Image Compression Based on 2D Dual Tree Complex Wavelet Transform (2D DT-CWT)

Transcription

1 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) D. Salh Husan Al * & Aymen Dawood Salman* Receved on:1/7/9 Accepted on:7/1/1 Abstact By emovng the edundant data, the mage can be epesented n a smalle numbe of bts and hence can be compessed. Thee ae many dffeent methods of mage compesson. Ths pape nvestgates a poposed fom of compesson based on D Dual Tee complex wavelet tansfom (D DT-CWT). Many wavelet coeffcents ae closed to zeo. Thesholdng can modfy the coeffcents to poduce moe zeos that ae allowed a hghe compesson ate. The wavelet analyss does not actually compess a sgnal. Theefoe Huffman codng s used wth a sgnal pocessed by the wavelet analyss n ode to compess data. Wde ange of theshold values s used n the poposed fom. Fom the esults the poposed fom gve hghe ate of compesson and lowe RMS eo compaed wth that foms based on Dscete Wavelet Tansfom (DWT), Dual Tee Real Wavelet Tansfom (DT-RWT) and the well known method based Dscete Cosne Tansfom (DCT). Keywods: Wavelet, Complex Wavelet Tansfom, Image Compesson, Dual Tee Complex Wavelet Tansfom. ضغط الصور بالا عتماد على التحويل المويجي المركب ثناي ي الا بعاد ذي الشجرة الثناي ية الخلاصة يمكن تمثيل الصورة با قل عدد من البايتات با زالة البيانات الفاي ضه. وبذلك سيتم ضغط الصورة. وهناك عدة طراي ق لضغط الصورة. استخدمت في هذا البحث طريقة مقترحة تعتمد على التحويل المويجي المركب ثناي ي الا بعاد ذي الشجرة الثناي ية DT-CWT) D) لضغط الصورة. العديد من المعاملات الخاصة بتحليل المويجي تقترب قيمتها من الصفر. من الممكن تعديل هذه المعاملات وتقريبها للصفر باستخدام العتبة (theshold) لنحصل على نسبة ضغط ا على. ولايعتبر التحليل المويجي هو كاف لضغط البيانات ويستخدم معه ترميز هوفمان Codng) (Huffman لتحسين نسبة الضغط وتم استخدام مدى واسع من قيم العتبة في الطريقة المقترحة وتبين من النتاي ج ا ن الطريقة المقترحة تعطي نسبة ضغط عالية ونسبة خطاء قليلة eo) (RMS بمقارنتها مع.(DCT) والطريقة المعروفة المعتمدة على (DT-RWT) وطريقة (DWT) *Compute Engneeng & Infomaton Technology, Depatment, Unvesty of Technology / Baghdad 19

2 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) 1. Intoducton A typcal stll mage contans a lage amount of spatal edundancy n plan aeas whee adjacent pctue elements (pxels, pels) have almost the same values. It means that the pxel values ae hghly coelated. The edundancy can be emoved to acheve compesson of the mage data. The basc measue fo the pefomance of a compesson algothm s compesson ato (CR), defned as a ato between ognal data sze and compessed data sze. In a lossy compesson scheme, the mage compesson algothm should acheve a tadeoff between compesson ato and mage qualty. Hghe compesson atos wll poduce lowe mage qualty and vce vesa. Qualty and compesson can also vay accodng to nput mage chaactestcs and content [1]. In ecent yeas, many studes have been made on wavelets. Image compesson s one of the most vsble applcatons of wavelets. The apd ncease n the ange and use of electonc magng justfes attenton fo systematc desgn of an mage compesson system and fo povdng the mage qualty needed n dffeent applcatons[1]. Although the dscete wavelet tansfom (DWT) has domnated the feld of mage compesson fo well ove a decade, DWTs n the tadtonal ctcally sampled fom ae known to be somewhat defcent n seveal chaactestcs, lackng such popetes as shft nvaance and sgnfcant dectonal selectvty [].. Complex Wavelet The ctcally sampled dscete wavelet tansfom (DWT) has been successfully appled to a wde ange of sgnal pocessng tasks. Howeve, ts pefomance s lmted because of the followng poblems []. Oscllatons of the coeffcents at a sngulaty (zeo cossngs). Shft vaance when small changes n the nput cause lage changes n the output. Alasng due to downsamplng and non-deal flteng dung the analyss, whch s cancelled out by the synthess fltes unless the coeffcents ae not alteed. Lack of dectonal selectvty n hghe dmensons, e.g. nablty to dstngush o o between + 45 and 45 edge oentatons. To ovecome the shft dependence poblem, they can explot the undecmated (ove-complete) DWT, howeve, wthout solvng the dectonal selectvty poblem. Anothe appoach s nsped by the Foue tansfom, whose magntude s shft nvaant and the phase offset encodes the shft. In such a wavelet tansfom, a lage magntude of a coeffcent mples the pesence of a sngulaty whle the phase sgnfes ts poston wthn the suppot of the wavelet. The complex wavelet tansfom (CWT) employs analytc o quadatue wavelets guaanteeng magntudephase epesentaton, shft nvaance and no alasng []. 191

