But for the work of making anything

you are a very very

very very very expert on
offering all things like

if you present them with
your wife while doing.

What.

Thank you thank you for
that tending Is that allowed is it OK.

I assume you can all hear me.

All right.
I understand that those of you who are

thinking of this for credit will not get
credit if you don't ask three questions.

So you have to work on that.

I wasn't sure what was
the nature of these or seminars.

So I made a decision which is to present

a large number of projects
twenty somethings.

And I would have about one slide each.

But we have time.

So if for you feel that more
interested in another done

hesitate to stop me to ask the day it's
questions make suggestions clarifications.

We don't have to cover them all and
I do my best.

There is a different type of work
that our community produces.

Some of it is very complex
mathematically complicated.

I'll go read McLean involved but

you know your head is wrapped
around in the it's very difficult.

Some of it is very system
engineering people are building

solution birth by scaffolding complicated
modules many theorists thinks that.

To be fine tuned parameter tweaking and

they produced something
incredible was great.

Labor and those are very important and
I do appreciate them but

my style is I'm more attracted towards

what I would call beautiful solutions and
to define them.

I would say that.

After you have done your research
it should boy down to something so

simple that you can actually explain
it to somebody in a couple of minutes.

The second requirement I have is that I
actually any of you could sit down and

implement a solution in a few days or
so or in this project.

I will be presenting to you you
will judge whether some of those

are beautiful by the definition or
not and you will see not all of them are.

That.

If there is any difficulty in
understanding what I'm saying or

terminology.

Please do not hesitate and intervene.

All right so we'll talk about shapes and
I made sense.

And I split them into things

like design analysis compression
simulations I have little modules.

So this is about designing shapes.

I may sions.

And this project started
several years ago we

wanted to have a nice an environment for
humans to sculpt shapes.

And I am reasonably
good with both hands so

I felt that for people who are somewhat
ambidextrous it would be nice

to be able to put your two hands in that
virtual space and do things with them.

So what we did is we
instrumented of these and

I need to point was
a pointer Here's our try.

We in straw man to ease devices or
of the previous

generation of those so you hold in
each hand a device that this capable.

Tracking six degrees of freedom.

Positions and orientations.

OK And

you know what we design is an interface
through which you would the form shapes.

And the principle there is quite simple.

Let's do one head.

So you put your hand somewhere and
you press a button.

At this moment will record the coordinate
system which I call a frame.

And then as you keep the button
pressed you move it and

that any time you have another frame.

So it's a starting fright and
ending frame.

And when you're done you're ready is
the bottom of the ending frame is frozen.

So how do we compute the shape
of the formation from those

given two of those frames.

There is a unique screw motion.

Which is like a corkscrew it has an axis.

It's a rotation a translation with us.

But that axis.

So that there is a unique screw
motion that takes one into the other.

And we say we're going to apply
that screw motion to the shape but

not the entire shape points
close to where we grab it.

The starting frame would
be transformed transformed

by the host promotion points
farther away will be transformed

by less of the screw motion maybe
one half maybe one quarter.

And that decay function is a cosigner
square function that you may

have seen previously.

OK So this is the principle
of what we call twister.

When you use both hands.

And the region of influence where
the decay function is not zero overlap

these cost and square functions give
you a very nice natural behavior so

with both hands you can
achieve the right result.

So that was very nice we discover
that even though it was into it live.

The user could and
dissipate what would be happening.

Doing things like this.

Take an ear of a bunny and twisted or

bend it was not that convenient
was those two hands.

Because you had a longer element that
you wanted to twist your band and

say it wasn't working very well.

So invented something else
which is with your two

hands now you don't control the shape
directly you control a ribbon.

Like a flexible.

Belts.

OK And then you put that belt in the shape
like if it were butter and your bed

is very hot and you sort of sort of divide
this and it freezes and you grab it.

So you grab the whole shape with a belt.

It's the same as if you had one
hundred little hands along the belt.

And you freeze them grab them and
you lift them see how

many come with two screws going on like
this and you apply the same principle.

