Modeling, Manipulating, and Visualizing Continuous Volumetric Data: A Novel Spline-based Approach

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1 Modelng, Manpulatng, and Vsualzng Contnuous Volumetrc Data: A Novel Splne-based Approach Jng Hua Center for Vsual Computng, Department of Computer Scence SUNY at Stony Brook Talk Outlne Introducton and Motvaton Splne-based Heterogeneous Volume Modelng Geometrc Modelng and Physcal Attrbute Modelng Feature-Senstve Volume Reconstructon Drect Manpulaton of Dynamc Volumetrc Models Scalar-Feld Guded Shape Deformaton (f tme permts) Concluson 1 2 Introducton Geometrc Desgn and Reconstructon (x, y, z) Physcal Attrbute Modelng and Reconstructon (d) Challenges n 3D Modelng and Vsualzaton There are no ntegrated approach and unfed paradgm to represent both geometry and attrbutes for both modelng and vsualzaton purposes. Geometrc Representatons Physcal Attrbute Representatons Lack of effectve, nteractve sculptng toolkts for the natural and ntutve manpulaton of geometrc objects and ther assocated attrbutes. Computer-Aded Dgtal Models for Real-world objects Smulaton Reconstructon geometry and ts assocated attrbutes smultaneously from an exstng dscrete model s under-explored. More dffcult for knematc & dynamc analyss of physcal objects 3 4 Talk Outlne Introducton and Motvaton Splne-based Heterogeneous Volume Modelng Geometrc Modelng and Physcal Attrbute Modelng Feature-Senstve Volume Reconstructon Drect Manpulaton of Dynamc Volumetrc Models Scalar-Feld Guded Shape Deformaton (f tme permts) Concluson Splne based Volume Modelng Our Approach A modeled object s defned as a splne-based analytc formulaton. Arbtrary Topology and complcated geometry Collson detecton and topologcal changes Arbtrary resoluton and compact storage Nce blendng propertes Ant-alasng Ease of manpulaton and deformaton 5 6

2 B-splne Modelng (Parametrc vs. Implct) B-splne based Volumetrc Implct Functons Scalar trvarate B-splne s( u, v, w) l1 m1 n1 0 j0 k 0 p jk B, r ( u) C j, s ( v) Dk, t ( w) d ( B C D) p T d [, s, ] ( [0, U ], j [0, V ], k [0, W ]) T p [, p, ] ( [0, l 1], j [0, m 1], k [0, n 1]) jk jk [Publshed at PG 2001, SMI 2002, VolVs 2002] 7 8 Hybrd, Herarchcal Scalar B-splnes Splne-based volumetrc mplct functons N F( x, y, z) s ( T ( x, y, z)) 1 Herarchcal Data Fttng for Rectlnear Volumes Create an octree for the entre workng space and subdvde the root node to eght chld nodes. Ft a sngle scalar, trvarate B-splne to the regon of each chld node usng the least-square technque. Evaluate the mean square error (MSE) at node, N 1 2 ( d f ( x )) N j 0 If s less than the user-specfed error bound, then mark the node as a leaf node. Otherwse, subdvde the node to eght chld nodes and repeat the above. j j 9 10 Fttng Examples Fttng Examples (Pont Clouds) 11 12

3 Smplex Splne based Heterogeneous Modelng An ntegrated approach for representng, modelng, and renderng of mult-dmensonal, physcal attrbutes across any volumetrc objects of complcated geometry and arbtrary topology. Our model makes use of a more general and flexble tetrahedral doman and offers a compact contnuous representaton at the same tme. It drectly facltates multresoluton modelng. The C n-1 contnuty and C 0 contnuty can both be modeled wth ease. Such flexblty also allows us to model contnuous or dscontnuous dstrbuton n the attrbute feld. Usng tme-varyng knots nstead of fxed knots offers more freedom and mproves accuracy for approxmaton. The knots are explctly and automatcally determned by optmzng certan objectve functon. [Submtted to ACM Sold Modelng 04] 13 Trvarate Smplex Splne Volumes Trvarate smplex splne s based on a tetrahedral doman, whch can be of complcated geometry and arbtrary topology. For a general trvarate smplex splne, each doman tetrahedron I has ts own set of control ponts. Mathematcal formulaton: s( u) I I n c ( u) I N I N ( u) d( p, q, r, s ) M ( u V ) j k l I 14 Couplng Sold Geometry and Physcal Attrbutes A control coeffcent (and possbly other vector-based quanttes wth n components) s assocated wth a correspondng control pont and evaluated wth the geometry smultaneously over the same tetrahedral doman. Modelng of Sold Geometry of Volumes I g g I ( x) N ( x) I s p Attrbute Edtng Usng Control Coeffcents Feature-Senstve Volume Reconstructon Problem Statement: m 4 Gven a set P { p} 1 of ponts p { x, y, z, d ) R, 3 4 fnd a trvarate smplex splne volume s : R R, that approxmates P. The fttng algorthm 1. Create a tetrahedral doman for the entre volume; 2. Mnmze the square error by treatng control vectors as free varables. 3. For each node of the tetrahedralzaton, f the fttng error n ts 1- rng neghborng tetrahedra s too large, mnmze the square error by treatng the knots assocated wth the node as free varables. 4. For each tetrahedron, f ts fttng error s too large, subdvde t nto four tetrahedra and repeat prevous two steps

