PowerPoint

An Application of Lie theory to Computer GraphicsApplied Topology25th July. 2013. Bdlewo

Shizuo KAJIYamaguchi UniversityJST/CREST

First of all, I would like to expressToday I am going to talk about how to make this kind of movie using an elementary Lie theory.

1Blend Shape - a problem in CGMathematical setupOur solution

OutlineHere is an outline:I will introduce a problem in CG, called BS.And then I will give a mathematical formulation of the problem, and a solution of it.

2Blend Shapelets see what it is.3

Statue of MoaiJapanese ex-prime minister MoriLook at this picture.He is an ex-prime minister of Japan, not a good one as usual.So I decided to transform him into a rock.

This is what we want to do !

More precisely.4Blend Shape The problemProblem:From (two or more) input images, generate interpolated imagesinput: A, B => (1-t)A + tB: outputInputOutput

t=0.5We want to generate interpolated images from input images.

5Michael Jackson Black or White Movie: Terminator 2Forest GumpCreating new industrial designVideo compression, High framerate TVAdobe Photoshop/After effects Puppet Warp

Where is it used ?

Igarashi et al. 2005

Beier and Neely 1992Here are the examples of this technique in use.As you can imagine, it is used in movies, but not only for that, it is also used for creating new industrial design or image processing engine in TVs at your home.

6Construct a series of homeos indexed by t R ft : R2 R2 t R

Blend Shape basic ideainput: A, B => (1-t)A + tB: output

It usually requires additional input which should be given manually.(ex. the arrows in the picture)The first algorithms appeared in 90s are very straightforward.They somehow construct a series of time varying self-homeos on the plane, and then interpolate the color of each pixel between the initial and the terminal pictures.

7Weakness of classical method

desirable

We want Geometry aware methodThose classical approaches give satisfactory results for producing special visual effects. Youve probably seen it in Terminator 2.However, it fails when we try to blend more sharp images..

8What is an appropriate setupfor this problem ?So we need a data structure that captures geometry of shapes,And then we translate the problem in terms of that mathematical structure.

9Simplicial complex

In CG, the most basic representation of geometric object is (triangulated) polyhedron, i.e., simplicial complex2D

3D10Simplicial complex

More precisely,Abstract simplicial complex (= set of tetrahedra)Piecewise Linear (PL) embedding (= vertex positions in Rn)

11Instead of considering objects themselves,Consider embeddings from a (reference) simplicial complexAnd find a suitable path in the space of embeddingsParaphrase of blend shape

fAfBfA/2+B/2

12Our algorithmBased on Joint works withK. Anjyo, S. Hirose, H. Ochiaiand K. Anjyo, S. Hirose, Y. Mizoguchi, S.SakataOur strategy is divided into three stepsExpress input shapes by PL-maps from a reference simplicial complex

For each t R;Construct interpolated affine maps independently for each simplex Assemble those maps into a global PL-immersionWorkflowProblemFrom given input shapes, generate interpolated shapesinput: A, B => (1-t)A + tB: output141. Isomorphic subdivisionFact: Homeo PL-isomorphic (dimension3)There are algorithms to give an explicit isomorphism.But none of them is perfect. This is still an open problem in computational geometry.Sub problemGiven homeomorphic simplicial complexes,subdivided them to make combinatorially isomorphicWe simply rely on existing methods here

1-to-115

fAfBH1/2

HtAssume we have PL-embeddings fA, fB of a reference simplicial complex to the shapes to be blended.We want to find a homotopy Ht connecting fA and fB.16Sub problemFor each simplex, construct a time varying series of interpolated affine maps,which preserve geometry locally.

First, we consider simplex-wisely

2. Local construction17More precisely,2. Local constructionSub problemGiven two affine maps X and Y,generate a blended affine map At := (1-t)X + tYThe first guess for the answer would be simply the linear combination of X and Y, but it produces bad results.

undesirabledesirableThe second guess would be using Lie correspondence

2. Local constructionThe above map is almost bijective and its continuous inverse can be computed.

Lie algebraLie groupBut the problem is the group of affine transformations is not compact.Based on the Cartan decomposition, we have the following surjection,which meets our need. Now we can blend affine maps linearly in the parameter space

Key points: is not surjectiveThere is an explicit and fast computation algorithm for , The blended map has geometric meaning: for example, the interpolated map stays as close as Euclideanmotion (In the sense of Frobenius norm)2. Local construction

Nice both geometrically and computationally !

Sub problemAssemble the locally constructed affine maps into a global PL-map

3. Patching local mapsLocaltoGlobalIt is done by finding the minimum of a certain energy functional on the space of PL-immersions.

21The energy functional Key points:* It measures the difference between the PL-map and the locally constructed maps up to PL-conformal equivalence* The minimizer is unique (modulo linear conformal equivalence) and varies smoothly with regard to tInvariant under conformal transformationUsers can easily direct the result by adding some terms to it: the simplex-wisely blended affine map

Visual consequenceLeft and center: user defined constraintsRight: effect of PL-conformal invariance Energy w/o invarianceEnergy w/ invariance

Trajectory of a pointIs specified by user

Applicationsinput: A, B, C, D => sA + tB + uC + vD + : outputOur algorithm directly generalizes to more than two inputs Furthermore, nice properties of our linear parametrization of the affine transformation group allow various applicationsshape deformer (addition/scalar product)Inverse kinematics (calculus/differential equation)Motion analysis/compression using PCA/ICA (linear algebra)DemoThree shapes are blended according to weights. Weights are specified by user through the ball controller

25Target shape (dino) is deformed according to users operation on the yellow cage.Demo

Shapes are deformed according to user manipulated handles (invisible)Demo

There will be a conference on Math for CG in OctoberGoogle MEIS2013 for detailAdvertisement