ReMESH USER MANUAL - imati-cnr - Consiglio Nazionale delle
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1. image sets is no longer constant it can assume a finite number of values Specifically the cardinality of the EE may be 2 or 4 the one of the ET may be 1 or 2 and the one of the TT may be 0 1 2 or 3 All the others are still properly constant 6 2 A non redundant data structure Clearly a data structure coding all the relations depicted in Figure 7 is redundant Conversely when dealing with manifold triangle meshes with boundary the scheme depicted in Figure 8 meets all of the requirements discussed above 4 T Figure 8 Scheme of relations from which it is possible to derive all of the other non stored relations in optimal time The dotted line representing the VE indicates that such a relation is only partially stored In Figure 8 the VE is indicated with a dotted line meaning that such a relation is only partially stored From now on we denote with VE such a restricted VE VE v maps v into one of its incident edges The complete VE v can be computed starting from the VE v by turning around v through successive applications of the coded relations keeping track of the already traversed triangles In particular the initial VE is initialized as the VE then choose one triangle of the ET VT v let it be and choose the edge e of TE such that v e EV e and e VE Now add e to the set VE and repeat the same operations by considering e as the new VE If v is not on the boundary the process terminates when e become2. Botsch M and Kobbelt L P 2001 4 Robust Procedure to Eliminate Degenerate Faces from Triangle Meshes In Proceedings of Vision Modeling and Visualization Bruzzone E and De Floriani L 1990 Two Data Structures for Constructing Tetrahedralizations The Visual Computer 6 5 266 283 De Floriani L Falcidieno B and Pienovi C 1985 Delaunay based Representation of Surfaces defined over Arbitrarily Shaped Domains Computer Vision Graphics and Image Processing 32 127 140 Forrest A R 1987 Computational geometry and software engineering Towards a geometric computing environment Techniques for Computer Graphics 23 37 Fortune S 1996 Robustness issues in geometric algorithms In Proceedings of the 1 Workshop on Applied Computational Geometry WACG 96 9 14 Garland M and Heckbert P S 1997 Surface simplification using quadric error metrics In Proceedings of ACM SIGGRAPH 97 209 216 Guibas L Salesin D and Stolfi J 1989 Epsilon geometry building robust algorithms from imprecise computations ACM Symposium on Computational Geometry 5 208 217 Guskov I and Wood Z 2001 Topological noise removal In Proceedings of Graphics Interface 01 19 26 Hatcher A 2002 Algebraic Topology Cambridge University Press UK Kobbelt L 2000 Sgrt 3 subdivision In Proceedings of ACM SIGGRAPH 00 103 112 Lee A W F Sweldens W Schr der P Cowsar L and Dobkin D 1998 MAPS Multiresolution adaptive
3. hence a 0 simplex is a vertex A d simplex o is said to be incident at a k simplex t if t c o Let Vs be a set of d 1 affinely independent points in the 3 dimensional Euclidean space R with d lt 3 The subset o of R formed by the points x that can be expressed as the convex combination of the points v of Vo d d x vv with VI 1 20 i 0 i 0 is called an Euclidean d simplex generated by Vo and the points of V are called the vertices of o Any Euclidean s simplex t generated by a set V c Vo of cardinality s lt d is called an s face of o A finite collection X of simplexes is an Euclidean simplicial complex iff the following conditions hold 1 For each simplex o e 2 all faces of o belong to X 2 For each pair of simplexes o and 1 either o n T Doro N t is a face of both o and 1 Let V be a nonempty set and let X be an abstract simplicial complex on V An embedding of Y into RP is a function f V gt R such that e For every k simplex o in X the set o generated by the vertices of is an Euclidean k simplex e The collection X of all the simplexes generated by f as described above fulfills condition 2 of the definition of Euclidean Simplicial Complex Thus an embedding of an abstract simplicial complex is fully defined by mapping its vertices into points in the Euclidean space In R an abstract simplicial complex X endowed with an embedding f describes a triangle mesh M V E T where V E and T ar
4. parameterization of surfaces In Proceedings of ACM SIGGRAPH 98 95 104 Loop C 1987 Smooth subdivision surfaces based on triangles Master s thesis University of Utah USA Department of Mathematics Rocchini C Cignoni P Ganovelli F Montani C Pingi P and Scopigno R 2001 Marching Intersections an efficient resampling algorithm for surface management In Proceedings of Shape Modeling International SMI 01 296 305 Schroeder W Zarge J and Lorensen W E 1992 Decimation of triangle meshes In Proceedings of ACM SIGGRAPH 92 65 70 Zorin D Schr der P and Sweldens W 1996 Interpolating subdivision for meshes with arbitrary topology In Proceedings of ACM SIGGRAPH 96 189 192 17
