A icle
E icien Low-Resou ce Comp ession o HIFU Da a
Pe Klepa nik ∗ID , Da id Ba ina ID , Pa el Zemcik ID and Ji i Ja os ID
Cen e o Excellence IT4Inno a ions, Facul y o In o ma ion Technology, B no Uni e si y o Technology,
Boze echo a 1/2, 612 66 B no, Czech Republic; [email p o ec ed].cz (D.B.); [email p o ec ed].cz (P.Z.);
[email p o ec ed].cz (J.J.)
*Co espondence: [email p o ec ed].cz
Recei ed: 10 May 2018; Accep ed: 24 June 2018; Published: 26 June 2018

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Abs ac :
La ge-scale nume ical simula ions o high-in ensi y ocused ul asound (HIFU), impo an
o model-based ea men planning, gene a e la ge amoun s o da a. Typically, i is necessa y o sa e
hund eds o gigaby es du ing simula ion. We p opose a no el algo i hm o ime- a ying simula ion
da a comp ession specialised o HIFU. Ou app oach is pa icula ly ocused on on- he- ly pa allel
da a comp ession du ing simula ions. The algo i hm is able o comp ess 3D p essu e ime se ies
o linea and non-linea simula ions wi h e y accep able comp ession a ios and e o s (o e 80%
o he space can be sa ed wi h an accep able e o ). The p oposed comp ession enables signi ican
educ ion o esou ces, such as s o age space, ne wo k bandwid h, CPU ime, and so o h, enabling
be e ea men planning using as olume da a isualisa ions. The pape desc ibes he p oposed
me hod, i s expe imen al e alua ion, and compa isons o he s a e o he a s.
Keywo ds:
comp ession; ul asound simula ion; high-in ensi y ocused ul asound; k-Wa e oolbox
1. In oduc ion
The applica ion o high-in ensi y ocused ul asound (HIFU) in human medicine is an eme ging
non-in asi e pe spec i e he apy. The ocused beam o ul asound, ypically gene a ed using a la ge
ansduce , is sen in o he issue. The cells o issue in a localised egion a e apidly hea ed, which
causes issue dea h while he su ounding issue is le unha med. In ecen yea s, many HIFU clinical
ials o he ea men o umou s in he p os a e, kidney, li e , b eas , o b ain ha e been pe o med.
The mos impo an issue is he p ecise placemen o he ul asound ocus and dosage assessmen [
1
].
The e a e some issue p ope ies ha can signi ican ly dis o ul asound dis ibu ion, o example, in
he skull. This is he main eason ha la ge-scale nume ical HIFU simula ions a e needed.
Howe e , wo key challenges mus be add essed. Fi s , i is necessa y o c ea e an acous ic
and he mal model ha is physically complex due o a he e ogeneous medium and non-linea wa e
p opaga ion. Second, he simula ions a e compu a ionally e y in ensi e and expensi e as hey mus
be execu ed on la ge domains wi h billions o g id poin s [
2
]. Al hough he compu a ional equi emen s
we e alle ia ed by a dis ibu ed mul i-g aphics p ocessing uni (GPU) implemen a ion [
1
,
2
], he amoun
o gene a ed da a quickly becomes unmanageable; mo eo e , he inpu /ou pu (I/O) ope a ions
become he domina ing ac o o he simula ion un ime [1,3].
Because o la ge dis ances a elled by he ul asound wa es ela i e o he wa eleng h o
he highes ha monic equency, and in o de o ob ain p ecise esul s and applica ions in medical
ea men s, e y la ge simula ions ha e o be pe o med. A he ime being, he esolu ions o he
ul asound simula ion g id each up o 4096
×
2048
×
2048 samples in 3D space. Usually, he limi ed
numbe o ime s eps in a small a ea encompassing he ocus (senso mask) o a discon inuous
ield o poin senso s ha e o be sampled and sa ed o u he p ocessing, o example, in he mal
modelling [
1
,
3
]. The ypical size o sampled da a o a clinically applicable simula ion eaches 0.5 TiB.
