Low-complexity Image and Video Coding Based on an Approximate Discrete Tchebichef Transform

Lo w-complexit y Image and Video Coding Based on an Appro ximate Discrete Tc hebic hef T ransform Paulo A. M. Oliveira ∗ Renato J. Cintra † F´ abio M. Bay e r ‡ Sunera Kulasekera § Arjuna Madanayak e § V ´ ıtor A. Coutinho ¶ Abstract The usage of linear transformations has great r elev ance for data decorrelation applications, li k e image and video compression. In that sense, the discrete Tch ebic hef transform (DTT) p ossesses useful co ding and decorrelation proper ties. The DTT transform k ernel does not dep end on the input data and fast algorithms can b e developed to real time applications. How eve r, the DTT fast algorithm presen ted i n literature possess high computational complexity . In this w ork, we in tro duce a new lo w-complexity appro ximation for the DTT. The fast algorithm of the prop osed transfor m is multiplication-free and requires a reduced num b er of additions and bit-shifting op er ations. Image and video compression simulations in p opular standards shows go o d p erformance of the prop osed transform. Regarding hardwa re r esource consumption for FPGA s ho ws 43.1% reduction of configurable logic blo ck s and A SIC place and route r eali zation shows 57.7% reduction in the area-time figure when compared wi th the 2-D version of the exact DTT. Keywords Approx imate transforms, discrete Tche bich ef transfor m, fast algori thms, image and video co ding 1 Intr oduction Discrete v ariable orthogonal p olynomials emerge as solutions of sever al hyp ergeometric d ifference equations [1 ]. Classic applica- tions of this class of orthogonal p olynomials include functional analysis [2] and graphs [3]. Additionally , suc h p olynomials are emplo yed in the computation of moment functions [4 ], which are largely used in image pro cessing [5–7]. F or instance, the discrete Tc h ebichef moments [8], whic h are derived from the dis- crete Tchebic h ef p olynomials, form a set of orthogonal moment functions. S uch functions are not discrete ap p ro ximation b ased on contin u ou s functions; they are n aturally orthogonal over the discrete domain. The Tc hebichef momen ts hav e been u sed for quantifying image block artifact [9], image recognition [10–12], blind integrit y ver- ification [13], and image compression [14–18]. In the d ata com- pression context, bi-dimensional (2-D) moments are computed by means of the 2-D discrete Tchebic hef t ransform (DTT). In fact, the 8-p oin t DTT can ac h ieve b etter performance when com- parison with t he discrete cosine transform (DCT) [19], in terms of av erage bit length as rep orted in [16 , 20, 21]. Moreo ver, the 8-p oint DTT-based embedded enco der prop osed in [18], shows impro ved image quality and reduced enco ding/decod ing t ime in comparison with state-of-the-art DCT-based embedded co ders. The 8-p oint DTT has also b een employ ed in blind forensics, as a tool to d etermine the integrit y of medical imagery sub ject to ∗ Paulo A. M. Oliveira was with the Sign al Pro ce ssing Group, Depar- tamento de Estat ´ ıstica, Universidade F ede ral de Pernambuco, Recife, PE , Brazil; M ultimed ia Communications an d Signal P roc essing, University of Erlangen–Nu rember g, Erlangen , BY, Germ any . † Renato J. Cintra i s with th e Signal Pro ce ssing Group, Universi- dade F ederal de Pernambuco, Caruaru, PE, Brazil. He w as with ´ Equip e Cairn , INRIA-IRISA, Universit´ e d e Rennes, Rennes, F ranc e; an d LIRIS, In stitut Nati on al de s Sc ience s Appl iqu´ ees, Ly on, F rance (e-mail: rjdsc@de . ufp e.b r). ‡ F´ abio M. Ba yer is with the Departame nto de Estat ´ ıstica and LAC ESM, Universidade F e deral de Santa Maria, Santa Maria, RS, Brazil (e-mai l : bay er@u fsm.br). § Sunera Kulasekera and Arj una Madanay ake were with the Departm ent of El ectrical and Compute r Engine ering at th e Un iversit y of Akron, OH. ¶ V ´ ıtor A. Coutinho w as with th e Signal Pro cessing Group, Departa- mento de E stat ´ ıstica, UFPE, Rec i fe, Brazil. filtering and compression [13]. How ever, the exact DTT p ossesses high arithmetic complex- it y , due to its significant amount of add itions and float-p oint multipli cations. Such multiplicatio ns are k now n to b e more de- manding computational structures th an additions or fix ed-p oint multipli cations, both in softwar e and har dwar e . Th us, the h igher computational complexity of t h e DTT p recludes its applications in lo w p ow er consumption systems [22, 23] and/or real-time pro- cessing, such as video streaming [24, 25]. Therefore, fast algo- rithms for the DTT could improv e its computational efficiency . A comprehensive literature search reveals only tw o fast algo- rithms for the 4-p