1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping...
-
Upload
jared-williams -
Category
Documents
-
view
219 -
download
0
Transcript of 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping...
![Page 1: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/1.jpg)
1
Optimal Transport, Conformal Mappings,
and Stochastic Methods for Registration and Surface
Warping
Allen TannenbaumGeorgia Institute of Technology
Emory University
![Page 2: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/2.jpg)
2
This lecture is dedicated to Gregory Randall, citizen of the world. Muchas gracias, Gregory (El Goyo).
![Page 3: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/3.jpg)
3
Collaborators:
Conformal Mappings: Angenent, Haker, Sapiro, Kikinis, Nain, Zhu
Optimal Transport: Haker, Angenent, Kikinis, Zhu
Stochastic Algorithms: Ben-Arous, Zeitouni, Unal, Nain
![Page 4: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/4.jpg)
4
Surface Deformations and Flattening Conformal and Area-Preserving Maps
Optical Flow
Gives Parametrization of SurfaceRegistration
Shows Details Hidden in Surface Folds
Path PlanningFly-Throughs
Medical ResearchBrain, Colon, Bronchial PathologiesFunctional MR and Neural Activity
Computer Graphics and VisualizationTexture Mapping
![Page 5: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/5.jpg)
5
Mathematical Theory of Surface Mapping
Conformal Mapping:One-oneAngle PreservingFundamental Form
Examples of Conformal Mappings:One-one Holomorphic FunctionsSpherical Projection
Uniformization Theorem:Existence of Conformal MappingsUniqueness of Mapping
GFEGFE ,,,,
![Page 6: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/6.jpg)
6
Deriving the Mapping Equation
L e t p b e a p o i n t o n t h e s u r f a c e . L e t
b e a c o n f o r m a l e q u i v a l e n c e s e n d i n g p t o t h e N o r t h P o l e .
I n t r o d u c e C o n f o r m a l C o o r d i n a t e s vu , n e a r p ,
w i t h 0 vu a t p .
I n t h e s e c o o r d i n a t e s ,
22 2,2 dvduvuds
W e c a n e n s u r e t h a t 1p . I n t h e s e c o o r d i n a t e s , t h e L a p l a c e B e l t r a m i o p e r a t o r t a k e s t h e f o r m
2
2
2
2
2,
1
vuvu.
2: Sz
![Page 7: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/7.jpg)
7
Deriving the Equation-Continued
Set ivuw . The mapping wzz has a simple pole at 0w , i.e. at p .
Near p , we have a Laurent series 2DwCBwA
wz
Apply to get
wAz
1.
Taking 21A ,
)2(2
1
log2
1
log2
1
1
2
1
pvi
u
wv
iu
wv
iu
wz
![Page 8: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/8.jpg)
8
The Mapping Equation
pvi
uz
.
S i m p l y a s e c o n d o r d e r l i n e a r P D E . S o l v a b l e b y s t a n d a r d m e t h o d s .
![Page 9: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/9.jpg)
9
Finite Elements-I i s a t r i a n g u l a t e d s u r f a c e . S t a r t w i t h
pvi
uz
M u l t i p l y b y a n a r b i t r a r y s m o o t h f a n d i n t e g r a t e b y
p a r t s . F o r a l l f w e w a n t :
pv
fip
u
f
dSfv
iu
dSfz p
L e t PLfz , , t h e s p a c e o f p i e c e w i s e l i n e a r f u n c t i o n s .
![Page 10: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/10.jpg)
10
Finite Elements-IIF o r e a c h v e r t e x P , l e t p b e t h e
c o n t i n u o u s f u n c t i o n s u c h t h a t :
gle.each trianon linear is
, vertexa ,,0
1
P
QPQQ
P
P
P
T h e s e f u n c t i o n s f o r m a b a s i s f o r t h e f i n i t e d i m e n s i o n a l s p a c e PL . T h e n PPP zz . A n d w e w a n t , f o r a l l Q ,
pvQ
pu
QQdSPPzP
T h i s i s s i m p l y a m a t r i x e q u a t i o n .