In o ma ion 2018,9, 155; doi:10.3390/in o9070155 www.mdpi.com/jou nal/in o ma ion
In o ma ion 2018,9, 155 2 o 14
In his pape , we p esen a no el comp ession me hod o ime- a ying HIFU simula ion da a
(acous ic p essu e and eloci y). Ou app oach is ocused on pa allel on- he- ly da a comp ession
du ing dis ibu ed simula ion, in which e e y g id poin o in e es in 3D space is p ocessed sepa a ely.
We comple ed an expe imen al implemen a ion o he comp ession me hod o o line da ase
p ocessing. We pe o med se e al expe imen s on da ase s ep esen ing clinical si ua ions wi h
he e ogeneous issue. The esul s showed e y p omising comp ession a ios wi h low da a dis o ion.
The es o his pape is o ganised as ollows. Sec ion 2b ie ly p esen s he exis ing comp ession
me hods. Sec ion 3p esen s he p oposed me hod. The subsequen Sec ions 4and 5discuss ou cu en
implemen a ion and he expe imen s pe o med. Finally, Sec ion 6concludes he pape .
2. Rela ed Wo k
In gene al, sophis ica ed comp ession me hods o a ious applica ion da a ypes exis . Howe e ,
no me hod has been ailo ed o HIFU simula ion da a. Mos o he ela ed pape s ocus on he
comp ession o da a ob ained by gene al ul asonic imaging.
Th ee-dimensional ul asound compu ed omog aphy (USCT) da a can be educed wi h a
comp ession me hod based on he disc e e wa ele ans o m [
4
]. Many echniques ha e been applied
o p ocess one-dimensional (1D) ul asonic signals, o example, ma ching pu sui , he 1D disc e e
cosine ans o m (DCT), and he Walsh–Hadama d ans o m (WHT) [
5
,
6
]. Radio- equency (RF)
signals a e analog- o-digi al con e sion be o e beam o ming a e comp essed wi h echniques such as
peak ga es, ace comp ession, la ges a ia ion (LAVA), and linea p edic i e coding (LPC), o also
wi h he well-known ZIP, JPEG, o MPEG me hods [
7
,
8
]. Some o he abo e-men ioned echniques
can be applied o HIFU simula ion da a; howe e , hey a e no ocused on on- he- ly pa allel da a
comp ession du ing la ge-scale simula ions. A ele an o e iew and compa ison o he comp ession
s anda ds o ul asound imaging can be ound in [
9
]. Un o una ely, all hese s anda ds p ocess he
da a as 2D slices, leading o unsui abili y o pa allel on- he- ly p ocessing. Plen y o coding me hods
ha e been de eloped o deal wi h he comp ession o audio and speech signals. Lossy comp ession
me hods p o ide highe comp ession a ios a he cos o ideli y. These me hods ely on human
psychoacous ic models o educe he ideli y o less audible sounds. In con as o his, lossless
me hods p oduce a ep esen a ion o he inpu signal ha decomp esses o an exac duplica e o he
o iginal signal. Howe e , he comp ession a ios a e a ound 50–75% o he o iginal size. Fo he
pu pose o ou compa ison, we selec ed se e al common audio coding me hods, implemen ed in
he FFmpeg amewo k. These a e he Ad anced Audio Coding (AAC), AC-3, Opus, adap i e
di e en ial pulse-code modula ion (ADPCM), F ee Lossless Audio Codec (FLAC), Apple Lossless
Audio Codec (ALAC), and uncomp essed pulse-code modula ion (PCM). The indi idual o ma s a e
b ie ly discussed below in his sec ion. The inqui ing eade is e e ed o [10] o u he de ails.
As men ioned ea lie , he ypical size o he ul asound da a o be comp essed eaches abou 0.5 TiB.
Because he exis ing audio coding me hods we e no designed o use in ex emely memo y-limi ed
en i onmen s, hey consume he signal samples in ames ypically con aining housands o samples.