oint DTT [14, 26] and one for the 8-p oint DTT [15]. Although th ese fast algorithms p ossess lo w er arith- metic complex ities when compared with the direct DTT calcu- lation, they still p ossess high arithmetic complexity , requ iring a significan t amount of additions and bit-shifting op erations. In a comparable scenario, th e computation of DCT-based transforms—whic h h as b een employ ed in several p opular co ding sc hemes suc h as JPEG [27], MPEG-2 [28], H.261 [29], H.263 [30], H.264 [31], H EVC [32, 3 3], and VP9 [34]—has profited from matrix appro x imation theory [35–41]. I n this context, discrete transforms are not exactly calculated, but instead an appro xi- mate, low-cos t comp u tation, is p erformed. The approximations are designed in such a w ay to allow similar sp ectral and co ding chara cteristics as w ell as lo wer arithmetic complexity . Usually , approximatio ns are m ultiplierless, requiring only ad d ition and bit-shifting operations for its computation. In [42], a multiplier- less appro ximation for the 8-p oint DTT i s prop osed. T o the best of our k n o wledge, this is the only DTT apro x imation arc hived in literature. The aim of this w ork is to introd uce an efficien t low-complexit y approximatio n for the 8-p oin t DTT capable of outperform- ing [42]. T o derive m ultiplierless approximate DTT matrix, a multicri teria optimization problem is sought, combining d iffer- ent co ding metrics: co ding gain and transform efficiency . Addi- tionally , a fast algorithm for efficient computation of t he sought approximatio n is also pursued. F or co ding p erformance ev alu- ation, we prop ose tw o computational exp eriments: (i) a JPEG image co mpression sim ulation and (ii) a v ideo co ding exp eriment 1 whic h consists of embedd ing the sought approximation in to th e H.264/A VC standard. The paper unfolds as follo ws. Section 2 reviews the mathemat- ical background of the DTT. Section 3 introduces a parametriza- tion of th e D TT to derive a family of DTT approxima tions and sets up an optimization problem to identify optimal approxima- tions. In Section 4 , w e assess th e obtained approximatio n in terms of co ding p erformance, proximit y with th e exact trans- form, and computation cost. Moreo ver, a fast algorithm for the prop osed approximate DTT is introdu ced. Section 5 show s the results of th e image and video compression simulatio ns. Sec- tion 6 shows hardware resource consumption comparison with the exact DTT for b oth FPGA and A SIC realizations. A d iscus- sion and final remarks are sho wn in Section 7. 2 Discrete Tchebiche f Tra nsfo rm 2.1 Discrete Tchebiche f Pol ynomials The d iscrete Tc hebichef p olynomials are a set of discrete vari able orthogonal p olynomials [43]. The k th order discrete Tc hebichef p olynomials are given by t h e f ollo wing closed form ex pres- sion [14]: t k [ n ] = (1 − N ) k · 3 F 2 ( − k , − n, 1 + k ; 1 , 1 − N ; 1) , where n = 0 , 1 , . . . , N − 1, 3 F 2 ( a 1 , a 2 , a 3 ; b 1 , b 2 ; z ) = P ∞ n =0 ( a 1 ) k ( a 2 ) k ( a 3 ) k ( b 1 ) k ( b 2 ) k z k k ! is th e generalized hypergeometric func- tion and ( a ) k = a ( a + 1) · · · ( a + k − 1) is t h e descend ant facto- rial. Tchebic hef p olynomials can be obtained according to the follo wing recursion [14]: t k [ n ] =  2 k − 1 k t k [1]  t k − 1 [ n ] −  k − 1 k  N 2 − ( k − 1) 2   t k − 2 [ n ] , for t 0 [ n ] = 1 and t 1 [ n ] = 2 n − N + 1. Ind eed , the set { t k [ n ] } , k = 0 , 1 , . . . , N − 1, is an orthogonal basis in resp ect with th e unit weigh t. Consequently , th e d iscrete Tchebic hef p olynomials satisfy the follo wing mathematical relation: N − 1 X i =0 t i [ n ] t j [ n ] = ρ ( j, N ) · δ i,j , where ρ ( k , N ) = ( N + k ) ! (2 k +1) · ( N − k − 1)! and δ i,j is the K ronec ker d elta function which yields δ i,j = 1, if i = j , and δ i,j = 0, otherwise. 2.2 2-D Discrete Tchebiche f Transform Let f [ m, n ], m, n = 0 , 1 , . . . , N − 1, be an intensit y d istribu tion from a discrete image of size N × N pixels. The 2-D DTT of f [ m, n ], denoted by M [ p, q ], p, q = 0 , 1 , . . . , N − 1, is given by [8, 14]: M [ p, q ] = N − 1 X m,n =0 e t p [ m ] · e t q [ n ] · f [ m, n ] , (1) where e t k [ n ], k = 0 , 1 , . . . , N − 1, are the orthonormali zed d iscrete Tc hebichef p olynomials given by e t k [ n ] = t k [ n ] / p ρ ( k , N ). Note that the transform kernel describ ed in (1) is separable. Hence, the follo wing is relation holds true: M [ p, q ] = N − 1 X m =0 e t p [ m ] N − 1 X n =0 e t q [ n ] · f [ m, n ] , for p, q = 0 , 1 , . . . , N − 1. Therefore, the t ransform-domain coef- ficients of f can b e calculated by the follo wing matrix op eration: M = T · f · T ⊤ , (2) where T is the N -p oin t unid imensional DTT