So that gives us the ability
to manipulate shapes.

Then I was at a conference and a colleague
from Georgia Tech actually from

the BE in need of Parkman
the biomedical engineering set all

well this is really cool because we
have a problem that this could solve.

So I was all excited and it happens that.

They have been working with surgeons
who do operations and heart surgery for

children born to single ventricle.

And these poor children undergo
a series of operations that

attempt to in some sense rearrange
the piping around the heart.

And so what the surgeon needs to
do is to plan for the operation.

Each operation could be done in
a couple of different ways maybe more.

There are some variations for each option.

So what the surgeon wants to do
is to actually take the scan of

the patient's anatomy.

And then work on the patients
anatomy grab the arteries

yank it from here stick it there.

Bend it like this and saying that's how I
could do it or I could do it from here.

And then they ask well
which one is better.

So the one that is better is the one
where the heart would have less

resistance pumping blood and

where the blood would go more evenly to
both lungs so that one lung would not be.

So the guys in B M E Do a fluid
simulation of the blood flow all

through these new German trees and

what they were missing is the interface
between the surgeon and the shapes.

They used to have the surgeon sketch it
on paper and then they would sit down for

a couple of weeks and
do it in a CAD system when here now.

The surgeon does it himself and

you discover very quickly that
surgeons are very good money

plating things like this with their
hands in fact they are perfect.

They are immediately on the stand.

What's going on and quite precise.

Of course that their job.

So we're working with
Dr Cantor at Emory and

he has been using the system to
prepare for some of the operations.

OK but that was quite nice.

Later here.

That is an excellent question and
let me let me phrase it for everybody.

Some of these operations
that we perform or

the volume warping but they do not
preserve the physical properties.

So the vessels the arteries
that they are moving

may undergo the formation which
are not physically correct.

And we do not attempt to
provide this simulator for

training surgeons nor to simulate reality.

This is more in the spirit
of a sketch book.

So the surgeon know what is possible or
not.

And yes the answer is if we have more time
and more energy we would have any effect.

We try to do some physical
based modeling of that but

in the present incarnation
which it is used.

It's a purity of design system without and

you know about an artery staying
roughly circular and cross sections.

Although often it does.

OK So

this is a slightly different area of work
very related which we haven't completed.

OK So this is not fair.

Sickly based.

On this problem really

like when very

very very.

Thank you for the question so
the day of the anatomy of

the patients often has you know it's not a
small shape it's an artery it has a lot of

intricate details bends and
different sharp shapes

and the transformation we do are in
some sense space warping transformation.

They take a chunk of space which
contain the shape and they the form it.

They need to do some stretching because
otherwise you know this stays here this

goes there in the middle.

Things have to stretch or compress.

So in that sense some of those details
get a long gaiters or get shortened.

And the question if I understand
correctly is well what could be done so

that these details are behaving more
physically in a more correct way and

there is a number of techniques
that have been studied

which try to capture the intrinsic
detail of a surface.

And then as you're doing as
the formation you constrain

the piece here in the piece
they're trying to solve for

the intermediate surface to
still recover this detail and

be nice to connecting to the rest so
that work is going on in progress but

we have not incorporated it here
in what I have been describing.

OK And it is a difficult word because
to be correctly you need to understand

the elastic physical properties of
the material and actually simulate them.

Given that we're doing that in real time
you need to make sure that your simulation

is a compromise between accuracy and
real time.

If this thing is not done something
like more than thirty frames a second.

It doesn't work.

I mean by that it doesn't war but the
surgeon or the user would be frustrated.

You would have the feeling that you
are over shooting them a system is lagging

immediately becomes unusable in the sort
of sense of not being nice to use so

you need to have about thirty
frames a second feedback.

In fact most applications require more.

The front application but
on the similar vein with Dr Sam and

I believe we are working at helping
him do grafts of Carthew large.

Here's what's going on.

On the tell us here in your in Europe and
call us.

You may have a lesion typically older
people have additions but sports or

sports might have lesions as well.

