Sean I Young, PhD — Fully Convolutional Slice-to-Volume Reconstruction for Single-Stack MRI

Abstract

In magnetic resonance imaging (MRI), slice-to-volume

reconstruction (SVR) refers to computational reconstruction

of an unknown 3D magnetic resonance volume from stacks

of 2D slices corrupted by motion. While promising, current

SVR methods require multiple slice stacks for accurate 3D

reconstruction, leading to long scans and limiting their use

in time-sensitive applications such as fetal fMRI. Here, we

propose a SVR method that overcomes the shortcomings of

previous work and produces state-of-the-art reconstructions

in the presence of extreme inter-slice motion. Inspired by the

recent success of single-view depth estimation methods, we

formulate SVR as a single-stack motion estimation task and

train a fully convolutional network to predict a motion stack

for a given slice stack, producing a 3D reconstruction as a

byproduct of the predicted motion. Extensive experiments on

the SVR of adult and fetal brains demonstrate that our fully

convolutional method is twice as accurate as previous SVR

methods. Our code is available at github.com/seannz/svr.

1. Introduction

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Figure 1: Method overview. Inspired by monocular disparity estimation (a), we tackle slice-to-volume reconstruction (b) by predicting

slice

motion from a single stack of 2D slices using a fully convolutional network model. ?e intensities of the 2D slices are splatted using the slice

motion predicted by our network and the splatting is then interpolated to produce an artifact-free 3D MR reconstruction.!

(a) Monocular Depth (Disparity) Estimation [38–41] (b) Single-Stack Slice-to-Volume Reconstruction (Proposed)

Splat and

Interpolate

Standard

U-Net

Splat–Slice

U-Net

Single View Disparity Map Single Slice Stack Motion Stack 3D Reconstruction

Fully Convolutional Slice-to-Volume Reconstruction for Single-Stack MRI

Sean I. Young* Yaël Balbastre*

Harvard Medical School Harvard Medical School

siyoung@mit.edu ybalbastre@mgh.harvard.edu!

Bruce Fischl Polina Golland Juan Eugenio Iglesias

Harvard Medical School MIT Harvard Medical School

fischl@nmr.mgh.harvard.edu polina@csail.mit.edu jei@mit.edu

*?ese authors contributed equally to this work.

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2. Related Work

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2.1. Optimization Approaches

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2.2. Learning-Based Approaches

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Figure 2: Supervised learning. We train our splat–slice U-Net (a) on paired (slice stack, motion stack) examples, which are generated by

slicing 3D volumes with randomized slice-wise motion (motion stack). We additionally train a fully convolutional interpolation network (b)

on paired (splat volume, 3D volume) examples, where splat volumes are generated by splatting slice stacks with ground truth motion.

(a) Single-Stack Motion Estimation (b) Supervised Interpolation

Slice Stack

Splat Volume

3D Volume

𝐲

ℒ

Interpolator

𝐱

𝐲

ℒ

𝐱

3D Volume

Slice Stack

Motion Stack

Splat–Slice

U-Net

Slice

Splat

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2.3. Pre- and Post-Processing

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01,!-,01$<+!$/!B*5619!,0!*9J!NY>P!*&<!H*9,1(!,0!*9J!NYFPG!2$01!

$/!31(61!4+,!#[\!-$<,9+!NYQP!0$!8%,<(60!/$%,.%$4&<!-*+E+!

<(%,6097!$&!+9(6,+J!#$%!8$+0'8%$6,++(&.!$/!01,!%,6$&+0%460,<!

F?!)$94-,G!j4!,0!*9J!NYYP!8%$8$+,!&,4%*9!)$94-,!%,&<,%(&.!

+(-(9*%!0$!\,B#!NYRP!0$!+48,%'%,+$9),!%,6$&+0%460($&+!*0!*&!

*%2(0%*%7! %,+$940($&J! C&! 01(+! 3$%EG! 3,! *<<(0($&*997! 0%*(&! *&!

(&0,%8$9*0($&!&,03$%E!0$!8$+08%$6,++!01,!%*3!%,6$&+0%460($&+!

*&<!(&0,%8$9*0,!$),%!%,.($&+!$/!-(++(&.!(&0,&+(0(,+!01*0!-*7!

*88,*%!(&!01,!+(&.9,'+0*6E!%,6$&+0%460($&!6*+,G!8%$<46(&.!*&!

*%0(/*60'/%,,!%,6$&+0%460($&!/$%!$01,%!<$3&+0%,*-!0*+E+J!

3. Mathematical Preliminaries

3.1. Slice Acquisition Model

! C&!1(.1'%,+$940($&!F?!ABC!$/!+425,60+!,:1(2(0(&.!+,),%,!

4&6$&0%$99*29,!-$0($&G!/*+0!+9(6,!*6D4(+(0($&!+,D4,&6,+!+461!

*+!+(&.9,!+1$0!/*+0!+8(&!,61$!*%,!4+,<!0$!l/%,,X,m!01,!-$0($&!

(&'89*&,G!+42+0*&0(*997!-(0(.*0(&.!-$0($&!*%0(/*60+!6$-8*%,<!

0$!-490('+1$0!-,01$<+!NYSPJ!HIB!8%$6,<4%,+!*(-!0$!*9(.&!*&!

*6D4(%,<!+0*6E!$/!+9(6,+!01,%,27!%,-$)(&.!01,!%,-*(&(&.!=(&'

89*&,!*&<!01%$4.1'89*&,@!-$0($&!*6%$++!+9(6,+J!

! k,0!4+!<,&$0,!01,!6$$%<(&*0,+!(&+(<,!01,!ABC!+6*&&,%!27!