matrix giv en by T =    e t 0 [0] e t 0 [1] ··· e t 0 [ N − 1] e t 1 [0] e t 1 [1] ··· e t 1 [ N − 1] . . . . . . . . . . . . e t N − 1 [0] e t N − 1 [1] ·· · e t N − 1 [ N − 1]    . The matrix op erations induced by (2) represents the 2-D DTT. Because of t he kernel separation prop erty , the 2-D DTT can b e calculated by means th e su ccessiv e applications of the 1-D DTT to the row s of f ; and then to columns of the resulting intermedia te matrix. The original intensit y distribution f can b e reco vered by the inverse p roced u re: f = T − 1 · M · ( T − 1 ) ⊤ = T ⊤ · M · T . The last equality abov e stems from the D TT orthogonalit y prop- erty: T ⊤ = T − 1 [14]. Therefore, th e same structure can b e u sed at t h e forw ard transform as w ell in the inv erse. F or N = 4 and N = 8, we hav e th e particular cases of interest in the context of image and video co d ing. Th us, the 4- and 8-p oint DTT matrices are, resp ectively , furnished by: T 4 = F 4 ·  1 1 1 1 − 3 − 1 1 3 1 − 1 − 1 1 − 1 3 − 3 1  and T 8 = 1 2 · F 8 ·     1 1 1 1 1 1 1 1 − 7 − 5 − 3 − 1 1 3 5 7 7 1 − 3 − 5 − 5 − 3 1 7 − 7 5 7 3 − 3 − 7 − 5 7 7 − 13 − 3 9 9 − 3 − 13 7 − 7 23 − 1 7 − 15 15 17 − 23 7 1 − 5 9 − 5 − 5 9 − 5 1 − 1 7 − 21 35 − 35 21 − 7 1     , where F 4 = diag  1 2 , 1 √ 20 , 1 2 , 1 √ 20  and F 8 = diag  1 √ 2 , 1 √ 42 , 1 √ 42 , 1 √ 66 , 1 √ 142 , 1 √ 546 , 1 √ 66 , 1 √ 858  . W e observe that T 4 and T 8 are written as a result from the pro duct of an integ er matrix and a diagonal matrix whic h requires fl oat-p oint representa tion. 3 DTT Appro xim a tions and Coding Optima lity In t h is section, we aim at prop osing an extremely lo w-complexity DTT approximation. Our metho dology consists of generating a class of p arametric approximate matrices and then identif y the optimal class member in terms of co ding p erformance. 3.1 Rela ted Work T o the b est of our knowledge, the only D TT approximation arc hived in literature was prop osed in [42]. That app ro ximation w as obtained by means of a parameterization of integer func- tions combined with a normalization of transformation matrix columns. The derived approximation in [42] furnishes go o d co d- ing capabilities, but it lac k s orth ogonality or near-orthogonality prop erties. As a consequence, the forw ard and invers e transfor- mations are qu ite d istinct and possess u nbalanced computational complexities. 2 3.2 P arametri c Lo w-complexity Ma trices In [35, 36, 38, 44], DCT approximations were prop osed according to follo wing op eration: int( α · C ) , where int( · ) is an integer function, α is a real scaling factor, and C ´ e is the exact DCT matrix. Usual intege r functions include the flo or, ceiling, signal, and round ing functions [38]. In this w ork, these functions op erate element-wise when applied to a matrix argument. A similar approach is sough t for t he prop osed DTT ap p ro x- imation. How ever, in con trast with the DCT, the rows of DTT matrix (basis vectors) ha ve a widely v arying dynamic range. Th u s, the integer fun ct ion ma y excessively p enalize the row s with small dynamic range. T o compensate t h is phenomenon, we normalize the ro ws of T 4 and T 8 accord- ing to left multiplicatio ns by D 4 = diag  2 , √ 20 3 , 2 , √ 20 3  and D 8 = diag  √ 8 , √ 168 7 , √ 168 7 , √ 264 7 , √ 568 13 , √ 2184 23 , √ 264 9 , √ 3432 35  , re- sp ectively . The sought approximations are required to p ossess extremely lo w complexit y . On e wa y of ensuring this prop erty is to a dopt an integ er function whose co-domain is a set of lo w-complexity in- teger. In the DCT literature, common sets are: P 0 = {± 1 } [35], P 1 = { 0 , ± 1 } [36], and P 2 = { 0 , ± 1 , ± 2 } [38]. Note that el- ements from these sets hav e very simple realization in hard- w are; implying multiplierless designs with only addition and b it- shifting op erations [45]. Adopting P 2 , we have that a suitable intege r function is given by: round : [ − 1 , 1] → P 2 = { 0 , ± 1 , ± 2 } , x 7→ round( α · x ) , 0 < α < 5 / 2 , where round( x ) = sign( x ) · ⌊| x | + 1 2 ⌋ is the rounding function as implemented in MA TLAB [46], Octave [47], and Python [48] programming languages. F ollo wing the method ology described in [38], we obtain th e follow ing parametric class of matrices: T N ( α ) = roun d( α · D N · T N ) , N ∈ { 4 , 8 } . (3) 3.3 DTT Appro xima tion A giv en lo w-complexity matrix T N ( α ) can b e u sed to approxi- mate the DTT matrix b y means of orth ogonalizatio n or quasi- orthogonalizatio n as describ ed in [36–38]. As a result, an ap- proximati on for T N , referred to as ˆ T N ( α ), can b e obtained by: ˆ T N ( α ) = S N ( α ) · T N ( α ) , (4) where S N ( α ) = q { ediag( T N ( α ) · T N ( α ) ⊤ ) } − 1 is a diago nal ma- trix, ediag ( · ) returns a diagonal matrix with the diagonal entries of its argumen t and √ · is the m atrix element-wise squ are root operator [38]. I f T N ( α ) · T N ( α ) ⊤ = [diagonal matrix] (5) holds true, then ˆ T N ( α ) is an orth ogonal matrix [49]. Otherwise, it is p ossibly a n ear orthogonal matrix [38]. An approximation is said q uasi-orthogonal when the deviation from diagonalit y of T N ( α ) · T N ( α ) ⊤ is considered small. Let A b e a sq uare real matrix. The d eviation from diagonalit y δ ( A ) is given by [50]: δ ( A ) = 1 − k ediag( A ) k F k A k F , where k · k F is the F rob enius norm for matrices [49]. In the context of ima ge compressio n, a deviation from diagonalit y v alue b elo w 1 − 2 √ 5 ≈ 0 . 1056 indicates qu asi-orthogonalit y [35, 38]. 3.4 Optimiza tion Pr oblem Now our goal is to identify in the family T N ( α ) the matrix that furnishes the b est approximatio n. W e adopted tw o metrics as figures of merit to guide the optimal choice: (i) th e unified co d- ing gain C g [51, 52] and (ii) the transform effici ency η [53]. These metrics are relev ant, b ecause they quantify the transform capac- it y of removing signal redundancy , as w ell as d ata compression and decorrelation [53]. Hence, follo wing the metho dology in [54 ], we prop ose the fol- lo wing multicriteria opt imization p roblem: α ∗ = arg max 0 <α< 5 / 2 n C g  ˆ T N ( α )  , η  ˆ T N ( α ) o , N ∈ { 4 , 8 } , where α ∗ is the scaling parameter that results in the optimal low complexity matrix T ∗ N , ˆ T N ( α ∗ ) according to (3). The ab o ve opt imization problem is not analytically tractable. Thus, we resort to exhaustive numerical search to obtain α ∗ . W e consider linearly spaced v alues of α with a step of 10 − 3 in the interv al 0 < α < 5 / 2. F or N = 4 and N = 8, we obtain th at op- timalit y is found in the in terv als ( 3 2 , 5 2 ) and ( 23 14 , 69 34 ), resp ectively . Therefore, any v alue of α in the aforementioned interv als effects the same appro x imations. F or op erational reasons, we selected α = 2. Thus, the resulting low-complexit y matrices are given b elo w: T ∗ 4 =  1 1 1 1 − 2 − 1 1 2 1 − 1 − 1 1 − 1 2 − 2 1  and T ∗ 8 =     1 1 1 1 1 1 1 1 − 2 − 1 − 1 0 0 1 1 2 2 0 − 1 − 1 − 1 − 1 0 2 − 2 1 2 1 − 1 − 2 − 1 2 1 − 2 0 1 1 0 − 2 1 − 1 2 − 1 − 1 1 1 − 2 1 0 − 1 2 − 1 − 1 2 − 1 0 0 0 − 1 2 − 2 1 0 0     . The associate optimal approximations are denoted by ˆ T ∗ N , ˆ T N ( α ∗ ) and can be computed according t o (4). Hence, w e obtain: ˆ T ∗ 4 = diag  1 2 , 1 √ 10 , 1 2 , 1 √ 10  · T ∗ 4 and ˆ T ∗ 8 = diag  1 √ 8 , 1 √ 12 , 1 √ 12 , 1 √ 20 , 1 √ 12 , 1 √ 14 , 1 √ 12 , 1 √ 10  · T ∗ 8 4 Ev alua tion and Comput a tional C omplexity 4.1 Discussion The obtained matrix T ∗ 4 satisfies (5) and therefore ˆ T ∗ 4 is orthog- onal. In fact, the prop osed matrix is iden tical to the 4-p oint in te- ger transform for H.26 4 enco ding in tro duced b y Malv ar et al. [44]. Therefore, it is also an optimal approximate DTT. Because the 4-p oint DCT approximatio n matrix b y Mal v ar et al . was sub mit- ted to in-dept h analyses in the context of video co ding [31], such results also app ly to T ∗ 4 . Therefore, h ereafter we focus the forth- coming discussions to the prop osed 8-p oint approximation T ∗ 8 . 4.2 Or thogonality a nd Inver tibility The matrix T ∗ 8 does not satisfy (5). Theref ore, the associate approximatio n ˆ T ∗ 8 is n ot orthogonal, i.e. ( ˆ T ∗ 8 ) − 1 6 = ( ˆ T ∗ 8 ) ⊤ . A s a consequ ence, the in verse do es not inherit the low-complexit y prop erties of T ∗ 8 . I n oth er words, t he entries of ( T ∗ 8 ) − 1 are n ot in P 2 . Hence, the prop osed transform p ossesses asymmetrical computational costs when comparing t h e direct and inv erse op- erations [5 5]. The in verse transformation is a m uch required tool, 3 sp ecially for reconstruction en coded images back to the sp atial domain [27, 56]. How ever, T ∗ 8 · ( T ∗ 8 ) ⊤ has a lo w deviation from diagonal- it y [38, 50]: only 0 . 024—roughly 4.4 times less than the devi- ation imp lied by the S DCT [35], which is t aken as the stan- dard reference. Therefore, the follo wing approximation is v alid: ( ˆ T ∗ 8 ) − 1 ≈ ( T ∗ 8 ) ⊤ · diag  1 √ 8 , 1 √ 12 , 1 √ 12 , 1 √ 20 , 1 √ 12 , 1 √ 14 , 1 √ 12 , 1 √ 10  . Moreo ver, since diagonal matrices can b e absorb ed into other computational steps [38, 40, 42], ( T ∗ 8 ) − 1 can b e replaced with the low-complexit y matrix ( T ∗ 8 ) ⊤ . This has th e adv antage of using the same algorithm for b oth forw ard and invers e app ro xi- mations. 