterate until convergence. Most of the examples presented in the paper converged in less than five rounds. Finally, we detect global reflective and (discrete) rotational symmetry in the input model [30]. When detected, we enforce symmetry in the curve network, minimally moving the curves to make them exactly symmetric using a symmetrization approach [31]. In the regularization steps, we only update positions and normals without changing connectivity or topology of the network. Hierarchical Simplification. Our original method for network topology extraction is fairly conservative, often resulting in more charts than strictly necessary for reconstruction purposes. Hence it is often possible to simplify the network further without compromising the abstraction quality. Simpli- fication can also be used to create higher levels of abstraction. To define the hierarchical simplification mechanism we reuse the smoothness and approx- imation criteria targeted during network geometry extraction. We compute a per-curve simplification error measuring the impact of remov- ing the curve from the network using two terms. We use integral dihedral angle along a curve to measure smoothness error. To quickly estimate the approximation or reconstruction error, we integrate the deviation of the av- erage normal along the curve from the arc-length interpolation of the curve end-point normals at that point. This measure provides a rough estimate of the difference between the reconstructed surfaces with and without the curve in question. To control the simplification process we set separate thresholds on the two terms. During simplification, rather than removing curves completely, we simply demote them to connectivity-only discarding all the geometric information associated with them (Figure 5.5). Leaving the network topology untouched, helps the reconstruction process while the overhead of storing connectivity- only curves is negligible. In order to maintain reconstruction consistency, once a curve is demoted, we update the curve network, re-assigning nor- 38 5.3. Network regularization and simplification Figure 5.5: Input model, processed segments, vectorized curve network, and reconstructed abstraction. Zoom panels show section of curve network, and normals along the curves. Connectivity-only or virtual edges are marked in brown. For ease of visualization, normals from same curve loops are marked in identical colors. mals along the affected edges using our chart optimization (see Section 5.2) and regularization procedures. Recomputing the surrounding geometry af- ter each edge removal is expensive. However, the number of curves in our curve networks is small, allowing such expensive iterations. Thus we get multiple curve networks and associated reconstructions or abstractions of the input (see Figure 5.1). 39 Chapter 6 Reconstruction The reconstruction process constructs the abstracted model shapes from the vector representation (Figure 3.1-right), establishing both the underlying mesh connectivity and the actual mesh geometry. Initial Reconstruction. The reconstruction starts by triangulating the faces of the curve network, aiming to construct a connectivity that reflects the target geometry. It first embeds the boundary of each face in the plane, using Multi-Dimensional Scaling (MDS) [28] aiming to preserve the shape of the boundary. The MDS embedding best preserves the Euclidean dis- tances between all pairs of vertices on the boundary, which in case of planar boundary loops results in preservation of the boundary shape and even for non-planar boundaries generates a reasonable embedding. As MDS allows for self intersections, we explicitly check for them. When an intersection is observed, we embed the loop on a circle using arc-length parameteriza- tion. Once a planar boundary is computed, its interior is triangulated [41], providing the desired connectivity reconstruction. To generate the geometry of the abstracted model we deform the computed planar meshes, enforcing the boundary vertex positions and normals pre- scribed by the input vector representation. We use the shape preserving de- formation method of Popa et al. [39] whose rotation propagation approach is well suited to our setting. The deformation is applied simultaneously to clusters of faces connected with connectivity-only network curves as we expect the normal impact to propagate across them. Improvement. Although for many surfaces the initial reconstruction pro- duces satisfactory results, it is not always the case. First, if the normal differences across a reconstructed face are fairly large, our deformation may “under-rotate” the triangles involved as it satisfies the normal constraints only in a least-squares sense. This problem is easily resolved by iterating 40 Chapter 6. Reconstruction through the deformation step using the current surface as the undeformed model. Second, if the initial embedding of the boundary differs significantly from the “natural” parameterization of the corresponding face, the mesh triangles can undergo significant deformation leading to visible artifacts. This is resolved by repeating the connectivity and geometry construction steps. Since we now have an interior triangulation, we use standard mesh parameterization methods to find the optimal planar domain, e.g ABF++ [40]. We then retriangulate the obtained boundary and repeat the deformation as above. 