OBJECTIVE QUALITY MEASURES FOR COMPRESSED IMAGES

Transcription

1 Nat.Lab. Unclassfed Report UR 000/818 Date of ssue: 08/00 Unclassfed Report OBJECTIVE QUALITY MEASURES FOR COMPRESSED IMAGES Grégory HAMON Konnkljke Phlps Electroncs N.V. 000

2 Contact: Konnkljke Phlps Electroncs N.V. 000 All rghts are reserved. Reproducton n whole or n part s prohbted wthout the wrtten consent of the copyrght owner Konnkljke Phlps Electroncs N.V. 000

3 Unclassfed Report: UR 000/818 Ttle: OBJECTIVE QUALITY MEASURES FOR COMPRESSED IMAGES. Author(s): Grégory HAMON Part of project: Customer: Keywords: Abstract: Image compresson, DCT, human vsual system, mage qualty. After a study and mplementaton of several objectve mage qualty measures, an mage compresson scheme has been modfed to mprove the vsual qualty of the compressed mages. The DCTbased compresson scheme produces a scalable bt stream, whch may be truncated at any pont. Ths enables trval bt rate control and easy adaptablty to transmsson channels wth varyng bandwdth. Prevously, the compresson scheme optmzed the sgnal-tonose rato (SNR) of the compressed mages, but ths measure does not correspond very well to the vsual qualty as t s perceved by human observers. By optmsng the compresson for a perceptually weghted SNR, a better mage qualty has been obtaned. Conclusons: In ths paper, the mage compresson has been mproved and we found an objectve qualty measure based on the human vsual system wth follows our subjectve evaluaton for ths compresson algorthm. Konnkljke Phlps Electroncs N.V. 000

4 Résumé: Après avor etudé et mplémenté pluseurs mesures de qualté d mages numérques, nous avons modfé un algorthme de compresson d mages pour amelorer la qualté vsuelle des mages compressées. L algorthme est basé sur la transformée DCT et génère un flux de bts qu peut-être tronqué a n mporte quel endrot: s le flux de bts est tronqué, l mage compressée sera reconstrute avec les bts dsponbles et elle contendra des erreurs plus ou mons vsbles. Par consequent, le but est de mettre l nformaton la plus mportante au debut du flux de bts. Cet algorthme permet faclement de gérer le nombre de bts par pxel que l on souhate pour reconstrure l mage et de s adapter aux canaux de transmssons avec des largeurs de bande dfferentes. Au debut, le shema de compresson optmsat le rapport sgnal sur brut des mages compressées mas cette mesure de qualté ne correspond pas très ben avec la qualté vsuelle telle qu elle est percue par l observateur human. En optmsant la compresson pour une autre mesure, un rapport sgnal sur brut pondéré perceptuellement, une melleure qualté de l mage a été obtenue. v Konnkljke Phlps Electroncs N.V. 000

5 Phlps profle. World-wde Company Profle. Phlps, one of the world s bggest electroncs companes, was founded n 1891 when Gerard Phlps establshed a company n Endhoven, the Netherlands, to manufacture ncandescent lamps and other electrcal products. The company ntally concentrated on makng carbon-flament lamps and by the turn of the century was one of the largest producers n Europe. Developments n new lghtng technologes fuelled a steady program of expanson, and, n 1914, t establshed a research laboratory to study physcal and chemcal phenomena, so as to further stmulate product nnovaton. In 190, Phlps began to protect ts nnovatons wth patents, for areas takng n X-ray radaton and rado recepton. Ths marked the begnnng of the dversfcaton of ts product range. Havng ntroduced a medcal X-ray tube n 1918, Phlps then became nvolved n the frst experments n televson n 195. It began producng rados n 197 and had sold one mllon by 193. One year later, t produced ts 100 mllonth rado valve, and also started producton of medcal X-ray equpment n the Unted States. Phlps frst electrc shaver was launched n 1939, at whch the Company employed 45,000 people worldwde and had sales of 15 mllons gulders. Scence and technology underwent tremendous development n the 1940s and 1950s, wth Phlps Research nventng the rotary heads whch led to the development of the Phlshave electrc shaver, and layng down the bass for later ground-breakng work on transstors and ntegrated crcuts. In the 1960s, ths resulted n mportant dscoveres such as CCDs (charge coupled devces) and LOCOS (local oxdaton of slcon). Phlps also made major contrbutons n the development of the recordng transmsson and reproducton of televson pctures, ts research work leadng to the development of the Plumbcon TV cameratube and mproved phosphors for better pcture qualty. It ntroduced the Compact Audo Cassette n 1963 and produced ts frst ntegrated crcuts n The flow of exctng new products and deas contnued throughout the 1970s : research n lghtng contrbuted to the new PL and SL energy-savng lamps, but t was n the processng, storage and transmsson of mages, sound and data that Phlps Research made key breakthroughs, resultng n the nventons of the Laser Vson optcal dsc, the Compact Dsc and optcal telecommuncaton systems. In ths perod, the feld of optcal recordng was opened up by Phlps Research gvng rse to such well-known products as Compact Dsc Dgtal Audo, CD-ROM and -more recently- DVD (the 'Dgtal Vdeo Dsc' or 'Dgtal Versatle Dsc'). Phlps Research was, by ths tme, also heavly nvolved n medcal systems such as magnetc-resonance magng and ultrasound. In moble telephony -where the smaller bandwdth and the requred error correcton ask for more economc speech coders than normal telephony- an mportant Phlps Research contrbuton, the full-rate GSM speech coder, found ts way nto all GSM basestatons and handsets n the nnetes. The same holds true for televson Konnkljke Phlps Electroncs N.V. 000 v