4.3 Perfo rmance Assessment The prop osed approximations w ere compared with their corre- sp on d ing exact DTT in terms of codin g p erformance as mea- sured according to the co ding gain and th e transform efficiency . F or N = 8, we also include in our comparisons th e DTT ap- proximati on prop osed in [42]. The co ding p erformance of the prop osed approximations was eval uated according to the fi gures of merit C g and η . T able 1 displays the results. The prop osed approximatio ns are capable of furnishing co ding measures very close to the exact transformations. F or comparison purp oses, the exact DCT has its co ding gain and t ransform efficiency of 8.83 dB and 93.99, resp ectively . Although our goal is to derive go o d approximations for coding, w e also analyzed the resulting approximati ons in terms of prox- imit y metrics. W e separated the mean square error ( MS E) [57], the total energy error ǫ [36], and the transform distortion d [58]. All these measures aim at quantifying the d istance b etw een the exact transformations and th eir resp ective approximations. An - alytic expressions fo r the MSE and ǫ are d et ailed in [54]. W e al so ev aluated the p ro ximity of the prop osed approximatio ns with re- sp ect to the ex act D TT according to the transform distortion measure su ggested in [58]. This metric was originally prop osed as the DCT d istortion in th e context of DCT appro ximations and quantifies in p ercentage a distance b etw een exact and appro xi- mate DCT. Adapting it for the DTT, we obtain t he transform distortion as follo ws: d ( ˆ T ∗ N ) =  1 − 1 N ·    ediag n T N · ( ˆ T ∗ N ) ⊤ o    2 2  × 100% , where k · k 2 is the eucli dean norm for matrices [49]. Low v alues of distortion ind icates pro ximity with the DTT. A s a comparison for N = 4, the 4-p oin t DCT app ro ximation prop osed in [59] has a d istortion of 7 . 32%. Proximit y results are also sho wn in T able 1. 4.4 F ast Algorithm and A rithmetic Complexity Now we aim at deriving fast algorithms for the obtained DTT ap- proximati ons. Being identical to the H.264 4-p oint DCT approx- imation, the d erive d matrix T ∗ 4 w as given fast algorithms in [44]. Although T ∗ 8 is multiplicati on free, without a fast algorithm, its direct implementation requires 44 additions and 24 bit-shifting operations. Th u s we focu s our efforts on the efficient computa- tion of T ∗ 8 . T o such end, a sparse matrix factorization is sought, where the num b er of add itions and bit-sh ifting op erations can b e significantly redu ced [45, 60]. The sparse matrix factorization prop osed in the manuscript w as derived from scratch b ased on usual butterfly structures [ 45]. W e obtained th e follow ing decomp osition: T ∗ 8 = P · A 2 · A 1 · B 8 , where B 8 =     1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 1 − 1 0 0 0 0 0 1 0 0 − 1 0 0 0 1 0 0 0 0 − 1 0 1 0 0 0 0 0 0 − 1     , A 1 =       0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 − 1 2 − 1 0 0 0 0 2 0 − 1 − 1 0 0 0 0 0 0 0 0 2 − 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 2 − 1 0 0 0 0 0 0 0 − 2       A 2 =     1 1 1 0 0 0 0 0 0 0 0 1 − 2 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 − 1 0 1 0 0 0 0 0 0 − 1 0 1 0     , P =     1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0     . Matrix B 8 represents a la yer of b u tterfly structu res, A 1 and A 2 denote additive m atrices with bit- shifting op erations, and P represents a fin al p ermutation, which is cost-free. The result- ing algorithm p reserv es all algebraic and co ding prop erties of the direct computation, while requiring less arithmetic op era- tions. Moreo ver, the factor of 2 of th e first matrix row can b e absorbed into the diagonal matrix. The obtained factori zation is represented by t he signal fl o w graph sho wn in Figure 1. Such algorithm reduces th e arithmetic cost of the proposed approxi- mation to only 24 additions and six bit-shifting op erations. T able 2 compares the arithmetic complexit y of the discussed metho d s ev aluated according to b oth their respective fast algo- rithms. The add itive and total arithmetic complexity of the prop osed approximation are 45.5% and 58.9% lo w er than the ex- act transform, respectively . Although, th e computational cost of the p rop osed approximation is slightly h igher th an th e f orwar d DTT approximation in [42], it is imp ortant to notice th at the inv erse t ransformation in [42] is relatively more complex. Con- sidering th e combination of forward and inverse transformation, the design in [42] requires 49 additions, whereas t h e prop osed design requires 48 additions. Bit-shifting costs are virtually null, b ecause in a hardware implementation t hey represent only wiring. F or comparison, the p opular L o effler DCT algorithm [61] requires 11 floating-p oin t m ultiplications and 29 additions. Al- though t he actual approximation consists of the multiplication of the lo w-complexity matrix and a d iagonal matrix as shown in (4 ), the multiplications introd uced by diagonal matrices rep- resen t n o additional arithmetic complexity in image compres- sion app lications. This is b ecause th ey can b e absorb ed into the image qu antiz ation step [27, 56] of JPEG-lik e image compres- sion [36–41]. F urthermore, t h e new approximation is capable of a b etter coding p erformance and p ossesses one order of magni- tude lo wer pro ximity measure as show n in T able 1 . Therefore , b oth performance and arithmetic complexity measures are fa vor- able to the prop osed approximation. 