![Page 11: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/11.jpg)
11
Finite Elements-III
S e t
PQDD , dSQPPQD .
D e f i n e v e c t o r s
p
uQ
Qaa
,
p
vQ
Qbb
.
O u r e q u a t i o n b e c o m e s s i m p l y ibaDz .
SRPQD cotcot21
,
PQDQPPPD .
N e e d f o r m u l a s f o r ., ba
![Page 12: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/12.jpg)
12
Finite Elements-IV
S u p p o s e t h e p o i n t p l i e s o n a t r i a n g l e w i t h v e r t i c e s ABC .
S i n c e
p
uQ
Qaa
,
a n d
p
vQ
Qbb
,
w e h a v e CBAQQibQa , , if 0 .
![Page 13: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/13.jpg)
13
Finite Elements-V
I f CBAQ , , , t h e n c o n s i d e r i n g t h a t Q i s
l i n e a r o n ABC :
, 1
,1
,11
:
CQEC
i
BQEC
iAB
AQEC
iAB
Qib
Qa
2
,
AB
ABAC
![Page 14: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/14.jpg)
14
Finite Elements-VI If we set iyxz , then our system
ibaDz becomes aDx and bDy .
D is sparse, real, symmetric and positive semi-definite. Its kernel is the space of constant vectors, and it is positive definite on the space orthogonal to its kernel.
These properties of D allow us to use the conjugate gradient method to solve the system.
![Page 15: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/15.jpg)
15
Summary of FlatteningF l a t t e n i n g :
C a l c u l a t e t h e e l e m e n t s o f t h e m a t r i c e s baD and , , .
U s e t h e c o n j u g a t e g r a d i e n t m e t h o d t o s o l v e bDyaDx and . T h e r e s u l t i n g iyxz i s t h e c o n f o r m a l m a p p i n g t o t h e c o m p l e x p l a n e .
C o m p o s e z w i t h i n v e r s e s t e r e o p r o j e c t i o n t o g e t a c o n f o r m a l m a p t o t h e u n i t s p h e r e .
![Page 16: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/16.jpg)
16
Cortical Surface Flattening-Normal Brain
![Page 17: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/17.jpg)
17
White Matter Segmentation and Flattening
![Page 18: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/18.jpg)
18
Conformal Mapping of Neonate Cortex
![Page 19: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/19.jpg)
19
Coordinate System on Cortical Surface
![Page 20: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/20.jpg)
20
Principal Lines of Curvature on Brain Surface-I
![Page 21: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/21.jpg)
21
Principal Lines of Curvatures on the Brain-II
![Page 22: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/22.jpg)
22
Flattening Other Structures
![Page 23: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/23.jpg)
23
Bladder Flattening
![Page 24: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/24.jpg)
24
3D Ultrasound Cardiac Heart Map
![Page 25: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/25.jpg)
25
High Intelligence=Bad Digestion
Low Intelligence=Good Digestion
Basic Principle
![Page 26: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/26.jpg)
26
Flattening a Tube(1) Solve
1 1
0 0
10\ 0
onuonu
onu
(2) Make a cut from 0 to 1 .
Make sure u is increasing along the cut.
![Page 27: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/27.jpg)
27
Flattening a Tube-Continued
( 3 ) C a l c u l a t e v o n t h e b o u n d a r y l o o p
00 1 cutcut
b y i n t e g r a t i o n
dsn
uds
s
vv
( 4 ) S o l v e D i r i c h l e t p r o b l e m u s i n g b o u n d a r y v a l u e s o f v .