5. switch to par face color mode from the Graphics window e Remove triangles Each triangle the user clicks on is removed from the mesh The process takes care of maintaining a coherent data structure possibly by duplicating non manifold vertices that can be created due to the removal e Swap edges When the user clicks on or nearby an edge that edge is swapped Also in this case the toolbox takes care of not performing illegal swaps i e boundary edges or resulting incoherent topology e Insert vertices When this toggle is set a new vertex is inserted in the point of the surface the user clicks on If such a point is inside a triangle the insertion is performed through a triangle split operation If it is on or nearby an edge an edge split operation subdivides that edge If it is on or nearby a vertex that vertex is snapped to the position of the point In all of the cases the term nearby is quantified by the Acos tolerance chosen by the user in the Check amp Repair window Such a threshold prevents the creation of nearly degenerate triangles that would be produced by splitting a triangle with a point too close to one of its edges e Triangulate Holes When this toggle is set the user can click on a boundary edge to start a triangulation routine that tries to fill the corresponding hole The triangulation approach is based on a number of heuristics inspired from 16 which try to minimize bad behaviors such as extreme dihed
6. M Attene ReMESH Istituto di Matematica Applicata e Tecnologie Informatiche Consiglio Nazionale delle Ricerche Genova ITALY ReMESH Version 1 0 An interactive and user friendly environment for remeshing surface triangulations MARCO ATTENE USER MANUAL ReMESH M Attene Index T VEO CU CEO Fc dene rara ei ara a 3 REMESAS ear 4 2 1 Pomt and Chck a os tent 4 2 2 Navisding dl do n epe 5 2 3 Overview OF the TS AUER oobis eue meas I Sti api at pad togam exa qp NU lara 5 24 Example of editing osos se eene ise tote A diee o tela be A eaten Vue IES dece deg 6 2 dbocal ODGtatfOls ctae daran Rae 7 2 6 Operations on selected regions aue e dta ecc bedient 9 Zed Global Operatiotis diues hec reo t aeg ee rede d dat rca tere utei Cate laden 9 2 8 Checking and Repairing MESA dille dcn uat plated opa opa ti 10 3 Graphical SEUNG Sool diem ceni eatem tees iD eum lalla 12 d Sag du m e m p m m E 12 5 Manifold triangle fnesh s ssh me prete e tee ote abl aded testa dia lira tate sid 13 6 A data structure for manifold triangle Meshes iii 13 6 1 Scheme OF the relationships os 14 6 2 A MOn tedundant data stFUCtUTe clash set nt eee oa sek a dod beo ail iena 15 Bibliography hs atin ail ali a MR 17 M Attene ReMESH Research and software development involving geometry processing are often slowed down by the absence of suitable models for testing and benchmark purposes In particular when dealing with triangle mes
7. Transform This button pops up a dialog containing a number of controls for setting affine transformations to the whole geometry While the visualization of the scene being transformed is interactive vertex coordinates are actually transformed only when the dialog is closed 5 Flip Normals When the user selects this button all the triangles of the mesh are inverted 6 Fill Holes When selected this button pops up a dialog where the user can select a threshold number n Then all the boundary loops made of at most n edges are patched using the same strategy as in the local operation Triangulate Holes 7 Sharp edge tagging This feature allows the user to automatically tag as sharp see the local operation Tag Sharp Edges all of the edges having an excessive dihedral angle The threshold value is selected through a dialog as in the above Fill Holes 8 Split all edges This button splits all the edges at their middle points The geometry of the mesh does not change but the number of triangles is quadruplicated 9 Loop subdivide This button performs a subdivision step using Loop s scheme 14 Boundary treatment is provided while no special rule is implemented to manage sharp edges 10 M Butterfly Sub This button performs a subdivision step using Zorin s modified Butterfly scheme 17 Both boundaries and sharp edges are treated properly 11 Sqrt 3 Subdivide This button performs a subdivision step based on Kobbelt s approximant sqr
8. ance of the scene The Modeling window provides controls for local tasks In contrast the A gorithms window allows the user to perform global tasks on the mesh The Check amp Repair window contains a number of controls to set tolerances to check the topological coherence of the data structure to remove possible degenerate elements and so on Besides navigation and visualization tasks the user can perform a number of editing operations that we conveniently subdivide into two classes Interactive operations This kind of actions can be performed through mouse clicks directly on the surface Notice that clicks and drags can also be used to navigate the scene In order to establish the behavior of mouse events in the canvas without ambiguity two buttons are provided the little 5 ReMESH M Attene arrow and the hand on the top right corner of the window for switching between visualization and interactive modes At start up the mode is set to visualization the pointer has the shape of a hand and mouse events are used for navigating the scene The user can switch to the interactive mode by clicking on the arrow button and can go back to visualization by clicking on the little hand Interactive operations include triangle removal vertex insertion edge swapping shell flipping and many others that can be selected from a list of toggle buttons in the Modeling window Clicking using the middle mouse button sets the center of a spherical selection th
