Image Morphing Based on Morphological Interpolation Combined with Linear Filtering

Transcription

1 Image Morphng Based on Morphologcal Inerpolaon Combned wh Lnear Flerng Marcn Iwanowsk Insue of Conrol and Indusral Elecroncs Warsaw Unversy of Technology ul.koszykowa Warszawa POLAND el e-mal: ABSTRACT Ths paper descrbes a novel approach o color mage morphng whch s based on he combnaon of morphologcal mage processng ools and lnear flerng. In he proposed mehod he morphng engne s provded by he morphologcal nerpolaon by means of he morphologcal medan. By he successve generaon of morphologcal medans usng an algorhm proposed n he paper, he sequence ransformng one npu mage no anoher one s produced. The algorhm makes use of he smlary measure beween wo successve frames. Two versons of he algorhm are proposed - n he frs one he requred parameer s he number of frames of he fnal sequence, n he second one he maxmal accepable error beween wo consecuve frames. Three lnear ools are proposed o mprove he vsual qualy of he morphng sequence: he emporal lnear flerng, he spaal lnear flerng and he auxlary cross-dssolvng. Conrary o he radonal approaches o mage morphng, he proposed mehod doesn requre any conrol pons. The human operaor s oblged o nroduce only a few npu parameers. Two examples showng he resuls are also presened n he paper. Keywords: Image morphng, color mage processng, mahemacal morphology, morphologcal nerpolaon. INTRODUCTION Alhough varous mage processng conceps are n common use n he compuer graphcs [Gomes97], an applcaon of mahemacal morphology [Serra83,88] o hs doman s a recen fled of research. I has proved s usefulness n varous areas of mage processng: mage flerng, segmenaon, granulomery, ec. One of he mos popular felds of s applcaons s mage nerpolaon a process of auomaed creaon of a sequence ransformng one gven mage (he nal mage) no he second (fnal) one. Ths knd of operaon s presen n compuer graphcs under he name of morphng. The applcaon of mahemacal morphology no mage nerpolaon s a recen feld of exploraon [Iwano]. The resuls of he very frs research works have been publshed n 996 and 998 [Beuch98, Meyer96, Serra98]. They concerned he nerpolaon of bnary, mosac and grayone mages. In [Beuch98, Serra98] an mporan operaon was nroduced: he morphologcal medan. Ths operaon - compleely dfferen from he medan fler wdely used n mage processng - resuls n a new mage, creaed on he bass of gven wo npu mages, locaed halfway beween hem. The erave generaon of medan mages allows producng he nerpolaon sequence ha ransforms he nal mage no he fnal one a morphng sequence. Imporan s ha, conrary o he well-known cross-dssolvng [Gomes97, Wolbe99], he morphologcal approach ransforms he shapes of objecs on he mages. Cross-dssolvng produces merely a knd of blendng of npu mages. The mehod proposed n [Beuch98] has been exended no color mages n [Iwano99]. Two verson of nerpolaon was proposed: he fully auomac and half-auomac nerpolaon. In hs paper we exend he fully auomac mehod. I allows deformng he mages n a auomac manner, whou seng up he conrol pons. The orgnal mehod has a dsadvanage: n

2 some cases, he vsual qualy of he sequence s no hgh, especally when he npu mages nclude he areas of dfferen colors on he nal and fnal mages. Frames of he morphng sequence conan ofen nclusons of hghly conrased pxels wh he hue and lumnance values dfferen from her background. Due o ha fac he vsual qualy of he morphng sequence s no hgh enough o use n an effcen way. In hs paper he qualy mprovemen s proposed. In order o mprove he smoohness of he sequence we use he lnear mage processng echnques. The combnaon of he morphologcal nerpolaon and he qualy mprovemen by lnear spaal and emporal flerng allows performng he mage morphng, whch produces neresng and good-lookng resuls. The paper presens also a new mehod for he producon of he morphologcally nerpolaed sequence. The algorhm s based on he smlary measure beween frames. The radonal mehod [Beuch98,Iwano99] s a blnd one nerpolaed frames are produced whou comparng he already nerpolaed ones. The new algorhm does and s more flexble allows producng he nerpolaon sequence usng dfferen crera, wo of whch are presened n he paper: he number of frames and maxmum error crera. The paper s dvded no 6 secons. Secon descrbes he morphologcal medan of color mages. New mehod of he nerpolaon sequence formaon s nroduced n secon 3. Secon 4 descrbes he mehod of he lnear qualy mprovemens. Secon 5 conans he resuls and examples, and fnally secon 6 summarzes and concludes he paper.. COLOR