Table Of ContentA Discrete Tchebichef Transform Approximation for Image and
Video Coding
PauloA. M.Oliveira Renato J.Cintra Fa´bio M.Bayer†
∗ ∗
5 SuneraKulasekera‡ ArjunaMadanayake‡
1
0
2
n
a Abstract
J
8 In this paper, we introduce a low-complexityapproximationfor the discrete Tchebicheftransform
2
(DTT).Theproposedforwardandinversetransformsaremultiplication-freeandrequireareducednum-
] ber of additions and bit-shifting operations. Numerical compressionsimulations demonstrate the effi-
E
ciencyoftheproposedtransformforimageandvideocoding.Furthermore,XilinxVirtex-6FPGAbased
M
hardwarerealizationshows44.9%reductionindynamicpowerconsumptionand64.7%lowerareawhen
.
t comparedtotheliterature.
a
t
s
[ Keywords
ApproximateDTT,fastalgorithms,imageandvideocoding.
1
v
5
5 1 Introduction
5
0
0 The discrete Tchebichef transform (DTT) is a useful tool for signal coding and data decorrelation [1]. In
.
2
recentyears,signalprocessingliteraturehasemployedtheDTTinseveralimageprocessingproblems,such
0
5 asartifact measurement [2],blindintegrity verification [3],andimagecompression [4–7]. Inparticular, the
1
8-point DTT has been considered in blind forensics for integrity check of medical images [3]. For image
:
v
i compression, the8-pointDTTisalsocapableofoutperformingthe8-pointdiscretecosinetransform(DCT)
X
intermsofaverage bit-length inbitstream codification [4]. Moreover, in[7]an8-point DTT-basedencoder
r
a
capable ofimproved image quality and reduced encoding/decoding timewasproposed; being acompetitor
tostate-of-the-art DCT-basedmethods. However,tothebestofourknowledge, literature archives onlyone
fast algorithm for the 8-point DTT, which requires a significant number of arithmetic operations [6]. Such
PauloA.M.OliveiraandRenatoJ.CintraarewiththeSignalProcessingGroup,DepartamentodeEstat´ıstica,Universidade
∗
FederaldePernambuco,Recife,PE,Brazil. R.J.CintraisalsowiththeLIRIS,InstitutNationaldesSciencesApplique´es(INSA),
Lyon,France(e-mail:[email protected]).
†Fa´bioM.BayeriswiththeDepartamentodeEstat´ısticaandLACESM,UniversidadeFederaldeSantaMaria,SantaMaria,RS,
Brazil(e-mail:[email protected]).
‡SuneraKulasekeraandArjunaMadanayakearewiththeDepartmentofElectricalandComputerEngineering,TheUniversity
ofAkron,Akron,OH,USA(e-mail:[email protected]).
1
high arithmetic complexity may be a hindrance for the adoption of the DTT in contemporary devices that
demandlow-complexitycircuitry andlowpowerconsumption [8–10].
Analternative to the exact transform computation is the employment of approximate transforms. Such
approach has been successfully applied to the exact DCT, resulting in several approximations [11,12]. In
general, an approximate transform consists of a low-complexity matrix with elements defined over a set
of small integers, such as 0, 1, 2, 3 . The resulting matrix possesses null multiplicative complexity,
{ ± ± ± }
becausetheinvolvedarithmeticoperationscanbeimplementedexclusivelybymeansofareducednumberof
additions and bit-shifts. Prominent examples ofapproximate transforms include: the signed DCT[13], the
seriesofDCTapproximationsbyBouguezel-Ahmed-Swamy[14–16],theapproximationbyLengwehasatit-
Ortega[17],andtheintegerbasedapproximations described in[11,12,18,19].
Inthiswork,weintroducealow-complexityDTTapproximationthatrequires54.5%lessadditionsthan
the exact DTT fast algorithm. The proposed method is suitable for image and video coding, capable of
processing data coded according to popular standards—such as JPEG[20], H.264 [21], and HEVC[22]—
at a low computational cost. Moreover, the FPGA hardware realization of the proposed transform is also
sought.
This paper unfolds as follows. Section 2 describes the DTTand introduces the approximate DTTwith
its associate fast algorithm. A computational complexity analysis is offered. In Section 3, we perform
numerical experiments; applying of the proposed transform as a tool for image and video compression.