9. at can be used for processing sub regions of the mesh Other operations Besides interactive operations there are several other actions available for the user in which some parameters are required In such a case a new dialog style window containing controls on the parameters is shown For example if the user wants to simplify the mesh the corresponding dialog shows a slider for the selection of the target number of vertices a toggle button for choosing whether the boundary has to be simplified or not and other controls When settled with the parameters the user clicks on the OK button to actually perform the simplification Also non interactive operations include all of the actions which do not require the specification of a particular element of the mesh such as global modifications or operations which rely on a prior specification of a selection 2 4 Example of editing Before describing the various functionalities in this section we show a typical example of use of the tool Our objective is to obtain a nice and smooth version of the teapot shown in Figure 2 in which we want to shorten the spout and remove the cap The first step of the editing stands at selecting the part of the spout to be removed To do this the user simply middle clicks on the tip of the spout and drags a slider The point of the surface on which the user clicked is the center of a semi transparent sphere see Figure 3 whose radius is interactively controlled throug
10. ation parameters of the scene being rendered e Material Editor The material editor is a widget through which the user can control a number of settings about the material properties of the mesh es being displayed According to OpenGL s definition of materials the user may control material reaction to ambient diffuse specular and emissive lightening as well as surface shininess and transparency Through the menu button Edit a material list is provided from which the user may choose a pre defined set of material properties While chaning parameters the effects are interactively shown in the main scene unless manual was chosen from the Edit menu e Background Color This is another set of controls to define the color of the background of the scene being rendered These controls are not interactive that is the user is required to select OK to show the result e Per face color This button toggles the status per face color which defines wether the mesh must be rendered with a single material color throughout the whole surface or with different colors for different triangles When this toggle is set ReMesh shows selected triangles using a brighter color with respect to the remaining surface Notice that this feature may slightly impact on the rendering performances e Per tag color As ReMesh supports edge tagging this button allows the user to display edges currently tagged as sharp using a different color red Notice that this fe
11. ature may slightly impact on the rendering performances e Hidden line This button toggles between the default flat shading and the hidden line wireframe shading More visualization options may be selected through the proper pull down menu by right clicking into the rendering area 4 Saving The file format chosen for the output meshes of ReMESH is VRML 1 0 ASCII There are two buttons that provide saving facilities 1 Save as out wrl The file out wrl is the default filename for work in progress models If ReMesh was launched from the directory MY PATH then this button creates the file MY PATH out wrl and saves the current mesh in it Notice that if the file already exists ReMesh automatically overwrites it without asking confirmation because again this file should be considered a work in progress 2 Save as This button pops up a file selector from which the user may choose the name of an existing file to overwrite or may specify a new file name for the current model If the file exists confirmation will be asked before overwriting M Attene ReMESH 5 Manifold triangle meshes Let V be a nonempty set An abstract simplicial complex 11 is a collection X of finite nonempty subsets of V such that every v e V belongs to X and if A is an element of Y then so is every nonempty subset of A Each element of V is called a vertex of X An element o of X of cardinality k 1 is called a k simplex and k is the order of o
12. ced more slowly but this rarely makes sense In contrast when dealing with 3D objects zooming a particular part of the model could be very important and maintaining an acceptable level of precision in details can be a difficult task In this scenario surface remeshing is becoming more and more important Using modern range scanning devices in fact it is possible to acquire real objects with a high level of precision which results in a huge number of triangles in the digital model On the other hand most applications cannot afford such a complexity some may also have strict requirements on the shape of the faces or on their connectivity some others may require the sample points to be in particular positions In other words a triangle mesh that is optimal in a particular context may be a bad input for applications belonging to some other contexts Figure 1 Hence it is imperative to continue improving the methods for the remeshing of surface triangulations Techniques for global remeshing i e polygonal simplification mesh optimization for FEA FEM are already included in the most diffused software tools for geometric modeling These tools however are mainly focused on simplifying the design of the overall shape and thus rarely provide facilities to interactively modify single elements of the triangle mesh such as swapping an edge removing a triangle and so on If a user must necessarily perform such low level operations he she mu