MORPHOLOGICAL MEDIAN. Defnon of morphologcal medan The morphologcal medan has been nroduced for bnary [Serra98], mosac and grayone [Beuch98] mages. I s based on he morphologcal operaons of eroson and dlaon [Serra88]. Eroson s defned as a mnmum operaor, whch assgns o every mage pxel a mnmum value from among her neghbors. The neghborhood s defned n mahemacal morphology usng a srucurng elemen. In he case consdered n hs paper an elemenary srucurng elemen s used. I conans he closes pxel s neghborhood. The eroson of he npu mage F s defned by: { F( p + q) } h G ε ( F) p Ρ: G( p) mn () q N ( p) where G represens he oupu mage, ε s he eroson operaor, P s an mage doman, N(p) represens he closes neghborhood of pxel p, and h s he hegh of he srucurng elemen. The operaon of dlaon s based on he maxmum value among he neghborng pxels and s wren as: { F ( p + q) } + h G δ ( F ) p Ρ: G( p) max () q N ( p) where δ represens he operaor of dlaon (he res of symbols has he same meanng as descrbed above). The eroson and dlaon of gven sze n of mage F are defned as, respecvely: ( n) ε ( F) ε( ε... ε( F)..) ; n mes δ ( n) ( F ) δ ( δ... δ ( F )..) n mes The medan mage [Beuch98] s defned as: ( λ ) ( λ ) { nf [ δ (nf( F, ), ε (sup( F, ) ]} M ( F, sup (3) λ where λ are ncreasng neger values. sup and nf symbols represen a supremum and nfnmum mages whch are defned as, respecvely: G sup( F, F ) p Ρ G( p) max{ F ( p), F ( p)} (4) G nf( F, F ) p Ρ G( p) mn{ F ( p), F ( p)} (5) Morphologcal medan of bnary mages. Fgure In case of he bnary mages he hegh of he srucurng elemen have obvously o be h. In he bnary case, however, here exss an addonal condon - he nersecon of he npu bnary mages mus be non-empy [Serra98,Iwano]. The example of bnary medan s presened on Fg.. When compung he morphologcal medan of he grayone mages, he hegh of he srucurng elemen h> [Beuch98,Iwano]. Formally sayng corresponds o he operaon performed wh he cylndrcal srucurng elemen. On he oher hand, he n case of mulvalued mages he condon of a non-empy nersecon doesn exs. In he color case

3 he hegh s represened by a vecor h[h h h 3 ] such ha h,h,h 3 > are n he RGB color space - he ncremenal values of red, green and blue componen respecvely.. Medan of color mages In case of grayone mages he compuaon of (3) s obvous - he mn and max operaors n (),(),(4) and (5) are beng compued on he scalar values. In he case of color mages funcons mn n () and (5) as well as max and () and (4) mus be calculaed on he values from color (vecor) space. In [Iwano99] a soluon based on lexcographc orderng and comparave color space was proposed. In every Caresan color space one always has o compare rples of numbers (hree-elemen vecors) when comparng pxels. One of he mos popular ways of comparng s a lexcographc orderng. I has been appled o color morphology n [Talbo98]. I s based on he successve comparsons of vecors componens begnnng wh componens wh he lowes ndexes. Ths approach has however one mporan dsadvanage - he a pror orderng of he mporance of he vecor componens. In he case of he RGB color space, he r-componen s consdered as more mporan han he g-componen, whle he b- componen s he less mporan one. Bu here s no reason o apply such an order of preference. To solve hs problem a new color space he comparave space [Iwano99] - s nroduced exclusvely for vecor comparson usng he lexcographc orderng. The nal RGB color space s convered o he comparave one by usng a 3x3 converson marx. The converson s based on he vsual mporance of color channels for he human percepon. The color componens are no of he same mporance for he human vson. The ransformaon o he comparave space sors ou he color componens and/or combnes hem by nroducng her lnear combnaons so ha he consecuve comparsons follow he vsual mporance of componens. Transformaon marx M of nal color space no comparave mulples he vecor of color componens: [ v v v ] T M [ r g b] T 3 where [ v v v ] T s a vecor n he comparave 3 s a vecor n he RGB vecor space and [ ] color space. Marx M s n hs paper compued accordng o lgr-orderng [Iwano99]:.3 M.6. In oher words, he order of comparsons s followng: he lumnance value, he g-value, and fnally he r-componen. The approach presened above allows comparng he vecors of RGB color space whle akng no accoun he mporance of colors for human vson. Due o ha fac allows also performng he morphologcal operaons on color mages. Consequenly enables he calculaon he morphologcal medan of color mages..3 The algorhm Equaon (3) s appled o consruc he erave algorhm of medan mage calculaons [Iwano99]. Sarng