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TENSOR PRODUCT VOLUMES AND MULTIVARIATE METHODSCAGD Presentation by Eric Yudin
June 27, 2012
MULTIVARIATE METHODS: OUTLINE
Introduction and Motivation Theory Practical Aspects Application: Free-form Deformation (FFD)
INTRODUCTION AND MOTIVATION
Until now we have discussed curves ( ) and surfaces ( ) in space.
Now we consider higher dimensional – so-called “multivariate” objects in
23
, 3n n
INTRODUCTION AND MOTIVATION
ExamplesScalar or vector-valued physical fields
(temperature, pressure, etc. on a volume or some other higher-dimensional object)
INTRODUCTION AND MOTIVATION
ExamplesSpatial or temporal variation of a surface
(or higher dimensions)
INTRODUCTION AND MOTIVATION
ExamplesFreeform Deformation
MULTIVARIATE METHODS: OUTLINE
Introduction and Motivation Theory Practical Aspects Application: Free-form Deformation (FFD)
THEORY – GENERAL FORM
Definition (21.1): The tensor product B-spline function in three variables is called a trivariate B-spline function and has the form
It has variable ui, degree ki, and knot vector ti in the ith dimension
1 2 31 2 3 1 1 2 2 3 3
1 2 3
1 2 3 , , 1 2 3, , , , , ,( , , ) ( ) ( ) ( )i i i i k i k i k
i i i
T u u u P B u B u B u
THEORY – GENERAL FORM
Definition (21.1) (cont.): Generalization to arbitrary dimension q: Determining the vector of polynomial degree
in each of the q dimensions, n, (?), forming q knot vectors ti, i=1, …, q
Let m=(u1, u2, …, uq)
Let i = (i1, i2, …, iq), where each ij, j=1, …, q q-variate tensor product function:
Of degrees k1, k2, …, kq in each variable
1 2 31 1 2 2 3 3
1 2 3
1 2 3, , , , , , , ,( ) ... ( ) ( ) ( )... ( )q
q qq
qi k i k i k i ki i i i
T PB u B u B u B u
i
q
THEORY – GENERAL FORM
is a multivariate function from to given that
If d > 1, then T is a vector (parametric function)
( )T q ddP i
THEORY – GENERAL FORM
Definition for Bézier trivariates : A Tensor-Product Bézier volume of degree (l,m,n) is defined to be
where
and are the Bernstein polynomials of degree , index .
Bézier trivariates can similarly be generalized to an arbitrary-dimensional multivariate.
, , ,0 0 0
( ) ( ) ( ) ( )l m n
ijk i l j m k mi j k
X u v w
u P
( , , ), , , [0,1]u v w u v w u
THEORY
From here on, unless otherwise specified, we will concern ourselves only with Bézier trivariates and multivariates.
THEORY – OPERATIONS & PROPERTIES
Convex Hull Property: All points defined by the Tensor Product Volume lie inside the convex hull of the set of points .
Parametric Surfaces: Holding one dimension of a Tensor Product Bézier Volume constant creates a Bézier surface patch – specifically, an isoparametric surface patch.
Parametric Lines: Holding two dimensions of a Tensor Product Bézier Volume constant creates a Bézier curve – again, an isoparametric curve.
THEORY – OPERATIONS AND PROPERTIES
Boundary surfaces: The boundary surfaces of a TPB volume are TPB surfaces. Their Bézier nets are the boundary nets of the Bézier grid.
Boundary curves: The boundary curves of a TPB volume are Bézier curve segments. Their Bézier polygons are given by the edge polygons of the Bézier grid.
Vertices: The vertices of a TPB volume coincide with the vertices of its Bézier grid.
THEORY – OPERATIONS AND PROPERTIES
Derivatives: The partial derivatives of order of a Tensor Product Bézier volume of degree ( at the point u = is given by
Where the forward difference operator is:
and
( )p q r
p q rX
u v w
u , , ,
0 0 0
! ! !( ) ( ) ( )
( )! ( )! ( )!
l p m q n rpqr
ijk i l p j m q k n ri j k
l m nu v w
l p m q n r
P
000
00 0 0 0,0, 1 0,0, 1, , 1[ ( )]
ijk ijk
pqr p q r rijk i j k ijk
P P
P P P
1
1 20
:
: ... ( 1)1 2
i i i
kk li i k i k i k i i k l
l
k k k
l
P P P
P P P P P P
THEORY – OPERATIONS AND PROPERTIES
Degree Raising: To raise a Tensor Product Bezier volume of degree to degree , then the new points
This works similarly for the other dimensions as well.
(0,1,... ),
all ,
J
iJk ijkj J
J m J
J mj m j
m i k
m
P P
THEORY – OPERATIONS AND PROPERTIES
Degree Reduction: To lower a Tensor Product Bezier volume of degree to one of degree , we need:
Then the new points are given recursively by:
Each iteration reduces the degree of the dimension of interest by 1. This works similarly for the other dimensions as well.