6 system research, wth emphass on dgtal standards and dgtal processng. Systems are made up of components and software. Research nto components has brought a great deal of success. World-class semconductor lasers from nfrared to red, yellow and green are good examples of ths. Also, research nto polymer Lght-Emttng Dodes (LEDs) and plastc electroncs show great prospects for useful nnovatve components. New dedcated mult-mllon-transstor ICs are desgned for dgtal vdeo codng and decodng (accordng to the MPEG standards), for the recepton of Dgtal Audo Broadcastng (DAB) and for speech recognton, to name a few applcatons. Programmable processors (lke TrMeda), however, make t attractve to realze ncreasngly more functons n software. Fndng the rght balance between dedcated and programmable solutons ( co-desgn ) s just one more example of the many actvtes n whch Phlps Research s nvolved today. Phlps Research. Founded n Endhoven, The Netherlands, n 1914, Phlps Research a part of Phlps Electroncs N.V. has expanded the scale and scope of ts actvtes to become one of the world s major prvate research organsatons. Wth laboratores n sx dfferent countres (The Netherlands, England, France, Germany, Tawan and the Unted States) and staffed by around 3,000 people, the common vson s to create technologes that wll lead to products for mprovng people s lves. Scentsts from a wde range of dscplnes, from electrcal engneerng and physcs to chemstry, mathematcs, mechancs, nformaton technology and software, work n close proxmty, nfluencng and broadenng each other's vews. In close co-operaton wth the Phlps Product Dvsons, the Phlps Research organsaton generates optons for new and mproved products and processes and produces mportant patents n many felds. I have performed my fnal year project at the Phlps Natuurkundg Laboratorum (Nat.Lab.) n Endhoven. It s the largest research centre, employng over 000 persons workng on ICs, electronc system, multmeda, optcs, chemstry My project s nvolved n the Dsplay Systems & Personal Care sector wthn the Vdeo Processng and Vsual Percepton group. The vdeo processng team tres to provde new features for hgh qualty televson recevers applyng recent developments n the feld of dgtal mage processng and works on programmable vdeo archtectures n order to employ sutable hardware archtectures and software support, wth respect to a wde range of vdeo applcatons. v Konnkljke Phlps Electroncs N.V. 000

7 1 Introducton 1 Compresson algorthm descrpton. 3.1 The prncple of the compresson. 3. The compresson scheme. 4 3 Dfferent measures to evaluate the qualty of a compressed mage The Peak Sgnal-to-Nose Rato (PSNR) Defnton Results The Sgnal Nose Rato (SNR) Defnton Results The Block Imparment Metrc (BIM) Defnton Results The vsual dstorton metrc developed n the EBCOT algorthm Defntons and explanatons Results. 1 4 Compresson algorthm mprovement Comparson between the Unform, the Spral, the Adaptve codec Subjectve evaluaton Objectve evaluaton 4 4. Algorthm modfcaton Comparson between the Adaptve, the Bt Plane and the Adaptve Bt Plane codec Subjectve evaluaton Objectve evaluaton Comparson between the Adaptve Bt Plane and the Energy Bt Plane Codec Subjectve evaluaton Objectve evaluaton 35 5 Suggestons for future work Subjectve evaluaton Objectve measures Algorthm mprovement 37 6 Concluson Appendx 39 8 References 47 Konnkljke Phlps Electroncs N.V. 000 v

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9 1 Introducton Dstorton measures, whch gve a numercal measure of pcture qualty, play an mportant role n many felds of mage processng and especally n mage codng. These dstorton measures can be used for example to provde a objectve crtera n the desgn of mage-compressng systems. Objectve measures are based on mathematcal formulas that somehow calculate the error between an orgnal mage and a compressed one. The dsadvantage s well known to anyone that has had to choose a compresson scheme for ther applcaton, mathematcal measures do not always ndcate whether an mage looks pleasng or whether t s acceptable for use snce the fnal user n mage compresson s the human observer. The need to have objectve measure that correlate well wth subjectve mage qualty has led to many measures. In ths report we are gong to determne a metrc qualty whch s close to our subjectve evaluaton as part to a partcular stll mages compresson algorthm: Low-complexty scalable DCT mage compresson [1] and []. After havng descrbed brefly ths compresson algorthm and studed several objectve mage qualty measures, the mage compresson scheme has been modfed to mprove the vsual qualty of the compressed mages by optmsng t wth a perceptual metrc. Konnkljke Phlps Electroncs N.V

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11 Compresson algorthm descrpton..1 The prncple of the compresson. Why do we need compresson? The examples, n Table 1 below, clearly llustrate the need for suffcent storage space, large transmsson bandwdth, and long transmsson tme for mage data. At the present state of technology, the only soluton s to compress (wth a encoder) multmeda data before ts storage and transmsson, and decompress t at the recever (wth a decoder) for play back. For example, wth a compresson rato of 10:1, the space, bandwdth, and transmsson tme requrements can be reduced by a factor of 10, wth acceptable qualty. Multmeda Data sze Bts /pxel Uncompressed sze Greyscale mage 51 * KBytes.1 Mb/mage Color mage 51 * KBytes 6.9 Mb/mage 640 * 480 Full-moton durng Vdeo 1 mn (30 frames/sec) Transmsson bandwdth Transmsson tme (usng a 8.8K Modem) 1 mn 13 sec 3 mn 39 sec GB 1 Mb/sec 5 days 8 hrs Table 1 Image and vdeo data types and uncompressed storage space, transmsson bandwdth, and transmsson tme requred. What are the prncples behnd compresson? A common characterstc of most mages s that the neghbourng pxels are correlated and therefore contan redundant nformaton. The foremost task then s to fnd less correlated representaton of the mage. Two fundamental components of compresson are redundancy and rrelevancy reducton. Redundancy reducton ams at removng predctablty from the sgnal source (here, mage). Irrelevancy reducton omts parts of the sgnal that wll not be notced by the sgnal recever, namely the Human Vsual System (HVS). In general, three types of redundancy can be dentfed: - Spatal redundancy or correlaton between neghbourng pxel values. - Spectral redundancy or correlaton between dfferent color planes or spectral bands. - Temporal Redundancy or correlaton between adjacent frames n a sequence of mages (n vdeo applcatons). Konnkljke Phlps Electroncs N.V

12 Image compresson research ams at reducng the number of bts needed to represent an mage by removng the spatal and spectral redundances as much as possble. Snce we wll focus only on stll mage compresson, we wll not worry about temporal redundancy. What are the dfferent classes of compresson technques? Two ways of classfyng compresson technques are mentoned here. - Lossless vs. Lossy compresson: In lossless compresson schemes, the reconstructed mage, after compresson, s numercally dentcal to the orgnal mage. However lossless compresson can only a acheve a modest amount of compresson. An mage reconstructed followng lossy compresson contans degradaton relatve to the orgnal. Often ths s because the compresson scheme completely dscards rrelevant nformaton. However, lossy schemes are capable of achevng much hgher compresson. Under normal vewng condtons, no vsble loss s perceved (vsually lossless). - Predctve vs. Transform codng: In predctve codng, nformaton already sent or avalable s used to predct future values, and the dfference s coded. Dfferental Pulse Code Modulaton (DPCM) s one partcular example of predctve codng. Transform codng, on the other hand, frst transforms the mage from ts spatal doman representaton to a dfferent type of representaton usng some well-known transform and then codes the transformed values (coeffcents). Ths method provdes greater data compresson compared to predctve methods, although at the expense of greater computaton.. The compresson scheme. In ths secton, we re gong to explan brefly how the Low-Complexty Scalable Image Compresson algorthm descrbed n detal n [1] and [], works n order to understand the qualty measures, explaned n the next secton, that we have mplemented n t. The man property of ths algorthm s that t s scalable. Scalable compresson refers to the generaton of a bt-stream whch represents an effcent compresson of the orgnal mage. A key advantage of scalable compresson s that the target bt-rate or reconstructon resoluton need not be known at the tme of compresson. Another advantage of practcal sgnfcance s that the mage need not be compressed multple tmes n order to acheve a target bt-rate, as s common wth the exstng JPEG compresson standard. The algorthm compresson s based on the transform codng and used the 8*8 block Dscrete Cosne Transform (DCT) defned as: 4 Konnkljke Phlps Electroncs N.V. 000