41 Chapter 7 Results We tested our abstraction algorithm on a variety of man-made models mostly described by polygon soups. The abstracted models are generated using a combination of local processing with subsequent consolidation of global re- lations common in man-made forms. The abstracted shapes, having high- lighted the mutual relations of parts or curves, are significantly more com- pact and lightweight compared to any low level representation. Here we take advantage of the observation that engineered forms are often defined by a few 1D curves, having relatively low information content compared to organic objects. The resultant models are extremely concise, while explic- itly encoding information about relations between parts (see table). For the models shown in the paper, the average time taken for generating an ab- straction was less than 2 minutes on a 3GHz machine with 2GB RAM, with the envelope surface creation phase taking the majority of computing time. Figure 7.1 indicates how the abstraction result degrades with increasing noise and ambiguity in the data. Since the input models consist of many disconnected components, little is achieved by local smoothing. In contrast, the abstraction results, which enforce local and global regularization, are stable and capture the characteristic features. Figure 7.1: Even for noisy input models, the regularization step allows creation of quality abstractions. We present abstractions with 2% and 5% (of bounding box diagonal length) noise added to the dome model. 42 Chapter 7. Results Model # triangles # components level #lines # arcs # curves Empire State 16k 17 low 34 0 4 high 146 2 4 Eiffel Tower 15.6k 2417 low 47 19 19 high 85 31 24 building 3.7k 89 low 70 0 12 high 288 2 9 Arc-Triomphe 13k 8 low 127 2 10 high 163 16 14 Pagoda 37k 1173 low 64 0 1 high 54 0 22 dome 3.8k 2 low 24 2 0 high 116 23 6 rocking chair 32k 22 low 26 33 39 high 55 35 32 baroque chair 164k 353 low 25 25 16 high 20 33 42 Table 7.1: Abstraction statistics. It has been hypothesized that abstract shapes are easier to recognize and classify compared to detailed ones [23]. Besides adding highlights for NPR applications, curves or feature lines are also used for detecting partial sym- metries [9], for shape editing [17], etc. Such methods usually work on man- ifold, connected geometric models or rely on edges with sharp dihedral an- gles to identify feature edges, and thus cannot be readily applied to polygon soups or noisy models. Our abstraction procedure, which robustly generates a curve network capturing the high-level features of the model while ignoring finer details or deviations, provides an abstraction network that can used as auxiliary information by the above mentioned algorithms. Limitations. Our method’s ability to remove topological details is cur- rently strongly linked to the resolution of the envelope. Once the envelope separates features like the fine diagonal bars in the water tower in Figure 7.2, they persist throughout the algorithm stages, independent of size. Topology level abstraction, removing such features is an important topic to address in future research. On a more perceptual note, not all objects around us have well-defined, rec- ognizable abstract representations. For instance, the Burj Al-Arab building in Figure 7.2 is not yet well-known enough for its abstract representation to 43 Chapter 7. Results be recognizable. Figure 7.2: (Left) Fine topological features are easily combined by our envelope construction stage. However, once such features are extracted by a finer grid resolution, we have no easy method to remove them, independent of their size. (Right) Some objects, perhaps those less familiar to us, have no obvious natural abstraction. 44 Chapter 7. Results Figure 7.3: Result gallery showing various input models, extracted curve networks (with normals), and reconstructed abstractions. The high- resolution abstractions are rendered in yellow, while the low-resolution ones are in blue. 45 Chapter 8 Conclusion We presented an algorithm for creating abstractions of 3D geometric shapes, specifically targeting man-made objects, using hierarchical curve networks that capture the defining characteristics of the inputs. 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