from a par (F, of nal mages, we nroduce he hree auxlary mages Z, W, M, whch are nally equal o: Z nf( F, ; W sup( F, ; M nf( F, The ndexes represen he number of eraon. Ieraed values n he -h eraon are compued usng he followng rules: Z δ ( Z ) ; W ε( W ) ; M sup[nf( Z, W ), M ] Ieraons are performed unl dempoence of M, whch means ha when he mage M sops o change he ask s accomplshed and fnally: M ( F, where M(F, s he morphologcal medan of mages F and G, and s he lowes eraon number such ha M M. + The above algorhm s convergen, n a sense ha he dempoence s always reached. I s guaraneed by he equaon (3) and he dscree naure of dgal mages. The only heorecal danger for he convergence are he oscllaons such ha M M + M + M. Snce however ha + 3 M M such a suaon canno happen. Examples of color medan mages obaned usng hs algorhm are presened on Fg. and Fg.3. Fg. shows a morphologcal medan of wo mages wh a smlar color palee conanng mosly green and yellow hue values. Fg.3 conans he medan of wo mages wh dfferen color palees wh domnan hues: red-yellow and green-blue. I s clearly vsble M +,

4 represen s ha he medan mage conans he objecs obaned by he shape-deformaon of he objecs on boh npu mages. 3. MORPHING SEQUENCE In he prevous secon he mehod of a sngle nerpolaed mage generaon was presened. Ths secon descrbes how o produce he complee sequence. Tradonal mehod of he producon of he nerpolaon sequence [Beuch98] s based on he successve producon of new medans beween pars of already generaed ones. Such an approach has, however, one dsadvanage. I doesn consder he conen of he mage - so s a knd of blnd operaon. Moreover doesn perm obanng he sequence of any gven lengh. We propose here a new mehod, whch generaes eravely he frames of he nerpolaon sequence one frame per eraon. New frames are nsered n dfferen posons n he sequence. The algorhm decdes, dependng on he dfference beween wo neghborng mages, wheher s necessary o generae a new medan or no. Ths decson s based on he dfference beween mages and s aken afer calculang he smlary measure beween wo frames. In he mehod from [Beuch98] he ask was o fnd an equdsan dsrbuon of nerpolaon levels whou akng no consderaon he conen of he mages. In our case nsead of he dsrbuon of levels, one opmzes he dsrbuon of measures beween every par of consecuve frames. The smlary measure s equal o an error e, whch s compued as he mean square error (MSE) beween he lumnance values of pxels belongng o consecuve frames P and Q: x y e( P, Q) xmax ymax (6) where funcon lum represens a lumnance value of s argumen, and x, y max max are he mage szes. max max [ lum( P(, j)) lum( Q(, j)) ] The algorhm makes use of wo vecors. The frs one S S S,, s a vecor of sequence frames [ ], he second one s a vecor of smlary measures beween every par of consecuve frames e. Boh vecors have he same number of elemens. The npu mages are X and Y. The emporary varable (couner) s equal o he number of already produced frames. The algorhm of morphng sequence producon: Le: (sar he couner) Le: S X, S Y Compue he error beween he npu mages: e e( S, S) Le: max (ndex of he hghes error value) Whle (no(sop-condon)) do:. Inser new, empy frame beween frames max and max+ (elemens of vecors S and e wh ndexes beween and max remans unchanged; elemens wh ndexes from max + o ge new ndexes from max + o + respecvely, ndex of a new elemen s max+). Calculae new medan: S M S, S ) max + ( max max+ e max e( Smax, S max+ e max + e( S max +, S max + 3. Le: ) 4. Le: ) 5. Le: + 6. Fnd max such ha: e max max{ e, e,, e } Sop he algorhm The fnal sequence of mages S [ S S, S ],, he morphologcally nerpolaed sequence, such ha S X, S Y ; frames S, S,, S he nerpolaed mages. The use of he srucurng elemen of a gven hegh h> durng he calculaons of sngle medans, guaranees he convergence of he above algorhm. I means ha when compung S k M ( S k, X ), afer ceran number of eraons Sk Sk X. The same resul would be obaned f, nsead of he nal mage X, he fnal one Y, would be used. Two versons of he algorhm are proposed. The dfference beween hem les n he sopcondon. Frs one generaes he sequence wh gven number of frames n. The algorhm s sopcondon n hs case s: ( n ). In he second verson s based on he hghes accepable error e MAX beween wo consecuve frames. The number of frames of he fnal sequence can vary dependng on he complexy and dfference beween npu mages. The eraons are performed unl he error (6) becomes smaller han e. In hs MAX case he sop-condon s: ( e < e ). The fnal max MAX value of ndcaes he number of frames of he nerpolaon sequence.