13 F( u, v) C( u). C( v) 4 = 7 7 = 0 j= 0 ( + 1). uπ ( j + 1). vπ cos.cos. f (, j) where 1 C(ε) = 1 for ε = 0 otherwse and f (, j) (respectvely, F (, j) ) s the mage n the spatal doman (respectvely, spatal frequency doman) as you can see n the fgure below. Fgure 1 From spatal doman to spatal frequency doman. In theory, the DCT transform ntroduces no loss to the source mage samples: t merely transforms them to a doman n whch they can be more effcently encoded. Furthermore, snce the DCT s a orthonormal transformaton, frst, the total energy n the coeffcent doman wll equal that of the orgnal data (n the spatal doman): t means that, f we ntroduce a error durng the encodng, n the frequency doman, ths error wll be conserved n the spatal doman. Secondly, DCT decreases consderably the correlaton between each coeffcent: t means that each coeffcent can then be treated ndependently. For a good analyss of orthogonal transforms for mage codng you can refer the Clarke s book [3]. In the present algorthm there s no addtonal quantzaton or entropy codng (such as Huffman or arthmetc codng). The DCT coeffcents are organsed n bt planes. There are 11 bt planes because the DCT coeffcents are coded wth 11 bts (the maxmum coeffcent value wll be 048, n accordance wth the DCT formula). The frst bt plane, as shown n the fgure, of a block represent the plane wth all the most sgnfcant bt of each coeffcent of ths block. Thus, the last bt plane s consttuted wth all the least sgnfcant bts of each of the 64 coeffcents of ths block. Bt-rate or qualty scalablty s enabled by encodng the DCT coeffcents bt plane per bt plane, startng at the most sgnfcant plane of each block, the frst one. The goal of scalable compresson methods s to generate a bt strng that can be truncated at any desred pont, whle always gvng the best possble qualty for the selected bt-rate. Therefore, snce a truncatable bt-strng s generated, the man goal s to put the most sgnfcant nformaton n the begnnng of the bt strng. Snce, the DC coeffcents (the coeffcents n the upper left corner of each DCT blocks) corresponds to the average lumnance of ts block, t means that they contan the man nformaton, they are processed separately from the AC coeffcents (all Konnkljke Phlps Electroncs N.V

14 coeffcents except the DC coeffcent). The DC coeffcents from all the blocks are collected and put nto the bt strng before any AC coeffcent data. They sent frst wthout any further encodng. The DC coeffcent 8 8 The 64 th coeffcent The 1 st bt plane The 11 th bt plane Fgure A DCT block wth 11 bt planes. So, snce each tme one bt plane of a block s put n the bt stream (each block s scanned or processed 11 tmes), there are dfferent way to put the data or to organse the bt strng n order to reconstruct the mage. Frst, we can put each bt plane of the processed blocks n the usual scan order, from the upper left corner to the lower rght of the mage, t means that we can put frst, n the bt strng, the bt plane blocks correspondng to those who are n the upper left corner of the mage untl those who are n the lower rght corner. We can call ths way unform so the assocated encoder-decoder wll called Unform codec. Secondly, nstead of the unform way, we can transmt at frst (after the DC coeffcents, of course) the processed blocks n the mddle of the mage and after, n a spral way towards the edge. So the last processed blocks n the bt stream are those whch are on the edge of the mage. It wll be called the Spral codec. Thrdly, we can send the processed blocks n accordance wth ther own energy. The human vsual system s less senstve when there are degradatons n areas where there are much detals, areas whch possess hgh energy, than the ones where there are less detals (low-contrast/low-texture areas), those whch have lower energy. So, we transmt, at frst, those wth low energy. As a contrast measure, we use the total number of sgnfcant coeffcents for a block(c.f. []). Durng the encodng of a certan bt plane, a sgnfcant coeffcent s a coeffcent whose magntude had a one n any of the hgher bt planes (whch have already been 6 Konnkljke Phlps Electroncs N.V. 000

15 encoded). An example s gven the fgure 3. Ths bt belongs to the 3 rd bt plane x x x x x x x x Ths bt belongs to the 1 st bt plane. Fgure 3 Example of a coeffcent whch wll become sgnfcant durng the encodng of the 3 rd bt plane (x = 0 or 1). Consequently, n ths case, the scan order s adapted to each ndvdual mage. In the mplementaton, both encoder and decoder re-adjust ther scannng orders, accordng to the block contrast, at the start of each new coeffcent bt plane. Wth ths adaptve scannng order, the perceptual mage qualty of ths scheme should be better as you could see n the secton 5-1. Ths codec wll be called the Adaptve codec. 3 Dfferent measures to evaluate the qualty of a compressed mage. The followng measures have been mplemented n the codec software related wth the Low-Complexty Scalable Image Compresson descrbed n the frst secton as the Spral codec. Most of the measures are n accordance wth the bt rate,.e. the number of bt per pxel from those, the compressed mage s reconstructed. So, more the bt rate wll be hgh more the reconstructed mage wll be closer to the orgnal one. As data, we have used the Lena mage (51*51 pxels) n color and n grey level whch you can look at n the appendx (fgure 36). 3.1 The Peak Sgnal-to-Nose Rato (PSNR) Defnton. 55 PSNR db = 10 *log wth the Mean Squared Error defned as * [ ] 10 MSE MSE = N = 1 ( X Y ) N wth - N s the number of pxel of the mage - X, the orgnal mage and Y, the reconstructed one Konnkljke Phlps Electroncs N.V