In conjunc ion wi h he p e ious poin , such la ge ame sizes make he common audio codecs unusable
o comp ession o he ul asound da a. Ano he issue o deal wi h is he sampling a e. Because
he human hea ing ange is commonly gi en as 20 o 20,000 Hz, he common audio codecs we e
designed o p ocess signals o compa able a es (e.g., 44.1, 48, 96, o 192 kHz). Howe e , ul asound
signals a e acqui ed (simula ed) a a much highe sampling a e. Fo ins ance, he a e o ou es ing
signals was 20 MHz. We we e he e o e o ced o slow down he ul asound da a o no mally audible
equencies. We chose he sampling a e o 48 kHz, as his was he g ea es common denomina o o all
he examined audio o ma s. This s ep did no in ol e any subsampling signal. I was me ely a ma e
o ins uc ing codecs. Ano he echnical issue was he app op ia e audio bi dep h selec ion. We used
16 bi s pe sample (bps), as his was he common bi dep h suppo ed by all he examined codecs.
AAC is an in e na ional s anda d o lossy audio comp ession. I was designed o be he successo
o he ubiqui ous MP3 o ma . Because AAC gene ally achie es be e sound quali y han MP3 a
In o ma ion 2018,9, 155 3 o 14
he same bi a e, we omi ed he MP3 o ma in ou compa ison. The comp ession p ocess can be
desc ibed as ollows. Fi s , a modi ied disc e e cosine ans o m (MDCT) based il e bank is used o
decompose he inpu signal in o mul iple esolu ions. A e wa ds, a psychoacous ic model is used
in he quan isa ion s age in o de o minimise he audible dis o ion. The ime-domain p edic ion
can be employed in o de o ake ad an age o co ela ions be ween mul iple esolu ions. The o ma
suppo s a bi a y bi a es.
Dolby AC-3 (also Dolby Digi al, ATSC A/52) is a lossy audio comp ession s anda d de eloped by
Dolby Labo a o ies. The decode has he abili y o ep oduce a ious channel con igu a ions om 1 o
5.1 om he common bi s eam. Du ing he comp ession, he inpu signal is g ouped in o blocks o
512 samples. Howe e , depending on he na u e o he signal, an app op ia e leng h o he subsequen
MDCT-based il e bank is selec ed. The o ma suppo s selec ed bi a es anging be ween 32 and
640 kbi /s.
Opus is an open lossy audio comp ession s anda d de eloped by he Xiph.O g Founda ion. I is
he successo o he olde Vo bis and Speex me hods. The codec is composed o a laye based on he
linea p edic ion and a laye based on he MDCT, bu bo h laye s can be used a he same ime (hyb id
mode). Opus suppo s all bi a es om 6 o 510 kbi /s.
ADPCM is a lossy comp ession o ma wi h a single ixed comp ession a io o abou 4:1. ADPCM
comp ession wo ks by sepa a ing he inpu signal in o blocks and p edic ing i s samples on he basis
o he p e ious sample. The p edic ed alue is hen adap i ely quan ised and encoded o nibbles,
gi ing ise o he 4:1 comp ession a io (192 kbi /s o 48 kHz sampling a e). A psychoacous ic model
is no used a all.
FLAC is an open lossless audio o ma now co e ed by he Xiph.o g Founda ion. FLAC uses linea
p edic ion ollowed by coding o he esidual (Golomb-Rice coding). Fo music, i was epo ed om
a ious benchma ks ha FLAC ypically educes he inpu signal size o 50–75% o he o iginal size.
ALAC is he open lossless audio o ma by Apple. Simila ly o FLAC, ALAC uses linea p edic ion
wi h Golomb-Rice coding o he esidual. Addi ionally, he achie able comp ession pe o mance is
simila o ha o FLAC.
As a e e ence o ma , an uncomp essed signal was used. Technically, his o ma is e e ed o as
linea PCM. Because we used 16 bps and a 48 kHz sampling a e, he uncomp essed signal esul ed in
a bi a e o 768 kbi /s.
