Wavelet txia: kev txiav txim rau daim ntawv thov piv txwv

Lub advent ntawm pheej yig cov koob yees duab tau txhais hais tias yog ib feem loj ntawm lub inhabitants ntawm cov ntiaj chaw, tsis hais hnub nyoog thiab pw ua ke, tau mas yuav kis tau tus cwj pwm mus ntes nws txhua txhua kauj ruam thiab muab tso rau lawv cov dluab nyob rau pej xeem cov zaub nyob rau hauv lub kev tes hauj lwm. Tsis tas li ntawd, yog hais tias tus ua ntej lawm tsev neeg yees duab archive twb muab tso nyob rau hauv tib album, niaj hnub no nws muaj pua pua ntawm cov duab. Nyob rau hauv thiaj li yuav pab txhawb cia thiab kis tau tus mob thoob plaws tes hauj lwm yuav tsum tau ib cov duab ntawm qhov ceeb thawj yuav txo tau. Yuav kom qhov no kawg, txoj kev yog siv uas yog raws li nyob rau hauv ntau yam algorithms, xws li ib tug wavelet txia. Yuav ua li cas yog nws, qhia rau peb cov tsab xov xwm no.

Yuav ua li cas yog ib cov duab

Nrig txog kev pom cov lus qhia nyob rau hauv lub computer yog sawv cev nyob rau hauv daim ntawv ntawm cov zauv. Nyob rau hauv tej yam yooj yim cov ntsiab lus, ib tug yees duab nrog ib cov ntaus ntawv, yog ib lub rooj nyob rau hauv uas lub hlwb yog nkag mus rau qhov tseem ceeb ntawm txhua tus ntawm nws cov pixel xim. Thaum nws los txog rau ib tug monochrome duab, ces lawv hloov los ntawm luminance qhov tseem ceeb ntawm lub luv [0, 1], qhov twg 0 yog siv los xa mus rau dub, thiab 1 - dawb. Lwm yam xim yog muab fractional tus xov tooj, tab sis nrog lawv ua tau txawv txawv rau koj khiav lag luam, ces tus ntau yog ncua thiab tus nqi xaiv los ntawm lub luv ntawm 0 thiab 255. Yog vim li cas yog no? Nws yog yooj yim! Nrog rau qhov no xaiv nyob rau hauv lub binary sawv cev rau encoding lub luminance ntawm txhua pixel yuav tsum tau raws nraim ib byte. Nws yog pom tseeb tias ib tug ntau ntawm lub cim xeeb yuav tsum tau muab khaws cia txawm ib tug me me daim duab. Piv txwv li, daim duab loj ntawm 256 x 256 pixels yuav siv sij hawm 8 Kbytes.

Ob peb lo lus hais txog cov duab compression txoj kev

Muaj tseeb txhua leej txhua tus tau pom cov neeg pluag zoo ntawm cov duab uas muaj distortions nyob rau hauv daim ntawv ntawm cov voos ntawm tib xim, uas yog hu ua artifacts. Lawv tshwm sim raws li ib tug tshwm sim ntawm lub thiaj li hu ua lossy compression. Nws yuav txo tau tus luj ntawm lub duab, Txawm li cas los, nws yuav inevitably kev cuam tshuam rau nws zoo.

Rau lossy compression algorithms muaj xws li:

• JPEG. Qhov no yog nyob deb ib qho ntawm feem nrov algorithms. Nws yog raws li nyob rau kev siv ntawm discrete cosine txia. Nyob rau hauv kev ncaj ncees nws yuav tsum tau muab sau tseg hais tias muaj kev xaiv rau JPEG ua tau zoo lossless compression. Cov no muaj xws lossless JPEG thiab JPEG-LS.
• JPEG 2000. Lub algorithm yog siv rau mobile platforms, thiab raws li nyob rau hauv daim ntawv thov ntawm ib tug discrete wavelet txia.
• fractal compression. Nyob rau hauv tej rooj plaub, nws tso cai rau koj mus muab tau cov dluab ntawm zoo heev zoo txawm muaj zog compression. Txawm li cas los, vim muaj teeb meem nrog rau cov patenting ntawm no txoj kev tseem mus yuav tau txawv.

