Chap 4 Interpolation-Based Animation Animation (U), Chap 4, Interpolation-based Animation 1 CS, NCTU, J. H.Chuang

2 Outline Key-Frame Systems Animation Languages Deforming Objects 3D Shape Interpolation 2D Morphing Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 3 Key-Frame Systems Hand-drawn animation Key Frames - defined and drawn by master animators Intermediate frames – drawn by assistant animators Computer animation Key Frames - be generalized to apply to any VARIABLE whose value is set at specific key frames Intermediate frames – values are interpolated according to some prescribed procedure Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 4 Key-Frame Systems Animation (U), Chap 4, Interpolation-based Animation Specify interpolation of key values and tangents at segment boundaries

CS, NCTU, J. H.Chuang 5 Key-Frame Systems What is the key? Difficult to interpolate hand-drawn images Different approach in computer animation Each key frame is described by a set of parameters Sequence of key frames = points in high-dimensional space Compute in-between by interpolating these points Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 6 Key-Frame Systems What is a key? For a bouncing ball 3D Positions Orientation? Squishedness? Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 7 Key-Frame Systems What is a key? For Shrek? 3D Position and orientation Joint angles of the skeleton Facial features Hair/fur? Clothing? Clouds? Scene components? Camera Lights Shrek (PDI/DreamWorks, 2001) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 8 Key-Frame Systems Key-framing Procedures Specify the key frames rigid transformation, forward/inverse kinematics Specify the type of interpolation linear, cubic, parametric curves Specify the speed profile of the interpolation constant velocity, ease-in/out, etc. Computer generates the in-between frames Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 9 Key-Frame Systems Pros and Cons Good control over motion Eliminates much of the labor in traditional animation, but still very labor-intensive Impractical for complex scenes water, smoke grass in the wind crowds Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 10 Key-Frame Systems Basic operation: Interpolating Curves Point-to-point basis: straightforward Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 11 Key-Frame Systems Basic operation: Interpolating Curves Point-to-point correspondence is not known Curve-to-curve correspondence is given Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 12 Key-Frame Systems Basic operation: Interpolating Curves Curve-to-curve correspondence is given What happen at intermediate points along the curve is left undefined Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 13 Key-Frame Systems Basic operation: Interpolating Curves If both curve are Bezier curves interpolating control points, or Generate curve points on both curves, followed by point-to-point based interpolation Moving Point Constraints approach [Reeve ’81] Allows users to specify more information about the point correspondence along the curves and the speed of interpolating these points Uses “patch technology” Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 14 Key-Frame Systems Basic operation: Interpolating Curves Moving Point Constraints approach [Reeve ’81] Defines a segment of the curve to interpolate, bounded on top and bottom by interpolation constraints Interpolation of the very top and very bottom of the curve Define an intermediate curve based on the constraints – C(t) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 15 Key-Frame Systems Basic operation: Interpolating Curves Animation (U), Chap 4, Interpolation-based Animation Moving points

CS, NCTU, J. H.Chuang 16 Animation Languages What is animation languages? A set of structured commands that can be used to encode information necessary to produce animations Script-based Text instructions Flowchart-like diagrams encode relationships between objects and procedures Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 17 Animation Languages Artist-oriented animation languages Full-featured programming languages for animation Graphical languages – dataflow network Actor-based animation languages Actor: a graphical object with its associated data and procedures, including geometric description, display attributes, and motion control. Communication between actors: message passing Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 18 Deforming objects Deforming and morphing an object is a visually powerful animation technique Flexible body animation makes the objects much more expressive and alive How? Physically based simulation Less control by animators Computationally expensive By animator’s direct manipulation Key and interpolation Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 19 Deforming objects Picking and Pulling (Editing) Displace one or more of object’s vertices Others are propagated with attenuated distances specified by a function of distance between the seed vertex and the vertex to be displaced Minimum number of edges connecting these two vertices Minimum distance traveled over the surface between these two vertices Geodesic distance Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 20 Deforming objects Picking and Pulling (Editing) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 21 Deforming objects Picking and Pulling (Editing) Animation (U), Chap 4, Interpolation-based Animation i: minimum of connecting edges n: maximum range of effect k: user-selected scale factor