(𝑥, 𝑦, 𝑧)!*&<!*++4-,!*!+0*6E!𝑓!$/!>?!+9(6,+!(+!*6D4(%,<!*9$&.!

01,!𝑧!*:(+!3(01!+8*6(&.!𝑠!/%$-!*&!4&<,%97(&.!)$94-,!𝑣J!M,!

-$<,9!/$%!01,!*6D4(+(0($&!$/!01,!𝑘 01!AB!+9(6,!(+!N""G!"TP`!

𝑓

𝑘

(𝑥, 𝑦) = 𝜀 + ∫ (𝑣 ∘ 𝑢

𝑘

)(𝑥, 𝑦, 𝑧)ℎ(𝑠𝑘 − 𝑧) d𝑧

∞

−∞

="@!

/$%!𝑘 = 1, . . . , 𝐾G! 31,%,!𝑢

𝑘

: (𝑥, 𝑦, 𝑧) ↦ (𝑥, 𝑦, 𝑧)

′

!<,&$0,+!

*&!4&$2+,%),<!+9(6,')*%7(&.!%(.(<!+8*0(*9!+425,60!-$0($&G!ℎ!

<,&$0,+!01,!8$(&0!+8%,*<!/4&60($&!$/!+9(6,!*6D4(+(0($&G!*&<!𝜀!

6*804%,+!*99!AB'(&<46,<!,%%$%+!+461!*+!&$(+,G!2(*+G!,06J!

! g$!<(+6%,0(X,!-$<,9!="@G!+488$+,!𝑣!(+!+*-89,<!$&!*!642(6!

9*00(6,!$/!𝑁

!8$(&0+!*&<!𝑓

𝑘

!$&!*!>?!9*00(6,!$/!𝑁

!8$(&0+J!^,!

6*&!3%(0,!01,!<(+6%,0(X,<!-$<,9!=48!0$!%*&<$-!&$(+,@!*+!

𝐟 =

[

𝐟

; . . . ; 𝐟

𝐾

]

∈ ℝ

𝑁

𝐾

, 𝐟

𝑘

= 𝐇

𝑘

𝐔

𝑘

𝐯 ∈ ℝ

𝑁

=>@!

(&!31(61!𝐔

𝑘

!*&<!𝐇

𝑘

!%,8%,+,&0!<(+6%,0(X*0($&+!$/!(∘ 𝑢

𝑘

)!*&<!

ℎ(𝑠𝑘 − ⋅)G!%,+8,60(),97J!C&!01,!+-*99!-$0($&!%,.(-,G!3,!6*&!

/4%01,%!*88,*9!0$!g*79$%n+!*88%$:(-*0($&!*&<!<,a&,!

𝐔

𝑘

𝐯 = 𝐯 +

[

𝐝𝐢𝐚𝐠(𝐯

𝑥

) , 𝐝𝐢𝐚𝐠(𝐯

𝑦

) , 𝐝𝐢𝐚𝐠(𝐯

𝑧

)

]

𝐒𝐮

𝑘

=F@!

(&!31(61!𝐒𝐮

𝑘

%,8%,+,&0+!01,!,:8*&+($&!$/!01,!6$,L6(,&0+!$/!

-$0($&!𝐮

𝑘

(&!*!2*+(+!𝐒 *&<!𝐯

{𝑥,𝑦,𝑧}

!*%,!01,!%,+8,60(),!+8*0(*9!

<,%()*0(),+!$/!01,!F?!)$94-,J!^,!*66$--$<*0,!<,/$%-*29,!

-$0($&!-$<,9+!27!+,00(&.!𝐒!0$!01,!(<,&0(07!-*0%(:G!(&!31(61!

6*+,G!𝐮

𝑘

<,&$0,+!01,!),60$%(X*0($&!$/!*!F?!-$0($&!a,9<J!

3.2. Classical SVR Methods

! #$%! 2%,)(07G! 9,0! 4+! 3%(0,!𝐔 =

[

𝐇

𝐔

; . . . ; 𝐇

𝐾

𝐔

𝐾

]

!*&<!

01,!*++$6(*0,<!-$0($&!8*%*-,0,%+!*+!𝐮 =

[

𝐮

; . . . ; 𝐮

𝐾

]

J!^,!

6*&!3%(0,!01,!HIB!8%$29,-!3(01!01%,,!$%01$.$&*9!+0*6E+!*+!

minimize 𝐷(𝐮

{1,2,3}

, 𝐯

{1,2,3}

) =

‖

𝐔

𝐯

− 𝐟

‖

𝐔

𝐯

− 𝐟

‖

𝐔

𝐯

− 𝐟

‖

subject to 𝐯

− 𝐯

= 𝐯

− 𝐯

= 𝐯

− 𝐯

= 𝟎!

=Q@!

(&!31(61!𝐮

{1,2,3}

<,&$0,!01,!8*%*-,0,%+!$/!-$0($&!(&!,*61!$/!!

01,!+0*6E+J![$&0%*+0'(&)*%(*&0!9$++!/4&60($&+!+461!*+!-404*9!

(&/$%-*0($&!9$++!NYWP!*%,!*9+$!8$849*%97!4+,<!(&!89*6,!$/!01,!

D4*<%*0(6!$&,!+1$3&!1,%,!/$%!+(-89(6(07J!Z,%,G!3,!(&0%$<46,!

$80(-(X*0($&! )*%(*29,+!𝐯

{1,2,3}

!3(01! ,D4*9(07! 6$&+0%*(&0+! *+!

$88$+,<!0$!*!+(&.9,!)*%(*29,!𝐯!0$!/*6(9(0*0,!+$9)(&.!=Q@!4+(&.!