5 Dependng on he verson of he algorhm appled eher he requred number of frames n, or he value e should be gven. of MAX 4. QUALITY IMPROVEMENTS The bgges problem ha occurs, whle observng he frames of he morphologcally nerpolaed sequence, are he nclusons of hghly conrased groups of pxels. They make he sequence look unnaural and dsurb s vsual qualy by decreasng he spaal smoohness of he sequence frames. The nex problem les n he emporal smoohness of a sequence. I happens ha he ransons beween he frames of he sequence somemes seem abrup. The soluon of boh problems s based on he lnear flerng. I allows mprovng boh he spaal and he emporal smoohness of a sequence. We propose hree qualy-mprovng ools ha can be appled eher separaely or jonly dependng on he parcular demand. Frs wo of hem are he lnear flers. The reason for nroducng he lnear flerng s no o remove he nclusons, bu o sofen hers color value and o oban smooher ransons beween sequence frames. The hrd ool s an auxlary cross-dssolvng operaon ha mproves he emporal smoohness of he sequence. 4. Temporal flerng The emporal flerng allows reducng he conras of he mage pxels. The emporal fler of sze s gven by: S S '( x, S + λ S + S λ + S + f f f < < n n (7) where S (x, s a pxel on he -h frame of he nal sequence, S (x, s pxel on he flered frame, n s a oal number of frames of he morphologcally nerpolaed sequence. I s a weghed mean value - he weghs of he pxels from he prevous and he nex frame are se o. The wegh of he curren pxel λ can be chosen manually dependng on he vsual qualy of he sequence. Snce he frs and he las sequence frame are equal o wo nal frames, hey are no flered. The parameer λ conrols he nfluence of he curren frame on he flered one. I can be equal o any posve neger. The growh of hs parameer reduces however he effec of flerng. The applcaon of emporal flerng sofens he color values of nclusons whou nroducng he spaal blurrng. I allows reducng he conras of he nclusons. I resuls n more naural vsual effec and smoohes ou he emporal ransons beween frame pxels. The emporal flerng offers he bes resuls for he nclusons presen a he curren frame beng a he same me absen on he precedng. In such a case an effec of blendng s vsble and conras reducon s remarkable. On he oher hand when small nclusons are no growng on several consecuve frames, hey sar o be oo conrased agan afer few successve frames. Such knd of nclusons canno be flered usng he emporal flerng. The example of emporal flerng s shown on Fg.4b. I conans he resul of he emporal lnear flerng (wh λ) of mage on Fg.4a. 4. Spaal flerng Anoher knd of lnear flerng proposed s he spaal flerng. I reduces he vsual mpac of he nclusons by nroducng he spaal blurrng. The lnear spaal flerng s expressed by usng he followng equaon: S '( x, s s S ( x +, y + j) m( + s +, j + s + ) (8) s j s s+ s+ j m(, j) where S (x, s a pxel on he -h frame of he nal sequence, S (x, s a pxel on he flered one, s s a sze of a lnear fler, and m s a mask of hs fler represened by a marx of sze s+. Smlarly o he case of he emporal flerng, n he curren one we also fler only he morphologcally nerpolaed frames.e. frames wh he ndexes: < < n. The frs and he las frame of he sequence are smply coped from he nal sequence o he flered one. Varous masks can be appled o perform he spaal flerng. The followng hree masks of sze s have been consdered: m m 4 m 3 4 (9) Mask m represens he mean fler 3x3, mask m s weaker verson and mask m 3 he Gaussan fler. Ths knd of flerng s necessary when he case, menoned a he end of he precedng secon, happens. I occurs when he conras of he nclusons s oo hgh and, a he same me, he nclusons have smlar shape on several consecuve frames. In such