Lossless compression algorithms ua los ntawm:

• RLE (siv raws li tus thawj txoj kev nyob rau hauv lub TIFF hom ntawv, BMP, TGA).
• LZW (siv nyob rau hauv GIF hom).
• LZ-Huffman (siv rau PNG hom).

Fourier transform

Ua ntej tig mus rau lub wavelet, nws yuav ua rau kev txiav txim zoo mus tshawb lwm zog, piav txog lub coefficients ntawm lub expansion ntawm thawj zaug ntaub ntawv mus rau hauv elementary Cheebtsam, piv txwv li. E. Harmonic deeg nrog txawv frequencies. Nyob rau hauv lwm yam lus, lub Fourier transform - ib tug tshwj xeeb lub cuab tam txuas discrete thiab nruam worlds.

Nws zoo li no:

Cov inversion mis yog sau raws li nram no:

Yuav ua li cas yog ib tug wavelet

Tom qab no lub npe hides xaam muaj nuj nqi, uas tso cai rau koj los soj ntsuam qhov sib txawv zaus Cheebtsam ntawm qhov kev kuaj ntaub ntawv. Nws teeb yog ib qho undulation uas nws amplitude txo rau 0 deb ntawm lub hauv paus chiv keeb. Nyob rau hauv kev txaus siab yog cov wavelet coefficients txiav txim ib lub teeb liab.

Wavelet spectrograms yog cov sib txawv ntawm cov pa Fourier spectra, txij li thaum ntau yam nta txuam spectrum Pib ntsais koj teeb nrog lawv sab nqaij daim tawv tivthaiv.

Wavelet transformation

Qhov no txoj kev ntawm lub teeb liab hloov dua siab tshiab (zog) tso cai rau nws mus txhais lus los ntawm ib lub sij hawm nyob rau hauv lub sij hawm-zaus sawv cev.

Yuav kom wavelet transformation yog ua tau, lub sib nug xov wavelet muaj nuj nqi, cov nram qab no tej yam kev mob yuav tsum tau ntsib:

• Yog hais tias rau ib co muaj nuj nqi ψ (t) -Fourier txia muaj rau hauv daim

uas mob yuav tsum tau txaus siab:

Nyob rau hauv tas li ntawd:

• Wavelet yuav tsum muaj ib tug finite zog;
• nws yuav tsum tau integrable tas mus li thiab muaj kev cog lus ntawm kev pab txhawb nqa;
• wavelet yuav tsum tau laus ob leeg nyob rau zaus thiab nyob rau hauv lub sij hawm (qhov chaw).

hom

Ib tug tas mus li wavelet txia yog siv rau cov duas paub Pib ntsais koj teeb. Ntau npaum li cas nthuav yog nws discrete analogue. Tom qab tag nrho, nws yuav siv tau rau cov lus qhia ua nyob rau hauv computers. Txawm li cas los, ib qho teeb meem tshwm sim nyob rau hauv hais tias lub mis rau ib tug discrete fiberboard tsis tau los ntawm tej yam yooj yim uas tsim nyog discretization qauv DNP.

Cov tshuaj rau qhov teeb meem no twb pom los ntawm Daubechies, uas muaj peev xwm xaiv ib txoj kev los tsim kom tau ib tug series ntawm orthogonal wavelets, txhua tus uas yog txhais los ntawm ib tug finite nab npawb ntawm cov coefficients. Tom qab ntawd ceev ceev algorithms tau tsim, xws li cov algorithm Malla. Nyob rau hauv nws daim ntawv thov rau decompose los yog los qhov yuav tsum tau thiaj li yuav ua hauj lwm CN, qhov twg N - qauv ntev, thiab nrog - lub xov tooj ntawm coefficients.

Vayvlet Haar

Yuav kom hlob ib tug duab, nws yog tsim nyog los nrhiav tau ib tug tej yam regularity ntawm nws cov ntaub ntawv, thiab txawm zoo dua yog hais tias nws yuav ua tau ntev chains ntawm zeros. Qhov no yog qhov uas nws yuav ua tau pab tau rau lub wavelet txia algorithm. Txawm li cas los, peb tseem los saib xyuas cov hauj lwm txoj kev nyob rau hauv kev txiav txim.