13. e mesh on the right are better shaped for further numerical processing Both the approximating triangle meshes have been created through ReMESH starting from a densely sampled version of the model 2 ReMESH ReMESH incorporates a wrapper for loading several file formats including the web standard VRML 2 0 the older VRML 1 0 the OpenInventor file format iv the Object File Format off and the SWM which is the compressed format produced by the SwingWrapper encoder 2 In all of the cases however the parser assumes that the file has one section describing the vertex coordinates and one section describing the triangles other polygonal faces are not allowed further information transformations colors is ignored While loading a data structure such as the one described in appendix B is initialized Some file formats however may represent R sets which are non manifold and or non orientable In this case one has two possibilities 1 Report an error saying that the tool cannot manage such a kind of input or 2 try to perform some topological corrections in order to find a manifold and orientable approximation of the input that can be encoded by the data structure We chose the second strategy and the loader adds one triangle at a time to the structure and just skips triangles if they would make the mesh non manifold and or non orientable Once the loader is settled with the topology and the structure is properly filled the tool pe
14. e the sets of 0 1 and 2 simplexes respectively and each element of V can be mapped to R through f 13 An element of V is called a vertex an element of E is called an edge and an element of T is called a triangle Furthermore we call connectivity of M the combinatorial structure of X while we call geometry of M the function f that associates a 3D point to each vertex of V The set E c R defined as the union of all the Euclidean k simplexes generated by the vertices of Ao for each o Y is called the geometric realization of Y We say that a triangle mesh is manifold possibly with boundary if X is a two dimensional manifold possibly with boundary 6 A data structure for manifold triangle meshes The definition of a data structure for boundary representations such as triangle meshes requires the coding of topological entities with the associated geometric information and of a suitable subset of the topological relationships between such entities In particular it is desirable that all of the following requirements are satisfied e The structure must be complete that is it must be possible to extract all of the entities and relationships which are not explicitly stored without ambiguity e The structure should be non redundant that is if an entity or a relationship can be computed in optimal time then it should not be explicitly coded in the data structure 13 ReMESH M Attene e Each relationship which is no
15. es throughout the whole surface Default values are safe for rough meshes obtained through dense and non adaptive samplings 17 Simplify This button pops up a dialog containing a set of parameters for the simplification The user can set the desired number of vertices choose whether boundaries must be simplified or not choose the quality of the simplification taking into account that as the quality increases the processing time grows prevent mesh inversion also in this case the process becomes slower It is important to consider that due to topological constraints it cannot be guaranteed that the target number of vertices is reached The simplification strategy implemented is a variation of Garland s method 8 18 Marching Cubes This operation remeshes the model using a marching cubes like pattern The user must select the size of the grid and the process is performed according to the Marching Intersection paradigm 15 This functionality is particularly useful to find a single component approximation of a set of nearly adjacent meshes Vertices of the new mesh can be forced to coincide with vertices of the grid In their turn vertices of the grid can be forced to be integer values 19 Normal noise This operation distributes Gaussian noise over the vertices in the normal direction The amount of noise can be selected by the user in the pop up dialog as a percentage of the model s bounding ball radius 2 8 Checking and Repairing a
16. f which e is an edge Notice that since we consider only manifold triangle meshes possibly with boundary each set ET e contains either two elements when e is an internal edge or one element when e is a border edge The set EE e is the set of the edges bounding the triangles of which e is an edge excluding e itself Thus if e is an internal edge the set EE e contains four edges while if e is on the boundary EE e 2 Let t be a triangle The set TV t is the set of the three vertices of t TE t is the set of the three edges bounding and finally TT t is the set of the triangles sharing an edge with Moreover a topological relation may map each element of its domain into a set of constant or variable cardinality Hence when dealing with closed triangle meshes one can classify the relationships in e Constant relations These relations map edges and triangles into sets of neighboring elements with constant cardinality Specifically EV e 2 EE e 4 ET e 2 TV t 3 TE t 3 and TT t 3 e Variable relations These relations are vertex based and the cardinality of the set may vary depending on which vertex is being mapped VV VE and VT In the design of a data structure however it is useful to extend the concept of constant relation to the case of manifold triangle meshes with boundary In fact although the cardinality of some The set VV v is also known as the ink of v M Attene ReMESH
