Nonorthogonal RGB-space decomposition

Transcription

1 onorthogonal R-space decomposton atala Vaganova nsttute of Computatonal Mathematcs and Mathematcal eophyscs ovosbrsk Russa Abstract hs paper deals wth some new deas of mage color models constructons he results of sphercal mage codng wth the dvson are also presented Keywords: mage compresson Color Space RODUCO At present tme the more or less mage data nterpretaton procedure presents sgnfcant nterest for the specalsts n mage processng and data transmsson by communcaton channel t s clear the man queston to decde what knd of nformaton we have to extract from huge volume data n order to catch sense thers he more common way for soluton of ths problem based on separaton of typcal mage propertes hs process corresponds to bran processng that s all of unnecessary or obscure nformaton s dscarded and the sketch s made up hs sketch then s used as mage syntheszng contaned only man propertes of concrete mage FA SASCS OF USE COLOR HE COD MAE Let us consder the mage palette π = { P = ( R ) = } n R color space and R are non-negatve ntegers We wll determne a gravty centre P ( R ) of the palette by next coordnates: R = R = = he man drecton n palette π by defnton s the drectng vector of straght lne L= { P R : P= P + t < t <+ } contaned the gravty center and has mnmal root-mean-square dstance for ponts of the palette We wll suppose vector has unt Eucldan length - = Hence we have to mnmze the functonal Φ ( ) = ρ( LP ) mn ; here ρ ( L P ) means the Eucldan dstance between the lne L and palette pont P After the smplest lnear transformaton P = P P we have ρ ( LP ) ρ ( L P ) P ( P ) we get the mnmzaton problem Φ ( ) = P ( P ) mn = = hen under the natural constran = or the maxmzaton problem arses ψ ( ) = ( P ) max Usng the Lagrange functon L( λ) = ψ( ) + λ( ) after the dfferentaton we obtan L ( k ) = ( x P ) λk = k = k () () () where x = R R x = x = = f we denote by the rectangular - ( k ) = x then relaton above can be wrtten matrx { } n the matrx form k= = λ (*) () e s the egenvector of and = t s also () () worth to menton that the egenvectors produce an orthonormal bass because the matrx Symmetrc Let = λ l λ λ λ l = Let us fnd ψ ( ) ( k) ψ = x k = k= ( ) ( ) l ψ ( ) = = λ hus the maxmal value of ψ ( ) or mnmal value of Φ ( ) take place on the egenvectors () whch corresponds to the maxmal egenvalue λ of the matrx P f we construct now the plane whch ncludes gravty center () of the palette π and s orthogonal to the egenvector and nternatonal Conference raphcon 5 ovosbrsk Akademgorodok Russa

2 fnd n ths plane the best drecton then t corresponds to the () second egenvector for the matrx he last () egenvector determnes the least mportant drecton n the () () () palette π hus the egenvectors determne three factors n the order of mportant decay wth respect to the a prory gven palette π and the levels of mportance as we suppose are regulated by the rato λ: λ : λ Accordng to ths -factors analyss of the palette the color Rmage can be decomposed nto components (FAdecomposton) F F F by the followng rule: () () () F R R () () () F = or the matrx () () () F form F = V( P P ) he nverse transformaton s also very P P V F = + because V s an orthogonal matrx smple hs knd of decomposton for the color mage we call FA standard FA-DECOMPOSO ASED O MAE PREPROCESS AD ERACVE USER ADDOAL FORMAO he algorthm descrbed n prevous secton may be really used on practce and can be appled for effectve decomposton of mages of average resolutons Also there are numbers of mages n that the color are not the man factors from the pont of the palette however one may be mportant for mage understandng For example t may be some dedcated mage detals captons geographc object on the map and so on n ths case for effectve mage analyss and decomposton the addtonal nformaton from mage assgned wth detals has been gven f there many mportant detals are on the mage the nformaton may be taken from mage analyss by hstogram constructon and we can say the color of gven number are very often meetng n the mage and t may be mportant for decodng of the mage User also may want mportance coeffcents assgn to colors n the palette hat s needs to propose algorthms where mage colors nclude n FA-analyss wth some weght he method of mage decomposton wth addtonal nformaton s smlarly to above descrbed method n contrast to problem (*) the color statstcs correspond to weght coeffcents Let us consder next problem: n color R space wth gven fnte palette π we take ntal mage as number of pont { P = ( α( R )) = } here R - ntenstes of Red reen lue color components corresponds to pont of the palette and α - weght e number of ponts of -th color n the mage herefore the ravty Center of π may be easy calculated: R αr = α = = α and (*) may be rewrte AL = λ where s rectangular matrx from prevous secton AL - vector at length consst of weght coeffcents All descrbed n prevous secton reasonng are smlarly performed Remark f reader not assgns colors from palette specal weght that s computer performs calculatons automatcally then palette from the mage are not pcked out snce t decreases tmes complexty of the method as for mage wth average resoluton as for hgh one 4 OORHOOAL DECOMPOSO OF R SPACE he ways descrbed above fnd three orthogonal vectors n real tme as we understand always the vectors has not been orthogonal Of course the prncpal drecton n ths case s found n a better way and corresponds to maxmal energy concentraton n the frst component but second and thrd vectors may be selected n the best of manner and tme complexon s naturally ncreased Well let us prncple drecton have been selected and frst component performed R R () () () F = ( ) hen nverse transformaton gves the components of the mages reconstructed by factor F : R = R + F F F F () () () = + = + Calculates the dfferences between ntal mage components R and reconstructed by factors F to get the second drecton R = R RF = F = F Further consderng R nstead of ntal mage components R selects the prncpal drecton R R () () () F = ( ) nternatonal Conference raphcon 5 ovosbrsk Akademgorodok Russa