CS, NCTU, J. H.Chuang 22 Deforming objects Picking and Pulling (Editing) Animation (U), Chap 4, Interpolation-based Animation K=0: linear attenuation K<0: more elastic K>0: more rigid displacement

CS, NCTU, J. H.Chuang 23 Deforming objects Deforming an Embedding Space Deforming an Embedding Space Establish a local coordinate system that encases the area of the object to be distorted Transform vertices to local coordinates The local coordinate system is deformed by users in some way – easier or more intuitive The local coordinate of the vertices are used to map their positions in global space Example: Free-Form deformation (FFD) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 24 Deforming objects Deforming an Embedding Space Deforming an Embedding Space Is easier or more intuitive than to manipulate vertices of the object Restricted to possible distortions of the local coordinate system Mapping should be continuous More powerful than affine transformations Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 25 Deforming objects 2D Grid Deformation Local coordinate system a 2D grid in which an object is placed, aligning with the global axes Local-to-global mapping translation and scaling Deformation Moving grid points to distort the local space Object’s vertices are relocated in the distorted grid by bilinear interpolation relative to the grid cell Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 26 Deforming objects Deforming an Embedding Space Animation (U), Chap 4, Interpolation-based Animation A: Global coordinate: (25.6, 14.7), Local coordinate : (5.6, 2.7)

CS, NCTU, J. H.Chuang 27 Deforming objects Deforming an Embedding Space Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 28 Deforming objects Deforming an Embedding Space Animation (U), Chap 4, Interpolation-based Animation P=(0.6)(0.7)P 00 +(0.6)( )P 01 +( )(0.7)P 10 +( )( )P 11

CS, NCTU, J. H.Chuang 29 Deforming objects Polyline Deformation Similar to grid approach object vertices are mapped to the polyline Polyline is modified Object vertices are mapped to the same relative location on the polyline Polyline system Polyline Boundary lines Bisectors Perpendicular lines at end points Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 30 Deforming objects Polyline Deformation Animation (U), Chap 4, Interpolation-based Animation For a given object vertex, we record 1.The closest line segment (L 2 ) 2.The distance (d) to L 2 3.The ratio r

CS, NCTU, J. H.Chuang 31 Deforming objects Polyline Deformation Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 32 Deforming objects Freeform Deformation (FFD) 3D extension of 2D grid deformation A localized coordinate grid is superimposed over an object For each object vertex, coordinate s relative to local grid are determined The grid is manipulated by the user Each object vertex is mapped back into the modified grid Cubic interpolation is typically used with FFD Bezier interpolation in Sederberg’s paper Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 33 Deforming objects Freeform Deformation (FFD) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 34 Deforming objects Freeform Deformation (FFD) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 35 Deforming objects Freeform Deformation (FFD) To facilitate the modification of local coordinate system, a grid of control points is created Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 36 Deforming objects Freeform Deformation (FFD) As the control points moved, the point P(s,t,u) moves Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 37 Deforming objects Freeform Deformation (FFD) Multiple FFD control grids can be joined with continuity constraints across the boundaries Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 38 Deforming objects Freeform Deformation (FFD) Other FFD control grids Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 39 Deforming objects Freeform Deformation (FFD) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 40 Deforming objects Freeform Deformation (FFD) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 41 Deforming objects Freeform Deformation (FFD) Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 42 Deforming objects Composite FFD – Sequential An object is modeled by progressing through a sequence of FFDs, each of which imparts a particular feature to the object Various detail elements can be added to an object in stages as opposed to trying to create on complex FFD designed to do everything at once Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 43 Deforming objects Composite FFD – Sequential Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 44 Deforming objects Composite FFD – Hierarchical Allows the user to work at various levels of detail Finer-resolution FFDs are embedded inside FFDs higher in hierarchy As a coarser-level FFD is used to modify object’s vertices, it also modifies the control points of any children FFDs that are within space affted by the deformation Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 45 Deforming objects Composite FFD – Hierarchical Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 46 Deforming objects Animated FFD FFDs can be used to control the object’s animation By deformation tools By animating the FFD control points Deformation tools – a composition of An user-defined initial lattice A final lattice – modified from initial lattice by the user Object’s animation can be driven by Moving the tool Moving the object Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 47 Deforming objects Animated FFD Animation (U), Chap 4, Interpolation-based Animation Deformation tool applied to an object