3. P oposed Me hod
A ypical ou pu o he HIFU simula ion con ains wo basic ypes o da a—3D spa ial da a, o
example, maximum p essu e a de ined g id poin s, and 4D spa ial– empo al da a wi h ime- a ying
p essu e se ies a de ined g id poin s. Conside ing he 3D da ase s, inal simula ion alues ac oss he
whole simula ion domain, o a leas agg ega ed alues (such as he minimum, maximum, o oo
mean squa e), ha e o be s o ed in pe manen s o age. In he case o ime- a ying 4D da ase s, an
acous ic p essu e, acous ic eloci y, and in ensi y ha e o be s o ed ac oss he de ined se o loca ions
(senso mask) [11,12].
Ou comp ession me hod is ocused on ime- a ying signals and is designed o 1D da a.
Because he shape o he senso mask can be an a bi a y and spa se se o loca ions [12], we decided
no o deal wi h spa ial cohe ence. Mo eo e , by ea ing e e y spa ial g id poin independen ly, he e
was no need o addi ional communica ion on he dis ibu ed clus e s. The goals o he comp ession
me hod a e low memo y complexi y and high p ocessing speed; negligible da a dis o ion is accep able.
These in en ions di e en ia e ou me hod om o he s a e-o - he-a comp ession echniques.
Du ing HIFU simula ion, he sou ce ul asound signal is emi ed om a ansduce in o he issue
wi h which i in e ac s. Mul iple in e sec ing beams o ul asound a e concen a ed on he a ge ( ocus
poin ). Some phenomena, such as a enua ion, ime delay, sca e ing, o non-linea dis o ion, may
occu du ing he ul asound p opaga ion [
13
–
15
]. The sou ce ul asound signals a e always de ined by
In o ma ion 2018,9, 155 4 o 14
a known ha monic unc ion o a gi en equency, usually a p essu e sinusoid o equency anging
om 0.5 o 1.5 MHz [15].
We can assume ha he ime- a ying quan i ies, such as p essu e and eloci y a e e y g id poin ,
also ha e a ha monic cha ac e , wi h only a small ampli ude and phase de ia ions. These quan i ies
a e usually ampli ude-modula ed and can be composed o he undamen al and se e al ha monic
equencies. The numbe o ha monics depends on he size o he simula ion g id and a ac o o
non-linea i y. Cu en ly, we use up o i e o six ha monics o eal simula ions, which co esponds o
he HIFU in he mal mode. Howe e , o his o ipsy, as many as 50 ha monics may be needed.
A ypical p essu e signal eco ded a a single g id poin is shown in Figu e 1. An ou pu signal
(con aining, e.g., ime- a iable acous ic p essu e) can be subdi ided in o h ee s ages acco ding o
he ampli ude magni ude and i s changes. Du ing he i s s age, he signal esembles noise wi h an
ampli ude close o ze o. The leng h (in samples) o his s age depends on he dis ance o he g id poin
om he ansduce . The second s age can be cha ac e ised by a la ge inc ease in ampli ude (a leading
edge). This ansien s age is usually e y sho , o example, a ound 5% o he o al simula ion leng h.
The hi d s age ca ies a ela i ely s able ampli ude. This pa o he signal is he mos impo an pa
o he calcula ing o a hea deposi ion [
15
]. The s able s age usually s a s a he momen when all
wa es emi ed om he ansduce a i e a he ocus poin .
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Time s ep
-2
0
2
4
P essu e (Pa)
106
01234567
F equency (Hz) 106
0
0.5
1
1.5
2
2.5
P essu e (Pa)
106
Figu e 1.
The illus a ion o a high-in ensi y ocused ul asound (HIFU) simula ion signal a one poin
in 3D space.
The p oposed app oach is o modelling an ou pu signal, such as he decomposi ion o a 1D
signal (one poin in 3D space), as a sum o o e lapped exponen ial bases mul iplied by a window
unc ion. Each base is de ined by i s complex coe icien s (ampli ude and phase). We decided o use
complex exponen ial bases, because such bases can ep esen undamen al ha monic unc ions well,
including phase changes and highe ha monics, as well as window unc ions whose sum is cons an i
hey a e hal -o e lapped, because o on- he- ly p ocessing by pa s o he signal.
