A Performance Comparative Analysis of Block Based Compressive Sensing and Line Based Compressive
Sensing
Mansoor Ebrahim Department of Computer Science
Iqra University Karachi, Pakistan [email protected]
Syed Hasan Adil Department of Computer Science
Iqra University Karachi, Pakistan [email protected]
Daniyal Nawaz
Department of Computer Science Iqra University
Karachi, Pakistan [email protected]
Abstract—Compressive sensing (CS) is an innovative idea that has opened new areas for viable communication of correlated data. In this paper, a comparative performance analysis of two different variants of compressive sensing i.e. block based compressive sensing (BCS) and line based compressive censing (LCS) schemes is performed for natural images. The idea is to evaluate which variant performs better in terms of reconstruction quality and provides easy initial solution. The experimental analysis demonstrates that LCS scheme can enhance the image reconstruction at lower subrates by 0.5 dB to 2.5 dB, when compared to the BCS scheme.
Keywords-compressive sensing; block based approach; line based approach; reconstruction; image
I. INTRODUCTION
Compressive sensing (CS) is one of the latest techniques that have achieved popularity recently and is applied to various imaging applications [1], such as magnetic resonance imaging (MRI) [2, 3] and seismic identification [4]. CS [5, 6] idea is to express a signal (image/video) with sample estimation rate much lower than that of the Nyquist rate which is essential for the recovery of the signal. One of the innovative of this is the single-pixel camera [1] that openly condenses the sampling and amount of data that will be transmitted, but increases the difficulty of recovering the original signal. In other words, the recovery of the original signal from small number of sample measurements is difficult. Numerous earlier researches have already examined the utilization of CS in image compression [3-8]. Many researchers concentrated their work on proposing how natural images can be compressed utilizing CS and the recovery of the encoded signal (image) from such little estimation. In CS we concentrate mostly towards discrete signs instead of constant time space signals. In addition, compressive sensing of natural images is exposed to few issues such as computationally complex reconstruction algorithm and requirement of large memory to store the random sampling operator Φ. In order to encounter the aforementioned issue different approaches were developed i.e. block based compressive sensing and line based compressive sensing. The block-based approach is far more developed and widely
employed as compared to the newly developed line based approach to reduce of the computationally complexity of CS scheme [9-14]. Such methodologies value CS because (i) block/line based estimation is more convenient for applications where the sample image data don't need to be encoded completely in a block/line form until the point when the estimation of the whole image is completed, (ii) the application and capacity of the estimation operator are straightforward, (iii) the individual handling of each block/line of image data brings about simple initial solution with considerably quick and better recovery process.
In this paper, a performance comparison of two different variants of compressive sensing i.e. block based compressive sensing and line based compressive sensing for images is performed. The purpose is to find out which variant works best to solve the issues related to CS. The variants are evaluated based on computational complexity and reconstruction quality at various subrates for different test image datasets. In addition, block based and line based approaches are also compared at various block/line sizes to validate the effectiveness of each scheme. The paper is structured as follows. The fundamentals of the CS along with block based and line based approaches are presented in section II. Section III presents and discusses the simulation results. Section IV concludes the paper.
II. THEORY OF COMPRESSIVE SENSING
The CS theory [5] states that a sparse (original or some transform domain) signal can be completely recovered from a number of samples below than that stated by the Nyquist theorem. The CS scheme efficiently eases the computational prerequisites, for example, memory, processing power, and transmission data transfer capacity at the encoder by relating signal acquirement and dimensionality reduction into a distinct stage. CS is effective in two circumstances. When direct measurements of a high-resolution signal are hard to attain and when multiple high-resolution signals are complex to encode.