16 j * We can also defne the Coeffcents PSNR lke the PSNR but wth, n ths case, X (respectvelyy ) represent the th coeffcent of the j th frequency band of the orgnal j (respectvely reconstructed) transformed mage n the frequency doman (the DCT doman). 55 PSNR db = 10 *log. Coeffcents [ ] 10 MSE / N j j MSE = ( X Y ) N j= 1 = 1 It gves the same result as the PSNR because of the DCT property, the conservaton of the energy. We have defned ths measure n order to ntroduce the followng measure: * The Weghted PSNR s defned lke the Coeffcents PSNR but wth a dfferent MSE. We use a perceptually weghted MSE defned as: / N j j Weghted MSE = α * ( X Y ) N j= 1 = 1 α s a weght n order to gve more mportance to the lower spatal frequences because the human eye s less senstve to hgher frequences (frequency senstvty). So we can defne t wth the help of the lumnance quantzaton matrx and the chromnance quantzaton matrx. These quantzaton matrces have been specfed by the JPEG standard and based on the human vsual percepton [4]. These matrces are defned as -for the lumnance: -for the chromnance: The quantzaton values can be set ndvdually for each coeffcent, usng crtera based on vsblty of the bass functons. If we measure the threshold for vsblty of a gven bass functon (the coeffcent ampltude that s just detectable by human eye) we can dvde (quantze) the coeffcents by that value (wth approprate roundng to nteger values). If we multply (dequantze) the scaled-down coeffcent by that value before 8 Konnkljke Phlps Electroncs N.V. 000

17 reconstructng, we create a condton n whch the eye should not be able to detect any dfference between quantzed and unquantzed DCT coeffcents. If we are wllng to tolerate some vsble artfacts n the reconstructed mage, we mght dvde by a value larger than the vsblty threshold. But, as we read before, there s no quantzaton durng the compresson but we can use these values n order to gve more or less mportance to any coeffcents, n weghtng them. We can, n fact, consder α as a detecton threshold. Wthout weghtng, t s lke that α and Q (the th element of the quantzaton matrx whch contan of course 64 coeffcents) are equal to 1 for { 1;64} so we choose α 64 c such as α = 64. Consequently, we can defne α n ths way: α =, wth c a Q = 1 constant (whch depends on the quantzaton matrx) and stll α = 64.So c s such as N 1 c * = 64. = 1 Q Consequently, c= for lumnance quantzaton matrx and c=537.8 for chromnance quantzaton matrx. We can also above what the 64 coeffcentsα look lke. - for the lumnance: 64 = for the chromnance: Konnkljke Phlps Electroncs N.V

18 3.1. Results. As you we wll see on the followng plots, for the color mage the hghest bt rate s 13.7 nstead of 4 bts (8 bts for each component: Y, the lumnance one, U and V the two chromnance ones), t means that t s the lowest bt rate wth whch, we get a lossless compresson (.e. when all the bt plane are decoded, when all the generated bt-strng s decoded). For the grey level mage t s 4.9 nstead of 8 bts (8 bts only for the lumnance). The Y,U and V components come from the R,G, B (Red, Green, Blue) components whch are use to represent a color mage for a dsplay CRT (usual computer screen). The relatons between them are: Y= 0.R+0.587G+0.114B; V=R-Y; U=B-Y; The lumnance provdes a greyscale verson of the mage, and the chromnance components provde the extra nformaton that converts the greyscale mage to a color mage. Below a bt rate of, n accordance wth the fgure 4, the Y component qualty s smaller than the other ones so that, there s more error n the Y component. Consequently, snce all the components are encoded n the same way, we can affrm that the lumnance component contans more nformaton than the chromnance ones. Fgure 4 PSNR of the Y,U and V components 10 Konnkljke Phlps Electroncs N.V. 000

19 Wth the fgure 5 and 6, we can say the same remark because the Coeffcent PSNR and the Weghted PSNR have approxmately the same behavour that the PSNR. Fgure 5 Coeffcent PSNR of the Y,U and V components. Fgure 6 Weghted PSNR of the Y, U and V components. Konnkljke Phlps Electroncs N.V

20 In fgures 7, 8, 9 and 10, there s a lttle dfference between the PSNR and the coeffcents PSNR because the DCT coeffcents are rounded durng the calculaton n the encoder. They are also rounded durng the Inverse DCT (IDCT) n the decoder. In fgures 7, 8, 9 and 10, we can also note that the Weghted PSNR s lower than the Coeffcents PSNR and the PSNR below a bt rate of. Snce, the quantzaton matrx gve more mportance to the low frequences (as you can see above, the coeffcents α are bgger for the low frequences than for the hgh frequences), we confrm that the low frequences contan the man nformaton of the mage. We can also remark the dfferent evoluton between the PSNR of the lumnance component n the grey level mage (fgure 7) and the PSNR of the lumnance component n the color mage (fgure 8). Ths dfference s due to the way each components are encoded and decoded n the software. The floor effects n the fgure 8 (related wth the color mage) s due to the fact that when a component (for example Y), s encoded and decoded durng a certan bt rate the other ones (for example U and V) can t be processed too n the same tme. Consequently, the PSNR of those components s constant. Of course we don t have ths phenomena for the grey level mage because there s only one component the lumnance one, Y. Fgure 7 PSNR of Grey level Lena. 1 Konnkljke Phlps Electroncs N.V. 000

21 Fgure 8 PSNR of the lumnance component, Y. Fgure 9 PSNR of the chromnance component, U. Konnkljke Phlps Electroncs N.V

22 Fgure 10 PSNR of the chromnance component, V. 3. The Sgnal Nose Rato (SNR) Defnton. The SNR s defned almost n the same way than the PSNR. The dfference les n the fact that the PSNR takes the maxmum power of any mages ( 55 ) whle n the SNR, t s the real power of the consdered mage. Consequently, σ [ ] S SNR db = 10 *log 10 wth σ S, the mage power s defned as, MSE N 1 σ S = ( X mx ), where X s the value of the th pxel of the orgnal mage and N = 1 N mx the average value of the pxels of the mage. So, mx = X N = 1 For the transformed mage (DCT) the power whch s equal to power of the orgnal mage, because of the DCT property, s defned as: σ 64 C = σ = 1, wthσ, the power of the th coeffcent of the all DCT blocks (8 * 8), Konnkljke Phlps Electroncs N.V. 000