*90,%&*0(&.!$80(-(X*0($&!+0%*0,.(,+!+(-(9*%97!0$!N"SG!"TPJ!

! U!+(-89,!*90,%&*0(&.!$80(-(X*0($&!+0%*0,.7!31(61!<$,+!&$0!

(&0%$<46,!<4*9!)*%(*29,+!(+!0$!%,9*:!01,!,D4*9(07!6$&+0%*(&0+!0$!

01,!D4*<%*0(6!8,&*907!

𝑅

(

𝐯

{

1,2,3

}

)

‖[

𝐯

, 𝐯

]

−

[

𝐯

, 𝐯

]‖

=Y@!

*&<!$80(-(X,!01,!%,9*:*0($&!

𝐽(⋅) = 𝐷(𝐮

{1,2,3}

, 𝐯

{1,2,3}

) + 𝜆𝑅(𝐯

{1,2,3}

),!

=R@!

(&!31(61!𝜆!(+!*!8*%*-,0,%!6$&0%$99(&.!01,!%,9*0(),!3,(.10+!$/!

01,!<*0*!a<,9(07!0,%-!𝐷!*&<!01,!6$489(&.!0,%-!𝑅J!_25,60(),!

=R@!(&!/*60!6$%%,+8$&<+!0$!01,!$25,60(),!(-89(6(097!$80(-(X,<!

27!B$4++,*4!,0!*9J!N"TP!31,&!01,!-404*9!(&/$%-*0($&!9$++!(+!

4+,<!(&!01,!<*0*!a<,9(07!0,%-!𝐷J!_80(-(X(&.!=R@!4+(&.!29$6E!

6$$%<(&*0,!<,+6,&0!7(,9<+!48<*0,+!/$%!𝐯

, 𝐮

, 𝐯

, 𝐮

, 𝐯

!*&<!

𝐮

G!31,%,!01,!F?!%,6$&+0%460($&+!=𝑣'48<*0,@!*&<!01,!-$0($&!

8*%*-,0,%+!=𝑢'48<*0,@!$/!,*61!+0*6E!*%,!48<*0,<!(&!04%&!3(01!

01,!$01,%!03$!%,6$&+0%460($&+!a:,<!*0!,*61!04%&J!

! C&!01,!𝑛01!(0,%*0($&!$/!01,!29$6E!6$$%<(&*0,!<,+6,&0G!01,!𝑣'

48<*0,!+0,8!/$%!*!.(),&!+0*6E!*-$4&0+!0$!6$-840(&.!

𝐯

𝑛

← (𝐔

𝑛∗

𝐔

𝑛

+ 2𝜆𝐈)

−1

(𝐔

𝑛∗

𝐟

𝑛

+ 2𝜆𝐯







𝑛

),!

(&!31(61!𝐯

𝑛

!*&<!𝐟

𝑛

<,&$0,!01,!%,6$&+0%460($&!*&<!>?!+9(6,+!

$/!01,!𝑛%3%<!+0*6EG!%,+8,60(),97G!*&<!𝐯





𝑛

<,&$0,+!01,!*),%*.,!

%,6$&+0%460($&!/%$-!01,!03$!+0*6E+!$%01$.$&*9!0$!𝐯

𝑛

J!^,!6*&!

(&0,%8%,0!01(+!48<*0,!*+!*!%,6$&+0%460($&!$/!*!&,3!F?!)$94-,!

/%$-!*!3,(.10,<!+4-!$/!01,!9*+0!*),%*.,!F?!%,6$&+0%460($&!

*&<!3*%8,<!+9(6,+G!/$99$3,<!27!^(,&,%'<,6$&)$940($&J!M(+!

(&0,%8%,0*0($&!$/!01,!𝑣'48<*0,!3(99!.4(<,!$4%!6$&+0%460($&!$/!

*!/4997!6$&)$940($&*9!&,03$%E!9*0,%!(&!H,6!FJFJ!!

! U9.,2%*(6*997G!(𝐔, 𝐔

∗

) <,a&,!*&!*<5$(&0!8*(%!$/!3*%8(&.!

$8,%*0$%+G!31(61!3,!3(99!%,/,%!0$!*+!l+9(6(&.m!*&<!l+89*00(&.m!

%,+8,60(),97!NYTPJ!C&!01,!6*+,!31,%,!01,%,!(+!(&5,60(),!-$0($&!

%,9*0(&.!*!+9(6,!+0*6E!𝐟

𝑛

!0$!*!F?!)$94-,!𝐯G!3,!6*&!$20*(&!𝐟

𝑛

27!l+9(6(&.m!𝐯 3(01!𝐔

𝑛

!*&<!$20*(&!𝐯 27!l+89*00(&.m!𝐟

𝑛

!3(01!

𝐔

𝑛∗

J!#(.4%,!F!(994+0%*0,+!01,!03$!*<5$(&0!3*%8(&.!$8,%*0$%+!

(&!01,!<(+6%,0,!6*+,!31,&!-490('9(&,*%!(&0,%8$9*0($&!3,(.10+!

*%,!4+,<!/$%!/%*60($&*9!+9(6(&.!*&<!+89*00(&.J!

! M,!𝑢'48<*0,!/$%!01,!-$0($&!$/!01,!𝑘01!+9(6,!(&!*!+0*6E!(+!

𝐮

𝑘

𝑛

← (𝐒

∗

𝐕

𝑘

𝑛∗

𝐕

𝑘

𝑛

𝐒)

−1

𝐒

∗

𝐕

𝑘

𝑛∗

(𝐇

𝑘

𝐯

𝑛

− 𝐟

𝑘

𝑛

),!