In literature, CS is a standard and isn’t stated for any particular signal other than underlying sparsity assumptions. CS permits great prospect of signal reconstruction by utilizing least amount of unsystematic measurements, as long as the signal/image is
spa sch me red
me tra as
Fig me
vec ma ma est X=
by com can
the CS sim sig sam iss ma A.
the red div sep con
sam enc ma app bri pri rec pro
arse and inc hemes, in C easurements r ducing the com
Consider a easurementsis ansformation d shown in Figu
Y=Φ·X
g. 1. Compres asurement matrix
X∈RN and Y ctor respectiv atrix Φ is orth atrix. In addi timation is no
=Φ-1 Y is ill-po CS ought to mpleted by ta n be expressed x argmin However, mo eoretical and a S requires a multaneously gnals difficult
mpled signals sue, various res ainly include b Block Based Block based e sparse natur duce storage s vided in term parately based
nstrained (bloc
Φ =
Φ ⋯
⋮ ⋱
0 ⋯
The key plu mpled image coder until the aking block-ba plications, (iii) ing informatio imarily quick construction i ojection land
coherent. Un CS the enco rather than th mputational co
real-valued si to be recove domain Ψ with ure 1. The set o
ssive sensing a x Φ and transform
Y∈RM are the vely. It is pre honormal i.e. Φ
ition, the rec ot straight for osed [6]. But s o be sparse aking care of d as:
‖x‖ , s.t. y=Φ ost of existing are exposed to accessing the that makes i t. At decode
is generally searchers prop block based CS
Compressive CS [9] techni re of images space. In block ms of blocks d on the simila ck-diagonal) s
⋯ 0
⋱ ⋮
⋯ Φ
uses of using data do not e measuremen ased measurem
) each image b on about simp
reconstructio implicates two d-weber (SPL
nlike convent oder only ca he whole sig omplexity and ignal X with ered, has cert h random mea of measureme
acquisition proc mation matrix Ψ [6
e input signal esumed that th
Φ ΦT=A, whe covery of X rward, i.e. inv since the signa in nature, th the ℓ0 optimiz
ΦΨ-1X’
g works [3-8]
o few issues, e whole sig its real time er, the recon very expensiv posed different S, and line bas
Sensing ique has been
in transform k based CS th s and each b ar measuremen structure as in
g BCS schem need to be nt of the whole ment more exp block is proce ple preliminary
n process [10 o different v L) and tota
tional compre aptures the gnal. This hel bandwidth.
length N fro tain sparsity i asurement mat ents y is given
(1)
cess with a r 6]
l and measur he random se ere A is the id from such verse projecti al to be compr e recovery ca zation problem
(2) in CS remain like that at en gnal for sam sensing of n nstruction of
ve. To counte t variants of C sed CS.
proposed to e domain in ord he original im block is proc nt matrix “Φ” w
(3).
(3)
me are that ( transmitted b e image is don pedient for pra essed individua y arrangement 0]. The BCS ariants i.e. sm al variation
ession signal lps in
om M in the trix Φ as:
random
ement ensing dentity small ion of
ressed an be m that
at the ncoder mpling natural such er this CS that
exploit der to mage is cessed with a
i) the by the ne i.e.
actical ally to t with based mooth (TV)
min as i the of p find The 1
(I=Ieac sign com
sam con com enc
is p
B.
orig rath usin becbe ind resu imp bas reco reco 1 valu
nimization. Th it provides bet evenest cours piece-wise sm ding the sparse e reconstructio 1) Encoder Suppose, an i IR·IC). The im ch block havin
nal of the mpressed outpu
YB=Φ·XB
For the comp mpled by the nstrained arran mmunicated to coding steps ar
Input ima
Convert i
Sampled
Finally, Y The encoding presented in Fi
F
Line Based C Line based C ginal image is her than in blo ng the measur cause in line ba
encoded as dependently. In
ults in easy proved recove sed on TV min
onstruction onstruction pr 1) Encoder
Consider an I ues in each ro
his paper focu tter reconstruc se within the p mooth properti
e solution with on process of B r
image (I) has I mage I is divi ng size of B·B.
b-th block ut for this b-th
plete image, e e similar me ngement. The o the decoder re:
age I of size IR
image I into m each block X Y will be the e g process of b igure 2.