In Section 4, we provide very large scale integration (VLSI) realizations of the exact DTT and proposed
approximation. Conclusions andfinalremarksareinSection5.
2 Discrete Tchebichef Transform Approximation
2.1 Exact DiscreteTchebichef Transform
The DTT is an orthogonal transformation derived from the discrete Tchebichef polynomials [23]. The
entriesoftheN-pointDTTmatrixarefurnished by[1]:
(2k+1)(N k 1)!
t = − − (1 N) F ( k, n,1+k;1,1 N;1), k,n=0,1,...,N 1, (1)
k,n k 3 2
s (N+k)! · − · − − − −
where F (a ,a ,a ;b ,b ;z) = (cid:229) ¥ (a1)k(a2)k(a3)k zk is the hypergeometric function and (a) = a(a+
3 2 1 2 3 1 2 n=0 (b1)k(b2)k · k! k
1) (a+k 1) is the ascending factorial. Therefore, the analysis and synthesis equations for the DTT
··· −
are given by X = T x and x = T 1 X = T X, where x = x x x ⊤ is the input signal,
− ⊤ 0 1 N 1
· · · ··· −
h i
X= X X X ⊤ isthetransformed signal, andTistheN-pointDTTmatrixwithelementst ,
0 1 N 1 k,n
··· −
k,n=h0,1,...,N 1, i
−
In particular, the 8-point DTT matrix T can be described by the product of a diagonal matrix F and an
2
integer-entry matrixT [6],resulting in: T=F T ,where
0 0
·
1 1 1 1 1 1 1 1
7 5 3 1 1 3 5 7
−7 −1 −3 −5 5 3 1 7
T = 7 5 −7 −3 −3−7 5 7, (2)
0 −7 13 3 9 −9−3 −13 7
7−23 −17 15 15−17−23 7
−1 5− 9 − 5 5 9− 5 1
1 −7 21 −35 −35 21 −7 1
− − − −
and F= 1 diag 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 . A fast algorithm for the above integer matrix
2· √2 √42 √42 √66 √154 √546 √66 √858
T0 = F−1 T wa(cid:16)s derived in [6] requiring 44 additions (cid:17)and 29 bit-shifting operations. Such arithmetic
·
complexity is considered excessive, when compared to state-of-the-art discrete transform approximations
whichgenerally requirelessthan24additions [12,13,16,17].
2.2 DTT ApproximationandFastAlgorithm
In[12],aclassofDCTapproximationswasintroducedbasedonthefollowingrelation: round(a C),where
·
round()istheroundfunctionasdefinedinCandMatlablanguages[12],a isarealparameter,andCisthe
·
exact DCT matrix. We aim at proposing a similar approach to obtain an 8-point DTT approximation. The
scale-and-round approach is particularly effective when discrete trigonometric transforms are considered.
Thisisbecause theentriesofsuchtransformation matriceshavesmallerdynamicrangeswhencomparedto
theDTT.Incontrast, theDTTentrieshavevalueswithadynamicrangeroughlyseventimeslargerthanthe
DCT,forexample. Thustheapproximation errorimpliedbytheroundfunction islessevenlydistributed in
non-trigonometrictransformmatrices,suchastheDTT.Tomitigatethiseffect,weproposeacompading-like
operation [24], consisting of arescaling matrix Dthat normalizes theDTTmatrix entries. Thus, according
theformalismdetailedin[12],weintroduceaparametricfamilyofapproximateDTTmatricesT(a ),which
aregivenby:
T(a )=round(a T D ), (3)
0
· ·
whereD =diag( 6,√154,√66,√858,√858,√66,√154, 6).