=W@!

(&!31(61!𝐕

𝑘

𝑛

= 𝐇

𝑘

[

𝐝𝐢𝐚𝐠(𝐯

𝑥

𝑛

) , 𝐝𝐢𝐚𝐠(𝐯

𝑦

𝑛

) , 𝐝𝐢𝐚𝐠(𝐯

𝑧

𝑛

)

]

G 3(01!

01,!𝐯

{𝑥,𝑦,𝑧}

𝑛

<,&$0(&.!01,!%,+8,60(),!+8*0(*9!<,%()*0(),+!$/!01,!

.(),&!F?!)$94-,!%,6$&+0%460,<!(&!01,!(𝑛 − 1)01!(0,%*0($&J!

! C&!8%*60(6,G!+425,60!-$0($&!(&!*&!AB!+9(6,!9(E,97!,:6,,<+!

$&,!)$:,9!(&!-*.&(04<,!240!01,!9(&,*%!-$<,9!$/!3*%8(&.!=F@!

+0*7+!)*9(<!$&97!(/!01,!+9(6,!(+!a%+0!294%%,<!*0!01,!+6*9,!$/!01,!

-$0($&J!?4,!0$!01(+G!-$0($&!(+!$/0,&!,+0(-*0,<!$&!*!87%*-(<!

(&!*!6$*%+,'0$'a&,!-*&&,%G!4+(&.!01,!-$0($&!a,9<!,+0(-*0,<!

*0!6$*%+,%!9,),9+!0$!(&(0(*9(X,!-$0($&!*0!a&,%!$&,+!NRVPJ!ABC!

*6D4(+(0($&+!*9+$!+4e,%!/%$-!*!2(*+!a,9<!<4,!0$!&$&'4&(/$%-!

-*(&!-*.&,0(6!a,9<!*&<!%*<($/%,D4,&67!6$(9+G!(&0%$<46(&.!*!

.9$2*9!(&0,&+(07!+1(/0!2,03,,&!01,!4&E&$3&!)$94-,!*&<!01,!

+9(6,+J!U!E,7!61*99,&.,!(&!$80(-(X*0($&'2*+,<!HIB!-,01$<+!

(+!<,0,%-(&(&.G!*0!,*61!+6*9,G!-$0($&!(&!01,!8%,+,&6,!$/!2(*+!

*&<!,:0%,-,!+9(6,!*%0(/*60+J!A(+0*E(&.!,(01,%!/$%!-$0($&!6*&!

D4(6E97!9,*<!0$!8(0/*99+!$/!9$6*9!-(&(-*!(&!01,!$80(-(X*0($&!

$25,60(),!,+8,6(*997!*0!6$*%+,!+6*9,+J!Z$3,),%G!(0!(+!<(L6490!

0$!1*&<'6%*/0!*!%,6$&+0%460($&!8(8,9(&,!31(61!6*&!$),%6$-,!

(-*.,!&$&'6$&),:(0(,+!<4,!0$!+1,,%!&4-2,%+!$/!*6D4(+(0($&'

+8,6(a6!<,+(.&!61$(6,+!0$!2,!-*<,!=,J.JG!&$&'9(&,*%!a90,%+@J!

4. Fully Convolutional SVR

! K+(&.!*!#[\!-$<,9!/$%!HIB!,&*29,+!4+!0$!278*++!1*&<'

6%*/0,<!*6D4(+(0($&'+8,6(a6!%,6$&+0%460($&!8(8,9(&,+!*+!3,99!

*+!&4-,%(6*9!$80(-(X*0($&J!B*01,%!01*&!/$%-49*0(&.!HIB!*+!

01,!8%,<(60($&!$/!*2+$940,!+9(6,!6$$%<(&*0,+!(&!F?!+8*6,!*+!(+!

078(6*997!<$&,!(&!$01,%!9,*%&(&.'2*+,<!-,01$<+!N>WOF>PG!3,!

6*+0!HIB!*+!01,!%,.(+0%*0($&!$/!+$-,!4&$2+,%),<!F?!)$94-,!

0$!*&!$2+,%),<!+0*6E!$/!>?!+9(6,+J!^,!0%*(&!*!#[\!-$<,9!0$!

8%,<(60!-$0($&!%,9*0(&.!01,!+9(6,+!0$!*!F?!)$94-,!.(),&!$&97!

01,!+9(6,+!*+!(&840G!8%$<46(&.!*!F?!)$94-,!*+!*!27'8%$<460!

$/!%,.(+0%*0($&J![$&6,804*997G!01(+!(+!+(-(9*%!0$!01,!8%$29,-!

$/!-$&$649*%!<,801!,+0(-*0($&!NQVOQFPG!31,%,!$&,n+!.$*9!(+!

0$!8%,<(60!<(+8*%(07!=-$0($&!*9$&.!01,!,8(8$9*%!9(&,@!%,9*0(&.!

*!+(&.9,!>?!(-*.,!0$!01,!<,801!$/!01,!4&<,%97(&.!F?!+6,&,J!

4.1. Neural Network Architecture

! [\\!*%61(0,604%,+!+461!*+!01$+,!4+,<!(&!+,-*&0(6!(-*.,!

+,.-,&0*0($&!6*&!8%$<46,!+42$80(-*9!$406$-,+!/$%!-$0($&!

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0,:04%,!NFFOFTPJ!^1(9,!/4997!6$&)$940($&*9!&,03$%E+!+461!*+!

01,!K'\,0!NYQP!*%,!,&<$3,<!3(01!*!-490('+6*9,!*%61(0,604%,!