3 nverse transformaton performs the mage components reconstructed by factor F () RF = R + F F = + = + F () () At last thrd drecton s selected by analogue wth second one R = R R = = ntal only for mage Orgnal mage s restored by formulas: F F F R = RF + R F + R F = F + F + F = + + F F F 5 COMPARSO OF PAL AD FA SADARDS WH RESPEC O FRACAL MAE COMPRESSO ASED O SPHERCAL OJEC CLASSFCAO PAL s a famous televson orented standard [4 ]based on transformaton Y 59 R U = V 6 5 whch tends to forthcomng compresson As the compresson algorthm we use here so called fractal mage compresson based on sphercal classfcaton he algorthm conssts of specal classfcaton of greylevel square regons wth -splnes approxmaton as postprocessng of the mage and s detaled descrbed n [] We execute the comparson of PAL and FA standards n ths context wth three knds of mages: Lena usual lfe color photos Cheetah and frame from televson broadcastngs vdeo [5] Unfortunately t s mpossble here to prnt color pctures and we descrbed half-tone components only he mage n FAdecomposton and n PAL-decomposton wll be compressed wth the same compresson rato Example CHEEAH Fgure Second decoded component of CHEEAH ntal mage has resoluton wth 8-bt color depth Fgure Frst decoded component of CHEEAH Fgure hrd decoded component of CHEEAH hree factor analyss of the mage gves the followng results: Example LEA P = ( ) λ λ λ = V = ntal mage has resoluton 5 5 wth 6-bt color depth hree factor analyss of the mage gves the followng results: nternatonal Conference raphcon 5 ovosbrsk Akademgorodok Russa

4 Example VDEOFRAME ntal mage has resoluton wth 4-bt color depth Fgure 4 Frst decoded component of LEA Fgure 6 Second decoded component of LEA hree factor analyss of the mage gves the followng results: Fgure 7 Frst decoded component of VdeoFrame Fgure 5 Second decoded component of LEA P = (889479) λ λ λ = V = P = (564587) λ λ λ = V = nternatonal Conference raphcon 5 ovosbrsk Akademgorodok Russa

5 whch provde the beautful colourng of the word around us are not essentally mportant he expermental compresson rato results wth respect to JPE (the compresson ratos are about 5- tmes better then JPE n the same PSR of decoded mages ) algorthm are comprehensvely descrbed n [] he author s supported by Russan Scence Support Foundaton by ntegraton project 6 he development approxmaton theory by splnes trgonometrc polynomals and fractals wth applcaton to constructon of water turbne models and broadcastng mage compresson by nterschool Scentfc- echncal Program Fundamental hgher-school research n the feld of natural scences and humantes Unverstes of Russa under rand 95 Fgure 8 Second decoded component of VdeoFrame Fgure 9 hrd decoded component of VdeoFrame he carry out experments have shown that the applcaton of FA decomposton ncreases the mage qualty (PSR) of decodng mages by -4 d wth respect to PAL standard he results of FA non-orthogonal decomposton are presented at fgure -9 he expermental data about orthogonal usng may be found n [5] 7 REFERECES [] Vaganova A he algorthm of Varance and Sphercal fractal base constructon for mage compresson Vestnk of ovosbrsk State Unversty seres: mathematcs mechancs and nformatcs () SU page -9 [n Russan] [] ruzman S Krchuk VS Kosyh VP Peretyagn Spector AA Dgtal mage proceedn n the nformaton system SU [] Zubarev Yu lorozova L he broadcastng technque ude Moskow:-Rado and communcaton 994 [4] rodsky MA he Color broadcastng second edton Moskow:-Hgh school994 [5] Vaganova A he fractal vdeo compresson wth sphercal classfcaton and FA-orthogonal preprocessng Vestnk of ovosbrsk State Unversty seres: mathematcs mechancs and nformatcs page 5 [n Russan n prnt] About the author atala Vaganova s a scentfc researcher at nsttute of Computatonal Mathematcs and Mathematcal eophyscs Department of umercal analyss and Computer raphcs Head researcher of fractal group Enerroup nc (USA) atala Vaganova has receved her PhD n December Her contact emals are vaganova@oapmgssccru nvaganova@enersoftcom 6 COCLUSO n our experments wth decomposton and compresson of the color mages we have not the am to get the exact answer to the queston: what s the better the well-known PAL-standard or proposed FA-method? n fact our experment has shown that FA s usually better and t s not surprsng because the addtonal nformaton on the palette was used n decomposton On the other hand PAL-standard s more unversal and has the same coeffcents for all pctures and palettes t s dangerous to compress the man components F and Y both n FA and PAL but rest components can be essentally F and Y compressed wthout loss of qualty he components are qute nformatve n the color mage and the rest components nternatonal Conference raphcon 5 ovosbrsk Akademgorodok Russa