In o ma ion 2018,9, 155 5 o 14
I is impo an ha he inpu equency emi ed by he ansduce is known and ha i can be
used by he comp ession algo i hm. A complex exponen ial unc ion wi h he angula equency
ω
can be de ined o one ime s ep nas
(n) = e−ihωn, (1)
whe e
h
is he numbe o he ha monic equency (wa e numbe ), which can be gene a ed by non-linea
wa e p opaga ion. By mul iplying by a shi ed window unc ion wde ined as
w(n,a) = (0ad >n>ad +2d,
w0(n−ad)o he wise, (2)
whe e
w0
is a window unc ion (e.g., iangula o Hann window) and
d
is he hal -wid h o he
window, we ob ain linea ly independen sliding-window basis ec o s
b(n,a) = wa(n,a) (n)(3)
wi h 2
d
wid h, whe e
a
is he basis ( ame) index. The hal -wid h
d
should be an in ege mul iple o he
inpu pe iod
(
1
/ )
, as such a wid h ypically leads o mo e p ecise esul s. The whole econs uc ed
signal can hen be exp essed as
s(n) =
N/d
∑
i=0
b(n,i)bk(i), (4)
whe e
N
is he leng h o he signal,
I
is he numbe o complex ames
I=bN/dc
, and
bk(i)
a e
esul ing complex coe icien s.
An illus a ion o h ee hal -o e lapped windows and he eal-pa sum o window unc ions
wi h 2d=4/ is shown in Figu e 2.
50 100 150 200 250 300 350 400
Time s ep
-1.5
-1
-0.5
0
0.5
1
1.5
Ampli ude
Th ee Hann windows
Sum o windowed basis
Figu e 2. The illus a ion o h ee Hann window bases.
The complex coe icien s a e compu ed app oxima ely by co ela ion (do p oduc ) o he o iginal
signal xand he windowed exponen ial basis ec o o a selec ed ame ias
bk(i) =
N/d
∑
i=0
b(n,i)x(i). (5)
The coe icien s o o he ha monic equencies can be compu ed independen ly (as hey a e
linea ly independen ) and a e summed in he econs uc ion phase. E e y poin in 3D space can be
p ocessed sepa a ely and in pa allel wi hin bo h he encoding and decoding phases.
Wi hin he coding phase, wo do p oduc s a e g adually compu ed o a single ime s ep (one
signal sample) as a esul o wo o e lapped window basis ec o s. The numbe o memo y cells
c

In o ma ion 2018,9, 155 6 o 14
(single-p ecision loa ing-poin numbe s) equi ed o compu ing in e media e esul s in one ime s ep
depends on he numbe o ha monics H, and i can be e alua ed as
c=4H, (6)
as wo complex numbe s a e needed pe e e y ha monic equency.
Wi hin he decoding phase, he memo y equi emen s emain he same as in he encoding phase.
Two complex coe icien s a e needed o e e y ha monic equency; howe e , he decoded samples a e
compu ed independen ly, which can be use ul, o example, o as da a isualisa ion.
The comp ession a io o he me hod can be exp essed as
leng h(x)/(leng h(bk)2H). (7)
The a io is p opo ional o he wid h 2
d
o he o e lapping window bases. This signal
app oxima ion me hod yields a e y small dis o ion o he s able pa s o he signal whe e he
dis o ion depends only on he numbe o conside ed ha monic equencies. The dis o ion is highe
in he ansien pa s o he signal.
4. Implemen a ion
Cu en ly, wo baseline implemen a ions o he p oposed me hods a e a ailable—an implemen a ion
in Ma lab and FFmpeg o p ocessing 1D signals and a second implemen a ion in C++ o p ocessing
la ge 4D da ase s. Bo h o hese e sions p ocess he simula ion da a sequen ially and o line by
eading da ase s in HDF5 o ma . Ano he pa allel on- he- ly implemen a ion wi h CUDA and wi h
he OpenMP e sion is also a ailable.