CS, NCTU, J. H.Chuang 48 Deforming objects Animated FFD Animation (U), Chap 4, Interpolation-based Animation Deformation by moving the deformation tool relative to an object

CS, NCTU, J. H.Chuang 49 Deforming objects Animated FFD Animation (U), Chap 4, Interpolation-based Animation Deformation by moving the object through FFD space In logical FFD space In distorted FFD space

CS, NCTU, J. H.Chuang 50 Deforming objects Animated FFD Animating the FFD control points using, e.g., key-frame animation or by the result of physically based simulation Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 51 Deforming objects Problems of Editing and FFD Details are hard to be preserved Animation (U), Chap 4, Interpolation-based Animation Figure 3: Detail preservation is exhibited using Green Coordinates (on the right), where the details adhere to the surface deformation and rotate accordingly. In the middle, the MVC result is depicted where the details maintain their original orientation and therefore shear. From [Green Coordinates SIGGRAPH08]

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation 52 Interactive surface decomposition for polyhedral morphing Arthur Gregory et al. The Visual Computer 15(9), 1999

CS, NCTU, J. H.Chuang 53 3D Shape Interpolation Surface-based Vertex-to-vertex correspondence Interpolation between corresponding vertices Limitations on topological consistence volume-based Problems Surface representation -> volumetric representation More computationally expensive Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 54 3D Shape Interpolation Topology of surface or object Surface topology Connectivity Manifold vs. non-manifold Object topology Genus - hole Topologically equivalent A doughnut and a teacup is topologically equivalent Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 55 3D Shape Interpolation Animation (U), Chap 4, Interpolation-based Animation

CS, NCTU, J. H.Chuang 56 3D Shape Interpolation For Meshes With Same Topology Animation (U), Chap 4, Interpolation-based Animation Vertex-to-vertex correspondence problem Genus 0 Spherical parameterization Merging Find the vertex-to-vertex corresponding Genus >= 0 Consistent Dissection Parameterize patches in correspondence to planar domains Merging or re-meshing Derive vertex-to-vertex correspondence Vertex-to-vertex interpolation problem

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Spherical Parameterization 57 Spherical parameterization in [Zwicker and Gotsman 2004]

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Spherical Parameterization 58

CS, NCTU, J. H.Chuang 59 3D Shape Interpolation For Meshes With Same Topology Animation (U), Chap 4, Interpolation-based Animation Vertex-to-vertex correspondence problem Genus 0 Spherical parameterization Merging Find the vertex-to-vertex corresponding Genus >= 0 Consistent Dissection Parameterize patches in correspondence to planar domains Merging or re-meshing Derive vertex-to-vertex correspondence Vertex-to-vertex interpolation problem

CS, NCTU, J. H.Chuang 60 3D Shape Interpolation For Meshes With Same Topology Animation (U), Chap 4, Interpolation-based Animation Genus 0 Genus 1

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation User Guided Common Dissection 61 Interactive surface decomposition for polyhedral Morphing, by Arthur Gregory et al. The Visual Computer 15(9), 1999

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation User Guided Common Dissection 62 Input polyhedral with user-specified correspondences User interface for igloo-house morph showing completed feature net (red) with morphing patches

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Patch Parameterization 63

3D Shape Interpolation Parameterization Overlaying Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

3D Shape Interpolation Patch Parameterization Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

3D Shape Interpolation Patch Parameterization Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Example 67 Interactive surface decomposition for polyhedral morphing Arthur Gregory et al. The Visual Computer 15(9), 1999

3D Shape Interpolation Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Automatic Consistent Dissection 69

