Deforming Objects. Deformation Techniques. Deforming Objects. Examples

Transcription

1 Deforming Objects Deformation Techniques CMPT 466 Computer Animation Torsten Möller Non-Uniform Scale Global Deformations Skeletal Deformations Grid Deformations Free-Form Deformations (FFDs) Morphing 3D Shape Interpolation Examples Deforming Objects!"#$%&'()#$*#+$*$,,*-.%$/$%1",*#'&'(',2 Changing an object s shape Non-simulated algorithms User guidance Easiest when topology is preserved Topology = number of faces, vertices, edges, holes! Define the movements of vertices

2 Non-Uniform Scale Non-Uniform Scale (2) sx sy sz 1 Transformation matrix - diagonal elements Global Deformations Moving Vertices Transformations elements - functions of coordinates , Cannot define trajectory of all vertices Work on seed vertices Effect nearby vertices

3 Moving Vertices Defining Vertex Functions If vertex i is displaced by (x, y, z) units Displace each neighbor, j, of i by (x, y, z) * f(i, j) f(i,j) is typically a function of distance Euclidean distance Number of edges from i to j Distance along surface Vertex Displacement Function Vertex Displacement Function i is the (shortest) distance (in number of edges) from j n is the max number of edges affected (k=) = linear; (k<) = rigid; (k>) = elastic Figure 3.55 & i # f ( i) = 1. ' $! % n + 1" k + 1 & & i ## f ( i) = $ 1. ' $!! % % n + 1" " ; k ( ' k + 1 ; k <

4 Global Deformations Global Deformations - taper Alan Barr, SIGGRAPH 84 A 3x3 transformation matrix affects all vertices P =M(P)P M(P) can taper, twist, bend Figure Global Deformations - taper Global Deformations - Twist x = x*cos(f(y)) z*sin(f(y)) y = y z = x*sin(f(y)) + z*cos(f(y))

5 Global Deformations - Bend Global Deformations - Bend Global Deformations - Compound Grid Deformation 2D technique used in the film HUNGER Overlay 2D grid on top of object Map object vertices to grid cells (create local coordinate system) User distorts 2D grid vertices Object vertices are remapped to local coordinate system of 2D grid by using bilinear interpolation

6 Grid Deformation (2) Grid Deformation (3) For each vertex Identify cell Local u,v coordinate.8.5 Grid Deformation (4) P1 Pu1 P11 Grid Deformation (5) P Bilinear interpolation Pu = (1-u)*P + u*p1 Pu1 = (1-u)*P1 + u*p11 Puv = (1-v)*Pu + v*p1u Pu P1

7 Grid Deformation (5) Skeletal Deformation Skeletons - NOT these type! Skeletons

8 Skeletal Deformation (2) Skeletal Deformation (3) Interior angle bisectors Perpendiculars at end points Get object Draw polyline Map vertices to polyline Warp polyline Reposition vertices s d L Skeletal Deformation (4) Free-Form Deformation (FFD) Sederberg, SIGGRAPH 86 Position geometric object in local coordinate space Build local coordinate representation Deform local coordinate space and thus deform geometry

9 FFD (2) Similar to 2-D grid deformation Define 3-D lattice surrounding geometry Move grid points of lattice and deform geometry accordingly Use local coordinate system FFD (3) Define local coordinate system for deformation (not necessarily mutually perpendicular) T U FFD - register point in cell Trilinear Interpolation 9. : ; Let S, T, and U (with origin P ) define local coord axes of bounding box that encloses geometry A vertex P s coordinates are: P # P s = ( T " U )! ( T " U )! S P # P t = ( U " S)! ( U " S)! T P # P u = ( S " T )! ( S " T )! U P = P + s! S + t! T + u! U ((TxU). S) P TxU S T P (TxU). (P-P) U

10 Volumetric Control Points S, T, and U axes subdivided by control points Lattice of control points constructed Bezier interpolation of CPs define new vertex positions i j k P ijk = P +! S +! T +! U l m n l & l & m & n # l' i i $ & m# m' j j & n# P ( s, t, u) = $!(1 ' s) s ( $! ' ( $ ) (1 t) t ) $!(1 ' u) i= % i " % j= % j " % k = % k " n' k k ) u Pijk ##!! "" FFD - move and reposition Move control grid points Usually tri-cubic interpolation is used with FFDs Originally Bezier interpolation was used. B-spline and Catmull-Rom interpolation have also been used (as well as tri-linear interpolation) Examples FFD - extensions Hierarchical FFDs Animated FFD Static FFD that object moves through Non-parallelpiped FFD

11 Using FFDs to Animate Work Control point lattice smaller than geometry Move lattice through geometry so it affects different regions in sequence Creepy crawlers under your skin Figure 3.74 Work Using FFDs to Animate Build FFD lattice that is larger than geometry Translate geometry within lattice so new deformations affect it with each move Change shape of object to move along a path Figure 3.75

12 Animating the FFD Create interface for efficient manipulation of lattice control points over time Connect lattices to rigid limbs of human skeleton Figure 3.77 Physically simulate control points Animating the FFD FFD - films and video examples Boppin in Bean Town by John Chadwick Facit demo by Beth Hofer Balloon Guy by Chris Wedge

13 (2D) Morphing image post-processing technique transfer source into target image user need to specify corresponding elements Morphing 2D image metamorphosis Coordinate Grid Approach Feature-Based Approach Coordinate Grid Morph create intermediate grid warp source images to intermediate grid first in x then in y cross-dissolve images Coordinate Grid Source and destination images Overlay upon both a 2-D lattice of points Points along edges must remain on edges Internal points can be in different positions Same number of points in both Points define movement of pixels

14 Figure 3.79 Two-pass Rendering Overview Morphing Images to Intermediate First stretch in x direction and then in y direction Auxiliary lattice has x coordinates from source and y coordinates from intermediate lattice Morphing Images Intermediate Use scanline method to compute what pixels from source image map to a particular pixel of intermediate image Feature-Based Morphing Simplest case: user draws one ray on source and destination images to define morph

15 Local Coordinate Systems Local Coordinate Systems Root of feature line in source image is coordinate system origin Feature line in source image correspond to v axis (unit length) Construct perpendicular to this ray for u axis Every pixel (x, y) of image now mapped to (u, v) using projection to u/v axes Local Coordinate Systems Perform similar local coordinate system computation for destination image Build s/t axes Mapping Destination to Source (x, y) dest --- (s, t) (s, t) --- (u, v) (u, v) --- (x, y) source Color (x, y) dest with (x, y) source

16 Necessary Details More than one line Perform mapping for all line segments Weight each line segment s contribution to averaged color value Q 2 Q 1 = distance of line segment dist is distance from pixel to line a and b are user specified & Q Q w $ 2 ' 1 = $ % a + dist p #!! " b Necessary Details Mapping from destination to pixel to source pixel will not land on pixel centers Aliasing results Use quadrilateral centered at source (u, v) location to sample multiple pixels and average 3D Shape Interpolation still being researched map between topologically equal objects [easier] map between objects with same (geometric) topology Matching Topology same vertex-edge topology easy problem, just interpolate the (x,y,z) position of corresponding vertices

17 Star-Shaped Polyhedra find a vertex in the core of each polyhedra send rays / discretize polyhedra Axial Slices do the same as before but for 3D object on a per slice basis Map To Sphere works for genus topologies create mapping to sphere sphere is an intermediate surface, which lets us find a correspondence between the 2 objects