And so he's going to fix this lesion
on the card which has to be very smooth

because that's where the bones are or
sliding by stealing from

the femur or
some other bone a piece of card.

Cutting it as it was sitting there plug
making a hole here and put it there.

So his job is to find the dumber site
such that when he plugs it in it.

Not only connects muesli to
the surrounding bone around the lesion but

also has the same perverts are correct
wrist excess because it was manufactured

by nature to have these characteristics
for functioning correctly.

So imagine that doing that manually or

visually by hand from C.T.
scans it's impossible.

So what we do is we actually.

Recover a surface model from the scan.

Dr Levy paints Zedong a region that he's
comfortable with yanking something out

because he doesn't want to destroy your
card he lives in places where it's useful.

And then we automatically find that
there's a strong by color coding here.

The pieces of that surface which
are the best matches in terms of dramatic

curvature distribution continuity.

And then we position a virtual
cylinder we compute that position we

compute the orientation that would
be best here for them to match and

not only we do that calculation but
also help him during the operation

do the positioning of the tool for
proper insertion the extraction.

So this is not an easy problem for job
metry let's find the best match in shape.

It is difficult here because
these safe are very smooth.

So it's not your typical vision problem or
you have sharp features to match.

It's sort of there's nothing
interesting in the thieves.

OK so it's an optimisation.

And it's roughly five
degrees of freedom there.

So it's a huge space.

So what we do to accelerate it is
we compute some signatures of these

elements using a four year transform
of of core of a tree circles and

we try to match those before we
actually compute how good they are.

This is a work by my student just in
Jan who is finishing his beauty and

the idea here is I mean
architecture in mechanical CAD.

In other applications people design
irregular patterns of things here.

We made a restaurant made
of a number of levels.

Each level of rows of Fables each day its
role has tables each table has chairs.

OK.

You can represent visit
by the sort of a graph.

You say I have a florist and rolls and.

Tables and chairs I have five chairs on
the table these little letters define

transformations from one chair to
the next the chairs on the table.

This is sort of a seam graph.

Now what happens if you
go to an airplane for

example you'll notice that they
have the seats aren't like this.

Except some seats are a little
bit off some seats are missing.

So you would like to preserve
this graph structure but

you want to indicate exceptions.

You're saying are good grass but
this should not be here and

there should be moved.

How would you define those exceptions How
would you store them so that if you were

to Changi There's a graph maybe have
fewer rows your exceptions may survive.

So this is what's going on for example you
may want to say I want the second chair of

the third row of both floors to be read or
upon them removed or transformed somehow.

How would the user specify that.

HOW WOULD BE store it.

HOW WOULD BE use it.

The idea is very simple to cliques.

So every specific ation that
Justin does can be done was just

two clicks of the mouse.

And the idea is the following
When I click a single chair.

I am specifying a path from
the root to that instance.

And each time I got for
a brand that says repeat three times that

passes which of those three times did it
take to reach that share and not this one.

So a path is a sequence of numbers
go left left right but when you have

three proceed of going left you say take
the first one will second or third one.

Which two clicks you have to pass.

And you do a bully in them.

You do explore and when they disagree.

You put a star which means do
anything you want or do all three.

And it was this little combination
basically you can represent not open

the Will selections.

But a wide variety.

It which is sufficient for
a good number of applications.

OK So there's a very simple interface for
designing regular patterns and then

designing exceptions in them and storing
them so that the part a metric model could

rigs they could the whole thing or it.

Let's start with this little
rectangle this little

black square here were pulled
out the comptroller polygon.

And the person the designer
who has created this square

really wanted to have a nice curve
that is controlled by this control.

For example she made like the red curve
here which interplay the vertices or

maybe the blue curve which is a little
bit inside but it looks smoother.

To do so we could use a recursive
subdivision process which

doubles the number of vertices
each time one of these is saying

I'm going to take midpoint of each edge
here I'm working on the top as an example.

I'm going to find a better place for
it and bulge it out.

The better place here is it sits on
a cubic that interprets the four

points of the two edge
vertices of the neighbors.

