
varnakramiya aakar vishleshan

varnakramiya aakar vishleshan jyaamitiya aakaron kii tulanaa aur vishleshan karane key liye lapless-beltramy prachaalak key spectrum (igenvalue aur/yaa eigenfunctions) parr nirbhar kartaa haye. choonki lapless-beltram operator kaa spectrum samamitiyon key tahat aparivartaniya haye, isaliye yeh gair-kathor aakaron key vishleshan yaa punarpraapti key liye achchi tarah seey upayukt haye.ee. maanav, pashu, paudhe aadi jaisi modane yogya vastuein.
lapless.
lapless-beltramy prachaalak kaee mahatvapurna vibhedak samikaranon mein shamil haye, jaise kii ooshmaa sameekaran aur tarang sameekaran. ise ack remanian manifold parr ack vaastavik-moolya vaale falan "f" key pravanata key vichalan key roop mein paribhaashit kiya jaa sakata haih
isake varnakramiya ghatakon kii ganana helmholtz sameekaran (yaa laplacian igenvalue samasya) koo hull karke kii jaa sakati haih
samadhan eigenfunctions sootra _ 3 (mode) aur sambandhit eigenvalues sootra _ 4 hain, joe sakaratmak vaastavik sankhyaon key ack vichalan anukram kaa pratinidhitva karate hain. bund domen key liye yaa neuman seema sthiti kaa upayog karate samay pahala igen value shoonya hotaa haye. kuchh aakaron key liye, varnakram kii ganana vishleshanatmak roop seey kii jaa sakati haye (ee.jee. aaytaakaar, sapaat torus, cylinder, disk yaa golaa). udaaharan key liye, gole key liye, eigenfunctions golakar samanjasya hain.
igenvalue aur eigenfunctions key sabse mahatvapurna guna yeh hain kii way isometry aparivartaniya hain. dusare shabdon mein, yadi aakar faila naheen haye (ee.jee. kagaj kii ack sheet teesare ayam mein mudee hui haye), varnakramiya moolya naheen badalenge. modane yogya vastuein, jaise janwar, paudhe aur manushya, jodon parr kewal nyuntam khinchaav key saath shareer kii vibhinn mudraon mein jaa sakate hain. parinami aakritiyon koo nikat-samamit kahaa jaataa haye aur varnakramiya aakar vishleshan kaa upayog karke unaki tulanaa kii jaa sakati haye.
vivechana.
jyaamitiya aakritiyon koo aksar 2dee ghumaavdaar satahon, 2dee satah jaal (aamtaur parr tribhuj jaal) yaa 3dee thhos vastuon (ee.jee. voxel yaa tetrahedra mesh kaa upayog karna). inn sabhi mamalon key liye helmholtz sameekaran koo hull kiya jaa sakata haye. yadi koee seema maujood haye, tou ee.jee. ack varg, yaa kissy bhee 3 dee jyaamitiya aakar kii matra, seema sharton koo nirdisht karane kii aavashyakta haye.
vibhinn prakaar key jyaamiti pratinidhitva key liye lapless prachaalak key kaee vivechanaaen maujood hain (discreet lapless prachaalak dekhein). inmein seey kaee prachaalak antarnihit nirantar prachaalak kaa anuman achchi tarah seey naheen lagaate hain.
varnakramiya aakar vivaranak.
aakar aur isake roop.
aakar kaa aakar pahale varnakramiya aakar vivaranakon mein seey ack haye. yeh lapless-beltramy prachaalak key igenvalue kaa saamaanyikrit praarambhik anukram haye. isake mukhya laabh saral pratinidhitva (sankhyaon kaa ack sadish) aur tulanaa, paimaane kii aparivartaniyata hain, aur isaki sadgi key bavjood gair-kathor aakaron key aakar punarpraapti key liye ack bahut achchha pradarshan haye. 
aakar key pratiyogiyon mein bhooganitiya doori matricks (aiss. dee.-jee. dee. aim.) aur cumm dwi-harmonic doori matricks (aar.-bih. dee. aim.) key ekvachan moolya shamil hain.