Fig. 2. Encod
Compressive Se CS [15, 16] is first divided i ocks. Each line rement operato
ased approach a whole bu n addition, eac initial solutio ery process. T nimization sch of the e rocess of LCS
r
IR·IC image, wh ow and colum
uses on TV mi ction compare potential spac ies of typical hin the transfo BCS is discuss
IR (Rows) x IC
ded into diffe . Let XB repre through raste h block will be
each block is t easurement m e encoded sam for recovery.
R·IC.
multiple blocks
i with Gaussia encoded sampl block based co
ding process of BC
ensing a recent appr into N multiple e is then proce
or Φ. This app h sampled sign
ut in a line ch independen on with cons The LCS based
eme as it prov encoded me
is discussed in
here IR and IC
mn respectively
inimization sch d to SPL and e by making u
signals instea ormation doma
sed in section I
C (Columns) p erent small bl esent the vecto er scanning.
e:
(4) then independ matrix Φ wi mpled Y are
Summarizing
s X.
an Matrix Φ.
le.
ompressive sen
CS
roach in whic e lines of same essed independ proach benefit nal does not ne e by line fa nt line of the s
siderably fast d reconstructi vides fast and b
asurements.
n section II.C.
represent the y. At encoder
heme finds usage ad of ain Ψ.
II.C.
ixels, locks, orized The
dently ith a
then g, the
nsing
ch the e size dently ts CS eed to shion signal t and ion is better The
pixel side,
the lin pro CoTh
uti stru for sen
C.
pri ele of X=
CS by in wit the ma sig ins mi
pro the and rec pro con spl
e image is fir ne based CS.
ocessed indep onsider Xi as t hen the compre
YL=Φ·XL
For the entir ilizing a simi ructure. The es r recovery. Th nsing is presen Input image I Convert imag Sampled each Finally, Y wi
Decoding The reconst imary challeng ements is high X∈RN from
=Φ-1·Y is ill-po S ought to be s
taking care o transformed d th total variati e smoothest aking utilizati gnals as oppo side the tra inimization fun TV(X) =∑i,j
X E|| y- However, th oblem in (7) i e non-different
d non-linear construction.
oposed to so nventional au litting and t
rst distributed Each line w pendently usin the vectorized essed CS vecto
re image, each ilar estimation stimations Y a he encoding pr nted in Figure I of size IR x I ge I into multi h line Xi with ill be the enco
Fig. 3. Enco
truction of th ge of utilizing her than the n its relating Y osed [6]. Sinc sparse in natur of convex opti domain with ion (TV) norm
arrangement on of piece-w osed to findi ansformation nction is given
||Xi+1,j – Xi,j Θ X|| + λ TV(
he TV minim is exposed to tiable (presenc r properties,
In [17], a s olve (7). The ugmented Lag the alternatin
into N multip will have a s ng the measur d signal of i-th or output Yi w
h line is then n matrix Φ w are then transm
rocess of line 3 and its steps
C.
iple lines X.
Gaussian Mat oded sample
oding process of L
he encoded g CS. As the n number of obs Y∈RM i.e. inv ce the signal to re, the recover
imization prob either ℓ-norm m [7]. The TV inside the p wise smooth ing the inade
domain. T n as:
j||+||Xi,j+1– X X) s.t. Θ = Φ Ψ mization based
extra computa ce of the abso
restricting scheme name e scheme is grangian met ng direction
ple lines, usin size of 1·L a rement operat h line of the i will be:
(5) separately sam with a constr mitted to the de based compr s are.
trix Φ.
LCS
estimations i number of unk servations, rec verse projecti o be compress ry can be comp
blem using sp m or image gr V minimization potential spac qualities of n equate arrang The essential
Xi,j|| (6) Ψ (7) CS reconstru ational burden olute value fun its use for ed as TV-AL
a combinatio thod with va
method. TV ng the and is tor Φ.
mage.
mpled rained ecoder essive
is the known covery ion of sed by
pleted parsity adient n finds ce by natural
ement TV
uction ns, i.e.
nction) r CS L3 is on of ariable V-AL3
gen TV and distterm alte com met by pen mu reco LC
and eva this from var text thei ame ima charan per var wer resp the two 64x (PS of b the sch
nerates reconst V, but reduces t d alternating tinct the non ms, so as to f ernating mini mputational bu
thod of TV-A adding a squa nalty method b ultiplier λ [18
onstruction of S is presented Input the enco size/line size, Reconstruct e parameter by Lagrangian (T Finally, I’ wi image I.