So that's easy to calculate in fact
there's a very simple recipe for

presenting this point.

If you repeat this process for each.

And then repeated again for
the whole thing and again for

the whole thing you converge to this red
curve which is called a four point curve.

And it is nice.

It goes through the vertices but
it's only see one

continuous which means it's continues
in position and derivative.

But not the second derivative and that is
a problem in many applications as the.

Rick functional some people really
need a second derivative of or

better continue it.

Or if you don't like the red curve.

You say I want a better curve.

So you could say well
I am going to split it

into two was before by
inserting a midpoint.

And I'm going to round the corners.

Removing the old vertices and
stopping them through this blue parable.

You can always fit two parabola like this.

And the center of the parabola will be
the new position of the old vertices.

If you repeat that process for
each corner.

And again and again you end up with what
is called a cubic beast blind curve.

Now we said OK well why
don't we combine them.

We'll do a little bit of the red and
the one minus a little bit of the blue.

And the S. parameter defines how
much of the little bit you want.

So we call that the J. spines.

And they give you a whole
family of curves.

The beauty of them is that they have
these continuities property for

example if you constrain.

Has to be between zero and
four they are C two.

Even though at zero you get the four
point which is only see one.

If you push a little bit
farther you can even get a C.

for continuous curve which is marvelous.

So we use that and we worked with Scott
Shafer on retrofitting them so that you

can force them to go through the vertices
and still have this beautiful property.

Now.

So what did we say you are going to
double the number of vertices each time.

So your storage your temporary storage
before your end of the curve is doubling

you can use the same principle
to do surfaces in which.

As you do curves in one direction and

the curves in the other
the storage quadruples.

If you want a new mated surfaces you can
do the same thing in the third dimension

and the problem is that you know if you
want to subdivide five six seven eight

times to get a very smooth surface.

You'll start requiring a lot of storage.

So we developed arranging approach which
instead of this exponential storage cost

has a very small but in our cost our is
the number of a course and that's it.

So for a curve you need four times our
if our is six in twenty four registers.

That's it.

So it's very good for G.P.U. rendering and

the ringing basically works by saying
why I'm doing my job here on this curve.

I don't need the rest.

If I am going to do a subdivision once and
then I have enough information

to do it again and again and
again and then this guy slide by.

I want the next guy slide
by two megs five by four.

So it's like that the rings that
are smaller and smaller and

go faster and faster.

So this idea is very good for
running this on the C.P.U.

on the G.P.U. and the work free or crime
to do is to be able to stream an animation

where you send the control frame and
then the G.P.S. doing all sort of

rendering of the intermediate frames and
then you stream another country frame and

is doing all this rendering in the on
the intermediate frames to between them.

And this paper would appear next year I'm.

But you can have it on the on the website.

All right so now we are in
the business of processing geometry.

By process and mean to measure things
to compare things to change it

very often the measures on

processing John that we were based on
the short was distance to something.

For example if I wanted.

Establish a correspondence
between points on the green curve

through corresponding
points on the blue curve.

I could take a green point P.
and say Give me the closest point and

Q Nice easy to do.

Not is that if this is smooth.

This is the right angle because of
convincers of that because it's a minimum

of the core of the quote
of the distance function.

But this is typically
not the right angle so

that correspondence is by
definition monster metric.

And somehow when you see that is it.

It cannot be optimal.

OK You want a correspondence between P.

and Q Why should it be
a symmetric that correspondence.

So if you reverse the role of can peel
you get a different course point but

it's not good.

So instead of this big circle
here centered there and

touching here we make a smaller circle
that is Ted Nugent kissing at both ends.

It's an escalating US collating circle and

Mike has the same angles here
at least was attended plains.

So this is what we called the ball map and
this one is symmetry of course.

And it has very nice properties which
would investigate a bit more it reduces

distortion it doesn't have that much
distortion as this one through the map.

And actually surprisingly even though.

Q. is the closest point to P. and you
expect that this is the sort of distance.

If you integrate the distances
between these points and

their closest points you discover the sum
of those distances of the integral of

those distances is actually
longer than if you use this map

because you have to be careful
what you integrate over that.