23 t means the coeffcent of the th frequency band of the coeffcents mage. N / 64 1 j Consequently, σ = ( Y my N / 64 j= 1 j And Y represents the coeffcent of the mage. 3.. Results. j ) th wth N / 64 1 j = Y Y N / 64 j= 1 m j frequency band of the th j block of the DCT Seeng that the SNR takes account of the power of the mage tself, these curves (fgure 8, 9, 10 and 11) show better that the lumnance component possesses more nformaton that the chromnances ones because the SNR of the lumnance component s hgher than the SNR of the chromnance components. They also show, fortunately, lke the PSNR, that the low frequences contan the man nformaton of the mage. Fgure 11 SNR of the lumnance component, Y. Konnkljke Phlps Electroncs N.V

24 Fgure 1 SNR of the chromnance component, U. Fgure 13 SNR of the chromnance component, V. 16 Konnkljke Phlps Electroncs N.V. 000

25 Fgure 14 SNR of the grey level Lena. 3.3 The Block Imparment Metrc (BIM) Defnton. As t s well known, DCT codng scheme causes blockng artfacts n mage codng and PSNR or SNR are neffectve n quantfyng ths knd of artfacts. The BIM s a quanttatve dstorton measure for these blockng artfacts. The algorthm s based on the nterpxel dfference between each of the horzontal (vertcal edge artfacts ) and vertcal (horzontal edge artfacts) block boundares. There also parameters whch take nto account the lumnance maskng effects n extreme brght as well as extreme dark areas n the reconstructed mage. Consequently, ths dstorton measure does not requre the orgnal mage as a comparatve reference. BIM s detaled n [5]. Konnkljke Phlps Electroncs N.V

26 3.3. Results. Fgure 15 BIM of the components Y, U and V for Lena. In accordance wth the fgure 1, the BIM s smaller for the lumnance component than the chromnance ones consequently the chromnance blocknesses s more promnent that the lumnance blockness. The BIM for the orgnal mage s 0.41 db for the lumnance component and 0.17 and 0.14 db for the chromnance ones. We can compare now the PSNR wth the BIM n order to see f BIM measure brngs more nformaton than the PSNR (fgure 16 and 17). 18 Konnkljke Phlps Electroncs N.V. 000

27 Fgure 16 BIM and PSNR for Lena grey level. Fgure 17 (39dB-BIM) and PSNR for Lena grey level. Konnkljke Phlps Electroncs N.V

28 Between 0.3 and 1 bt per pxel, PSNR and (39dB-BIM) evolve approxmately n the same way. Consequently, at frst sght n ths case, the BIM does not brng more nformaton on the qualty of the compressed mage but referrng to [5] the BIM seems to be a better qualty measure than the PSNR for a POCS flterng whch s a postflterng of reconstructed vdeo mages algorthm to reduce the blockng artfacts. 3.4 The vsual dstorton metrc developed n the EBCOT algorthm Defntons and explanatons. As t s descrbed n [6], EBCOT s a new mage compresson algorthm. The acronym means Embedded Block Codng wth Optmzed Truncaton. It s another scalable mage compresson but rather than focusng on generatng a sngle scalable bt-stream to represent the entre mage, EBCOT parttons each subband nto relatvely small blocks of samples and generates a separate hghly scalable bt-stream to represent each so-called code-block. The EBCOT algorthm was adopted for ncluson n the evolvng JPEG000 mage compresson standard. Our goal here, t s not to descrbe the algorthm, but to adapt the dstorton metrc developed n [6] to our compresson algorthm. Consequently, here s the th frequency band dstorton formula: k k ( X Y ) k σ ( V ) D k = + where - k X ( respectvely, k Y ) represent the th k coeffcent of the th frequency band of the orgnal mage ( respectvely, the reconstructed mage). -σ s defne n [6] as the vsblty floor term whch establshes the vsual sgnfcance of dstorton n absence of maskng. In fact, n our case, we re gong to take σ 1 = α, α the weght defned n the secton s called the vsual maskng strength operator, defned as: k V - The frst adaptaton. V k k Φ = 1 10 k Φ [ k ] Φ k X k Here, denotes the sze of the neghbourhood. k Φ denotes a neghbourhood of the coeffcent In our case we have taken a neghbourhood of 1. It means that k Φ s 9. k X and 0 Konnkljke Phlps Electroncs N.V. 000

29 k We dvde V by 104 n order to scale t compare wth α because the DCT coeffcents are coded wth 11 bts so that the maxmum value wll be 048. When we dvdev by 048, ts nfluence s too small so that we dvde by 104. k We have adapted ths measure n ths way to take account the maskng nsde the same frequency band. Indeed, nteractons appear between coeffcents whch belong to the same frequency band. Such nterference results n a modfcaton of the detecton threshold α and t s stronger f the coeffcent s hgh and f hs neghbourhood V the same energy. -the second adaptaton. In ths case, we take the current 8*8 DCT block of the current coeffcent k has about k X as hs neghbourhood except the DC coeffcent. It means that ths neghbourhood represent n fact, the energy of the current block, so t won t depend on the coeffcent anymore but only on the block k. We use ths because on the block wth hgh energy, the dstorton wll be smaller snce the human eye s less sensble wth ths knd of regons. Referrng to the frst adaptaton, t s, n fact, a maskng due to the presence of the others coeffcents nsde a block, a nteracton between dfferent frequency bands. We can call that a contrast maskng referrng to [7]. So that, the neghbourhood s no more dependant of the current frequency band and t s defned as: V k = 10 1 * k ( X ) = The scalng factor has the same goal than n the frst adaptaton Results. -for the frst adaptaton. We can see, below, the remarks we dd above. In accordance wth the fgure 18, the error n the low frequences s dfferent than the weghted MSE: the error s weghted by the neghbourhood of the coeffcent of the orgnal coeffcent mage. In the hgh frequences the error s the same as the weghted MSE. In ths fgure, we plot the error n fact n accordance wth the DCT coeffcents from the left to the rght for each lne. Snce the low frequences are concentrated n the upper left corner, that s explaned the wave phenomena of the curve. In order to compare ths measure m wth the Weghted SNR we have computed t n the p same as the SNR, t means as 10 *log 10 where p s the coeffcents mage power. m In accordance wth the fgure 19, the dfference between ths dstorton metrc and the weghted SNR s hgher n the low bt rates than n the hgh bt rates. Indeed, error n the low frequences s reduced frst and contan the man nformaton. Konnkljke Phlps Electroncs N.V

30 Fgure 18 Lena grey level dstorton n accordance wth the frequency at 0.1bt per pxel. Fgure 19 Lena grey level weghted SNR and the frst adapted EBCOT dstorton metrc. Konnkljke Phlps Electroncs N.V. 000