Fig. 4
I In this sectio d line based CS aluate the sche s paper six im m [19-21]. T rious character tured surface, ir performance enable by bo ages size to 51 allenging in co nge and high
rformance com riants of com
re implemente pectively. We image recons o different x64/1x4096) b SNR) at variou block/line size encoded me hemes (BCS, L
tructed image the computatio
approaches. T -differentiable facilitate low- mization way urden. Moreo AL3 varies from
are penalty te by the presence
8]. The gene f encoded mea d in Figure 4 an
oded sample Y etc.)
encoded samp y solving (5) u
TV-AL3).
ll be the final
4. Decoding p
III. EXPERIM
on the perform S are presente emes perform mages, shown The selected d ristics such as , variations an es in different oth schemes, 12×512 pixels ontrast to stand her percentage mparison inv mpressive sens ed by using th e also investig struction quali block/line by measuring us subrates. T e and subrate easurements o LCS). Addition
with the high onal burden by The purpose e terms from
-complexity s y [18], resul over, the augm
m the typical L erm, whereas
e of the linear eric decoding asurements pro nd its steps are Y and paramet
ple Y with t using total va
reconstructed
process of BCS an
MENTAL RESUL
mance evaluati ed. A set of tes mance. For the in Figure 5, dataset consis high and low nd disparity r t conditions. T we down sam s. The down s
dard due to th e of un-textu volves using sing. The var he algorithms p gated the effec
ity. The evalu sizes (32 g the peak-sig The idea is to
on the recons obtained by b nally, as the m
quality of stan y applying spl
of splitting the different ub-problems lting in decre mented Lagran Lagrangian me from the quad r term involvin g process for oduced by BCS
e:
ters (Φ, R, C, b
the help of p ariation augm
d view of the a
nd LCS
LTS
ion of block b st images is us e work report are used, obt st of images w percentage o
ranges to eva To make the im
mpled the ori ampled image heir larger disp ured surface.
the two diff riants (BCS, L
proposed in [9 ct of block siz uation is carrie 2x32/1x1024
gnal to noise analyze the im struction quali both the discu measurement m
ndard litting is to tiable in an eased ngian ethod dratic ng the r the S and
block
plenty ented
actual
based sed to ted in ained
with of un- aluate mages iginal es are parity The ferent LCS) 9, 16]
ze on ed on and -ratio mpact ity of ussed matrix
Φ var val (0.
A.
LC per PS me sub BC Fro sho ov var sta rec and LC abo
F
Fig of 3
Fig of 6
is random in ry. Hence, an lues is presen .05-0.3) with a BCS vs LCS In this sectio CS. The compa rformance of SNR (dB), su
entioned imag brates. Table CS for the im
om the simula ows an averag
er BCS sche riation image ated that la construction q
d 7 show the CS and BCS
ove.
Fig. 5. Severa
g. 6. Average 32x32/1x1024
g. 7. Average 64x64/1x4096
nature, the im n average of 5 nted. All imag
an interval of 0
on, the perform arison is to inv
BCS and LC ubrate) result es by using di I presents the mages and the ation results, it ge gain 1dB-3
me and the g , Middle-burr arger measu quality at the e e average PSN at two differe
al standard graysc
PSNR results of
PSNR results of
mage reconstru 5 independent
es are sample 0.05.