I f y o u w a n t , s c a l e s o 2h , t a k e ivue
t o g e t a n a n n u l u s .
v = g ( u ) + h
u = 1u = 0
v = g ( u )
![Page 28: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/28.jpg)
28
Flattening Without Distortion-I
In practice, once the tubular surface has been flattened into a rectangular shape, it will need to be visually inspected for pathologies. We present a simple technique by which the entire colon surface can be presented to the viewer as a sequence of images or cine. In addition, this method allows the viewer to examine each surface point without distortion at some time in the cine. Here, we will say a mapping is without distortion at a point if it preserves the intrinsic distance there. It is well known that a surface cannot in general be flattened onto the plane without some distortion somewhere. However, it may be possible to achieve a surface flattening which is free of distortion along some curve. A simple example of this is the familiar Mercator projection of the earth, in which the equator appears without distortion. In our case, the distortion free curve will be a level set of the harmonic function (essentially a loop around the tubular colon surface), and will correspond to the vertical line through the center of a frame in the cine. This line is orthogonal to the “path of flight” so that every point of the colon surface is exhibited at some time without distortion.
![Page 29: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/29.jpg)
29
Flattening Without Distortion-II
S u p p o s e w e h a v e c o n f o r m a l l y f l a t t e n e d t h e c o l o n s u r f a c e o n t o a r e c t a n g l e
,,0 max uR . L e t F b e t h e i n v e r s e o f t h i s m a p p i n g , a n d l e t vu ,22 b e t h e a m o u n t b y W h i c h F s c a l e s a s m a l l a r e a n e a r vu , , i . e . l e t 0 b e t h e “ c o n f o r m a l f a c t o r ” f o r F . F i x 0 w , a n d f o r e a c h max0 ,0 uu d e f i n e a s u b s e t
RwuwuR ,, 000 w h i c h w i l l c o r r e s p o n d t o t h e c o n t e n t s o f a c i n e f r a m e . W e d e f i n e a m a p p i n g
u
u
v
dvvudvvuGvu0 0
0 ,,,,ˆ,ˆ .
![Page 30: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/30.jpg)
30
Flattening Without Distortion-III
W e h a v e
vu
dvvu
vv
uuvudG
u
u
v
vu
vu
,0
,,ˆˆ
ˆˆ,
0
0
,
10
01,, 00 vuvudG .
This implies that composition of the flattening
map with G sends level set loop 0uu on the surface to the vertical line 0u in the vu plane without distortion. In addition, it follows from the formula for dG that lengths measured in the u direction accurately reflect the lengths of corresponding curves on the surface.
![Page 31: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/31.jpg)
31
Introduction: Colon Cancer
US: 3rd most common diagnosed cancerUS: 3rd most frequent cause of deathUS: 56.000 deaths every year
Most of the colorectal cancers arise from preexistent adenomatous polyps
Landis S, Murray T, Bolden S, Wingo Ph.Cancer Statitics 1999. Ca Cancer J Clin. 1999; 49:8-31.
![Page 32: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/32.jpg)
32
Problems of CT Colonography
Proper preparation of bowelHow to ensure complete inspection
Nondistorting colon flattening program
![Page 33: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/33.jpg)
33
Nondistorting colon flatteningSimulating pathologist’ approachNo Navigation is neededEntire surface is visualized
![Page 34: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/34.jpg)
34
Nondistorting Colon Flattening
Using CT colonography dataStandard protocol for CT colonographyTwenty-Six patients (17 m, 9 f)Mean age 70.2 years (from 50 to 82)
![Page 35: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/35.jpg)
35
Flattened Colon
![Page 36: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/36.jpg)
36
Polyps Rendering
![Page 37: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/37.jpg)
37
Finding Polyps on Original Images
![Page 38: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/38.jpg)
38
Polyp Highlighted
![Page 39: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/39.jpg)
39
Path-Planning Deluxe
![Page 40: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/40.jpg)
40
Coronary Vessels-Rendering
![Page 41: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/41.jpg)
41
Coronary Vessels: Fly-Through
![Page 42: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/42.jpg)
42
Area-Preserving Flows-IL e t M b e a c l o s e d , c o n n e c t e d n - d i m e n s i o n a l m a n i f o l d . V o l u m e f o r m :
0)(
...,)( ,1
xg
dxdxdxdxxg n
T h eo rem (M o ser): M , N com pac t m an ifo ld s w ith vo lu m e fo rm s and . A ssu m e th a t M an d N a re d iffeo m o rp h ic . If
NM ,
then there exists a diffeomorphism of M into N taking into .