0 7 13 9 35 35 9 13 7
Weaimatidenqtifying aparticular optimal parameteqr a such thatT =T(a )results inamatrixsatis-
∗ ∗ ∗
fyingthefollowingconstraints: (i)theentriesofT mustbedefinedover 1,0,1 and(ii)T mustpossess
∗ ∗
{− }
lowarithmetic complexity. Constraint (i)implies thesearch space (0,3/2). Although theaboveproblem is
notanalyticallytractable,itssolutioncanbefoundbyexhaustivesearch[12]. Bytakingthevaluesofa over
the considered interval in steps of 10 3, above conditions are satisfied for 0.931 a 0.957. All values
− ∗
≤ ≤
ofa inthislatterintervalimplythesameapproximate matrix. Thus,theobtainedlow-complexity forward
∗
3
DTTapproximation isgivenby:
1 1 1 1 1 1 1 1
1 1 0 0 0 0 1 1
−1 −0 0 1 1 0 0 1
T = 1 1 1−0−0 1 1 1 (4)
∗ −0 1 0 1 1−0−1 0
0 −1 1 1 1 1−1 0
0 1−1−0 0 1−1 0
0 −0 1 1 1 1−0 0
− −
anditsinverseT isgivenby: (T ) 1=T D where
∗ ∗ − 1 1
·
1 3 3 2 1 1 1 1
1−2 1−2 1−1 −1−1
1−1 −1 1 −1 2−3 2
T =1−1 −1 1 −1 −2 1−3, (5)
1 1−1 −1 1 1 −2 −1 3
1 1 −1−1 1 2 −3 −2
1 2 −1−2−1 1 1 1
1 3 −3 −2 −1 −1 −1−1
−
and D =diag 1, 1 ,1, 1 ,1, 1 ,1, 1 . Considering the total energy error [13,18] between the exact and
1 8 10 8 10 4 10 8 10
approximate matrices, we obtained 3.32 and 4.86 as the error values for the direct and inverse transforma-
(cid:0) (cid:1)
tions,respectively. Sucherrorsareconsidered verysmall[19].
Thus, employing the orthogonalization procedure described in [12], we obtain the follow-
ing expression for the DTT approximation: Tˆ = D T , where D = ediag(T (T ) ) =
∗ ∗ ∗ ∗ ∗ ⊤
· ·
d d d d d d d d ⊤ is a diagonal matrix and ediag() returns apdiagonal matrix with
0∗ 1∗ 2∗ 3∗ 4∗ 5∗ 6∗ 7∗ ·
thhe diagonal elements of its matrixiargument [12]. The inverse transformation is (Tˆ) 1 =(D T ) 1 =
− ∗ ∗ −
·
(T ) 1 (D ) 1 =T D (D ) 1. Therefore, the analysis and synthesis equations for the proposed trans-
∗ − ∗ − 1 1 ∗ −
· · ·
formaregivenbyXˆ =Tˆ xandx=T D (D ) 1 Xˆ,whereXˆ = Xˆ Xˆ Xˆ ⊤ istheapproximate
1 1 ∗ − 0 1 7
· · · · ···
transformed vector. h i
However, in several contexts, diagonal matrices—such as D and D —represent only scaling factors
1 ∗
and may not contribute to the computational cost of transformations. For instance, in JPEG-based im-
age compression applications, diagonal matrices can be embedded into quantization block [6,11,12] and,
when the explicit transform coefficients are needless, a scaled version of the transform-domain spectrum
is sufficient [25]. Therefore, hereafter, we disregard the diagonal matrices and focus our analysis on the
low-complexity matrices T and T . A fast algorithm based onsparse matrix factorization [11,12,14]was
∗ 1
derived for the proposed forward and inverse approximations. InFigure 1, the signal flowgraph (SFG)for
the direct transformation is depicted. TheSFGfor the inverse transformation can be obtained according to
themethodsdescribed in[26]. Moreover, Table1summarizesthearithmetic complexityassessment forthe
proposed transformations. Thefastalgorithms forT andT demand 54.5%and 34.1%lessadditions than
∗ 1
theDTTfastalgorithm (ITT)proposed in[6],respectively.
4
ˆ ∗
x0 X0/d0
ˆ ∗
x1 X6/d6
ˆ ∗
x2 X4/d4
ˆ ∗
x3 X2/d2
ˆ ∗
x4 X7/d7
ˆ ∗
x5 X5/d5
ˆ ∗
x6 X3/d3
x7 Xˆ1/d∗1
Figure 1: Signal flow graph for T . Input data x , n=0,1,...,7, relates to the output Xˆ , k =0,1,...,7.
∗ n k
Dashedarrowsrepresent multiplications by 1. Scalingbyd ,k=0,1,...,7,canbeignored andabsorbed
− k∗
intothequantization step.