*&<!6$49<! 01,$%,0(6*997!,+0(-*0,!-$0($&!(&!*! 6$*%+,'0$'a&,!

-*&&,%G!4+(&.!6$&)$940($&!E,%&,9+!0$!8*%*-,0,%(X,!3*%8(&.!

Figure 3: Slicing and splatting. In slicing (top row), the endpoints

of motion vectors deﬁne coordinate locations in a moving image to

pull data from and construct an image. In splatting (bottom), the

same motion vector endpoints deﬁne coordinate locations in a ﬁxed

image to push data to. Multi-linear weights are used for fractional

splatting and slicing, guaranteeing diﬀerentiability w.r.t grid data.

Moving Image Motion Field Slicing Sliced Image

1¼

1½

Fixed Image Motion Field Splatting Splat Image

Slice (Warp)

Splat (Warp

)

=+8*6,')*%7(&.!0%*&+/$%-+@!(+!(&1,%,&097!(&,L6(,&0G!2$01!(&!

0,%-+!$/!8*%*-,0%(X*0($&!*&<!6$-840*0($&o!+,,G!,J.JG!NFFG!FTPJ!

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4+,<!(&!8*(%3(+,!(-*.,!%,.(+0%*0($&!N>WOF>PG!31(61!8%,<(60!

(-*.,!-$0($&!27!8%$6,++(&.!01,!(-*.,!8*(%!*6%$++!-490(89,!

9,),9+!$/!6$&)$940($&+J!M,+,!&,03$%E!-$<,9+!*%,!078(6*997!

,D4(88,<!3(01!3*%8(&.!9*7,%+!2,03,,&!*<5*6,&0!9,),9+!+461!

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9,),9J!\$0($&*997G!01,!+9(6,!+0*6E!*++4-,+!01,!%$9,!$/!*!a:,<!

(-*.,!31(9,!01,!F?!)$94-,G!31(61!3,!+,,E!0$!,+0(-*0,G!6*&!

2,!+,,&!*+!*!-$)(&.!(-*.,J!U+!<(+64++,<!(&!H,6J!FJ>G!3,!6*&!

$20*(&!01,!/,*04%,+!$/!01,!>?!+9(6,+!=a:,<!(-*.,@!27!+9(6(&.!

01,!/,*04%,+!$/!01,!F?!)$94-,!3(01!01,!,+0(-*0,<!-$0($&J!g$!

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Figure 4: Splat–Slice U-Net. ?e stack of 2D slices goes through

2D feature extraction (downward path) and reconstruction (upward

path) with 2D skip features. 3D volume features are reconstructed

by splatting 2D skip features with previous motion to form 3D skip

features. At each level, the sliced 3D feature volume and the slice

feature stack are convolved jointly to extract residual slice motion.!

Splat

Concat

Splat

Concat

2D Feat

3D Feat

Motion

Conv

Add

Slice-Conv

Figure 5: Rigid motion-compensated loss. Validation MSE and

EPE (end-point error) [33] of the predicted motion shown. Our loss

(blue), which compensates for oﬀsets in global rigid motion, trains

faster and attains higher ﬁnal prediction accuracy than the MSE loss

(orange) and the loss that compensates for translations only (gray). !

Rigid-Comp

No Comp

Trans-Comp

Motion MSE (voxels)

Motion EPE (voxels)

Rigid-Comp

No Comp

Trans-Comp

Epochs

Y!89$0+!01,!)*9(<*0($&!64%),+!/$%!$4%!&,03$%E!-$<,9!0%*(&,<!

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4.3. Interpolating Reconstructions

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5. Experimental Results

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=/$%!31(61!.%$4&<!0%401!AB!)$94-,+!*%,!*)*(9*29,@!*&<!/,0*9!

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NFFP!0$!-,*+4%,!8%,<(60($&!,%%$%o!+,,!U88,&<(:!UJ!

5.1. Single-Stack SVR of Adult Brains

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Slicing Directions and Reconstruction. H(&.9,'+0*6E!HIB!

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(&6%,*+,!(&!01,!8%,<(60($&!,%%$%!/$%!*:(*9!+0*6E+!-$+0!9(E,97!

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+9(6,+G!9,*<(&.!0$!*!9,++!8%,6(+,!*9(.&-,&0J!

Motion Prediction Error.!^,! 89$0!<(+0%(240($&+!$/!01,!8,%'!

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*901$4.1!01,!hhAC!<*0*+,0!,:1(2(0+!+9(.1097!1(.1,%!,%%$%+J!

Reconstruction and Interpolation.!#(.4%,!W!)(+4*9(X,+!01,!

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Figure 6: Accuracy of predicted motion. Slicing direction has an

impact on the accuracy of predicted motion. We plot the validation

MSE (left), and EPE (end-point error, right) of the predicted motion

on adult brain slices for sagittal, axial, and coronal acquisitions.

Sagittal

Axial

Coronal

Motion MSE (voxels)

Motion EPE (voxels)

Sagittal

Axial

Coronal

Epochs

Motion MSE (voxels)

All AB2 AB ADH B39 CO GSP MC OAS PP UKB

Figure 7: Motion accuracy across datasets. We plot the MSE of

the motion stacks predicted on 500 held-out slice stacks (1mm

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Axial

Coronal

5.2. Single-Stack SVR of Fetal Brains

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*6D4(+(0($&+!/$%!31(61!&$!.%$4&<!0%401!,:(+0+!*&<!-$0($&!(+!

+-*99,%J!K&9(E,!HIB&,0G!01,!8%$8$+,<!-,01$<!.,&,%*9(X,+!0$!

%,*9!*6D4(+(0($&+!,),&!31,&!01,!&,03$%E!-$<,9!(+!0%*(&,<!$&!