The Ma lab and FFmpeg implemen a ions we e de eloped speci ically o p ocessing 1D da a
( ime se ies a a single g id poin ). The FFmpeg implemen a ion media es simple ways o he
comp ession me hod’s debugging, compa ison wi h o he coding me hods (e.g., audio codecs), and
he isualisa ions o p ocessing s eps and esul s. The C++ implemen a ion di e s om he Ma lab
implemen a ion in he abili y o p ocess a la ge amoun o 4D da a. Indi idual HDF5 da ase s a e
loaded by 3D blocks depending on he main memo y size. Bo h implemen a ions we e es ed on
a desk op compu e wi h 24 GiB memo y and In el Co e i5-6500 p ocesso . Fo expe imen s wi h
ex ensi e HDF5 iles, he Salomon clus e wi h 128 GiB memo y, wo In el Xeon E5-2680 3 p ocesso s,
and Lus e sha ed s o age space wi h a maximal heo e ical h oughpu o 6 GiB/s o one compu ing
node was used. We will p o ide he link o c oss-pla o m sou ce code on eques by e-mail.
We plan o deploy he comp ession algo i hm in he pa allel en i onmen du ing a HIFU
simula ion. The main ad an age is he possibili y o a oid s o age o he whole ou pu simula ion
da a. This would sa e conside able s o age space and ime as a esul o omi ed I/O ope a ions.
Expe imen s wi h he e alua ion o compu a ional esou ces will ollow in u u e wo k.
5. Expe imen s and Resul s
Two se s o expe imen s we e pe o med. The i s se was ocused on es ing he comp ession
me hod on 1D signals, and he second se was conduc ed o la ge 4D da ase s. A iangula window
was used as a window unc ion o he p oposed comp ession me hod o all expe imen s. The main
pu poses o he expe imen s we e o de e mine he peak signal- o-noise a io (PSNR) wi h espec o
he bi a e and o compa e he p oposed me hod wi h o he comp ession me hods, pa icula ly wi h
audio codecs.
Fo 1D signal comp ession es s, h ee di e en ypes o signals we e selec ed. We always chose 50
andom 1D signals (poin s wi hin he senso mask) om he gi en da ase s. The schema ic o e iew
o he es signals is shown in Table 1.
In o ma ion 2018,9, 155 7 o 14
Table 1. The ypes o es ing signals. All signals ep esen an acous ic p essu e.
Name Desc ip ion Simula ion Size Senso Mask Size Samples Pe iod
Linea One ha monic 256 ×256 ×350 55 ×55 ×82 3301 15
Non-linea 2 Max. wo ha monics 512 ×384 ×384 101 ×101 ×101 10,105 35
Non-linea 6 Max. six ha monics 1536 ×1152 ×1152 101 ×101 ×101 30,604 106
The signals di e mainly in he simula ion size, senso mask size, numbe o sampled simula ion
s eps, inpu pe iod, and ype o he simula ion. The dense senso mask was always de ined wi hin he
highly ocused HIFU a ea in which he highes ampli ude and mos o he ha monics a e obse ed.
All signals con ain acous ic p essu e s o ed in 32-bi loa ing-poin o ma . The da ase e e ed o as
Linea is he ou pu o a linea simula ion; hus, he e a e no ha monics. The o he da ase s comp ise
he ou pu s om non-linea simula ions, and he numbe o ha monics depended, among o he
ac o s, on he simula ion esolu ion. Fo a be e connec ion wi h eali y, a ealis ic simula ion ha
was ep esen a i e o he clinical si ua ion wi h he e ogeneous issue used a simula ion g id size o
1536
×
1152
×
1152 and con ained abou six signi ican ly s ong ha monics ( e e ed o as Non-linea 6).
The same simula ion (Non-linea 2) bu wi h a smalle simula ion g id size (512
×
384
×
384) con ained
only wo ha monics.
The comp ession expe imen s compa ed he esul s o he p oposed HIFU comp ession me hod
(HCM) wi h se e al audio codecs implemen ed in he FFmpeg amewo k (ei he na i ely o h ough
an ex e nal lib a y), speci ically wi h ADPCM, FLAC, ALAC, AAC, AC-3, Opus, and PCM. The main
p ope ies and se ings o hese codecs, including he p oposed me hod, a e shown in Table 2.