Table1: Arithmeticcomplexity oftheproposed 1-Dtransforms
Method Mult. Additions Shifts Total
ExactDTT[6] 0 44 29 73
ProposedTˆ 0 20 0 20
∗
ProposedT 0 29 8 37
1
3 Experimental Results
3.1 ImageCompression
In order to assess the proposed transform in image compression applications, we performed a JPEG-like
simulationbasedon[6,11,12]. Asetof45512 5128-bitgrayscaleimagesobtainedfromastandardpublic
×
image bank [27] was considered. Each image was subdivided into 8 8 size blocks A , i,j=1,2,...,64.
i,j
×
Each block is submitted to two-dimensional (2-D) versions of the discussed transformations according to:
B =M A M ,whereB isthetransform-domainblockandM T,T Theresulting64spectralco-
i,j i,j ⊤ i,j ∗
· · ∈{ }
efficientsofeachblockwereorderedinthestandardzigzagsequence. Subsequently,therinitialcoefficients
in each block were retained and the remaining coefficients were discarded [12]. We adopted 1 r 45.
≤ ≤
Finally, each transform-domain subimage wassubmitted toinverse 2-D transformations and the full image
was reconstructed. Image quality measures were employed to assess the degradation between original and
reconstructed images. The considered measures were the structural similarity index (SSIM) [28] and the
spectral residual base similarity (SR-SIM) [29]. These measures have the distinction of being consistent
with subjective ratings [29,30]. The peak signal-to-noise ratio (PSNR) was not considered as a figure of
meritbecause ofitslimitedcapability ofcapturing thehumanperception ofimagefidelityandquality [31].
Foreachvalueofr,weconsideredaveragemeasuresacrossallconsideredimages. Suchmethodologyisless
pronetovarianceeffectsandfortuitousdata. Figure2showstheresultingSSIMandSR-SIMmeasurements.
Theproposed transform performed veryclosely totheexactDTT.Forqualitative purposes, Figure3shows
compressed images according to the DTT and the proposed approximation for r =6; images are visually
5
1.0
0.9
M 0.8
I
S
S
e
g 0.7
a
r
e
v
A
0.6
DTT
0.5 Tˆ∗
0 10 20 30 40
r
(a) SSIM
1.00
0.98
M
I
S 0.96
-
R
S
e
g
a 0.94
r
e
v
A
0.92
DTT
Tˆ∗
0.90
0 10 20 30 40
r
(b) SR-SIM
Figure 2: Quality metrics considering (a) SSIM and (b) SR-SIM for the exact DTT and the proposed ap-
proximation intermsofr.
6
(a) DTT,r=6 (b) Tˆ∗,r=6
Figure3: Compressed‘Lena’imageforr=6bymeansofthe(a) DTTand(b)theproposedapproximation.
indistinguishable.
3.2 Video Compression
With the objective of assessing the proposed transform performance in video coding, we have embedded
the proposed DTT approximation in the widely employed software library x264 [32] for encoding video
streams into the H.264/AVC standard [21]. The 8-point transform employed in H.264/AVC is an integer
approximation of the DCT that demands 32 additions and 14 bit-shifting operations [33]. In comparison,
the proposed 8-point direct transform requires 38% less additions and no bit-shifting operations, while the
proposed inverse transform requires 9% less additions and 43% less bit-shifting operations. We encoded
eleven CIF videos with 300 frames at 25 frames per second from a public video database [34] with the
standard and the modified libraries. In our simulation, we employed default settings and controlled the
video quality by two different approaches: (i) target bitrate, varying from 100 to 500 kbps with a step of
50 kbps and (ii) quantization parameter (QP), varying from 5to 50 withsteps of5units. Forvideo quality
assessment, we submitted the luma component of the video frames to average SSIM evaluation relative
to the Y component (luminance). Results are shown in Figure 4. Even in scenarios of high compression
(lowbitrate/high QP),thedegradation relatedtotheproposed approximation isintheorderof0.01unitsof
SSIM; therefore, very low. Figure 5 displays the first encoded frame of a standard video sequence at low
targetbitrate(200kbps). Theresulting compressed framesarevisually indistinguishable.