3D Shape Interpolation Patch Parameterization and Re-meshing Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang Modify the sampling and connectivity of a geometry Convert a irregular mesh to a (semi-)regular mesh ParametrizeRe-meshing

3D Shape Interpolation Patch Parameterization and Re-meshing Example Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 4 basic points Additional points User-specified feature points 71

3D Shape Interpolation Patch Parameterization and Re-meshing Example Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang Consistent common dissection of pig and triceratops models

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Patch Parameterization and Re-meshing Example M 0 : 164 facesM 2 : 2,624 facesM 4 : 41,984 faces M 0 : 164 facesM 2 : 2,624 facesM 4 : 41,984 faces

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Patch Parameterization and Re-meshing Example Another example w.r.t corresponding features

Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang 3D Shape Interpolation Example for More than 2 Meshes 75 Applied to more than two meshes

3D Shape Interpolation Example for More than 2 Meshes Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang Morph from the source image to the destination image Specifying corresponding elements in the two images Coordinate grid approach Feature-based approach

Image Morphing Coordinate grid approach User-defined curvilinear grid over each image Make sure corresponding elements in the images are in the corresponding cells Locate the same number of grid intersection point on both images Connecting curves are generated using intersection points as control points for a spline curve, e.g., Catmull-Rom spline Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Coordinate grid approach Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Coordinate grid approach Morphing from source to destination Generate an intermediate grid Linearly – two adjacent key frames Higher-order interpolation – more than two adjacent key frames Warp the source pixels and destination pixels to the intermediate grids Perform a cross dissolve in pixel-by-pixel basis to generate the final images Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Coordinate grid approach Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Coordinate grid approach Two-pass warping from source to intermediate Source grid to an auxiliary grid (for x direction) auxiliary grid to the intermediate grid (for y direction) Source grid to an auxiliary grid (for x direction) Based on scan line For each pixel in the auxiliary grid Find the range of pixel coordinates in the source image Use fractional coverage to effect anti-aliasing Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Coordinate grid approach Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Coordinate grid approach Once both images have been warped to the intermediate grid, cross-dissolve on a pixel-by- pixel basis is applied. Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Coordinate grid approach Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach Establish correspondence using feature lines Feature lines are drawn to identify features in correspondence Feature lines are interpolated to form an intermediate feature line sets Based on interpolating endpoints or interpolating center points and orientation Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach On the intermediate image Establish a mapping for each pixel in the intermediate image to each interpolated feature line Find a relative weight indicating the amount of influence that feature line should have on the pixel On the source image Use the mapping to locate source image pixel that corresponds to the intermediate image pixel Use the relative weight to average the source image locations generated by multiple feature lines into a final source image location Use the final location to determine the color of intermediate image pixel Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach Feature line coordinate for a feature line defined by P 1 and P 2 on the intermediate image Define a local coordinate system (U, V) For a pixel P, its coordinate (u, v) is Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach Feature line Q 1 and Q 2 on source image that corresponds to feature line defined by P 1 and P 2 on the intermediate image Local coordinate system (S, T) How to find pixel Q that corresponds to P? Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach How to find pixel Q that corresponds to P? Coordinates of Q are floating numbers Pixels corresponding to P are in an area Requires some kind of filtering Nearest neighbor Linear Interpolation Quadrilateral formed by mapping corners of P Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach Multiple feature lines In addition to mapping, each pixel P is associated with a weight based on P’s position relative to the a feature line in the intermediate image Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach (a, b, p) parameters If a ~ 0, the mapping is a rigid transformation When a increases, makes the effect of lines over the image smoother. Increasing p increases the effect of longer line. Increasing b makes the effect of a line fall off more rapidly (a, b, p) parameters can be global or on a feature-line-by-feature-line basis. Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang

Image Morphing Feature-based approach How to scale the displacement by using weights? For a given pixel in the intermediate image The displacement indicated by each feature line pair is scaled by its weight Final displacement is the weighted sum of all displacements for each feature line pair This gives the displacement from the intermediate pixel to its corresponding position in the source image. Animation (U), Chap 4, Interpolation-based Animation CS, NCTU, J. H.Chuang