Now before we go to the next
slide look at this scenario here.

The green curve and the blue curve are not
compatible in the following sense.

I can hear.

I have for each point P.
I have a unique point Q.

were if I grow this disc tangent here.

I touched you.

At one point here.

I don't have this relation there
are some branches of code that cannot be

reached by this disc
that also touches people.

So we say that this part
here is not compatible.

So some shapes are not compatible
Mind you they are not compatible for

the cause of projection either.

In fact.

It gets worse if you use
the cause of projection.

So here is an example where the blue and
orange curves are both compatible.

They are not closest
production compatible.

So the book about it will
give you a bit more power.

How many of you know what
the house door distance is.

All right.

I tell you my love story then.

So when I was young I used to love her and

she loved me too and we always have
been in this funny situation where

she would live on one island and I would
live in another and because we loved each

other we would come from as close to the
edge possible to each other and so that

this and that were separating us was the
minimum distance between the two islands.

And then you know I did something
wrong I still don't know what.

But now she hates me.

But I still love her.

So I want to be as close as possible and
because she hates me and

she knows where I'm going to she's going
to the other extreme of the island as far

as possible from I could
get close to it and

now what separates us is the House
of distance in fact to be precise.

She can choose which island she prefers.

So that measure is measure.

Drink.

How different two shapes
are it's not a very good measure.

But you know that's what people use often.

And it's reasonably
complicated to compute.

It's not very good because I
don't have a good example.

But let's suppose one curve is a circle.

The other one is sort of doing a little
thing like this comes back and

finishes a circle.

That little thing could be quite close to
the circle and the House of distance would

say well they're almost the same little
distance but they're very different.

So a better distance is
the fresher distance.

The Frechette distance
as you walk on one curve

your dog very smart dog
walks on the other curve.

None of you is allowed to backtrack.

You always move forward or
stop if you need to but never go back.

And the dog isn't a leash.

If the dog is very smart tries to
minimize the lens of the dish because it

hurts if it pulls the lens of
the least that is maximum at any point

that the maximum length that you
need is the fresher distance and

this one is even worse to compute it
requires an optimisation of the parameters

ation So we have shown one that when
the two shapes are both compatible.

This guy and this guy are the same and in
fact they are quite easy to compute they

are the diameter of the biggest disk
that touches both rapes in that sense.

All right so this is some nice terrific
a result which we did was colleagues from

the in the in France in their
system which is a cad vendor of

mechanical and architectural cat systems.

So we looked at this and said.

It's funny because if we remove that.

Incompatible region the dotted
lines above in some sense we are.

Rounding one shape and the radius of
the round is controlled by where the other

shape is so we should look at the other
safe will be now the control shape.

And we have a new way of specifying.

Roundings or
specifically the radius for rounding.

So I don't know if you can see but
these are true blue sit in.

There's a union.

There's a sharp concave edge here quite
complicated it's a quartic curve and

we would like to obtain a nice
blend of the surface here.

And we would like to control the edges
of the blending not as a constant but

in some more controllable way.

So to do so
we define a red shape a control shape.

We start maybe it was the blue shape the
Soviet Union or some grabbing and pulling.

And now we say let's do this business of
running the ball between the two shapes

and every place that it cannot roll.

Will add as a fillets to the shape or

if one is inside the other it
will remove that rounding.

So for example here one shape was blue
the conscious control shape was red.

We run this ball which does a little

strange torus that girls in red use and
comes back.

And this is what you get if you
move there on the controller shaped

to be more central this is what
you get if your control shape is

a plane on this very strands
of four Cuba arrangements.

You can be a rounding convex sets and
concave sets with the same operation.

So it's a nice operator.

And the most important thing the beauty
of this is that it's not only

doing its job but we have a mathematical
formulation for what it's doing.

And it's expressed here
in terms of billions.

So we can say what is produced
very often when you do out.

Goritz for computer graphics you say well
you know that's what the album does and

you ask the people who doesn't produce
Well it's what they are going that does.