17. h the slider labeled Region radius in the Modeling window Also ReMESH provides a more interactive way to perform spherical selections that is the user middle clicks on the sphere s center and without releasing the middle button moves the mouse pointer away from the center selected Figure 3 Example of spherical selection The selection includes all of the triangles having their three vertices in the sphere Among the controls provided in the Modeling window there is a button labeled Remove Region that when clicked eliminates all of the triangles contained in the sphere specified above After this operation the sphere vanishes and the model appears as depicted in Figure 4 a Since the user may be unhappy with the zig zag effect of the new boundary the toolbox provides the possibility to manually remove one triangle at a time simply by clicking on it So after the selection of the toggle button labeled Remove triangles in the Modeling window the user starts to click on unwanted triangles until the resulting boundary becomes satisfactory Figure 4 b M Attene ReMESH Figure 4 a Result of removal of a spherical selection b Refinement of the boundary through manual removal of some triangles Following a similar procedure the user middle clicks on the top of the model and selects a new spherical region removes inner triangles as described above and obtains the result depicted in Figure 5 Figure 5 Rem
18. hes a researcher may need to check the behavior of a new algorithm on several particular cases In most situations the test model is easily conceivable in mind but at actual design time its formalization turns out to be a much harder task than expected Also simple modifications over an existing triangle mesh may become a tedious work without a suitable interactive environment In order to simplify the remeshing of existing models we have developed ReMESH to interactively edit manifold triangle meshes mostly through user friendly actions such as mouse clicks and drags 1 Introduction Due to their simplicity and to the increasing support of hardware producers triangle meshes are becoming a de facto standard in most application areas A triangle mesh may be produced starting from another representation such as a CAD model or it may approximate a real object which was captured through modern digitization devices Digitizing shapes has some interesting analogies with capturing other sorts of media such as sound waves and images As far as sound waves are concerned however there is an important difference that comes from human s perception In fact our ears cannot perceive frequencies above a given threshold and this guarantees that a uniform dense enough discretization of the sound wave is indistinguishable from the original Differences would become perceivable if both the original and the discretized waves were pitched down i e reprodu
19. ident vertices coincident edges degenerate triangles and overlapping adjacent triangles according to the current value of e If one of these situations is verified the camera is automatically moved to a viewpoint suitable for showing the flaw so that the user can analyze what went wrong and choose to perform some manual editing for a possible correction Also the user can rely on the automatic correction approaches provided 3 Glue Boundaries This action looks for coincident edges and if possible merges them This is particularly useful to topologically join several connected components that are usually created by geometric modeling tools 4 Remove Smallest Components This action scans the data structure and removes all of the connected components but the one with the largest number of triangles This is useful for eliminating typical tiny disconnected sheets from models coming from marching cubes or other sorts of polygonization algorithms 5 Orient Normals This action scans the data structure and for each connected component orients the triangle normals consistently If a connected component is closed triangle normals are oriented so that the outer surface is visible i e normals point outwards the enclosed solid 6 Duplicate non manifold vertices This action checks for non manifold vertices and for each one of them creates a copy of the vertex for each connected component in its original VT The incidence relations of the ne
20. ighboring edges are then updated with the new vertices so as to have a single component VT for each copy Notice that this operation is purely topological that is the geometry of the surface is not modified As for the Check Connectivity button however this should have no effect in normal conditions 7 Remove Degenerate Triangles This button performs a removal of possible degenerate triangles According to the current value of e all of the degenerate triangles are removed from the mesh unless their removal would corrupt the topology in this case the user must manually remove these elements see Check Geometry above Note that in extreme cases very large or wire like cylinders connecting two bodies this action may also cause topological modifications 8 Remove Overlapping Triangles This action removes all of the triangles which form an excessive dihedral angle with one of the neighbors A dihedral angle is excessive if it is less than e of more than 7 11 ReMESH M Attene 9 Remove Tiny Handles This action looks for small handles and tunnels referred to as topological noise in 10 and removes them from the mesh The threshold size for handles to be removed is controlled through the spherical selection tool which helps the user in having a visual idea of that size Resulting holes can be patched through the Fill Holes button 3 Graphical Settings ReMesh provides some facilities for interactively setting visualiz