+7&01,0(6!+9(6,!+0*6E+!3(01!<(e,%,&0!-$0($&!*&<!*6D4(+(0($&!

8*%*-,0,%+G!,J.JG!hH#J!H,,!U88,&<(:!]!/$%!-$%,!,:*-89,+J!

Quantitative Results.!g*29,! "!9(+0+!01,!8%,<(60($&!*664%*67!

$/!<(e,%,&0!-,01$<+!$&!)*9(<*0($&!+425,60+!3(01!Q!/$9<+J!C&!

*<<(0($&!0$!-$0($&!phpG!3,!%,8$%0!01,!+9(6,!hH\BG!$20*(&,<!

27!+9(6(&.!01,!0%4,!)$94-,!3(01!01,!8%,<(60,<!-$0($&G!*&<!01,!

hH\B!$/!01,!a&*9!%,6$&+0%460($&J!M,!01%,,'+0*6E!%,+490+!*%,!

/%$-!N>YPG!31(61!9(+0+!01,!*&61$%!8$(&0!,%%$%+!=Uhp@J!C&!01,!

+(&.9,'+0*6E!6*+,G!$4%!-,01$<!%,<46,+!,%%$%!(&!01,!8%,<(60,<!

-$0($&!27!SSJRs!*&<!QQJRs!%,9*0(),!0$!01,!HIB&,0!N>VP!*&<!

HI$Bg!=)>@!8%,<(60($&+G!%,+8,60(),97J!M,!+9(6,!*&<!)$94-,!

hH\B!1*),!*9+$!(-8%$),<!240!4&9(E,!01,!phpG!01,+,!-,0%(6+!

<,8,&<!$&!01,!hH#!$/!*6D4(+(0($&!*&<!%,6$&+0%460($&!*&<!<$!

&$0!<,a&(0(),97!-,*+4%,!*9(.&-,&0!*664%*67J!

ADHD200 8628223

ABIDE 50975

MCIC A00036476

Buckner39 990921

(a) Slice Stack

(b) Splat (Ours, True Motion)

(d) True Volume

Figure 8: SVR of adult brain scans. <$"6+#219+=$"*2."8>?".$#29%#"*,"#17+%%19"@.*A#"4BC'!"1&+19"@.*A"D'!"1,/"0*.*,19"@91#%".*A'"#9+0$"#%10E#"

#F,%;$#+=$/"2#+,7".1,/*5"#9+0$"5*%+*,"@1'("G#+,7"%;$"5*%+*,"#%10E"-.$/+0%$/":F"*2.",$%A*.E!"A$"#-91%"#9+0$"/1%1"%*".$0*,#%.20%"%;$"2,/$.9F+,7"

DH"6*925$"@:'("G#+,7"*2."-.$I%.1+,$/"+,%$.-*91%*.!"A$"%;$,"+,%$.-*91%$"%;$"5+##+,7"+,%$,#+%+$#"@;*9$#'"+,"*2.".$0*,#%.20%+*,"@0'("J$".$#29%"+#"

#+5+91."%*"%;$"%.2$"DH"6*925$"@/'("<$"1//+%+*,199F"6+#219+=$"+,"@:'"1,/"@0'"%;$"#-91%"1,/"+,%$.-*91%$/".$#29%#"A;$,"%;$"%.2$"5*%+*,"+#"2#$/(""

Sagittal Acquisition EPE: 1.66mm EPE: 0.00mm Sagittal View

Axial Acquisition EPE: 2.20mm EPE: 0.00mm Coronal View

Sagittal Acquisition EPE: 2.04mm EPE: 0.00mm Axial View

Coronal Acquisition EPE: 3.36mm EPE: 0.00mm Sagittal View

6. Discussion

Limitations.!_4%!/,0*9!HIB!&,03$%E!(+!64%%,&097!0%*(&,<!$&!

*40$-*0(6*997!+,.-,&0,<!2%*(&!+9(6,+J!g%*(&(&.!3(01!$%(.(&*9!

4&+,.-,&0,<!+9(6,!+0*6E+!6*&!(-8%$),!01,!%$24+0&,++!$/!$4%!

*88%$*61!*.*(&+0!(-8,%/,6097!.,&,%*0,<!+,.-,&0*0($&+J!

Future Work.!c(),&!01*0!$4%!*88%$*61!8%,<(60+!<,&+,!+9(6,!

-$0($&!/$%!%,6$&+0%460($&G!3,!89*&!0$!,:0,&<!$4%!*88%$*61!0$!

8%$29,-+!31,%,!01,!*6D4(%,<!+9(6,+!6*&!4&<,%.$!<,/$%-*29,!

-$0($&;,J.J!(&!/,0*9!0$%+$!$%!89*6,&0*9!%,6$&+0%460($&!N"WP;

*&<!6*&&$0!2,!0*6E9,<!4+(&.!%(.(<!-$0($&!HIBJ!U9+$G!2,00,%!

(&0,%8$9*0($&!&,03$%E+G!+461!*+!01$+,!01*0!*%,!,D4()*%(*&0!0$!

%(.(<!-$0($&G!6*&!/4%01,%!(-8%$),!%,6$&+0%460($&o!+,,!NSQPJ!!

7. Conclusion

C&!2%*(&!(-*.(&.G!+9(6,'0$')$94-,!%,6$&+0%460($&! =HIB@!

(+!*&!(-8$%0*&0!6$-840*0($&*9!0,61&(D4,!/$%!01,!(-*.(&.!$/!

+425,60+!31$+,!-$0($&!6*&&$0!,*+(97!2,!6$&0%$99,<J!i,0G!01,!