![Page 43: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/43.jpg)
43
Area-Preserving Flows-IIThe basic idea of the proof of the
theorem is the contruction of an orientation-preserving automorphism homotopic to the identity.
As a corollary, we get that given M and N any two diffeomorphic surfaces with the same total area, there exists are area-preserving diffeomorphism.This can be constructed explicitly via a PDE.
![Page 44: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/44.jpg)
44
Area-Preserving Flows for the Sphere-I
F i n d a o n e - p a r a m e t e r f a m i l y o f v e c t o r f i e l d s
1,0, tu t a n d s o l v e t h e O D E
ttudt
d t
t o g e t a f a m i l y o f d i f f e o m o r p h i s m s t s u c h t h a t
id0 a n d
)det()det(1det DftDftD t .
fS2
S2 f o
N
![Page 45: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/45.jpg)
45
Area-Preserving Flows for the Sphere-IIT o f i n d tu , s o l v e
)det(1 Df ,
t h e n
tDftu t
)det(1
![Page 46: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/46.jpg)
46
Area-Preserving Flows of Minimal Distortion
L e t M a n d N b e t w o c o m p a c t s u r f a c e s w i t h R i e m a n n i a n m e t r i c s h a n d g r e s p e c t i v e l y , a n d l e t b e a n a r e a p r e s e r v i n g m a p . T h i s
m e a n s i f g a n d h a r e t h e a r e a f o r m s t h e n
.)(*hg
M a n y o t h e r a r e a p r e s e r v i n g m a p s f r o m NM ( j u s t c o m p o s e w i t h a n y o t h e r a r e a p r e s e r v i n g m a p ) . W h i c h o n e h a s t h e s m a l l e s t d i s t o r t i o n ? M i n i m i z e t h e D i r i c h l e t i n t e g r a l w i t h r e s p e c t t o a r e a - p r e s e r v i n g m a p s :
J ( þ ) = 1 = 2R
Mj D þ j 2 Ò h
T h i s l e a d s t o e x p l i c i t g r a d i e n t d e s c e n t e q u a t i o n s . M e t h o d w i l l b e d i s c u s s e d w h e n w e d e s c r i b e M o n g e - K a n t o r o v i c h a l g o r i t h m s .
![Page 47: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/47.jpg)
47
Registration and Mass Transport
Image registration is the process of establishing a common geometric frame of reference from two or more data sets from the same or different imaging modalities taken at different times.
Multimodal registration proceeds in several steps. First, each image or data set to be matched should be individually calibrated, corrected from imaging distortions, cleaned from noise and imaging artifacts. Next, a measure of dissimilarity between the data sets must be established, so we can quantify how close an image is from another after transformations are applied to them. Similarity measures include the proximity of redefined landmarks, the distance between contours, thedifference between pixel intensity values. One can extract individual featuresthat to be matched in each data set. Once features have been extracted from each image, they must be paired to each other. Then, a the similarity measure between the paired features is formulated can be formulated as an optimization problem.
We can use Monge-Kantorovich for the similarity measure in this procedure.
![Page 48: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/48.jpg)
48
Mass Transportation ProblemsOriginal transport problem was proposed
by Gaspar Monge in 1781, and asks to move a pile of soil or rubble to an excavation with the least amount of work.
Modern measure-theoretic formulation given by Kantorovich in 1942. Problem is therefore known as Monge-Kantorovich Problem (MKP).
Many problems in various fields can be formulated in term of MKP: statistical physics, functional analysis, astrophysics, reliability theory, quality control, meteorology, transportation, econometrics, expert systems, queuing theory, hybrid systems, and nonlinear control.
