TENSOR PRODUCT VOLUMES AND MULTIVARIATE METHODSCAGD Presentation by Eric YudinJune 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 MOTIVATIONExamples

Scalar or vector-valued physical fields (temperature, pressure, etc. on a volume or some other higher-dimensional object)

INTRODUCTION AND MOTIVATIONExamples

Spatial or temporal variation of a surface (or higher dimensions)

INTRODUCTION AND MOTIVATIONExamples

Freeform Deformation

MULTIVARIATE METHODS: OUTLINE Introduction and Motivation Theory Practical Aspects Application: Free-form Deformation (FFD)

THEORY – GENERAL FORMDefinition (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 ut t t

THEORY – GENERAL FORMDefinition (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 ut t t t

m i

q

THEORY – GENERAL FORM is a multivariate function from to

given that If d > 1, then T is a vector (parametric

function)

( )T m q ddP i

THEORY – GENERAL FORMDefinition 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 PROPERTIESDerivatives: 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 r Xu 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 n u v wl 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 kl

P P P

P P P P P P

THEORY – OPERATIONS AND PROPERTIESDegree 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 JJ mj m j

m i km

mm

mm

m

P P

THEORY – OPERATIONS AND PROPERTIESDegree 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 mj m q

i k

m P

, 1,all ,1 ( ),

0,1,..., 1ijk i j kijk

i km j

j mm j

P P P

THEORY – TERMINOLOGYIsosurfaces: 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 – CONSTRUCTORSExtruded Volume & Ruled Volume

THEORY – CONSTRUCTORSExtruded 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 – CONSTRUCTORSRuled 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 DEFORMATIONIntroduction & 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 DEFORMATIONProcess (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 DEFORMATIONObtain/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

0ijki j kP X S T Ul m n

0X

APPLICATION: FREE-FORM DEFORMATIONTransform 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 Xs t u

T U S S U T S T U

APPLICATION: FREE-FORM DEFORMATIONEmbed 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 DEFORMATIONEmbed 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 DEFORMATIONVolume 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 Fx y zG G GJx y zH H Hx y z

F

APPLICATION: FREE-FORM DEFORMATIONVolume 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 DEFORMATIONExamples 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 – VISUALIZATIONMarching 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 – VISUALIZATIONMarching Cubes Algorithm: 2D case

With ambiguity in cases 5 and 10

PRACTICAL ASPECTS – VISUALIZATIONMarching Cubes Algorithm: 3D case. Generalizable by 15 families via rotations and symmetries.