6 a case he effec of locally hgh conras canno be removed by emporal flerng and he spaal one mus be appled. The choce of fler depends on he vsual qualy of he fnal sequence. If one prefers o ge a sequence more myserous one may use sronger flers lke e.g. m. If one wans o have sharper nerpolaed mages weaker fler lke e.g. m s more suable. The example of spaal flerng s presened on fg.4c. The flerng (8) has been performed usng fler m from (9). Afer boh knds of flerng he arfacs caused by he morphologcal medan (clearly vsble on Fg.4a), has been eher removed (by emporal flerng) or sofened (by he spaal one). 4.3 Cross-dssolvng In order o make he ransons beween morphologcally nerpolaed frames smooher, he addonal cross-dssolvng s appled. Ths operaon can be wren usng he followng equaons: S' S' S' α S, α+ β α ( + ) ( S + < β < α; < n β α ) S + β α S +, () where α s gven number of new frames ncluded beween every par of consecuve sequence frames, S s a frame of cross-dssolved sequence, S s a frame of he nal one. Afer he cross-dssolvng operaon he lengh of a sequence S grows α+ mes by nserng α crossdssolved frames beween every par of consecuve frames of he nal morphologcally nerpolaed and lnearly flered sequence S. Cross dssolvng s especally useful n he ransformaon of he nal, gven mage o he closes morphologcally nerpolaed one. I allows human percepon o adap o he nerpolaed mage. An example of he crossdssolvng s presened on Fg.5a-5g. Fg.5a conans he npu, gven mage. Fg.5g he frs frame of he morphologcally nerpolaed sequence. Fg.5b-5f show he cross-dssolved frames (α5). 5. RESULTS Two morphng sequences are presened as an example. Boh have been obaned usng he proposed mehod. All hree proposed mprovemens have been appled: emporal flerng, spaal flerng and cross-dssolvng. The frs sequence conans he ransformaon of an mage of flowers no anoher one presenng a bole of beer. Boh mages have smlar color palees. Fg.6 presens he frames of a sequence, whch were produced usng he morphologcal nerpolaon and he lnear flerng. The morphologcal medan has been calculaed usng he dlaons () and erosons () usng a srucurng elemen of hegh h[,,]. The sequence was generaed usng he second rule. The gven maxmal error was equal o e MAX. I resuled n 8 morphologcally nerpolaed frames. Each nerpolaed frame was flered usng he emporal fler (7) wh λ and spaal fler (8) wh mask m from (9). The error values and order of producon of new medans s shown n Table. To creae he fnal sequence he cross-dssolved frames was nsered beween every par of consecuve frames accordng o (). The number of cross-dssolved frames was α. The second example shows he ransformaon of he nal mage presenng a ree n summer no he fnal mage wh a fores n auumn. The color palees n boh npu mages are dfferen. Fg.7 presens he frames of a sequence, whch were produced usng he morphologcal nerpolaon and he lnear flerng. The morphologcal medan has been usng a srucurng elemen of hegh h[,,]. The gven lengh of a sequence was: n3. Each frame was flered usng he emporal fler (7) wh λ and spaal fler (8) wh mask m 3 from (9). Fg. 4 shows he resul of cross-dssolvng produced afer he consrucon of morphologcal nerpolaon followed by emporal and spaal flerng. In hs case he number of cross-dssolved frames nsered beween every par of consecuve ones was α5. Frame Prod. order Error S Inal e(s, S )998 S 7 e(s, S )8 S 6 e(s, S 3 )6 S 3 4 e(s 3, S 4 )45 S 4 e(s 4, S 5 )76 S 5 e(s 5, S 6 )8 S 6 3 e(s 6, S 7 )6 S 7 5 e(s 7, S 8 )455 S 8 8 e(s 8, S 9 )883 S 9 Fnal Frs example - ls of errors Table 6. CONCLUSIONS The mehod of auomac morphng has been presened. I combnes he morphologcal nerpolaon and lnear flerng. The mehod s able o produce he nerpolaon sequence usng exclusvely he mage processng ools. Sarng from wo nal mages he morphng sequence