3 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) Recently, complex-valued wavelet tansfoms CWT have been poposed to mpove upon these DWT defcences, wth the dual-tee CWT (DT-CWT) [3] becomng a pefeed appoach due to the ease of ts mplementaton. In the DT-CWT, eal valued wavelet fltes poduce the eal and magnay pats of the tansfom n paallel decomposton tees, pemttng explotaton of wellestablshed eal-valued wavelet mplementatons and methodologes. A pmay advantage of the DT-CWT les n that t esults n decomposton wth a much hghe degee of dectonalty than that possessed by the tadtonal DWT. Howeve, snce both tees of the DT-CWT ae themselves othonomal o bothogonal decompostons, the DT- CWT taken as a whole s a edundant tght fame []. An analytc wavelet ψ (t) s composed of two eal wavelets ψ (t) and ψ (t) fomng a Hlbet tansfom (HT) pa whch means that they ae othogonal,.e. shfted by π / n the complex plan [4]. ψ ( t) = ψ ( t) + jψ ( t).. (1) c 1 ψ ( t) 1 ψ ( t) = HT{ ψ ( t)} = dτ ψ ( t) π = t τ π t.() And fo the Foue tansfom pas H and H (ω). H (ω) ( ω) = FT{ HT{ ψ ( t)}} = j.sgn( ω) H ( ω c (3) The same concept of analytc o quadatue fomulaton s appled to the flte bank stuctue of standad ) DWT to poduce complex solutons and n tun the CWT. The eal-valued flte coeffcents ae eplaced by complex-valued coeffcents by pope desgn methodology that satsfes the equed condtons fo convegence. Then the complex flte can agan be decomposed nto two eal-valued fltes. Thus, two eal-valued fltes that gve the espectve mpulse esponses n quadatue wll fom the Hlbet tansfom pa. The combned pa of two such fltes s temed as an analytc flte. The fomulaton and ntepetaton of the analytc flte s shown n fgue (1) [4]..1 1D Dual-Tee CWT Smlaly to the DWT, the DT- CWT s a multesoluton tansfom wth decmated subbands povdng pefect econstucton of the nput. In contast, t uses analytc fltes nstead of eal ones and thus ovecomes poblems of the DWT at the expense of modeate edundancy. Runnng n two DWT tees a and b (eal and magnay) of eal fltes, ths tansfom poduces eal and magnay pats of the coeffcents. The wavelet functons ψ (t) and ψ (t) poducng the complex wavelet ψ c (t) fom an appoxmate HT pa. The same apples to the scalng functons φ (t) and φ (t). The wavelet and scalng functons ae elated to the coespondng fltes though the dlaton and wavelet equatons [5]. ψ ( t ) = h1 ( n) φ(t n)..(4) n φ ( t ) = h ( n) φ(t n) (5) n 19