![Page 49: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/49.jpg)
49
Monge-Kantorovich Mass Transfer Problem-I
W e c o n s i d e r t w o d e n s i t y f u n c t i o n s
R
ö 0 ( x ) d x =R
ö T ( x ) d x W e w a n t
M : R d ! R d w h i c h f o r a l l b o u n d e d s u b s e t s A ú R d R
x 2 A ö T ( x ) d x =R
M ( x ) 2 A ö 0 ( x ) d x F o r M s m o o t h a n d 1 - 1 , w e h a v e ( J a c o b i a n e q u a t i o n ) )())(())((det 0 xxMxM T W e c a l l s u c h a m a p M m a s s p r e s e r v i n g ( M P ) .
![Page 50: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/50.jpg)
50
MK Mass Transfer Problem-II
J a c o b i a n p r o b l e m h a s m a n y s o l u t i o n s . W a n t o p t i m a l o n e ( L p -K a n t o r o v i c h - W a s s e r s t e i n m e t r i c )
d p ( ö 0 ; ö 1 ) p : = in f M
Rj M ( x ) à x j p ö 0 ( x ) d x
O p t i m a l m a p ( w h e n i t e x i s t s ) c h o o s e s a m a p w i t h p r e f e r r e d g e o m e t r y ( l i k e t h e R i e m a n n M a p p i n g T h e o r e m ) i n t h e p l a n e .
![Page 51: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/51.jpg)
51
Algorithm for Optimal Transport-I
Ò0;Ò1 ú R d
Subdomains with smooth boundaries and positive densities:
RÒ0
ö0 =R
Ò1ö1
We consider diffeomorphisms which map one density to theother:
öo = det(Duà)ö1 î uà
We call this the mass preservation (MP) property. We let u be ainitial MP mapping.
![Page 52: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/52.jpg)
52
Algorithm for Optimal Transport-II
We consider a one-parameter family of MP maps derived as follows:
uà := u î sà 1; s = s(á;t); ö0 = det(Ds)ö0 î s
Notice that from the MP property of the mapping s, and definition of the family,
uàt = àö0
1Duà áð; ð = ö0st î sà 1
div ð = 0
![Page 53: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/53.jpg)
53
Algorithm for Optimal Transport-III
M(t) =R
Ò0Ð(uà(x; t) à x)ö0(x) dx
=R
Ð(u(y) à s(y; t))ö0(y) dy; x = s(y; t); sã(ö0(x)dx) = ö0(y)dy
M 0(t) = àRhÐ0(u à s); stiö0dy
= àRhÐ0(uà(x;t) à x); ö0st î sà 1i dx
= àR
Ò0hÐ0(uà(x;t) à x); ði dx
We consider a functional of the following form which we infimize with respect tothe maps :
Taking the first variation:
uà
![Page 54: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/54.jpg)
54
Algorithm for Optimal Transport-IV
ð = Ð0(uà à x) + r p
div ð = 0
ðj@Ò0 tangential to @Ò0
É p+ div (Ð0(uà à x)) = 0; on Ò0
@n~@p + n~áÐ0(uà à x) = 0; on @Ò0
First Choice:
This leads to following system of equations:
uàt = à 1=ö0Duà á(Ð0(uà à x) + r p)
![Page 55: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/55.jpg)
55
Algorithm for Optimal Transport-V
Duà á(I à r É à 1r á)Ð0(uà à x)@t@uà = à
ö0
1
This equation can be written in the non-local form:
At optimality, it is known that
Ð0(uà à x) = r ë
where is a function. Notice therefore for an optimalsolution, we have that the non-local equation becomes
@t@uà = 0
![Page 56: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/56.jpg)
56
Solution of L2 M-K and Polar Factorization
uà = à 1=ö0Duà(uà à r É à 1 div(uà))
Ð(x) = 2jxj2
uà = u î sà 1 = r w + ÿ; div(ÿ) = 0 H elmholtz decomp:
For the L2 Monge-Kantorovich problem, we take
This leads to the following “non-local” gradient descent equation:
Notice some of the motivation for this approach. We take:
The idea is to push the fixed initial u around (considered as a vectorfield) using the 1-parameter family of MP maps s(x,t), in such a manneras to remove the divergence free part. Thus we get that at optimality
u = r w î s P olar factorization
![Page 57: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/57.jpg)
57
Example of Mass Transfer-I
We want to map the Lena image to the Tiffany one.