21. mesh The problem of robustness of geometric algorithms has received a lot of attention from more than 15 years 6 7 and it remains a vibrant area of research When designing a geometric algorithm in fact a researcher uses the theory of real numbers and their exact arithmetic Such a framework is often referred to as the Real Arithmetic Model or more compactly the RAM At implementation time however real numbers have to be approximated with finite precision representations and the error introduced cannot always be neglected In the context of our remeshing toolbox such an error may perturb the geometry and thus cause the creation of degenerate elements self intersecting surfaces and a number of other flaws Although a slight error in the geometry may be simply interpreted as noise when performing computations such a slight modification may cause a wrong branching in the process pipeline and as a consequence it may cause a topological inconsistence or a failure of the algorithm In order to deal with robustness we chose to implement a strategy based on the Epsilon Geometry introduced in 9 The toolbox provides the possibility for the user to choose a threshold angle e In all of the internal computations including the loading step the toolbox prevents the creation of triangles with angles smaller than e or bigger than n e Such a prevention is carried out through swapping and contraction of short edges inspired from ideas of 3 The defa
22. of the facilities for navigating the scene have been inherited from it For self containedness however we briefly sketch them here The canvas and all of the controls in its window are part of an Examiner viewer component of Open Inventor It allows the user to rotate the view around a point of interest using a virtual trackball This viewer also allows the user to translate the camera in the viewer plane as well as dolly move forward and backward to get closer to or further away from the point of interest The viewer also supports seek to quickly move the camera to a desired point With reference to Figure 2 the three wheels labeled Rotx Roty and Dolly control respectively the rotation around the X axis the one around the Y axis and the dolly These actions however can also be carried out through direct interaction of the mouse in the canvas Further navigation parameters can be controlled through the buttons on the right of the window Some aspects of the scene such as the background color and pictorial properties of the surface material shading can be selected by clicking the button labeled Graphics 2 3 Overview of the features At start up the toolbox shows three windows one containing the canvas one showing some information about the model and one containing some buttons In the latter there are four buttons that when clicked pop up new windows with additional controls The Graphics window contains controls for the appear
23. oval of the cap from the teapot To smoothen the whole model the user can perform either subdivision or smoothing In this example we apply both of them in sequence by clicking first on the button labeled M Butterfly Sub in the Algorithms window and then on Laplacian smoothing This latter button pops up a dialog window with a slider to be used to select the number of smoothing iterations We chose to perform two iterations and obtained the final model depicted in Figure 6 Figure 6 Final model resulting from subdivision and smoothing 2 5 Local Operations In this section we describe all of the local operations that can be performed interactively on the triangle mesh Sometimes it is easier to interact with the mesh if its edges are explicitly rendered to switch to such a kind of rendering the Graphics window includes a toggle button labeled hidden 7 ReMESH M Attene line To select a local operation the user must simply click on the corresponding toggle button in the Modeling window the viewer is automatically switched to the interactive mode The operations provided are e Select triangles When this toggle is set each triangle the user clicks on is marked as selected and subsequent operations requiring a selection will act only on these triangles If the user clicks on a previously selected triangle it changes its state to unselected In order to display selected triangles using a different color it is possible to