4 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) whee t denotes contnuous tme and n the dscete tme ndex The half-sample delay condton s deved fom a stategy of desgnng fltes so that the wavelets geneated by them fom the Hlbet tansfom pa. Ths happens when the scalng fltes h and g ae offset fom one anothe by a half sample. g ( n) = h ( n.5) (6) As a esult, the lowpass fltes of one tee ntepolate mdway between the lowpass fltes of the othe tee and the DT-CWT s shft nvaant. In the fequency doman, the magntude condton s gven by. jω jω G e ) = H ( e ).(7) ( And the phase condton uns []. G jω e ) = H jω ( e ).5ω (8) ( The fltebank stuctues fo both DT-CWT ae dentcal. Fgue () shows 1-D analyss and synthess fltebanks spanned ove thee levels. It s evdent fom the fltebank stuctue of DT-CWT that t esembles the fltebank stuctue of standad DWT wth twce the complexty. It can be seen as two standad DWT tees opeatng n paallel. One tee s called as a eal tee and othe s called as an magnay tee. Sometmes n futue dscussons the eal tee wll be efeed to as tee-a and the magnay tee as tee-b. The fom of conjugate fltes used n 1-D DT-CWT s gven as ( h ) x + jgx whee, h x s the set of fltes { h, h 1}, and g x s the set of fltes g, } both sets n only x- { g 1 decton (1-D). The fltes h and h 1 ae the eal-valued lowpass and hghpass fltes espectvely fo eal tee. The same s tue fo g and g 1 fo magnay tee. Though the notaton of h and h 1 ae use fo all level n the eal pat of analyss tee, h and h 1 of fst level ae numecally dffeent then the espectve fltes at all othe levels above level-1. The same noton s appled fo magnay tee fltes g and g 1. The synthess flte pas ~ ~ h, h 1,, and g ~ ~, g 1 as shown n fgue ( b) fom othogonal o bothogonal pas wth the espectve countepat fltes of analyss tee as shown n fgue ( a).. D DT-CWT Apat fom that, the D DT- CWT also moe selectvely dscmnates featues of vaous oentatons. Whle the ctcally decmated D DWT outputs thee oentatonal selectve sub-bands pe level conveyng mage featues o o oented at the angles of 9, ± 45, o and, the D DT-CWT poduces sx dectonal subbands pe level to eveal o the detals of an mage n ± 15, o ± 45 and o ± 75 dectons wth 4:1 edundancy [6]. The mplementaton of -D DT- CWT conssts of two steps. Fstly, an nput mage s decomposed up to a desed level by two sepaable -D DWT banches, banch a and banch b, whose fltes ae specfcally desgned to meet the Hlbet pa equement. Then sx hgh-pass subbands ae 193