4 Feature-Senstve Volume Reconstructon Fttng Examples 1. Fndng a good ntal tetrahedralzaton Geometrc features Attrbute feld features The orgnal dataset 2. Adjustment of Free knots accordng to features 3. Local adaptve refnement accordng to features Fttng wth control vectors only Fttng wth both control vectors and knots Fttng Examples Vsualzaton of Smplex Splne Volumes Drect Volume Renderng X-Ray volume renderng General volume renderng Multresoluton Marchng Tetrahedra Algorthm Talk Outlne Introducton and Motvaton Splne-based Heterogeneous Volume Modelng Geometrc Modelng and Physcal Attrbute Modelng Feature-Senstve Volume Reconstructon Drect Manpulaton of Dynamc Volumetrc Models Scalar-Feld Guded Shape Deformaton (f tme permts) Concluson Interactng wth Dynamc Volumetrc Models Integrate volumetrc splne functons and physcsbased modelng nto one sngle framework: Haptcsbased Dynamc Volumetrc Modelng Physcs-based technques provde a natural, forcebased nterface to facltate drect manpulaton and ntutve shape desgn Haptc nterface permts users to nteractvely sculpt vrtual materals and feel the physcal presence wth force feedback Varous sculptng tools are avalable [To appear n the journal: IEEE TVCG 2003] 23 24

5 Integratng Elastcty wth Geometry Integratng elastcty wth geometry s straghtforward. Allocate mass-ponts on samplng ponts, sprngs on edges of cells. In order to prevent shearng, angular sprngs may be used to preserve the regular geometry. Integratng Elastcty wth Attrbutes Densty sprngs Densty Sprngs do not change the geometrc postons of the mass ponts. Instead, they permt the densty change of mass ponts. When manpulatng the attrbute feld, the densty values are changed by the mass-sprng system. d x,y,z Dynamc Volumetrc Model (Vdeo) Smulaton Lagrangan dynamcs: Md Dd Kd f d M s a mass matrx, D s a dampng matrx, K s a stffness matrx. The appled force at every mass-pont s the summaton of all possble external forces Force Mappng & Force Tools Deformaton wth pont-based force tools Deformaton wth curve-based and area-based force tools Vrtual Scene Chsel and Squrt operaton Jonng wth a force tool Force tools allow users to sculpt sold nteror wthout breakng the outer materal 29 30

6 Talk Outlne Introducton and Motvaton Splne-based Heterogeneous Volume Modelng Geometrc Modelng and Physcal Attrbute Modelng Feature-Senstve Volume Reconstructon Drect Manpulaton of Dynamc Volumetrc Models Scalar-Feld Guded Shape Deformaton (f tme permts) Concluson SFD Algorthm Embed an entre model or a part of the model nto a scalar feld and calculate the scalar values at all the vertces of that embedded part. Constran the vertces on the level sets where they orgnally resde by enforcng vertex-flow constrants durng the deformaton process. [Publshed at SM 2003 and To appear n the journal: The Vsual Computer 2003] SFD of the Mannequn Polygonal Model SFD of the Car Polygonal Model SFD Appled to Pont Set Objects SFD Appled to Volumetrc Objects 35 36

7 Concluson A novel splne-based approach for multresoluton modelng, manpulatng, and vsualzng volumetrc data. Employ the trvarate B-splne to model rectlnear volume, and use the trvarate smplex splne to model unstructured grd volume. The data fttng algorthms reconstruct a compact contnuous representaton and offer a data reducton capablty. Our splne-based paradgm has the unque advantages. Enhancng the volume graphcs wth dynamcs and haptcs, and offer a natural nteracton. Scalar-feld guded shape deformaton supports large scale deformaton for arbtrary geometrc and volumetrc objects. Acknowledgements U.S. Natonal Scence Foundaton through grants: DMI , CAREER Award CCR , IIS , IIS , ITR IIS , ACI Honda Amerca, Inc. Alfred P. Sloan Fellowshp Thank my advsor: Prof. Hong Qn Thank my colleagues n the Vsualzaton lab at Stony Brook

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