The abo e-lis ed me hods we e selec ed o compa ison pu poses only. Excep o PCM, none o
hem mee he memo y equi emen s needed o implemen a ion in a pa allel dis ibu ed en i onmen .
Because we a ge a memo y-limi ed en i onmen , we se he ame size o he minimum size suppo ed
by he speci ic sampling a e, sample o ma , and codec. We no e ha as a esul o he ame-based
sys em, a delay is in oduced a e he encoding and decoding p ocesses. This delay is shown in he
las column o he e e enced able.
Table 2. O e iew o es ed o ma s and codec se ings.
Codec Long Name Class F ame Size Delay
PCM Pulse-code modula ion Lossless 1 0
FLAC F ee Lossless Audio Codec Lossless 16 0
ALAC Apple Lossless Audio Codec Lossless 4096 0
ADPCM Mic oso ADPCM Lossy 2036 0
Opus Opus Lossy 120 120
AAC Ad anced Audio Coding Lossy 1024 1024
AC-3 Dolby Digi al (ATSC A/52) Lossy 1536 256
HCM HIFU comp ession me hod Lossy 1 0
Because se e al audio codecs equi e he signal o be no malised on he
h
0, 1
i
in e al, we used
he maximal absolu e signal alue o he no malisa ion. In some cases, he inpu signal had o be
s o ed in 16-bi PCM o ma be o e he codec could be applied (ADPCM and Opus). This con e sion
was pe o med wi h he use o he maximum 16-bi in ege alue.
By de aul , he HCM expec s an unknown maximum absolu e signal alue and he e o e uses
32-bi loa ing-poin numbe s o s o e comp ession coe icien s. I he dynamic ange o alues is
known in ad ance, he me hod can con e 32-bi loa ing-poin numbe s in o 16-bi o e en 8-bi
in ege s wi h negligible loss o in o ma ion (4 o 2 by es o one complex numbe ). Thus, o no malised
signals, ano he wo modi ica ions o he HCM we e implemen ed. As shown in all igu es and ables
excep Figu es 3, 7 and 8, he HCM used 8-bi in ege s o coe icien s and hus 2 by es o one complex
numbe . Fo a be e unde s anding, a compa ison o h ee ypes o HCM coding is shown in Figu e 3.
In o ma ion 2018,9, 155 8 o 14
0
10
20
30
40
50
60
0.1 1 10
PSNR [dB]
bi - a e [bps]
HCM, 2 by es
HCM, 4 by es
HCM, 8 by es
(a) Linea .
0
10
20
30
40
50
60
0.01 0.1 1 10
PSNR [dB]
bi - a e [bps]
HCM, 2 by es
HCM, 4 by es
HCM, 8 by es
(b) Non-linea 2.
0
5
10
15
20
25
30
35
40
45
50
0.01 0.1 1 10
PSNR [dB]
bi - a e [bps]
HCM, 2 by es
HCM, 4 by es
HCM, 8 by es
(c) Non-linea 6.
Figu e 3.
Di e en a ia ions o he high-in ensi y ocused ul asound (HIFU) comp ession me hods
(HCMs). The a e age bi a e and he co esponding peak signal- o-noise a io (PSNR) o es ing
da ase s: (a) Linea , (b) Non-linea 2, and (c) Non-linea 6.
A dependency o he dis o ion on he bi a e is shown in Figu e 4 o he da ase s Linea ,
Non-linea 2, and Non-linea 6. The lossless me hods a e ep esen ed by a single poin , as hey do
no ha e, by hei na u e, he abili y o choose any cus om bi a e. Excep o he ADPCM, he lossy
me hods a e ep esen ed as a cu e o which he independen a iable is he cus om bi a e. Al hough
he ADPCM is a lossy me hod, i has no abili y o choose a bi a e. Thus, in his case, he me hod is
also ep esen ed by a single poin . Looking mo e closely a he e e enced igu e, we can see ha he
In o ma ion 2018,9, 155 9 o 14
HCM exhibi ed a leas compa able comp ession pe o mance. Mo e p ecisely, he PSNR o he Linea
case, as shown in Figu e 4a, eached compa able alues o all lossy me hods excep he ADPCM
and Opus, which had signi ican ly wo se esul s. The p oposed HCM eached he lowes bi a e.