5 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) geneated: HLa, LH a, HH a, HLb, LHb, and HH b, at each level. Secondly, evey two coespondng subbands whch have the same pass-bands ae lnealy combned by ethe aveagng o dffeencng. As a esult, subbands of -D DT-CWT at each level ae obtaned as LH + LH ) /, ( LH LH ) / b, a ( a b HL + HL ) /, ( HL HL ) / b, a ( a b HH + HH ) /, ( a b ( HH a HH b ) /. The sx wavelets defned by oented, as llustated n fgue (3). Because the sum/dffeence opeaton s othonomal; ths consttutes a pefect econstucton wavelet tansfom. The magnay pat of D DT-CWT has smla bass functon as the eal pat [8]. The -D DT-CWT stuctue has an extenson of conjugate flteng n -D case. The fltebank stuctue of - D dual-tee s shown n fgue (4). -D stuctue needs fou tees fo analyss as well as fo synthess. The pas of conjugate fltes ae appled to two dmensons (x and y) dectons, whch can be expessed as: ( h x + jgx)( hy + jgy) = ( hxhy gxgy) + j( hxg y + gxhy)..(9) The fltebank stuctue of teea, smla to standad -D DWT spanned ove 3-level, s shown n fgue (5). All othe tees-(b,c,d) have smla stuctues wth the appopate combnatons of fltes fo ow- and column- flteng. The oveall -D dual-tee stuctue s 4-tmes edundant (expensve) than the standad -D DWT. The tee-a and tee-b fom the eal pa, whle the tee-c and tee-d fom the magnay pa of the analyss fltebank. Tees-(a ~, b ~ ) and tees- ( c ~, d ~ ) ae the eal and magnay pas espectvely n the synthess fltebank smla to the coespondng analyss pas [4]. 3. Image Compesson Images eque much stoage space, lage tansmsson bandwdth and long tansmsson tme. The only way cuently to mpove on these esouce equements s to compess mages, such that they can be tansmtted qucke and then decompessed by the eceve. Thee ae many dffeent methods of data compesson. Ths nvestgaton wll concentate on tansfom codng and then moe specfcally on Wavelet Tansfoms. Image data can be epesented by coeffcents of dscete mage tansfoms. Coeffcents that make only small contbutons to the nfomaton contents can be omtted. Usually the mage s splt nto blocks (submages) of 8x8 o 16x16 pxels, then each block n dscete cosne tansfom s tansfomed sepaately. Howeve ths does not take nto account any coelaton between blocks, and ceates "blockng atfacts", whch ae not good f a smooth mage s equed. Howeve wavelets tansfom s appled to ente mages, athe than submages, so t poduces no blockng atfacts. Ths s a majo advantage of 194

6 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) wavelet compesson ove othe tansfom compesson methods. Fo some sgnals, many of the wavelet coeffcents ae close to o equal to zeo. Thesholdng can modfy the coeffcents to poduce moe zeos. In Had thesholdng any coeffcent below a theshold λ, s set to zeo. Ths should then poduce many consecutve zeo s whch can be stoed n much less space, and tansmtted moe quckly by usng entopy codng compesson. An mpotant pont to note about Wavelet compesson s "The use of wavelets and thesholdng seves to pocess the ognal sgnal, but, to ths pont, no actual compesson of data has occued". Ths explans that the wavelet analyss does not actually compess a sgnal, t smply povdes nfomaton about the sgnal whch allows the data to be compessed by standad entopy codng technques, such as Huffman codng. Huffman codng s good to use wth a sgnal pocessed by wavelet analyss, because t eles on the fact that the data values ae small and n patcula zeo, to compess data. It woks by gvng lage numbes moe bts and small numbes fewe bts. Long stngs of zeos can be encoded vey effcently usng ths scheme. Theefoe an actual pecentage compesson value can only be stated n conjuncton wth an entopy codng technque. To compae dffeent wavelets, the numbe of zeos s used. Moe zeos wll allow a hghe compesson ate, f thee ae many consecutve zeos, ths wll gve an excellent compesson ate. Thee ae two thesholdng methods fequently used. The softtheshold functon (also called the shnkage functon) s shown n fgue (6 a). η ( x) = sgn( x).max( x T,) T..(1) takes the agument and shnks t towad zeo by the theshold T. The othe popula altenatve s the hadtheshold functon s shown n fgue (6 b). ψ ( x) = x{ x T} (11) T > whch keeps the nput f t s lage than the theshold T ; othewse, t s set to zeo. The wavelet thesholdng pocedue emoves nose by thesholdng only the wavelet coeffcents of the detal subbands, whle keepng the low esoluton coeffcents unalteed [9]. 3.1 Compesson Algothm The dffeent method fo compesson, nvestgate dffe only n the selecton of the method. The basc pocedue emans the same: (1) Dgtse the souce mage nto a sgnal. () Decompose the sgnal s nto wavelet coeffcents usng DT-CWT. (3) Modfy the coeffcents fom w, usng thesholdng. (4) Apply vaable length codng (Huffman codng) to compess. (5) Reconstuct usng the ognal appoxmaton coeffcents usng nvese DT-CWT. Fgue (7) llustates that the compesson algothm usng DT- CWT. 195