So this of a chicken and egg problem you
cannot say this is correct because it is

correct in producing wood it produces OK
here we have a mathematical formulation of

the result and nice increment an algorithm
can check whether it is correct or not.

So it's always better to be able to
formulate what you're trying to compute

instead of just saying we
get whatever we get it.

Well so for example in this situation
these two blue things are not touching.

And in the end they will be touching.

So yes there are political
differences may occur and

hopefully you desire them but
sometimes you may not under subpoena.

So this is trying to work with
the Jason Williams who is also graduating

the idea is the following
start with the blue curve.

Take a coin of a given radius and
roll it over the place

the coin is not allowed to go
through to go over the blue curve.

So it will leave a green region
that it cannot reach without

getting over the curve.

We give this this region a name
we call that the mortar or the R.

mortar if you have
specified the value of our.

This is a morphology called Operation
you can construct this mortar by

doing some growing interest for example.

Now we say OK.

The intersection of
the curve was the mortar

can be simplified by
putting by tightening it.

So instead of the first in the mortar.

You know this by construction.

Although it doesn't show very well here.

It's mostly joined with
the rest of the curve and

it gives us of a minimum length
curve which we call a tightening.

Which is a very nice way of making
a compromise that does rounding in Philip

doing very very pleasant way.

So Jason has implemented that in two D.
and we know what you're getting we're

getting the shortest curve in
the mortar and the rest is on touch.

So the advantage of this is not
that smoothing filtering operation.

It doesn't change the shape elsewhere.

It just changes the shape where it's
needed in the mortar He has also

implemented that in three D. using voxels
but theoretically we have this sort of

debate about what should be the nature of
that surface in three D. mathematically.

It's not a minimum surface it.

If we try to compute the minimum
surface it wouldn't do the job but

you know the minimal surface for

example between two spheres would tend to
do this and be just a straight line break.

OK so it's a slightly different beast.

So we're have approach some mathematician
friends to help us with that but

it's a little bit difficult.

OK or this so sir you may have
seen a work in medical area

where you would take two
scans say of a body part and

somebody automatic your money would
highlight or would trace a curve.

So you have two slices and you want to
construct a surface that intra plates.

So we decided that if we do our
border map as shown on top here.

We're not is that actually you can
construct a sun in black here.

Always a circular trajectory
between one point and

the corresponding point on the other.

By the ball map.

Such that the circle trajectory
is not only a circle but

it is orthogonal to both
curves of the contact points.

So now what you want is to
do a surface reconstruction

from the top curve to the bottom curve.

If you look from the top and
project them both on the same plane.

This is the situation.

So I can't see this here but
there's a bunch of arcs here.

So this point correspond to
that point as the arc there.

If you travel on that arc in the plane and

at the same time you're raised
was elevator to the next level to

synchronize both things you're tracing a
circle in projection and you're going up.

You're tracing the hell it's the same
thing as a screw motions we have

seen at the beginning.

So each trajectory of each point is ahead.

Ix.

So this is a pencil of Calyx and
it gives very nice surfaces much

better than using the cause of projection
from one to the other or vice versa.

So they look very nice but

we said well you know look is looks
can we measure how nice they are.

And for.

Brian.

Has implemented a dozen techniques for
doing the morphing between one curve and

another he has I'm showing here eight or
so.

And these three

are the board map based This one is

the one that has a circular trajectories
this one is a huge propagation you

put this curve very cold this one very hot
and you trace the temperature gradients.

These to the heat propagation and
the circular

ball more are the best in terms of
the different measures we have tried.

And there is now be similar
not quite the same but

similar the advantage of
the circle ball more is that

it's trivial to compute compared to
the heat propagation which seems to be

solved in this domain by adding probably
trait of process for a fifty process.

So we're pretty happy with this and
we have shown that we have

the you know in these results
are unreadable on the right.

But we have compared the things was
different measures travel distance.

Amount of stretch acceleration surface
area and the mean curvature squared.