24. ragging or by releasing the button and using the Region Radius slider 2 6 Operations on selected regions The toolbox provides the possibility to act on limited regions of the mesh defined through spherical selections or triangle by triangle These operations are 1 Re triangulate region This button can be selected from the Modeling window and causes a re triangulation of the region using a Delaunay like approach Specifically a common plane is computed as the average of the planes of the triangles selected then the vertices of the region are projected on the plane and an iterative 2D Delaunay optimization is performed as described in 5 Finally the vertices are moved back to their original positions This operation is particularly useful to improve the quality of nearly flat regions 2 Remove region As the above one this button can be selected from the Modeling window and it causes the removal of all the triangles belonging to the selected region 2 7 Global operations The biggest set of actions provided belongs to the category of global operations Each one of these tasks acts on the whole mesh 3 Normalize This operation computes the bounding box of the model and normalizes it Specifically it performs a uniform scaling and a translation of the mesh so that it fits into the cube 0 1 0 1 0 1 This operation is particularly useful to avoid numerical overflows due to extreme values of the original coordinates 4
25. ral angles degenerate triangles and so on No new vertex is inserted e Flip shell When the user clicks on a triangle that triangle along with all the other triangles in the same connected component are reversed that is their orientations and therefore their normals are inverted e Remove shell When this toggle is set the connected component containing the triangle clicked is removed from the mesh e Tag Sharp Features When this toggle is set the behavior of the clicks is similar to the case of select triangles except for the fact that here the user selects edges instead of triangles To visualize the selected edges it is necessary to turn on the per tag mode from the Graphics window These edges will receive particular treatment during some other processes e Region radius To obtain a spherical selection click on the little arrow top right of the render area to pass to interactive mode Then middle click on the surface to set the center of the spherical selection and while keeping the middle button pressed drag the mouse to control the radius of the selection The selection can be remove by middle clicking again in the render area The last radius selected is remebered by ReMesh so after a first selection if the user simply middle clicks another point of the surface the selection inherits the same M Attene ReMESH radius as before This radius however can be modified either by keeping the middle button pressed and d
26. rforms various geometrical checks and corrections removal of isolated vertices degenerate triangles These corrections can be prevented by passing the additional parameter nocheck to the command line Finally the graphical user interface GUI starts up and shows the model encoded in the data structure some information about the model number of vertices handles boundaries and a set of controls and parameters to be used for the editing see Figure 2 If the user wants to operate on more models at the same time the Append button may be used to load a triangle mesh from file and to insert it in the current scene without removing the existing mesh es Notice that no geometrical check is performed on the newly loaded mesh that may consequently have degenerate elements To automatically remove all the degeneracies of the current meshes use the buttons provided in the Check amp Repair window 2 1 Point and Click Help To obtain information about a widget of the toolbox first click on Help and while the cursor looks M Attene ReMESH like a hand click on the desired widget to pop up a help dialog If no dialog is displayed it means that no help is available for that widget ssi a m El El ET Figure 2 Screenshot of the graphical user interface of the remeshing tool 2 2 Navigating the scene The graphical part of the toolbox is built upon the SGI Open Inventor toolkit and most
27. s equal to the original VE v If v is on the boundary e may become a boundary edge In this case the process continues by considering the unconsidered triangle of ET VE v and turns in the opposite sense An example of this process is depicted in the following Figure 9 NE NG b e sac Si Figure 9 Reconstruction of the VE relation starting from the VE Note that this process requires a number of operations that is linearly proportional to the number of elements of the final VE therefore it is optimal All the other relations which are not explicitly stored in the data structure may be derived in optimal time as follows 5 ReMESH M Attene e VE v construction described above e VV v we EV e e e VE v and wz vj e VT v fte ET e ee VE v e EE e fe TE t te ET e and f e e TV t ve EV e e e TE t e TI t ye ET e e e TE t and y t M Attene ReMESH Bibliography 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Attene M Falcidieno B Rossignac J and Spagnuolo M 2003 Edge Sharpener Recovering sharp features in triangulations of non adaptively re meshed surfaces Proceedings of the 1 Eurographics Symposium on Geometry Processing 63 72 Attene M Falcidieno B Spagnuolo M and Rossignac J 2003 SwingWrapper Retiling triangle meshes for better EdgeBreaker compression ACM Transactions on Graphics 22 4 982 996