31 -for the second adaptaton We have computed ths new measure lke n the frst adaptaton. Fgure 0 Lena grey level weghted SNR and the second adapted EBCOT dstorton metrc. We can remark that n the low bts rates the dfference between these two measures s bgger than n the hgher bt rate. Indeed, n a very low bt rate only the DC coeffcents are transmtted and so that decoded. But these coeffcents ntroduced no error, so the MSE, n ths case, s equal to the energy, the neghbourhood. So, the blocks wth hgh energy have the hghest error n the begnnng and later, all the blocks have about the same error. Konnkljke Phlps Electroncs N.V

32 4 Compresson algorthm mprovement. In order to evaluate the best measure, t means the one whch s closer to our subjectve judgement, descrbed before, we are gong to compare (objectvely and subjectvely) dfferent mages of Lena processed by the dfferent codecs explaned before n secton 1. After that, we wll explan the changes we dd n the algorthm and check them wth our subjectve evaluaton and wth the objectve measures. 4.1 Comparson between the Unform, the Spral, the Adaptve codec Subjectve evaluaton As we explaned before, the Unform codec processed the DCT blocks n the usual scan order (.e. left to rght and top to bottom of the mage). However, at lower bt rates (for example at 0.6 bt per pxel) ths gves annoyng artfacts (see the fgure 37 Lena_UN.pgm n the appendx), because of a clearly vsble qualty dfference between the blocks above and below the truncaton pont, we can also see that n the error mage (the dfference mage between the orgnal and the reconstructed one) n the fgure 39. So, the Spral codec, the one whch sends the DCT blocks n the mddle of the mage and sprallng out towards the edges, gve a hgher perceved qualty than the Unform codec (see the fgure 38 Lena_SP.pgm and fgure 40 n the appendx ). Snce t s closer to the human vsual system, the Adaptve codec seems n fact better at lower bt rate (see fgure 41 Lena_AD.pgm n the appendx). In the error mage (fgure 4) we can see that the error s dstrbuted n the areas wth hgh energy (edges and the har) Objectve evaluaton We can see below, that at 0.6 bt per pxel all measures are correlated wth our judgement except for the SNR. But the measures are not so well correlated when we compare Lena_UN.pgm and Lena_SP.pgm (except for the SNR!): t s logcal because they don t take account the dfference between the two codecs. You can see t also n the followng plots. Images SNR Weghted BIM EBCOT1 EBCOT SNR Lena_UN.pgm Lena_SP.pgm Lena_AD.pgm Table Objectve results of Lena at 0.6 bt per pxel. Let s see now, n a accordance wth the bt rate. Wth the fgure 1, we can see that most of tme the SNR of the Spral Codec s hgher or equal than the SNR of the Adaptve one. In the fgure, t s almost the opposte compare wth the fgure 1 and we know that the Adaptve codec gves better results so we can say, n ths case, that the weghted SNR s close to the subjectve evaluaton. 4 Konnkljke Phlps Electroncs N.V. 000

33 SNR (db) SNR for the Unform codec SNR for the Spral codec SNR for the Adaptve codec bt rate Fgure 1 SNR for the Unform, Spral and Adaptve codec Weghted SNR (db) Weghted SNR for the Unform codec Weghted SNR for the Spral codec Weghted SNR for the Adaptve codec bt rate Fgure Weghted SNR for the Unform, Spral and Adaptve codec. Konnkljke Phlps Electroncs N.V

34 1 BIM for the Unform codec BIM for the Spral codec BIM for the Adaptve codec 10 8 BIM (db) bt rate Fgure 3 BIM for the Unform, Spral and Adaptve codec EBCOT1 (db) EBCOT1 for the Unform codec EBCOT1 for the Spral codec EBCOT1 for the Adaptve codec bt rate Fgure 4 EBCOT1 metrc for the Unform, Spral and Adaptve codec. 6 Konnkljke Phlps Electroncs N.V. 000

35 Accordng to the fgure 3, we can note that n the low bt rates the BIM of the Adaptve codec s lower that the BIM of the other ones. It means that the Adaptve codec brngs less blockng artfacts. But we can check ths property wth our judgement, so we can notce that the BIM measure s also, n ths case, close to our subjectve evaluaton. In the fgure 4, the frst adapted EBCOT dstorton metrc (EBCOT1) has almost the same behavour than the weghted SNR. In the fgure 5, we can see that the second adapted EBCOT dstorton metrc (EBCOT) gves better results for the Adaptve Codec than the other measures because as we have seen, ths codec s lnked wth the energy of each block and the second adapted EBCOT metrc takes also account the energy of the block of the consdered pxel. So we can say that t s the best measure related wth our subjectve judgement but only for ths partcular codec. So, we can conclude frst that the Spral codec optmse the SNR when we compare t wth the Unform codec, and secondly, the Adaptve codec whch s the best for the moment optmse the EBCOT metrc. Seeng that we can expose now the changes we dd n order to mprove the algorthm EBCOT (db) EBCOT for the Unform codec EBCOT for the Spral codec EBCOT for the Adaptve codec bt rate Fgure 5 EBCOT metrc for the Unform, Spral and Energy codec Konnkljke Phlps Electroncs N.V

36 4. Algorthm modfcaton. We can mprove the codecs descrbed n the secton 1, n makng them closer to the human vsual system (HVS). For the moment these codecs don t gve more mportance to any coeffcents except for the DC coeffcent. All the DC coeffcents are sent frst. The dea, here, s dong the same for the others coeffcents, the AC coeffcents. Those whch are more mportant for the HVS,.e. those whch correspond to the low frequency are put frst n the bt stream. To determne the most mportant coeffcents we used agan the quantzaton matrx descrbed n the frst secton. We can dvded t n dfferent areas. We have found that the best dvson s defned n ths such a way (c.f. fgure 6). The frst regon, so, the most mportant covered the coeffcents from 1*10 to *10, the second one from *10 to 4*10, the thrd one, from 4*10 to 8*10, and the fourth one, the least mportant one, from 8*10 to 16*10 as you can see below Fgure 6 The dfferent areas of the lumnance quantzaton matrx. We can see now there a rato of two between each regon. So, we re gong to multply the coeffcents (except the DC one) whch belong to the most mportant regon by 8 n shftng them by 3, those of the second regon by, those of the thrd one by 1 and those of the last regon wll be no shfted. Consequently, snce the maxmum shft s 3, we need 3 more bt planes for the AC coeffcents. So, now we have 14 bt planes for the AC coeffcents and stll 11 for the DC coeffcent snce t s processed separately (c.f. fgure 7). But now, we have more nformaton to transmt but we know where s the useless nformaton. In fact the useless nformaton s the zeros we ntroduced when we shft a coeffcent (c.f. fgure 8): for example, f we shft a coeffcent by 3 (we multply t by 8), the useless nformaton s the 3 zeros at the end (those of the 3 last bt planes) so we don t need to transmt them. It s the same also for the coeffcent whch are no shfted, the useless nformaton s the 3 frst zeros (those of the 3 frst bt planes). 8 Konnkljke Phlps Electroncs N.V. 000