All right let's suppose you recall over

a triangle mash frog scanning
process laser scanning or

some other process that tries to do
reverse engineering of the existing shape

your laser would be
probably touching I mean

measuring distances not
knowing what is measuring so

sharp features like these
would typically be missed.

As a consequence.

When you recover a triangle mesh.

Near sharp features.

You will have this mess
here that you can maybe

understand it was the samples
are all over the place and

try to triangulate to get this time for
us which urges very ugly.

So the question is under which
circumstances and by which algorithm.

Could you recover the sharp features
many techniques have been proposed we

came up with one which has these six very
trivial little steps that the coloring of

edges vertices triangles to identify
the triangles that are on this some verse.

And then we basically extend
the a building surfaces using the.

Normal Intendant information and computer
intersections and produce these sharp

these sharpened features in
addition we apply a subdivision

process earth a bit like what we discussed
earlier but we have to modified so

that instead of producing
piecewise flat triangle measures.

It produces smooth surfaces.

But we wanted to preserve the sharp
features that we have recovered.

So we don't want to smoothing process
to go and sort of blend them.

OK so we had to modify
the subdivision process to respect

the sharp features which we have so
carefully identified.

It works beautifully well.

Even in surfaces which are not mechanical
but sort of you know sculpture is the or.

I mean models or whatever.

So it's a robust than very nice and
simple process for

recovering features in under
sampled representations.

This is a work done with
my friends in Barcelona.

Sort of similar.

But here instead of starting was
a triangle mesh like a laser scanner.

We start with the voting metric model for
example an M.R.I. or C.T.

scan and so we have then cities
are let's suppose your threshold them.

And we'll call green the Sam
the Vox of course Center is inside

below some threshold or above and
read the other ones outside.

So we have some edges which are red green
and then what you want is to break over

a model where we have plains blends
this sort of roundings sharp edges and

to do so what we need is to grow
planner representations that

stab as many of the red
green edges as possible and

then we to extend them into
second make decisions.

And finally the remaining surfaces.

To be smooth and from this ugly nonsense
to some small surfaces that would connect

to the planes nicely and so that was
a sort of a tedious much more complicated

than Origen expected amount of
work to get from here to there.

This is an optimisation for
dimensional space.

There's a lot of delicate guessing
of how to rearrange the surface so

it's moves and doesn't violate
the red green labeling here.

So it's constrained by the samples
you want to grow a nice curve

like here constrained by the red and
green samples.

Somewhat involved.

So I would guess this is less
elegant than some of the others.

This is a very nice piece of work
that we did together was immense.

Brian Whited was again
the primary developer here and

inventor of those things.

So what we want is to create
a representation of a river vessel.

Strong drawn by hand with
the centerline and with the thickness.

So what we do is we let the user
select the good color by clicking and

dragging little circle for alleged
about color get started and when we go.

And in real time you get
this series of disks that

are computed each one from the previous
one by wiggling it computing the best

center of the best read years so that
you have a lot of good material inside.

And a little bit of bad
material on both ends.

So it's a little local
optimization process.

So if you had this disk before you'd get
this mom you move the center at all.

Bit you move the red just to have some
good material and some by the material.

It works in real time into
a deep it takes about the so.

There are so are two seconds to
do this in three D. from three D.

scans of real patients data.

It's very robust and
it gives you a very nice flexible tool for

the user to say I want to cut it here of
the extended their what it produces is

a bunch of balls and
we call them Pearl strings.

Now what you would like is to get
a nice simple surfaces from those.

So the first thing of
course we did was OK.

Each ball is represented by X.Y.Z.
center and radius.

So it's a point in four dimensional
space we said let's use the j splines of

the vision to remember the skirts of that
isn't technique to construct a smooth or

smooth or curved a recursive subdivision
in a four dimensional space and

that's what we get this is another example
and not is that although it's very nice.

It has at some point.

Sometimes sharp sort of folds or
curvature.

So we said could we do better and
that we worked with our colleagues

from Siemens in fact Greg slabber was
a student here it is student here.