28. st rely on a tedious manual editing of the file describing the mesh assuming that the file format is human readable ReMESH provides a user friendly environment for this kind of triangle mesh editing As an example it gives the possibility to split a triangle into three by simply mouse clicking on it or to automatically find and visualize degenerate elements so that the user can interactively modify the local geometry and connectivity again through simple mouse clicks and drags For completeness however it also provides a number of high level operations such as triangle mesh simplification topological and geometrical correction subdivision and many other functionalities There are several reasons for low level editing meshes the most significant being the production of test cases for example the designer of a new algorithm may want to test it on a particular configuration of the connectivity graph of a mesh or may need to evaluate the numerical robustness 3 ReMESH M Attene under certain circumstances or may need to establish an invariant behavior of the method under well defined transformations of the geometry and or of the connectivity graph and so on i y AMA Lu RATS ON NAN SA SINARARA Figure 1 An original object and two triangle meshes approximating it The two models have about the same number of faces 5000 but while the rendering of the leftmost one is more faithful to the original object the faces of th
29. t 3 scheme 12 No support is provided for boundaries and sharp edges 12 Laplacian Smooth This button pops up a dialog where the user can select the number of Laplacian smoothing iterations to be performed on the mesh At each iteration each vertex is moved towards the center of mass of its neighbors The amount of movement is controlled by the parameter Neighbors weight 100 means that in one iteration each vertex is placed exactly in the center of mass of its neighbors Tagged sharp edges are smoothened using a one dimensional support ReMESH M Attene 13 Uniform Remesh This button performs an approximated uniform remeshing of the model The number of vertices in the new mesh can be specified in the dialog along with the number of relaxation steps Sharp edges and boundaries are handled properly 14 Open to Disk This operation performs a topological cut along edges in order to make the mesh homeomorphic to a disk Edges and vertices belonging to the cuts are properly duplicated 15 Spherize Iterative flattening of highly curved regions Tips and sharp concavities are greedily smoothened 16 Feature Recover When selecting this button an Edge Sharpener filter is run 1 A dialog is popped up from which the user may select threshold values default values are provided Note that a zero value of the parameter threshold normal angle implies that the algorithm automatically computes this value based on an averaging of the angl
30. t explicitly stored must be computable in optimal time that is the number of operations required must be linear in the number of elements of the relationship 6 1 Scheme of the relationships A topological relation is essentially a function which associates to each element o of a given type a vertex an edge or a triangle the set of all the elements of another given type having a topological connection with o For example if V is the set of vertices of a triangle mesh M V E T then the set of pairs VE v Y ve V Y CE and V Y v c qj is the topological relationship which relates each vertex with the set of all the edges incident at it Clearly the set of pairs VE may be viewed as a function VE V gt fo E which maps each element of V into a subset of E Let M V E T be a manifold triangle mesh The following Figure 7 depicts all of the possible relationships between elements of M Notice that a good data structure should not store them all in order to avoid redundancy Cy A V d Figure 7 A scheme of all of the possible relationships between topological entities in a triangle mesh Let v be a vertex VV v is the set of all the vertices which are connected to v through an edge VE v 1s the set of all the edges which are incident at v VT v is the set of all the triangles of which visa vertex Let e be an edge EV e is the set containing the two ending vertices of e ET e 1s the set of the triangles o
31. ult value of M Attene ReMESH is arcsin 10 by experiment this value proved to be a good compromise between precision and robustness Even if the strategy implemented does not guarantee robustness in all of the cases we have found that it avoids nearly all of the most common problems when dealing with remeshing Furthermore it is worth to be considered that if the input model has many degenerate faces according to the value of the corrected mesh may be strongly distorted To cope with this cases the toolbox provides a command line option to set the value of to zero but subsequent processing must take into account possible failures All of the check and repairing tasks can be performed by the user after selecting a new value for e This can be useful when the model being edited is known to become the input for a less robust system At run time the Check amp Repair window provides an interactive slider to set the value for e The same window provides a number of buttons for further operations dealing with robustness issues in particular the following controls are provided 1 Check Connectivity This action performs a sequence of consistency tests on the connectivity graph of the mesh Notice that in normal conditions it should always pass all of the tests However it has been incorporated to help the developer of a new plug in to check the status of the structure at any time 2 Check Geometry This action looks for coinc
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