5 Ima ge and Video Compression Expe riments In this section, w e p erform tw o computation exp erimen ts. The first one consists of still image compression considering a JPEG 4 T able 1: Perfo rmance assessment N Method C g (dB) η MSE ǫ d δ 4 Exact DTT [26] 7.55 97.25 - - - 0 Proposed 7.55 97.33 0.001 0.13 0.29% 0 8 Exact DTT [15] 8.68 92.86 - - - 0 DTT Approx. [42] 5.51 83.51 0.015 3.32 12.61 % 0.09 Proposed 9.25 92.71 0.002 0.77 3.03% 0.024 x 7 x 6 x 5 x 4 x 1 x 0 x 2 x 3 X 2 X 0 X 4 X 6 X 5 X 3 X 1 X 7 Figure 1: Signal flow graph for T ∗ 8 . I nput d ata x n , n = 0 , 1 , . . . , 7, relates to output X m , m = 0 , 1 , . . . , 7. Dashed arrows and blac k no des represent multiplications by − 1 and 2, respectively . T able 2: F ast algorithm arithmetic complexity comparison Method Mult. Adit. Shifts T otal Exact DTT [15] 0 44 29 73 F orw ard DTT approx. [42] 0 20 0 20 Inv erse DTT approx. [42] 0 29 8 37 Proposed 0 24 6 30 5 proced ure. The second sim ulation assess the effectiveness of the prop osed ap p ro ximations under realistic v ideo en cod in g condi- tions. 5.1 Image Compressi on W e adopted the image compression metho d according to the JPEG standard [27 , 56]. A set of 45 512 × 512 8-bit images were obtained from a p ublic image bank [62] and submitted to process- ing. The selected images encompass a wide range of categorie s, including 13 textures, 12 satellite images, three human faces and sever al other miscellaneous scenarios. F or each image, the lumi- nance comp onent w as ex tracted and sub divided into 8 × 8 blocks, I i,j , i, j = 1 , 2 , . . . , 64. After prepro cessing, each block was sub- mitted to the the follo wing operation: M i,j = P · I i,j · P ⊤ , where M i,j is t he 2-D transform-domain data and P is a given 1-D transformation matrix, such as the ex act or approximate DTT. Then each subblo ck M i,j w as element-wise divided by a quantization matrix, yielding the q uantized JPEG coefficients J i,j , as follo ws: J i,j = round ( M i,j ⊘ Q ) , where Q = ⌊ ( S · Q 0 + 50) / 100 ⌋ is the q uantizatio n matrix, ⌊·⌋ denotes the flo or function, the default quantization table is Q 0 =     16 11 10 16 24 40 51 61 12 12 14 19 26 58 60 55 14 13 16 24 40 57 69 56 14 17 22 29 51 84 80 62 18 22 37 56 68 1 09 103 77 24 35 55 64 81 1 04 113 92 49 64 78 87 1 03 121 120 1 01 72 92 95 98 1 12 100 103 99     , S = 5000 /QF , if QF < 50, and 200 − 2 · QF otherwise, and QF is the q uality factor [56]. I f QF = 50, then Q = Q 0 . Decreasing v alues of QF lead to higher compression ratios (with image total destruction at QF = 0); whereas increasing val ues leads to lo w er compression ratios (with b est p ossible quality at QF = 100). I n our exp eriments, we adopt ed QF varying from 10 to 90 in steps of 5. In the JPEG d ecoder, each sub-blo ck is initially arithmetic decod ed and dequantized according to: ˆ M i,j = J i,j ⊙ Q . Then, the sub-blo cks are inv erse transformed: ˆ I i,j = P − 1 · ˆ M i,j · P −⊤ . Original and compressed images w ere compared for image degradation. The structural similarity ind ex (S SIM) [63] and the sp ectral residual based similarit y (SR-SIM) [64] w ere separated as image q ualit y measures. The SS I M takes into account lumi- nance, contras t, and the image structure to qu antify th e image degradation, b eing consistent with sub jectiv e quality measure- ments [65]. On its turn, the SR- S IM is based on the hyp othesis that th e visual saliency maps of n atural images are closely re- lated to their p erceived q ualit y . This measure could outp erform sever al state-of-the-art fi gu res of merit in exp eriments wi th stan- dardized datasets [64]. W e did n ot consider the p eak signal-to- noise ratio ( PS NR) as a q ualit y measure, b ecause it is n ot a su it- able metric to capt ure the human p erception of image fidelity and q ualit y [57 ]. F or eac h va lue of QF , w e considered a verage measure va lues instead of values from particular images. This ap- proac h is les s prone to vari ance effects and fortuitous data [36, 66]. F or direct compariso n, we selected the exact DTT [15], the ap- proximati on p roposed in [42], and the prop osed appro ximation. As an ext ra reference, we