We worked with with him
defining of people eat problem

basically to arrange for
the surface to be optimized so

as to minimize a compromise between the
curvature measure and the surface area.

And so this is an example of
the result of the position not there is

that instead of having these sharp things.

It is nicer here it is
a complicated slow process and

the result is still I think not optimal.

I still think that this is
an open problem given the set of

bolts how to find the skin that.

Is the smoothest possible and most truly
represents what these balls are sampling.

All right.
I teach a lot of yes.

So the question is.

I could rom the surface
extraction by saying Give me

a point of a certain density and
I could fit a triangle

measured to that we were interested in
extracting actually the tubular structure.

Not thresholding Furthermore
the thresholding

is extremely difficult because if you do
the marking cube like approaches because

of the knowledge of the acquisition
process you get a lot of little

components which you have to filter out by
morphological operators so it's difficult.

Plus you don't have a higher level
representation of a spine and

original function which
is very valuable for

example if the edges goes like this
you say that's bad for the artery.

OK maybe there's a little constriction
here that we need to take care of.

So it gives us segmentation
an interpretation a more abstract

representation of the sicko triangle mesh.

The disadvantage is that this is not
good for things that are not long and

should be lower.

It's not going to do a good job for
your liver maybe but

it will do a decent job for your
arteries your fingers maybe your arms.

We have done some comparison Brian
spanned the couple of summers and

a few months in addition to that at
Siemens and here they already had some

of the state of the Arctic Nique And
so he did some comparison with this.

This seems to be extremely fast and.

Effective.

However.

It is not optimal.

And it just doesn't work when

you're not tracing something which
can be measured with his balls crap.

I didn't discuss the branching
structure but when the ball comes to or

going to a branch.

We take decisions that we need to
follow both branches or just one.

And so there are some interesting
issues there as well but I don't have

a comparison numbers to report because
there's no ground truth in some sense.

So the best you could do is to have
a bunch of humans construct this and

measure maybe the House of distance
between the two recover and

what they recover from so I'm not sure.

So in the last few minutes.

Let me just teach you a corner table.

So you want to represent a triangle mesh.

You have a list of vertices X.
Y. Z. maybe color if you want.

And then you need to
capture the connectivity.

That is each trap to know that reverses.

The way I like to represent that
is by two tables the N O V.

RI presents the vertex for each corner for
a triangle has three corner.

Let's suppose that this corner
here is called Corner C.

This is the vertex that vertex has an ID
Maybe vertex three hundred seventy five.

So this corner nice to know I'm sitting
on vertex three hundred seventy five

the Table V. at that entry for
that corner will have that number.

In addition in the table.

This corner would know would find
the ID of the opposite corner.

That's it.

So from these two tables of integers
which is very nice to manage

you can reconstitute next previous
corner as the idea of the triangle.

The left and the right ideas and
all sorts of cool.

Thanks.

And that's what I do with my students
we use this representation to

do all sorts of processing on the mesh and
recently.

What we discovered with Proust Durand.

Is that not only this can be
extended to tetrahedra measures

which we have done before and
other people have done as well but

also we can actually get rid of the V.
table completely.

So we don't need to represent
the view table any more

amazing because the table was what
tells us we diverge is our sitting on.

So we reduce the storage by fifty percent.

I'm still glossing over
the detail of a few extra bits.

Furthermore without any
additional storage for free.

We give you a pointer from each vertex
to one of the corners sitting on it.

So we can use that corner
to visit the neighbors.

So we have extended D.
the content of this data structure and

reduces storage by half and have applied
that not only to triangle measure but

also to try to draw matches and
so this is called the S O T.

It's a sort of all table.

So we only use the table to be sorted in
some smart way and you can read that paper

if you want to get more details because
I am afraid I'm running out of time but

I have some time for questions but.

Thank you.

Well you get credit for this class.

They were all supposed to
have three questions each.

OK if there is no question.

Well I invite you to take a listen.

Going to if you have some doubts or
interesting ideas e-mail me or

come and talk to me.

Thank you very much for attending.

Thank you thank you.