CS 551/651Advanced Computer Graphics

Warping and Morphing

Spring 2002

Deforming Objects

• Changing an object’s shape– Usually refers to non-simulated algorithms– Usually relies on user guidance

• Easiest when shape number of faces and vertices is preserved– Define the movements of vertices

Moving Vertices

• Time consuming to define the trajectory through space of all vertices

• Instead, control a few seed vertices which in turn affect nearby 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

• i is the (shortest) number of edges between i and j

• n is the max number of edges affected• (k=0) = linear; (k<0) = rigid; (k>0) = elastic• Figure 3.55

0;1

0.1)(

0;1

0.1)(

1

1

kn

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2-D Grid Deformation

• 1974 film “Hunger”

• Draw object on grid

• Deform grid points

• Use bilinear interpolation to recompute vertex positions on deformed grid

• Draw example on board

• Figure 3.57

Polyline Deformation

• Draw a piecewise linear line (polyline) through the geometry

• For each vertex compute– Closest polyline segment– Distance to segment– Relative distance along this segment

• Deform polyline and recompute vertex positions

Global Deformations

• Alan Barr, SIGGRAPH ’84

• A 3x3 transformation matrix affects all vertices– P’=M(P) .dot. P

• M(P) can taper, twist, bend…

• Figure 3.65 3.66

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

FFD

• Similar to 2-D grid deformation

• Define 3-D lattice surrounding geometry

• Move grid points of lattice and deform geometry accordingly

• Local coordinate system is initially defined by three (perhaps non orthogonal) vectors

Trilinear Interpolation

• Let S, T, and U (with origin P0) define local coord axes of bounding box that encloses geometry

• A vertex, P’s, coordinates are:

UTS

PPTSu

TSU

PPSUt

SUT

PPUTs

)()(

)()(

)()(

0

0

0

Volumetric Control Points

• Each of S, T, and U axes are subdivided by control points

• A lattice of control points is constructed

• Bezier interpolation of move control points define new vertex positions

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Examples

Using FFDs to Animate

• Build control point lattice that is smaller than geometry

• Move lattice through geometry so it affects different regions in sequence

• Animate mouse under the rug, or subdermals (alien under your skin), etc.

• Figure 3.74

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

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

Morphing

• 2D image metamorphosis

– Coordinate Grid Approach

– Feature-Based Approach

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

Two-pass RenderingOverview

• Figure 3.79

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 to 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

Local Coordinate Systems

• Let the root of the ray in source image be the coordinate system origin

• Let the ray in source image correspond to the 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

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

Necessary Details

• More than one line– Perform mapping for all line segments– Weight each line segment’s contribution to

averaged color value• Q2 – Q1 = distance of line

segment• dist is distance from pixel

to line• a and b are user specified

bp

dista

QQw

12

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

Assignment 2

• Due 2 weeks from today

• Implement Beier-Neely morphing using fltk and OpenGL– http://www.hammerhead.com/thad/morph.html

• Details online tonight