0 0
( 1 ,...,
0, 0,1,...,
all ,
qijk
q m m
j m q
i k
P
, 1,all ,1
( ),0,1,..., 1
ijk i j kijk
i km j
j mm j
P P P
THEORY – TERMINOLOGY
Isosurfaces: In constrast to isoparametric surfaces, an isosurface, or constant set is generated when the Tensor Product Bezier Volume function is set to a constant:
Data-wise, this might represent all of the locations in space having equal temperature, pressure, etc.
An isosurface is an implicit surface.
THEORY – CONSTRUCTORS
Extruded Volume & Ruled Volume
THEORY – CONSTRUCTORS
Extruded Volume: A surface crossed with a line.
Let and be a parametric spline surface and a unit vector, respectively. Then
represents the volume extruded by the surface as it is moved in direction It is linear in .
THEORY – CONSTRUCTORS
Ruled Volume: A linear interpolation between two surfaces.
Let and be two parametric spline surfaces in the same space (i.e., with the same order and knot sequence). Then the trivariate
constructs a ruled volume between and .
MULTIVARIATE METHODS: OUTLINE
Introduction and Motivation Theory Practical Considerations Application: Free-form Deformation
(FFD)
APPLICATION: FREE-FORM DEFORMATION
Introduction & Motivation: Embed curves, surfaces and volumes in the
parameter domain of a free-form volume Then modify that volume to warp the inner
objects on a ‘global’ scale [DEMO]
APPLICATION: FREE-FORM DEFORMATION
APPLICATION: FREE-FORM DEFORMATION
Process (Bézier construction):1. Obtain/construct control point structure (the
FFD)2. Transform coordinates to FFD domain3. Embed object into the FFD equation
From the paper:Sederberg, Parry: “Free-form Deormation of Solid Geometric Models.” ACM 20 (1986) 151-160.
APPLICATION: FREE-FORM DEFORMATION
Obtain/construct control point structure (the FFD):
A common example is a lattice of points P such that:
Where is the origin of the FFD space S, T, U are the axes of the FFD space l, m, n are the degrees of each Bézier dimension i, j, k are the indices of points in each dimension Edges mapped into Bézier curves
0ijk
i j kP X S T U
l m n
0X
APPLICATION: FREE-FORM DEFORMATION
Transform coordinates to FFD domain:Any world point has coordinates in this system such that:
So X in FFD space is given by the coordinates:0 0 0( ) ( ) ( )
, ,T U X X S U X X S T X X
s t uT U S S U T S T U
APPLICATION: FREE-FORM DEFORMATION
Embed object into the FFD equation:
The deformed position of the coordinates are given by:
0 0 0
(1 ) [ (1 ) [ (1 ) ]]l m n
t i t m j j n k kffd ijk
i j k
l m nX s s t t u u P
i j k
APPLICATION: FREE-FORM DEFORMATION
Embed object into the FFD equation:
If the coordinates of our object are given by:
and
then we simply embed via:
( ( (s f t g u h
( , , )ffdX X s t u
( ( , ( , ( )ffdX X f g h
APPLICATION: FREE-FORM DEFORMATION
Volume Change: If the FFD is given by
and the volume of any differential element is , then its volume after the deformation is
where J is the Jacobian, defined by:
( , , ) ( ( , , ), ( , , ), ( , , ))x y z F x y z G x y z H x y zF
( ( , , ))J x y z dx dy dz F
( )
F F F
x y z
G G GJ
x y z
H H H
x y z
F
APPLICATION: FREE-FORM DEFORMATION
Volume Change Results:
If we can obtain a bound on over the deformation region, then we have a bound on the volume change.
There exists a family of FFDs for which , i.e., the FFD preserves volume.
APPLICATION: FREE-FORM DEFORMATION
Examples Surfaces (solid modeling) Text (one dimension lower): Text Sculpt
[DEMO]
MULTIVARIATE METHODS: OUTLINE
Introduction and Motivation Theory Practical Considerations Application: Free-form Deformation (FFD)
PRACTICAL ASPECTS – EVALUATION
Tensor Product Volumes are composed of Tensor Product Surfaces, which in turn are composed of Bezier curves.
Everything is separable, so each component can be handled independently
PRACTICAL ASPECTS – VISUALIZATION
Marching Cubes Algorithm Split space up into cubes For each cube, figure out which points are
inside the iso-surface 28=256 combinations, which map to 16
unique cases via rotations and symmetries Each case has a configuration of triangles
(for the linear case) to draw within the current cube
PRACTICAL ASPECTS – VISUALIZATION
Marching Cubes Algorithm: 2D case
With ambiguity in cases 5 and 10
PRACTICAL ASPECTS – VISUALIZATION
Marching Cubes Algorithm: 3D case. Generalizable by 15 families via rotations and symmetries.