A new Algorithm for the Computation of the group logarithm ...

A new Algorithm for the Computation of the group logarithm of Diffeomorphism

Journal CLUB 4 December 2014

Mathias Bossa and Salvador Olmos

Presenting:


Journal CLUB 4 December 2014

Mathias Bossa and Salvador Olmos

Presenting:

Sebastiano Ferraris- [email protected] - University College London 3

I : RD � ⌦ �! R

Reference Image Floating Image

⌦

⌦

0

R R

R F

'

Id

8x 2 ⌦ R(x) = F ('(x)) ' 2 Di↵(⌦,⌦0)'

Sim(R,F,') =X

x2⌦

||R(x)� F ('(x))||2

Group of

Diffeomorphism


'

log

exp


↵�

S1 = {e↵i | ↵ 2 (�⇡,⇡)}

↵u

�uS1

T0S1 ' R

e : T0S1 �! S1

↵ 7�! e↵i

log : S1 �! T0S1

e↵i 7�! log(e↵i) = ↵


p

x u

log

exp

…Ok, but why!?!

Log-Euclidean Framework

log ! ISS

exp ! SS

Tx

Di↵

Di↵


p1, p2 2 Di↵Tx

Di↵

Di↵

u1 u1,u2 2 Lie(Di↵)

p1p2

dist(p1, p2) = ||log(p1)� log(p2)||= ||u2 � u1||

x

Log-Euclidean Framework to compute statistics on diffeomorphism

�u1

(p1)�1

u1 = log(p1)

�u1 = log(p1�1

)

u2


Log-Euclidean Framework to compute compositions of diffeomorphismfor image registration algorithms

x

pi

pi

pi+1

�uiui

ui+1

ui+1 = BCH(ui, �ui)

= log(exp (ui) � exp (�ui))

BCH(u, �u) = u+ �u+1

2[u, �u] +

1

12([u, [u, �u]] + [�u, [�u,u]])� 1

24([. . .

Tx

Di↵ pi+1 = pi � pi

Di↵


Log-Euclidean Framework to compute statistics on diffeomorphism

Log-Euclidean Framework to compute compositions of diffeomorphismfor image registration algorithms

u = log(p)

Given p 2 Di↵, we want u 2 Lie(Di↵) such that

Given u1,u2 2 Lie(Di↵), we want u 2 Lie(Di↵) such that

u = log(exp(u1) � exp(u2)) = BCH(u1,u2)

u = log(p1 � p2) = BCH(u1,u2)


Theorem:BCH(u, �u) = u+ �u+ 1

2 [u, �u] + 112 ([u, [u, �u]] + [�u, [�u,u]])� 1

24 ([. . .

Given p 2 Di↵, the iterative method

(u0 = 0

un = BCH(un�1, �un�1)

where

�un�1 = exp(�un�1) � p� Id

converges to u = log(p) with error En 2 Lie(Di↵), given by

En := log(exp(u) � exp(�un)) 2 O(||p� Id||2n

)


Be ui 2 Lie(Di↵) near u then:

p = exp(u) = exp(ui) � (exp(�ui) � p)

We define �ui 2 Lie(Di↵) as �ui = exp(�u0) � p. Then

p = exp(ui) � exp(�ui)

exp(u) = exp(ui) � exp(�ui)

u = log(exp(ui) � exp(�ui))

u = BCH(ui, �ui)

We approach u using an iterative algorithm based on

exp(�ui) ⇡ Id + �ui =) �ui ⇡ exp(�ui)� Id

Having ui as our initial value we define

�ui := exp(�ui)� Id

Using p = exp(ui) � exp(�ui) we can say that exp(�ui) = exp(�ui) �p and then

�ui = exp(�ui) � p� Id

just by definition. Since p is known we can start our successive approximation,and if we set ui = we have the result.

Proof:



Now …Results!


(u = random vector field

p = expBE(u)

First set of results from toy examples: Ground Truth:

(upper limit, non iterative)uapp ' p� Id

ISS

Series Log BCH-0

Series Log BCH-1

Series Log BCH-2

EDi↵ = p� expBE(uapp)

eLie(Di↵) = u� uapp

The uapp = log(p) was evaluated in 5 di↵erent ways:

Solid lines

Dashed lines



Second set of results from 3D MRI data:



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Pag. 21:



The End

A new Algorithm for the Computation of the group logarithm ...

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