7
1.00
0.98
0.96
M
I
S
S
Y- 0.94
e
g
a
r
ve 0.92
A
0.90 H.264
proposed
0.88
100 200 300 400 500
Target bitrate (kbps)
(a)
1.00
0.95
0.90
M
I
S
S
0.85
Y-
e
g
a
r 0.80
e
v
A
0.75
H.264
proposed
0.70
10 20 30 40 50
QP
(b)
Figure4: Videoqualityassessment intermsof(a)fixedtargetbitrateand(b)quantization parameter.
8
(a) H.264/AVC (b) ModifiedH.264/AVC
Figure 5: First frame of the compressed sequence ‘Foreman’ according to (a) the original H.264/AVC and
(b)modifiedH.264/AVCwiththeproposed approximation.
4 VLSI Architectures
Tocompare hardware resource consumption ofthe proposed approximate DTTagainst the exact DTTpro-
posed in [6], the 1-D version of both algorithms were initially modeled and tested in Matlab Simulink and
thenwerephysicallyrealizedonaXilinxVirtex-6XC6VLX240T-1FFG1156fieldprogrammablegatearray
(FPGA)deviceandvalidatedusinghardware-in-the-loop testingthroughtheJTAGinterface. Bothapproxi-
mations wereverifiedusing morethan 10000 testvectors withcomplete agreement withtheoretical values.
Results areshown in Table2. Metrics, including configurable logic blocks (CLB)and flip-flop (FF)count,
critical path delay (CPD, in ns), and maximum operating frequency (F , in MHz) are provided. In addi-
max
tion, static (Q , in mW) and frequency normalized dynamic power (D , in mW/MHz) consumptions were
p p
estimated using the Xilinx XPower Analyzer. The final throughput of the 1-D DTT was 438.68 106 8-
×
point transformations/second, with a pixel rate of 3.509 109 pixels/second. The percentage reduction in
×
the number of CLBs and FFs was 64.7% and 71%, respectively. The dynamic power consumption D of
p
the proposed architecture was 44.9% lower. The figures of merit area-time (AT) and area-time2 (AT2) had
percentage reductions of66.1%and67.5%whencomparedwiththeexactDTT[6].
5 Conclusion
In this paper, a low-complexity approximation for the 8-point DTT was proposed. The arithmetic cost of
the proposed approximation are significantly low, when compared with the exact DTT. At the same time,
theproposed toolisveryclose totheDTTintermsofimagecoding forawiderange ofcompression rates.
In video compression, the introduced approximation was adapted into the popular codec H.264 furnishing
virtuallyidenticalresultsatamuchlesscomputational cost. Ourgoalwiththecodecexperimentation isnot
9
Table2: Resourceconsumption onXilinxXC6VLX240T-1FFG1156device
Method
Resource
ExactDTT[6] Proposed
CLB(A) 408 144
FF 1370 396
CPD(T)(ns) 2.390 2.290
F (MHz) 418.41 438.68
max
AT 975.1 329.7
AT2 2330.5 755.1
D (mW/MHz) 5.10 2.81
p
Q (W) 3.44 3.44
p
to suggest the modification of an existing standard. Our objective is to demonstrate the capabilities of the
proposed low-complexity transform in asymmetric codecs [35]. Such codecs are employed when a video
isencoded oncebut decoded several timesinlowpowerdevices [35,36]. Additionally, theproposed trans-
form canbeconsidered indistributed video coding (DVC)[36,37], wherethecomputational complexity is
concentratedinthedecoder. ArelevantcontextforDVCisinremotesensorsandvideosystemsthatarecon-
strained intermsofpower,bandwidth, andcomputational capabilities [36]. Theproposed approximation is
aviablealternative totheDTT;possessing low-complexity andgoodperformance according tomeaningful
image quality measures. Moreover, the associated hardware realization consumed roughly 1/3 of the area
required by theexact DTT;also thedynamic powerconsumption wasdecreased by44.9%. Future workin
thisfieldmayconsidertheevaluation ofDTTapproximations inquantization schemes[4,5].
Acknowledgments
ThisworkwassupportedbytheCNPq,FACEPE,andFAPERGS,Brazil;andtheUniversityofAkron,Ohio,
USA.
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