also included the results from the stan- dard JPEG, which is based on the ex act DCT. Figure 2 d ispla y s the results. F or both selected measures, th e prop osed app ro x- imation p erformed very closely to the ex act DTT, specially at high compression ratios. It could outp erform the DTT app ro x- imation in [42] in terms of SSIM an d SR -SIM for Q F < 80 and Q F < 55, resp ectively . I t shows th at the prop osed approx- imation is more efficient in the scenario of high and mo derate compression, which are the very common cases [67], suitable for lo w-p o we r d ev ices. The app ro ximation in [42 ] could attain a b etter p erformance at low compression ratios b ecause it satisfies the p erfect reconstruction p rop erty—alb eit at the exp ense of an inv erse tran sformation with h igher arithmetic complexit y . On the other h and, the prop osed appro ximation explores the near- orthogonalit y prop erty which could excel in mo derate to h igh compression scenarios—whic h are often more relev ant [67]. F or qu alitativ e ev aluation purp oses, Figure 3 sh o ws the compressed Lena image according t o the exact DTT and the prop osed app ro ximation. W e adopted the scenario of high/moderate compression with Q F = 15 and Q F = 50, respec- tively . In both cases, the approximate transform was capable of prod ucing comparable results to the exact DTT with visually similar images. 5.2 Video Compression Sim ula tion T o ev aluate t he prop osed transform p erformance in v ideo co d- ing, w e hav e embedded th e DTT app ro ximation in the widely used x264 softw are library [68] for encoding video streams into the H.264/A VC stand ard [31]. The default 8-point transform em- plo yed in H.264 is the foll ow ing integer DCT appro ximation [69]: ˆ C = 1 8 ·     8 8 8 8 8 8 8 8 12 10 6 3 − 3 − 6 − 10 − 12 8 4 − 4 − 8 − 8 − 4 4 8 10 − 3 − 12 − 6 6 12 3 − 10 8 − 8 − 8 8 8 − 8 − 8 8 6 − 12 3 10 − 10 − 3 12 − 6 4 − 8 8 − 4 − 4 8 − 8 4 3 − 6 10 − 12 12 − 10 6 − 3     . The fast algorithm for the abov e transf ormation requires 3 2 addi- tions and 14 bit-shifting op erations [69]. Therefore, the prop osed 8-p oint transform requires 25% less add itions and 57% less bit- shifting operations than the fast algorithm for ˆ C . Elev en 300-frame common intermediate format (CIF) videos obtained from an online test video database [70] w ere encod ed with t he standard softw are and then with th e mod ifi ed softw are. W e employ ed the softw are default settings and conducted the sim ulation und er tw o scenarios: (i) target bitrate v arying from 50 to 500 kbps with steps of 50 kbps and (ii) q uantizatio n pa- rameter (QP) v arying from 5 to 50 with steps of 5. Psycho visual optimization was disabled in order to obtain val id SSIM v alues. Besides PS NR ev aluation, the discussed softw are library [68] of- fers natively SSIM measurements for video quality assessment. Average SSIM of th e lu ma comp onent were compu ted for all re- constructed frames. The results are shown in Figure 4 in terms of the absolute p ercentage error ( A PE) [38] of the S SIM with re- sp ect to the standard DCT-based transformation in the original H.264/A VC co dec. This measure is given by: APE(SSIM) =     SSIM H.264 − SS IM P SSIM H.264     , where SSIM H.264 returns the SSIM figures as comput ed accord- ing to the H.264 standard and SSIM P represents the SSIM when the exact DTT, the approximation in [42], or the prop osed approximatio n are considered. SSIM curves for the DCT are absent, b ecause they were employ ed as p erformance references. The use of the proposed transform effects a minor degradation in the video q uality . It also could p erform b ett er than t h e previous approximatio n in all cases. 6 10 20 30 40 50 60 70 80 90 Quality factor 0 . 70 0 . 75 0 . 80 0 . 85 0 . 90 0 . 95 1 . 00 A verage Y -SSIM JPEG Exact DTT [15] Approx. [42] Proposed (a) 10 20 30 40 50 60 70 80 90 Quality factor 0 . 960 0 . 965 0 . 970 0 . 975 0 . 980 0 . 985 0 . 990 0 . 995 1 . 000 A verage Y -SR-SIM JPEG Exact DTT [15] Approx. [42] Proposed (b) Figure 2: Average SSIM ( top) and SR-SI M (b ottom) measurements for image compression for the considered transforms at several v alues of QF . (a) Exact DTT [15], QF = 15 (b) Exact DTT [15], QF = 50 (c) Prop osed, QF = 15 (d) Prop osed, QF = 50 Figure 3: Compressed ‘Lena’ image for QF = 15 and QF = 50. 100 200 300 400 500 T arget bitrate (kbps) 0 . 00 0 . 01 0 . 02 0 . 03 0 . 04 0 . 05 A verage Y -SSIM APE Exact DTT [15] Approx. [42] Proposed (a) 10 20 30 40 50 QP 0 . 000 0 . 005 0 . 010 0 . 015 0 . 020 0 . 025 0 . 030 0 . 035 0 . 040 A verage Y -SSIM APE Exact DTT [15] Approx. [42] Proposed (b) Figure 4: Video q ualit y assessmen t in t erms of target bitrate and QP . 