-*5$%(07!$/!HIB!0,61&(D4,+!8%,)($4+97!8%$8$+,<!<$!&$0!%,*8!

01,!2,&,a0+!$/!#[\!-$<,9+!01*0!1*),!.%*<4*997!2,6$-,!01,!

-*(&+0*7!(&!69$+,97!%,9*0,<!0*+E+G!+461!*+!(-*.,!%,.(+0%*0($&!

*&<!+,.-,&0*0($&J!M,!/,3!9,*%&(&.'2*+,<!HIB!0,61&(D4,+!

%,+$%0!0$!8%,0%*(&,<!C-*.,\,0!69*++(a6*0($&!*&<!,),&!)(+($&!

0%*&+/$%-,%!-$<,9+J!\$0!$&97!<$,+!$4%!3$%E!a99!01,!.*8!(&!

01,!HIB!-,01$<!<,),9$8-,&0!0(-,9(&,!240!(0!*9+$!+1$3+!01*0!

*!#[\!-$<,9!6*&!8%$<46,!+0*0,'$/'01,'*%0!HIB!$406$-,+J!

Acknowledgments. HCi!01*&E+!A*%.1,%(0*!#(%,&X,!/$%!1,%!

01$4.10'8%$)$E(&.!D4,+0($&+J!h%(-*%7!+488$%0!.(),&!27!01,!

\CZ!.%*&0+!dTTUcVW"QTF!*&<!B#"AZ">F"TYJ!H488$%0,<!

27!\CZ!BV"UcVRQV>SG!B>"UcVW>VW>!*&<!hQ"p]VFVVVRJ!

FeTA Sub 073

FeTA Sub 053

MIAL001

Run

MIAL001

Run 1

(a) Slice Stack

(b) SVRnet

†

[20]

(d) Interp (Ours)

(f) True Volume

Figure 9: Single-stack fetal SVR. <$"6+#219+=$"%;$"8>?".$#29%#"*,"619+/1%+*,"#2:K$0%#"L.*5"%;$"M$NO"/1%1#$%"PQDR"1,/"%A*".$19"10S2+#+%+*,#""

L.*5")TOU"P4VR("W**5$/"+,"X&"L*.":$%%$."6+#+:+9+%F("Y2.".$#29%#"09*#$9F".$#$5:9$"%;$"7.*2,/"%.2%;"6*925$#"A;+9$"8>*?N6C"1,/"8>?,$%"

.$0*,#%.20%+*,#"@A+%;"*2."+,%$.-*91%+*,'"$&;+:+%"#-1%+19"/+#%*.%+*,"L.*5"+,1002.1%$"#9+0$"19+7,5$,%("

†

Y2."+5-9$5$,%1%+*,Z"#$$"O--$,/+&"O(""

Random Acquisition EPE: 8.69mm EPE: 3.70mm EPE: 1.49mm Pathological

Sagittal Acquisition EPE: Unknown EPE: Unknown EPE: Unknown

Random Acquisition EPE: 5.72mm EPE: 2.61mm EPE: 1.45mm Neurotypical

Sagittal Acquisition EPE: Unknown EPE: Unknown EPE: Unknown

Unknown

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1.81

0.40

23.69c1.39

23.43c1.40

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PCdR("N+5$/"*,"?Ngbaaa("

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01,!*&61$%!8$(&0+!$/!01,!.%(<+!01*0!<,a&,!01,!)$:,9!9$6*0($&+!

$/!01,!%,+8,60(),!+9(6,+!(&!F?!+8*6,J!g78(6*997G!*&61$%!8$(&0+!

*%,!*++4-,<!0$!2,!*0!01,!6,&0,%G!2$00$-!9,/0!*&<!2$00$-!%(.10!

6$%&,%+!$/!*!.(),&!+9(6,J!U++4-(&.!01*0!+9(6,+!*%,!4&<,%.$(&.!

%(.(<!-$0($&G!01,!Uhp!(+!,D4()*9,&0!0$!01,!php!*),%*.,<!$&!

01,!%(.10'0%(*&.49*%!%,.($&!/$%-,<!27!01,!01%,,!*&61$%!8$(&0+!

$&!01,!%,/,%,&6,!+9(6,J!

A.3 SVRnet Model Implementation

! #$%!%,8%$<46(2(9(07G!3,!8$%0!01,!$%(.(&*9!g,&+$%#9$3!"J"F!

(-89,-,&0*0($&!$/!HIB&,0!N>VP!3(01!*&!C&6,80($&!2*6E2$&,!

0$!h7g$%61!"J"FG!31,%,!3,!4+,!*!B,+\,0'FQ!2*6E2$&,!*&<!*!

8%,<(60($&!1,*<! 6$&+(+0(&.!$/! *!512 × 9!<,&+,!9*7,%G!31(61!

8%,<(60+!01,!+9(6,!8$+(0($&!),60$%+!*0!01%,,!*&61$%!8$(&0+J!^,!

a&<!01*0!>?!2*061!&$%-*9(X*0($&!2*+,<!$&!6$99,60,<!+0*0(+0(6+!

<$,+!&$0!8,%/$%-!3,99!*0!0,+0!0(-,!*&<!$80!0$!&$%-*9(X,!,*61!

,:*-89,!2*+,<!$&!01,!+0*0(+0(6+!$/!,*61!+9(6,!+0*6EJ!^,!4+,!

+42+*-89,<!+9(6,+!$/! +(X,!128 × 128!8(:,9+! *&<! (&0,%8$9*0,!

01,!8%,<(60,<!*&61$%!8$(&0!8$+(0($&!),60$%+!0$!*!9(&,*%!-$0($&!