37 The DC coeffcent The 1 st bt plane The 64 th coeffcent The 14 th bt plane Fgure 7 A DCT block wth 14 bt planes for AC coeffcents and 11 for the DC coeffcent. A coeffcent shfted by 3: x x x x x x x x x x x MSB Useless 0. A coeffcent shfted by : 0 x x x x x x x x x x x 0 0 MSB Useless 0. Konnkljke Phlps Electroncs N.V

38 A coeffcent shfted by 1: 0 0 x x x x x x x x x x x 0 MSB Useless 0. A no shfted coeffcent (shfted by 0): x x x x x x x x x x x MSB Useless 0. Fgure 8 Shfted coeffcents and useless 0 (x = 0 or 1). We can also calculate the maxmum of the AC coeffcent and see f t s worth to use 14 bt planes. We can send ths nformaton n the begnnng n the bt strng, t takes only 4 bts. For, most our test mages, for example Lena, the maxmum of the AC coeffcent s 636, so n ths case 13 bt planes are enough for them. It wll be the Bt Planes codec (the BP codec). Consequently, we can defne a new codec whch s n fact a combnaton of the two last ones, the Adaptve and the Bt Planes codec: the Adaptve Bt Planes codec (the Adaptve BP codec). We can also calculate the energy of each block n another way than n the Adaptve codec. Instead of countng the total number of sgnfcant coeffcents for a block durng the encodng of a bt plane, we calculate before the encodng the real energy of the block k k, V defned n the second adapted EBCOT dstorton metrc (secton 3.4.1) After scalng t, we ntroduce ths new parameter n the method we choose the regons of the quantzaton matrx descrbed above. Gven MAX, the maxmum of the regon r, for the th coeffcent, f 1 + k α r c V ( ) MAX r then ths coeffcent wll be shfted by the shft whch defned the c regon r. We have stllα = ( c = for the lumnance quantzaton matrx ) Q 64 k 1 k and = ( X ) 10 V. * 63 = 30 Konnkljke Phlps Electroncs N.V. 000

39 For example, for the frst regon, MAX r =19, so, for the th coeffcent, 1 f + k c V then ths coeffcent wll be shft by 3. α 19 Consequently, wth ths method the blocks wth very low energy ( V[ k] 0) wll be put n the bt-strng before those whch have hgher energy. In the decoder, we can calculate teratvely the total energy of each block durng the decodng of a bt plane. Ths codec wll be called the Energy Bt Plane Codec. 4.3 Comparson between the Adaptve, the Bt Plane and the Adaptve Bt Plane codec Subjectve evaluaton As we can guess, the best codec s n fact, the Adaptve Bt Plane codec and n second poston s the Bt Plane codec. So, the modfcaton we dd n the last secton s n fact a real vsual mprovement. You can see the mages at 0.6 bt per pxel n the appendx: Lena_BP.pgm (fgure 43) and Lena_ABP.pgm (fgure 44) respectvely processed by the Bt Plane and the Adaptve BP codec. The vsual dfference at ths bt rate s not so obvous, so you can see also the fgure 46, 47, and 48 whch represent respectvely Lena_AD.pgm, Lena_BP.pgm, Lena_ABP.pgm, at 0.5 bt per pxel. Let s see now the qualty measures Objectve evaluaton Images SNR (db) Weghted SNR (db) BIM (db) EBCOT1 (db) EBCOT (db) Lena_AD.pgm Lena_BP.pgm Lena_ABP.pgm Table 3 Results at 0.6 bt per pxel for Lena_AD.pgm, Lena_BP.pgm, and Lena_ABP.pgm. Konnkljke Phlps Electroncs N.V

40 SNR (db) SNR for the Adaptve codec SNR for the BP codec SNR for the Adaptve BP codec bt rate Fgure 9 SNR for the Adaptve, BP and Adaptve BP codec. Wth the fgure 9, we can notce that from 0.5 bt per pxel the SNR for the Adaptve Bt Plane codec s lower than the others (see also table 3 at 0.6 bt per pxel) and we know that n fact t s the best one for the human eye, so we can conclude that, here, the SNR s not correlated wth our subjectve evaluaton. In the fgure 30 and n the table 3, we can also remark that the Weghted SNR for the Adaptve BP codec s almost the same as the one for the BP codec but the both codecs have a hgher Weghted SNR than the one for the Adaptve codec. Snce the BP codec gve more mportance to low frequences, t s logcal that Weghted SNR gves better result. But the fact that ths measure gve the same results n the lower bt rate for the Adaptve BP and BP codec, makes ths measure not very well correlated wth our subjectve evaluaton, n ths case. 3 Konnkljke Phlps Electroncs N.V. 000

41 Weghted SNR (db) Weghted SNR for the Adaptve codec Weghted SNR for the BP codec Weghted SNR for the Adaptve BP codec bt rate Fgure 30 Weghted SNR for the Adaptve, BP and Adaptve BP codec BIM for the Adaptve codec BIM for the BP codec BIM for the Adaptve BP codec BIM bt rate Fgure 31 BIM for the Adaptve, BP and Adaptve BP codec. Konnkljke Phlps Electroncs N.V

42 But t s dfferent for the BIM measure when we look at the fgure 31 and the table 3. Indeed, the BIM tells that there are more blockng artfacts for the best codec from 0.6 bt per pxel than for the Adaptve codec. Ths maybe true n the absolute but t s less vsble, so, n ths case, the BIM measure s not very well correlated wth our judgement EBCOT1 (db) EBCOT1 for the Adaptve codec EBCOT1 for the BP codec EBCOT1 for the Adaptve BP codec bt rate Fgure 3 EBCOT1 for the Energy, SBP and Adaptve BP codec. The frst adapted EBCOT dstorton metrc n the fgure 3 does brng much more nformaton than the Weghted SNR. The neghbourhood of the frequency band has not a lot of nfluence. But as we can see n the fgure 33 and the table 3, the second adapted EBCOT dstorton metrc s slghtly better. Consequently, we can conclude that ths qualty metrc s the best one compare wth our subjectve assessment. 34 Konnkljke Phlps Electroncs N.V. 000