7 Figure 5 displa ys the first encoded f rame of tw o stand ard video sequences at low bitrate (200 kbps). The compressed frames re- sulting from the original and mo dified co d ecs are visually indis- tinguishable. 6 Hardw are Section T o compare the h ardw are resource consumpt ion of the pro- p osed approximate D TT against the exact DTT fast algorithm prop osed in [15], algorithm p rop osed in [42] and t h e Lo effler DCT [61], the 2-D versio n of b oth algorithms were initiall y mo d- eled and tested in Matlab Simulink and t h en were physically realized on a Xilinx Virtex-6 XC 6VSX475T-1FF1759 Reconfig- urable Op en Architecture Computing Hardware -2 (ROA CH2) b oard [71]. The RO ACH2 board consists of a X ilinx Virtex 6 FPGA, 16 complex analog-to-digital converters (ADC), multi- gigabit transceivers and a 72-bit DDR3 RAM. The 1-D versions w ere initially mo deled and th e 2-D versions w ere generated using tw o 1-D designs along with a t ransp ose buffer. Designs were verified u sing more than 10000 test vectors with complete agreemen t with theoretical val ues. R esults are sho wn in T able 3. Metrics, includ ing configurable logic b locks (CLB) and flip-fl op (FF) count, critical p ath delay ( T cp d , in ns), and maxim um operating frequency ( F max , in MHz) are provided. The p ercentage reduction in the number of CLBs and FFs w ere 43.2% and 25.0%, resp ectively , compared with the exact D TT fast algorithm prop osed in [15]. In is imp ortant to emph asize that the approximation in [42] is asymmetric; th e f orw ard and in - verse transform p ossess differen t structures, being the inv erse op- eration more complex (cf. T able 2). F or comparisons, w e adopt the av erage measurement b etw een forw ard and inv erse realiza- tions. The proposed appro ximation could pro vide higher maxi- mum op erating frequency with improv ements of 85.9%, 43.5%, and 9.7% when compared to the Lo effler DCT [61 ], t h e exact DTT [15], and the design in [42 ], resp ectively . The ASIC realizatio n wa s done by p orting the hardw are de- scription language co de to 0.18 um CMOS technolog y an d was sub jected to syn thesis and place-and-route according to the Cadence Encounter Digital Implementation (EDI) for A MS li- braries. Libraries for the b est case scenario were emplo yed in getting the place-and-route results with gate voltage of 1.8 V . The adopted figures of merit for th e ASIC synthesis we re: area ( A ) in mm 2 , area-time complexity ( AT ) in mm 2 · ns, area-time- squared complexity ( AT 2 ) in mm 2 · n s 2 , dynamic ( D p ) p o w er consumption in mW/MHz, critical p ath delay ( T cpd ) in ns, and maximum operating frequency ( F max ) in MHz. Results are d is- pla yed in T able 4. The figures of merit AT and AT 2 had p er- centag e reduct ions of 57.7% and 57.4% when compared with the exact DTT. Th us, the prop osed design could attain reductions of 17.3%, 20.6%, and 82.1% for area, AT 2 , T cp d , and dynamic p o we r consumption, resp ectively , when compared to [42]. 7 Discussion and Conclusion In t h is work, a lo w-complexity near-orthogonal 8-p oint DTT approximatio n suitable for image and video co ding wa s pro- p osed. A fast algorithm for prop osed DTT approximati on whic h requires only 24 add itions and six bit-shifting op erations was also introduced. This fast algorithm can b e used for b oth for- w ard and near in verse transformations. The additive arithmetic cost of th e prop osed approximatio n is 45.5% and 2.05% low er when compared with th e exact DTT fast algorithm an d the DTT app ro ximation in [42], resp ectively . Moreo ver, the pro- p osed transform exhibited similar co ding p erformance with th e exact DTT and outp erformed previous app ro ximations [42] ac- cording to computational exp eriments with p opular visual im- age compression stand ards. In terms of video co ding, t he results from the p rop osed to ol w ere v irtually indistinguishable from th e ones furn ished by the app ro ximation in [42]. Thus, th e n ew to ol outp erform the comp eting meth od s b oth in compu tational cost and co ding p erformance. The prop osed metho d was embedd ed into the JPEG stand ard and th e standard soft wa re library for H.264/A VC v ideo co ding. Obt ained results show ed negligible degradation when compared to the standard DCT-based com- pression metho ds in b oth cases. The 2-D versions were realized in FPGA using RO ACH2 hardware platform and A S IC place and route was realized using Cadence encounter with AMS stan- dard cells and the results show a 43 . 1% redu ction in the number of CLB for the FPGA realization and a 57 . 7% reduction in area- time figure for the ASIC place and route realization when com- pared with the exact DTT. 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