haalaanki, igenvalue vaishvik vivarankarta hain, isaliye aakar aur anya vaishvik varnakramiya vivaranakon kaa upayog sthaaniya yaa aanshik aakar vishleshan key liye naheen kiya jaa sakata haye.
vaishvik bindu hastakshar (jee. pee. aiss.).
ack bindu sootra _ 5 parr vaishvik bindu hastakshar sootra _ 5 (i) parr ganana kiye gaye lapless-beltramy operator key scale kiye gaye eigenfunctions kaa ack vector haye.ee. aakar kaa varnakramiya antahsthaapit). jee. pee. aiss. iss arth mein ack vaishvik visheshta haye kii iska upayog aanshik aakar milan key liye naheen kiya jaa sakata haye.
kernell signature (aich. key. aiss.) koo germ karein.
ooshmaa kernell hastakshar ooshmaa kernell key igen-vighatan kaa upayog kartaa haih
satah parr pratyek bindu key liye ooshmaa kernell sootra key vikarn koo vishisht samay moolya sootra 9 parr namuna liya jaataa haye aur ack sthaaniya hastakshar praapt hotaa haye jiska upayog aanshik milan yaa samroopata kaa pataa lagaane key liye bhee kiya jaa sakata haye.
tarang kernell hastakshar (doubleu. key. aiss.).
doubleu. key. aiss. aich. key samaan vichaar kaa anusaran karate hain, joe ooshmaa sameekaran koo schroedinger tarang sameekaran key saath pratisthaapit karate hain.
unnat tarang kernell hastakshar (iwk).
iwk gair-kathor aakar punarpraapti key liye igenvalue mein ack nae scaling function koo pesh karke aur ack nae vakrata shabd koo ekatrit karke dablyuke mein sudhaar kartaa haye.
varnakramiya graph tarang hastakshar (aiss. jee. doubleu. aiss.).
aiss. jee. doubleu. aiss. ack sthaaniya discriptor haye joe naa kewal samamit aparivartaniya haye, balki compact bhee haye, ganana karane mein aasaan haye aur baind-paas aur low-paas filter donon key laabhon koo jodata haye. aiss. jee. doubleu. aiss. kaa ack mahatvapurna pahalu doubleu. key. aiss. aur aich. key. aiss. key laabhon koo ack ekal hastakshar mein sanyojit karane kii kshamata haye, jabaki aakaron key bahu-sankalp pratinidhitva kii anumati detaa haye.
varnakramiya milan.
jatil aakaron seey judey graph laplacian kaa varnakramiya apghatan (discreet lapless operator dekhein) eigenfunctions (mode) pradaan kartaa haye joe isometry key liye aparivartaniya hain. aakar parr pratyek sheersh koo pratyek bindu parr igenmodel moolyon key sanyojan key saath vishisht roop seey darshaaya jaa sakata haye, jise kabhi-kabhi varnakramiya nirdeshank kahaa jaataa haih
varnakramiya milan mein vibhinn aakaron parr sheershon koo jodkar bindu patrachar sthaapit karna shamil haye jinamen sabse samaan varnakramiya nirdeshank hain. praarambhik kaary stereoscopy key liye viral patrachar parr kendrit thaa. computational dakshata abb poorna jaal parr ghaney patrachar koo saksham banaati haye, udaaharan key liye cortical satahon key beach. varnakramiya milan kaa upayog jatil gair-kathor chhavi panjikaran key liye bhee kiya jaa sakata haye, joe vishesh roop seey taba kathin hotaa haye jaba chhaviyon mein bahut badee vikritiyaan hoti hain. varnakramiya igenmodel moolyon parr adharit iss tarah kii chhavi panjikaran vidhiyaan vaastav mein "vaishvik" aakar visheshtaon koo pakadati hain, aur paaramparik gair-kathor chhavi panjikaran vidhiyon key vipareet joe aksar sthaaniya aakar visheshtaon (jaise.jee., chhavi pravanata).