43 EBCOT (db) EBCOT for the Adaptve codec EBCOT for the BP codec EBCOT for the Adaptve BP codec bt rate Fgure 33 EBCOT for the Energy, SBP and Adaptve BP codec. 4.4 Comparson between the Adaptve Bt Plane and the Energy Bt Plane Codec Subjectve evaluaton Although the Energy BP codec takes account the real energy of each block, when we look at the mages at lower bt rate, the Adaptve BP codec seems to be the better one. You can compare, n the appendx, Lena_ABP.pgm (fgure 44) and Lena_EBP.pgm (fgure 45), processed by the Energy BP codec reconstructed from 0.6 bt per pxel Objectve evaluaton Images SNR (db) Weghted SNR (db) BIM (db) EBCOT1 (db) EBCOT (db) Lena_EBP.pgm Lena_ABP.pgm Table 4 Results at 0.6 bt per pxel for Lena_EBP.pgm and Lena_ABP.pgm. Konnkljke Phlps Electroncs N.V

44 In fact, referrng to the table 4, any measures don t brng more nformaton because they are all n accordance wth our subjectve evaluaton. Snce the EBCOT seems to be the best measure for the moment, we can see how ths measure evolve at lower bt rate n the fgure EBCOT (db) EBCOT for the Adaptve BP codec EBCOT for the Energy BP codec bt rate Fgure 34 EBCOT for Adaptve BP and Energy BP codec. Ths measure confrms that the Adaptve BP codec s the best codec at that tme. 36 Konnkljke Phlps Electroncs N.V. 000

45 5 Suggestons for future work. 5.1 Subjectve evaluaton. Most of the tme we dd our subjectve evaluaton on the Lena mage but we dd t also to others few mages but to valdate the subjectve assessment more serously, more evaluaton wth more mages and more people wll be welcomed. We can evaluate also these qualty measures for vdeo sequences not only for stll mages. 5. Objectve measures We can, n fact, mprove the adapted EBCOT dstorton metrc n optmsng the scalng factor of the respectve neghbourhoods, n order to make these two measures closer to our subjectve evaluaton. 5.3 Algorthm mprovement In order to mprove the compresson algorthm we can reduce the number of bts n the bt stream n usng less bt planes for the DC and for the AC coeffcents. For the AC we explaned that n calculatng the maxmum of the AC coeffcents most of the tme 10 bts are enough nstead of 11 bts (t means 13 bt planes nstead of 14) but we can extend ths dea for each AC coeffcents. Lets take a example: for Lena mage, you can see n the fgure 35, the maxmums of the all the DCT coeffcents Fgure 35 the 64 maxmums of all the DCT coeffcents. Now, t s very easy to see that usng 10 bts for a coeffcent s most of the tme too much. Consequently, 4*63 bts s the maxmum nformaton we have to send n the begnnng of the bt stream but we can do t wth less bts wth a clever method. So, the number of bts we add s n fact much smaller than the number of bts we economse f we take 10 bts for each AC coeffcents. Then t s worth. We can also optmse the method to send the DC coeffcents. For the moment, as we Konnkljke Phlps Electroncs N.V

46 told, they are put n the bt stream drectly wthout further encodng. But we can send them wth the DPCM method, t means that we just send the dfference from DC coeffcent of the prevous DCT block and the current one (except of course for the frst one) because although the DC coeffcent s large and vared, t s often close to prevous value. But n ths case, the algorthm cannot be 100 % scalable, t wll become t from the frst AC coeffcent. It can be nterestng to mplement ths dea because most of the tme we wll truncate the bt stream after that the DC coeffcents we wll sent to have not a too bad reconstructed mage. 6 Concluson. In ths paper, we have, frst, studed several objectve mage qualty measures such as PSNR, SNR, BIM, and several perceptually weghted SNR whch take account the human vsual system (frequency senstvty, maskng) and secondly, mplemented them n dfferent avalable mage compresson schemes related wth Low-complexty Scalable DCT Image Compresson. At ths pont we found among the measures one, the perceptually weghted SNR called n ths paper the second adapted EBCOT dstorton metrc (EBCOT), whch follows our subjectve evaluaton. Fnally, by optmsng the compresson, called Adaptve Bt Plane, for ths partcular metrc, a better mage qualty has been obtaned. At the begnnng, the compresson scheme such as, Unform and Spral, optmzed the sgnal-to-nose rato (SNR) of the compressed mages. Next, the Adaptve codec optmzed the weghted SNR, the Block Imparment Metrc (BIM), the frst and the second adapted EBCOT dstorton metrc (EBCOT1 and ). Fnally, we saw that ndeed t s the second adapted EBCOT dstorton metrc whch was the best measure for the best codec, the measure whch correspond the best to the vsual qualty as t s perceved by human observers. But we have to know that a qualty measure cannot use for any knd of artfacts due to the compresson scheme. The metrc qualty we found s the best only for ths codec. Snce ths measure takes account the human vsual system, ths measure can be use for others processed mages but we cannot guarantee that t wll be the best. 38 Konnkljke Phlps Electroncs N.V. 000

47 7 Appendx Fgure 36 Orgnal Lena mage Konnkljke Phlps Electroncs N.V

48 Fgure 37 Lena_UN.pgm at 0.6 bt per pxel. Fgure 38 Lena_SP.pgm at 0.6 bt per pxel. 40 Konnkljke Phlps Electroncs N.V. 000

49 Fgure 39 The error mage related wth Lena_UN.pgm at 0.6 bt per pxel. Fgure 40 The error mage related wth Lena_SP.pgm at 0.6 bt per pxel. Konnkljke Phlps Electroncs N.V

50 Fgure 41 Lena_AD.pgm at 0.6 bt per pxel. Fgure 4 The error mage related wth Lena_AD.pgm at 0.6 bt per pxel. 4 Konnkljke Phlps Electroncs N.V. 000

51 Fgure 43 Lena_BP.pgm at 0.6 bt per pxel. Fgure 44 Lena_ABP.pgm at 0.6 bt per pxel Konnkljke Phlps Electroncs N.V

52 Fgure 45 Lena_EBP.pgm at 0.6 bt per pxel. 44 Konnkljke Phlps Electroncs N.V. 000

53 Fgure 46 Lena_AD.pgm at 0.5 bt per pxel Fgure 47 Lena_BP.pgm at 0.5 bt per pxel. Konnkljke Phlps Electroncs N.V

54 Fgure 48 Lena_ABP.pgm at 0.5 bt per pxel. 46 Konnkljke Phlps Electroncs N.V. 000