7 ransformng he nal mage no he fnal one s creaed. The ransformaon s produced whou applyng he conrol pons. The only human asssance s requred n order o ndcae he number of morphologcally nerpolaed frames and - f needed - he parameers for he lnear flerng ools (number of cross-dssolved frames as well as he ype of emporal and spaal lnear fler). The proposed mehod can be compared wh anoher auomac mehod: he pure cross-dssolvng. Conrary however o he las one conans shape meamorphoss of he objecs on he mage. In comparson on he oher hand wh he classc morphng echnques [Wolbe99], doesn requre an nroducon of he conrol pons, whch s meconsumng and manual process. Bu, n fac, sn a real compeor of classc morphng echnques. Boh mehods are complemenary and her usage depends on he mages one s dealng wh. The proposed mehod can be employed o ransform he mages whou sensve areas, ransformaon of whch have o be conrolled precsely, lke eyes, mouh, nose ec. n he human face morphng. The mehod can be successfully appled o he ransformaon of mages where he correspondence of areas on boh mages s no crucal. I s useful, n parcular, when a leas one of wo mages conans a lo of small deals (lke flowers or rees on examples shown n he paper), whch should be deformed - no blended (blendng could be obaned usng he smple cross-dssolvng). The deformaon of such mages usng radonal mesh-warpng would requre a huge amoun of conrol pons whch have o be ndcaed manually by he operaor. The proposed mehod deforms hem whou nroducng he conrol pons. The resuls, obaned usng he proposed mehod, are neresng also from he arsc pon of vew he ransformaon of he shape looks myserous. The mehod can be successfully appled o he producon of specal vsual effecs n TV, flm and mulmeda ndusry. [Iwano] Iwanowsk M.: Applcaon of mahemacal morphology o nerpolaon of dgal mages, Ph.D. hess Warsaw Unversy of Technology, School of Mnes of Pars, Warsaw-Fonanebleau [Meyer96] Meyer F.: Morphologcal nerpolaon mehod for mosac mages, In P.Maragos, R.W.Schafer, M.A.Bu Mahemacal morphology and s applcaon o mage and sgnal processng, Kluwer, 996. [Serra83] Serra J.: Image Analyss and Mahemacal Morphology vol.. Academc Press, 983 [Serra88] Serra J.: Image Analyss and Mahemacal Morphology vol.. Academc Press, 988 [Serra98] Serra J.: Hausdorff dsance and nerpolaons In: H.Hemans and J.Roednk, edors, Mahemacal Morphology and s Applcaons o Image and Sgnal Processng. Kluwer, 998 [Talbo98] Talbo H., Evans C., Jones R.: Complee orderng and mulvarae morphology In: H.Hemans and J.Roednk, edors, Mahemacal Morphology and s Applcaons o Image and Sgnal Processng. Kluwer, 998 [Wolbe99] Wolberg G.: Dgal Image Warpng, IEEE Compuer Socey Press, Los Alamos CA, REFERENCES [Beuch98] Beucher S.: Inerpolaon of ses, of parons and of funcons In: H.Hemans and J.Roednk, edors, Mahemacal Morphology and s Applcaons o Image and Sgnal Processng. Kluwer, 998 [Gomes97] Gomes J., Velho L.: Image Processng for Compuer Graphcs, Sprnger-Verlag, 997 [Iwano99] Iwanowsk M., Serra J.: Morphologcal Inerpolaon and Color Images Proc. of h Inernaonal Conference on Image Analyss and Processng Sep. 7-9, 999 Vence, Ialy; IEEE Compuer Socey

8 Two nal mages (a) and (b) conanng colors of smlar hue values (green and yellow) and her morphologcal medan (c). Fgure Frs example Fgure 6 Inal mages (a) and (c) conanng colors of dfferen hue values and her medan mage (b). Fgure 3 Morphologcal medan frame of he morphng sequence (a); afer emporal lnear flerng (b); and afer spaal and emporal flerng (c). Fgure 4 Cross-dssolvng Fgure 5 Second example Fgure 7