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01,!-$<,9!3(01!01,!0$%61)(+($&!C-*.,\,0"dt)"!3,(.10+J!

A.4 SVoRT Model Conﬁguration

! #$%!6$-8*%(+$&!3(01!HI$Bg!=)>@G!3,!4+,!-$<,9!3,(.10+!

8%$)(<,<!27!j4!,0!*9J!N>YP!$&!01,(%!%,8$+(0$%7G!6$&a.4%,!01,!

-$<,9!0$!4+,!$&,!+9(6,!+0*6E!3(01!*!+9(6,!.*8!$/!FJ>--G!*&<!

$80(-(X,!01,!%,6$&+0%460($&!hH#!=+9(6,!01(6E&,++!$/!"JR--@!

/$%!)*9(<*0($&!*664%*67!=(J,JG!*),%*.,!-$0($&!,&<'8$(&0!,%%$%@!

$&!01,!">!#,gU!)*9(<*0($&!+425,60+!4+(&.!*&!,:8$&,&0(*9!.%(<!

+,*%61J!^,! a:,<! 01,! +0*6E! 8$+(0($&*9! ,&6$<(&.! $/! HI$Bg!0$!

V!=%*01,%!01*&!*!%*&<$-!(&0,.,%@!/$%!%,8%$<46(29,!%,+490+J!^,!

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6$-840,!01,!-$0($&!AHp!*&<!php!/$%!)*9(<*0($&!*664%*67J!

B. Additional SVR Results

! Z,%,G!3,!8%$)(<,!*<<(0($&*9!HIB!%,+490+!$&!$4%!*<490!*&<!

/,0*9!<*0*+,0+J!#(.4%,!]"!)(+4*9(X,+!%,6$&+0%460($&+!$/!*<490!

2%*(&!AB!)$94-,+!/$%!01%,,!$/!$4%!)*9(<*0($&!+425,60+G!3(01!

*99!01%,,!$%01$.$&*9!)(,3+!+1$3&!/$%!6$-89,0,&,++J!#(.4%,!

]>!+(-(9*%97!)(+4*9(X,+!/,0*9!%,6$&+0%460($&+J!^,!(&694<,!01,!

6$%%,+8$&<(&.!HI$Bg)>!%,+490+!/$%!6$-8*%(+$&G!&$0(&.!01*0!

HIB&,0!%,6$&+0%460($&+!*%,!.*%29,<!(&!-*&7!6*+,+!=+,,!a%+0!

%$3!$/!#(.4%,!T@!*&<!*%,!9,++!-,*&(&./49!0$!6$-8*%,!*.*(&+0J!

"F!

ABIDE 50975

(a) Slice Stack

(b) Splat (Ours, True Motion)

(d) True Volume

MCIC A00036476

(a) Slice Stack

(b) Splat (Ours, True Motion)

(d) True Volume

Coronal Acquisition EPE: 2.00mm EPE: 0.00mm

Sagittal Acquisition EPE: 1.70mm EPE: 0.00mm

Axial Acquisition EPE: 2.35mm EPE: 0.00mm

"Q!

Buckner39 990921

(a) Slice Stack

(b) Splat (Ours, True Motion)

(d) True Volume

Figure B1: SVR of adult brain scans.

<$" 6+#219+=$" *2." 8>?" .$#29%#" *," #9+ 0$" #%10E#" #F,%;$#+=$/" 2#+ ,7" .1,/*5" #9+0$" 5*%+*," @1'(" G#+,7" %;$"

-.$/+0%$/"5*%+*,"#%10E!"A$"#-91%"#9+0$"/1%1"%*".$0*,#%.20%"%;$"2,/$.9F+,7"DH"6*925$"@:'("<

$"+,%$.-*91%$"%;$"5+##+,7"+,%$,#+%+$#"@;*9$#'"+,"*2."

.$0*,#%.20%+*,"@0'("<$"1//+%+*,199F"6+#219+=$"+,"@:'"1,/"@0'"#-91%"1,/"+,%$.-*91%$/".$#29%#"*:%1+,$/"A;$,"%;$"%.2$"5*%+*,"#%10E"+#"2#$/("

FeTA Sub 0

(a) Slice Stack

(b) Splat (SVoRTv2, Ours)

(d) Interp (Ours)

(d) True Volume

FeTA Sub 010

Random Acquisition EPE: 3.51mm EPE: 1.97mm

Random Acquisition EPE: 4.78mm EPE: 1.64mm

"Y!

(a) Slice Stack

(b) Splat (SVoRTv2, Ours)

(d) True Volume

FeTA Sub 036

(a) Slice Stack

(b) Splat (SVoRTv2, Ours)

(d) Interp (Ours)

(d) True Volume

FeTA Sub 0

(a) Slice Stack

(b) Splat (SVoRTv2, Ours)

(d) Interp (Ours)

(d) True Volume

Figure B2: Single-stack fetal SVR. <$" 6+#219+=$"%;$"8>?".$#29%#"*,"619+/1%+*,"#2:K$0%#"L.*5"%;$"M$NO"/1%1#$%"PQDR("Y2."

.$#29%#"09*#$9F"

.$#$5:9$"%;$"7.*2,/"%.2%;"6*925$#"A;+9$"8>*?N6C".$0*,#%.20%+*,#"@A+%;"*2."+,%$.-*91%+*,"1--9+$/'"$&;+:+%"#-1%+19"/+#%*.%+*,"L

.*5"+,1002.1%$"

#9+0$"19+7,5$,%("

Random Acquisition EPE: 2.40mm EPE: 1.25mm

Random Acquisition EPE: 3.40mm EPE: 1.91mm