![Page 58: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/58.jpg)
58
Example of Mass Transfer-II
The first image is the initial guess at a mapping. The second isthe Monge-Kantorovich improved mapping.
![Page 59: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/59.jpg)
59
Morphing the Densities-I
V(t;x) = x + t(uopt(x) à x)
![Page 60: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/60.jpg)
60
Morphing the Densities-II (Brain Sag)
![Page 61: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/61.jpg)
61
Deformation Map
Brain deformation sequence. Two 3D MR data sets were used. First is pre-operative, and second during surgery, after craniotomy and opening of the dura. First image shows planar slice while subsequent images show 2D projections of 3D surfaces which constitute path from original slice. Here time t=0, 0.33, 0.67,and 1.0. Arrows indicate areas of greatest deformation.
![Page 62: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/62.jpg)
62
Morphing-II
![Page 63: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/63.jpg)
63
Morphing-III
![Page 64: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/64.jpg)
64
Surface Warping-I
M-K allows one to find area-correctingflattening. After conformally flatteningsurface, define density mu_0 to be determinant ofJacobian of inverse of flattening map, and mu_1 to be constant. MK optimal map is then area-correcting.
![Page 65: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/65.jpg)
65
Surface Warping-II
![Page 66: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/66.jpg)
66
![Page 67: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/67.jpg)
67
![Page 68: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/68.jpg)
68
![Page 69: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/69.jpg)
69
![Page 70: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/70.jpg)
70
![Page 71: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/71.jpg)
71
fMRI and DTI for IGS
![Page 72: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/72.jpg)
72
Data Fusion
![Page 73: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/73.jpg)
73
More Data Fusion
![Page 74: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/74.jpg)
74
Scale in Biological Systems
![Page 75: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/75.jpg)
75
Multiscale / Complex System Modeling (from Kevrekidis)
“Textbook” engineering modeling:macroscopic behavior through macroscopic models(e.g. conservation equations augmented by closures)
Alternative (and increasingly frequent) modeling situation: Models
at a FINE / ATOMISTIC / STOCHASTIC level Desired Behavior
At a COARSER, Macroscopic Level E.g. Conservation equations, flow, reaction-diffusion,
elasticity Seek a bridge
Between Microscopic/Stochastic Simulation And “Traditional, Continuum” Numerical AnalysisWhen closed macroscopic equations are not available in closed
form
![Page 76: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/76.jpg)
76
Micro/Macro Models-Scale I
![Page 77: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/77.jpg)
77
Micro/Macro Models-Scale II
![Page 78: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/78.jpg)
78
How to Move Curves and Surfaces
Parameterized Objects: methods dominate control and visual tracking; ideal for filtering and state space techniques.
Level Sets: implicitly defined curves and surfaces. Several compromises; narrow banding, fast marching.
Minimize Directly Energy Functional: conjugate gradient on triangulated surface (Ken Brakke); dominates medical imaging.