A i s highes bi a e, he HCM also exhibi ed he bes PSNR, which was abou 52 dB. I is possible
o ob ain be e bi a es wi h he p oposed me hod by se ing he mul iple o o e lap size (MOS) o
highe alues.
The esul s o he non-linea simula ion signals we e sligh ly di e en om hose o he Linea
da ase . In Figu e 4b, he PSNR and bi a e o he Non-linea 2 da ase a e shown. We ob ained a
sligh ly wo se PSNR, app oxima ely 48 dB, o he p oposed HCM, and be e alues o he o he
me hods. We belie e his was due o he p esence o mo e ha monics.
In he case o he Non-linea 6 da ase (la ge simula ion g id size and abou six ha monics), he
p oposed me hod ga e simila esul s as o he p e ious cases. ADPCM exhibi ed compa able PSNR,
and i was only sligh ly wo se in e ms o bi a e han he p oposed HCM. Mo eo e , he AAC codec
achie ed a sligh ly be e bi a e, ollowed by he compe i i e AC-3. Fu he de ails can be seen in
Figu e 4c.
Di e en a ia ions o he HCMs used o he Linea , Non-linea 2, and Non-linea 6 comp ession
da ase s can be obse ed in Figu e 3. Impo an ly, he PSNR eached compa able-quali y alues o all
cases. I can be no iced ha he bi a es con e ged o a single poin in he plo — his was caused by
he use o a WAVE ile heade o s o ing he comp essed da a. The WAVE o ma was used only o
es ing pu poses wi h he FFmpeg ool.
The pa icula esul s, including comp ession a ios, wi h he PSNRs se as close as possible o
50 dB a e lis ed in Tables 3–5. The alues co espond o Figu e 5. We no e he compa able bi a es o
he HCM, AC-3, and AAC.
A sho segmen o a signal in he 1D Non-linea 6 da ase is shown in Figu e 6. The indi idual
sub- igu es illus a e he o iginal, HCM- econs uc ed, and co esponding di e ence signals,
espec i ely. All o hese co espond o a MOS alue se o 1. We no e he maximal e o in a pa wi h
e y ansien signal alues.
Table 3. Resul s o Linea da ase . Bi a es aken as close as possible o 50 dB.
HCM AC-3 AAC Opus ADPCM ALAC FLAC PCM
Bi a e (bps) 1.28 1.33 5.33 6.38 4.27 7.54 43.42 16.19
Comp ession a io 25:1 24:1 6:1 5:1 7.5:1 4.2:1 0.7:1 2:1
PSNR (dB) 52.05 44.66 48.67 22.43 33.83 101.11 101.11 101.11
Table 4. Resul s o Non-linea 2 da ase . Bi a es aken as close as possible o 50 dB.
HCM AC-3 AAC Opus ADPCM ALAC FLAC PCM
Bi a e (bps) 0.98 1.33 2.03 5.65 4.12 6.69 27.04 16.06
Comp ession a io 32.6:1 24:1 15.8:1 5.7:1 7.8:1 4.8:1 1.2:1 2:1
PSNR (dB) 47.81 47.87 49.85 25.35 42.03 101.23 101.23 101.23
Table 5. Resul s o Non-linea 6 da ase . Bi a es aken as close as possible o 50 dB.
HCM AC-3 AAC Opus ADPCM ALAC FLAC PCM
Bi a e (bps) 0.92 1.00 0.69 5.55 4.05 4.53 19.41 16.02
Comp ession a io 34.6:1 32:1 46.7:1 5.8:1 7.9:1 7.1:1 1.6:1 2:1
PSNR (dB) 45.92 49.80 50.82 28.94 51.28 102.34 102.34 102.34