7 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) 4. Expemental Results and Dscussons Many standad fles of mages (lena, monach, lmess and stone) ae taken n ths pape to know the stong and weak sdes of the poposed method. Anothe methods such as mage compesson based DWT, D DT-RWT and the well known method (mage compesson based DCT) ae mplemented fo the standad fles of mages that mentoned above. Table (1) epesents the esult of compesson ato and RMS eo fo may standads mages, each mage s compessed by usng the compesson methods based on DWT,D-DTRWT, DCT and the poposed fom that based D DT- CWT. The poposed method gve hghe ate of compesson and lowe RMS eo compaed wth othe methods fo many standad methods. The vaous methods of compesson ae affected by theshold value. Hgh values of theshold gve hghe ate of compesson. Fom Table (1), t s clea that the dffeent method of compesson gves the same compesson ato fo dffeent values of thesholds. Fo example when lena mage s compessed the compesson ato s (5:1), the thesholds values equals to 65, 5, 35 and 3 fo the methods based DCT, DWT, D DT- RWT and D DT-CWT espectvely. Hence the lena compessed mage as shown n fgue (8) s moe smoothng and have lowe RMS eo when the method based D DT-CWT s used. The esults of vaous methods ae explaned n fgue (9) wth the same theshold value (T=4) fo lena mage. Fgue (1) llustates that the compesson ato s mpoved wth nceasng the theshold value. On the othe hand the RMS eo value s nceased when the compesson ate (CR) s nceased as shown n fgue (11). 5. Conclusons Vaous methods of mage compesson based wavelet technques ae affected by theshold values untl the poposed. Image compesson scheme usng D Dual Tee Complex Wavelet Tansfom (DT-CWT). Fo the same compesson ato the compessed mage usng poposed fom s moe smoothng and have lowe RMS eo compaed wth othe methods based wavelet technques and DCT technque. Fo cetan value of theshold the poposed fom based D DT-CWT gve hghe ate of compesson and lowe RMS eo compaed wth that foms based on DWT, DT-RWT and method based (DCT). Refeences [1] S. Ggc and M. Ggc, "Pefomance Analyss of Image Compesson Usng Wavelet", IEEE Tansacton on Industal Electoncs, Vol. 48, No. 3, June 1. [] I. W. Selesnck, R. G. Baanuk, and N. G. Kngsbuy, "The dualtee complex wavelet tansfom, IEEE Sgnal Pocessng Magazne, vol., no. 6, pp , Novembe 5. [3] N. G. Kngsbuy, "Complex wavelets fo shft nvaant analyss and flteng of sgnals," Jounal of Appled Computatonal 196

8 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) Hamonc Analyss, vol. 1, pp , May 1. [4] P. D. Shukla. "Complex Wavelet Tansfoms and The Applcatons," PhD thess, The Unvesty of Stathclyde n Glasgow, 3. [5] I. W. Selesnck. "Hlbet Tansfom Pas of Wavelet Bases," IEEE Sgnal Pocessng Lettes, 8(6):17-173, June 1. [6] E. Hostalkova, A. Pochazka "Hlbet Tansfom Pas of Wavelet Bases," ntenet. [7] J. Yang, J. Xu, F. Wu, Q. Da, and Y.Wang. "Image Codng Usng D Ansotopc Dual Tee Dscete Wavelet Tansfom" IEEE 7. [8] K. Lees. " Image Compesson Usng Wavelets" May. s/capstones/haar/u9kvl.pdf [9] N. Jacob and A. Matn, "Image Denosng In The Wavelet Doman Usng Wene Flteng", e533/poject/f4/jacob_matn.pdf 197

9 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) Image lena mona ch lmees stone Table (1) RMS values of compessed mages fo dffeent test mages and CR 8X8 DCT DWT DT-RWT DT-CWT CR Mm Mm Thes Mm Thes Mm Thes Thes RMS RMS hold RMS e hold RMS e hold hold e e 5: : ; : : : : ; : :1 5:1 1:1 15;1 :1 5:1 5:1 1:1 15;1 :1 5:

10 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) Real Flte h( n) = h ( n) In stande DWT Complex (analytc) Flte h ( n) = h ( n) jh ( n) a + In CWT Real Valued Real Flte h( n) = h ( n) Real Valued Imagnay Flte h( n) = h ( n) Fgue (1) Intepetaton of an analytc flte by -eal flte [4]. Level 1 Level Level 3 h h h f(t) Real Tee Tee a Tee b g Imagnay Tee g g g 1 g 1 g 1 (a) 199

11 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) Level 1 Level Level 3 h h h Imagnay Tee Tee a h Tee b Real Tee h h (b) Fgue () (a) Analyss (b) Synthess flte bank fo 1-D DT-CWT [4]. Fgue (3) Typcal wavelets assocated wth the -D dual-tee oented wavelet tansfom. The top ow llustates the wavelets n the spatal doman, the second ow llustates the (dealzed) suppot of the spectum of each wavelet n the -D fequency plane [8] 13

12 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) Analyss Synthess Tee a ( h xhy ) Tee a ~ ( h ~ xh ~ y ) Tee b ( g xg y ) Tee b ~ ( g ~ xg ~ y ) f (t) ~ f ( t ) Real Tee Tee c ( h xg y ) ~ Tee c ( h xg ~ y ) Tee d ( g xhy ) Imagnay Tee Tee d ~ ( g ~ xh ~ y ) Fgue (4) Flte bank stuctue fo -D DT-CWT [4] Level 1 Level Level 3 h y h x h y1 h y h x h y h y1 h x1 h y h y1 h x h y h y1 h x1 h y1 h y h x1 h y1 Fgue (5) Flte bank stuctue of tee-a of fgue (4) [4] 131

13 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) (a) (b) Fgue (6) (a) Soft Thesholdng (b) Had Thesholdng [9]. Ognal Image Dual Tee CWT Apply Had Theshold Functon Apply Huffman Codng Invese Dual Tee CWT Fgue (7) Image Compesson stuctue usng DT-CWT a)ognal mage

14 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) b)compesson mage usng DCT c)compesson mage usng DWT Compesson Rato 5 : Compesson Rato 5 : 1 d)compesson mage usng DT-RWT e)compesson mage usng DT-CWT Compesson Rato 5 : Compesson Rato 5 : 1 Fgue (8) The pefomance of the vaous methods on lena.tf wth compesson ato (5:1) (a) Ognal. (b) Compesson mage usng DCT. (c) Compesson mage usng DWT. (d) Compesson mage usng DT-RWT. (e) Compesson mage usng DT-CWT 133

15 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) a)ognal mage b)compesson mage usng DCT c)compesson mage usng DWT Compesson Rato 13 : Compesson Rato 18 : 1 d)compesson mage usng DT-RWT e)compesson mage usng DT-CWT Compesson Rato 31 : Compesson Rato 57 : 1 Fgue (9) The pefomance of the vaous methods on lena.tf wth theshold level (T=4). (a) Ognal. (b) Compesson mage usng DCT. (c) Compesson mage usng DWT. (d) Compesson mage usng DT-RWT. (e) Compesson mage usng DT-CWT 134

16 Eng. & Tech. Jounal, Vol. 8, No.7, 1 Image Compesson Based on D Dual Tee Complex Wavelet Tansfom (D DT-CWT) 3 Compesson ato vesus Theshold Pt. 5 8X8 DCT D Standad D Reduced D dual Complx D dual Compesson ato Theshold pt. Fgue (1) Illustates Compesson Rato vesus theshold ponts plot fo lena.tf. 1 RMS eo vesus Compesson ato RMS eo X8 DCT D Standad D Reduced D dual Complx D dual Compesson ato Fgue (11) Illustates RMS eo vesus Compesson Rato plot fo lena.tf 135