![Page 79: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/79.jpg)
79
Diffusions
Explains a wide range of physical phenomenaHeat flowDiffusive transport: flow of fluids (i.e., water,
air)
Modeling diffusion is important At macroscopic scale by a partial differential
equation (PDE)At microscopic scale, as a collection of
particles undergoing random walks
We are interested in replacing PDE by the associated microscopic system
![Page 80: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/80.jpg)
80
![Page 81: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/81.jpg)
81
Interacting Particle Systems-I
Spitzer (1970): “New types of random walk models with certain interactions between particles”
Defn: Continuous-time Markov processes on certain spaces of particle configurations
Inspired by systems of independent simple random walks on Zd or Brownian motions on Rd
Stochastic hydrodynamics: the study of density profile evolutions for IPS
![Page 82: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/82.jpg)
82
Interacting Particle Systems-II
Exclusion process: a simple interaction, precludes multiple occupancy--a model for diffusion of lattice gas
Voter model: spatial competition--The individual at a site changes opinion at a
rate proportional to the number of neighbors who disagree
Contact process: a model for contagion--Infected sites recover at a rate while healthy
sites are infected at another rate
Our goal: finding underlying processes of curvature flows
![Page 83: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/83.jpg)
83
Motivations
Do not use PDEs
IPS already constructed on a discrete lattice (no discretization)
Increased robustness towards noise and ability to include noise processes in the given system
![Page 84: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/84.jpg)
84
Construction of IPS-I
S : a set of sites, e.g. S= Zd
W: a phase space for each site, W={0,1}
The state space: X=WS
Process X
Local dynamics of the system: transition measures c
![Page 85: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/85.jpg)
85
Construction of IPS-II
Connection between the process and the rate function c:
Connection to the evolution of a profile function:
),(),( xtxtdt
d
)(),0( xmt
![Page 86: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/86.jpg)
86
Curvature Driven Flows
![Page 87: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/87.jpg)
87
Euclidean and Affine Flows
![Page 88: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/88.jpg)
88
Euclidean and Affine Flows
![Page 89: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/89.jpg)
89
Gauss-Minkowki Map
![Page 90: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/90.jpg)
90
Parametrization of Convex Curves
![Page 91: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/91.jpg)
91
Evolution of Densities
![Page 92: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/92.jpg)
92
Curve Shortening Flows
![Page 93: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/93.jpg)
93
Main Convergence Result
![Page 94: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/94.jpg)
94
Birth/Death Zero Range Processes-I
S: discrete torus TN, W=N
Particle configuration space: N TN
Markov generator:
)()()( 12 fLfLNLf o
![Page 95: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/95.jpg)
95
Birth/Death Zero Range Processes-II
Markov generator:
)()()( 12 fLfLNLf o
)](2)()())[((2
1)( 1,1,
0 fffigfL ii
Ti
ii
N
elsej
iijj
iijj
jii
),(
0)(,,1)(
0)(,11)(
)(1,
![Page 96: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/96.jpg)
96
Birth/Death Zero Range Process-III
Markov generator:
)()()( 12 fLfLNLf o
NTi
ii ffidffibfL )]()())[(()]()())[(()( ,,1
elsej
ijjji
)(
1)()(,
elsej
iijjji
)(
,0)(,1)()(,
![Page 97: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/97.jpg)
97
The Tangential Component is Important
![Page 98: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/98.jpg)
98
Curve Shortening as Semilinear Diffusion-I
![Page 99: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/99.jpg)
99
Curve Shortening as Semilinear Diffusion-II
![Page 100: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/100.jpg)
100
Curve Shortening as Semilinear Diffusion-III
![Page 101: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/101.jpg)
101
Nonconvex Curves
![Page 102: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/102.jpg)
102
Stochastic Interpretation-I
![Page 103: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/103.jpg)
103
Stochastic Interpretation-II
![Page 104: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/104.jpg)
104
Stochastic Interpretation-III
![Page 105: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/105.jpg)
105
Stochastic Curve Shortening
![Page 106: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/106.jpg)
106
Example of Stochastic Segmentation
![Page 107: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/107.jpg)
107
Stochastic Tracking
![Page 108: 1 Optimal Transport, Conformal Mappings, and Stochastic Methods for Registration and Surface Warping Allen Tannenbaum Georgia Institute of Technology Emory.](https://reader034.fdocuments.in/reader034/viewer/2022052702/56649e425503460f94b34191/html5/thumbnails/108.jpg)
108
ConclusionsStochastic Methods are attractive
alternative to level sets.
No increase in dimensionality.
Intrinsically discrete.
Robustness to noise.
Combination with other methods, e.g. Bayesian.