mance of BCS vestigate the ra CS. All R-D p ts are obtain ifferent block/
e comparison e two differen t is observed th 3dB for lower gain is more ry. Furthermo urement size
expense of com NR results of
ent block/line
cale test image dat
f six different ima
f six different ima
uction quality trials of all P ed at lower su
S is compared ate-distortion performances’
ned from th /line sizes at v results of LC nt sizes consid
hat the LCS sc r to higher su
prominent in ore, from [22]
provides mplexity. Figu
the six imag sizes as disc
tasets of size 512
ages with block/li
ages with block/li
might PSNR ubrates
d with (R-D)
’ (i.e., he six
arious S and dered.
cheme ubrates n low ] it is better ures 6 es for cussed
x512
ine size
ine size TA
B.
esse visu seleresu ima LC that effe enc reg per smo mea
Fig.
and
BLE I. PSN VARIOUS DATA
Subrate 0.0 BCS 24.3 LCS 24.8 Gain 0.5 Subrate 0.0
BCS 28.9 LCS 29.7 Gain 0.7 Subrate 0.0 BCS 25.9 LCS 26.7 Gain 0.8 Subrate 0.0 BCS 25.7 LCS 26.6 Gain 0.8 Subrate 0.0
BCS 24
LCS 25.8 Gain 1.1 Subrate 0.0 BCS 24.9 LCS 26.0 Gain 1.0
Visual Result Since LCS an ential to certif ually detectab ected represen ults shown in ages of the en S using two d t the reconstru ect present in coded measure ions (red dotte rformed by us
oother and p asurements.
8. Visual res LCS encoded me
NR ACCOMPLISHED ASET IMAGES AT B A 05 0.1 35 25.25 85 25.89 5 0.64
B 05 0.1 98 30.68 75 31.48 77 0.8 Mon 05 0.1 91 28.03 74 28.94 83 0.91 Middl 05 0.1 79 28.99 65 29.89 86 0.9 Co 05 0.1 .7 26.98 83 28.27 13 1.29 Te 05 0.1 96 26.99 02 28.18 06 1.19
t Comparison nd BCS are e fy that the sub ble images. Tw nting medium
n Figures 8 a ncoded measur different subrat ucted images f n the image ements. Also, ed boxes), it c sing LCS enco
piercing than
sults of the recons easurements at dif
D BY UTILIZING BC BLOCK/LINE SIZE Aloe
0.15 0.2 26.18 27.19 26.87 28.01 0.69 0.82 Baby
0.15 0.2 31.99 33.01
33 34.08 1.01 1.07 nopoly
0.15 0.2 29.66 31.74 30.69 32.84 1.03 1.1 le-burry
0.15 0.2 31.14 32.98 32.15 34.21 1.01 1.23 ones
0.15 0.2 28.57 29.84 30.12 31.76 1.55 1.92 eddy
0.15 0.2 29.05 29.77 30.29 31.34 1.24 1.57
evaluated at lo brates used are
wo different i and high textu and 9 are of rements obtain tes, 0.1 and 0.2 from LCS imp reconstructed , by comparin can be noticed oded measure n the recons
struction of Mono fferent subrates
CS AND LCS TO EN OF 32X32/1X1024
0.25 0.3 28.15 32.6 29.12 33.8 0.97 1.1 0.25 0.3 34.73 35.8 35.84 37.0 1.11 1.1 0.25 0.3 34.17 35.
35.57 36.5 1.4 1.4 0.25 0.3 34.54 35.6 35.8 36.9 1.26 1.3 0.25 0.3 31.08 32.0 33.31 34.6 2.23 2.5 0.25 0.3 30.89 32.0 33.16 34.5 2.27 2.4
ower subrates, e adequate to c image dataset ure variations f the reconstru
ned from BCS 2. It can be no prove the disto d using the ng the empha
that reconstru ements looks m
struction of
opoly image usin NCODE
3 69 87 8 3 89 03 4 3
1 57 47
3 66 98 2 3 07 65 8 3 07 51 44
, it is create ts are
. The ucted S and oticed orting BCS asized uction much BCS
g BCS
Fig BC
blo sen and app 1d
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10
[11 [12
[13
g. 9. Visual re CS and LCS encod
This article ock based com nsing for diffe
d sizes. The e proach perfor dB to 3dB at va
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