Thawj zaug nws yog tsim nyog los rov qab hais tias dluab lub brightness ntawm ib sab pixels feem ntau yog yus muaj los ntawm ib tug me me npaum li cas. Txawm yog hais tias muaj yog dluab rau ntawm qhov chaw uas ntse ntse, contrasting sib txawv ntawm brightness, lawv nyob xwb ib feem me me ntawm daim duab. Raws li ib qho piv txwv, coj dua lub npe hu xeem Lenna grayscale duab. Yog hais tias peb coj ib tug matrix ntawm luminance ntawm nws cov pixels, ces rau ib feem ntawm cov thawj kab yuav tshwm sim raws li ib tug sib lawv liag ntawm tus xov tooj 154, 155, 156, 157, 157, 157, 158, 156.

koj yuav thov li no-hu ua delta txoj kev kom tau zeros rau nws. Ua li no, kom tsuas yog thawj tus xov tooj, thiab rau lwm tus noj xwb qhov sib txawv ntawm txhua tus ntawm yav tas los ib tug nrog tus kos npe rau "+" los yog "-".

Cov no ib tug ib theem zuj zus 154,1,1,1,0,0,1, -2.

Ib tug disadvantage ntawm delta-encoding yog nws uas tsis yog-tas. Nyob rau hauv lwm yam lus, nws yog tsis yooj yim sua rau noj xwb ib tug hlais cov ib theem zuj zus thiab paub dab tsi ci ci nws yog kho, decoded, yog hais tias tsis tag nrho cov tseem ceeb nyob rau hauv pem hauv ntej ntawm nws.

Yuav kom kov yeej no disadvantage, tus xov tooj yog muab faib ua officers thiab txhua yog ib nrab ntawm cov sum ntawm (v. A) thiab ib nrab ntawm cov sib txawv (v. D), m. F. Rau (154,155) (156,157) (157,157) (158,156) muaj (154.5, 0,5) (156.5,0.5) (157,0.0), (157, -1.0). Nyob rau hauv cov ntaub ntawv no, nws yog ib txwm tau mus nrhiav tau cov nqi ntawm cov ob tug zauv nyob rau hauv ib tug khub.

Nyob rau hauv dav dav, lub discrete wavelet txia ntawm lub teeb liab S, peb tau:

Qhov no txoj kev li nram no los ntawm lub discrete cov ntaub ntawv ntawm tas mus li wavelet txia, Haar thiab dav siv nyob rau hauv ntau yam teb ntawm cov ntaub ntawv ua thiab compression.

compression

Raws li twb tau hais, ib tug ntawm qhov kev siv ntawm wavelet txia algorithm yog lub JPEG 2000 compression txoj kev siv Haar raws li nyob rau hauv cov neeg txhais lus vector ntawm ob pixels nyob rau hauv lub X thiab Y vector (X + Y) / 2 thiab (X - Y) / 2. Nws yog txaus kom muab thawj zaug vector nyob rau hauv lub matrix hauv qab no.

Yog hais tias cov ntsiab lus ntau, noj ntau matrix, uas yog cov txheej txheem ntawm nyob rau hauv ib tug kab pheeb ces kaum matrix H. Yog li ntawd, cov thawj vector nws tus kheej ntawm nws kav ntev npaum yog tiav nyob rau hauv officers.

cov ntxaij lim dej

Cov uas ua "ib nrab-sum" - yog qhov nruab nrab luminance qhov tseem ceeb ntawm pixels nyob rau hauv officers. Qhov ntawd yog tus nqi thaum hloov dua siab tshiab rau daim duab yuav tsum tau muab nws ib daim qauv, txo nyob rau hauv 2 lub sij hawm. Nyob rau hauv no ib nrab-sum sim brightness, t. E. "lim" random bursts ntawm lawv qhov tseem ceeb thiab ua raws li zaus cov ntxaij lim dej.

Tam sim no cia tus nrog cov neeg uas qhia tau qhov txawv. Lawv yog "cais" interpixel "bursts", tshem cov qhov tivthaiv, piv txwv li. E. "lim" qhov tseem ceeb thaum uas tsis muaj frequencies.

Txawm los ntawm cov saum toj no Haar wavelet txia rau lub "dummies" nws yuav cuab kev hais tias nws yog ib tug khub ntawm cov ntxaij lim dej uas faib ib lub teeb liab mus rau hauv ob lub Cheebtsam: lub siab zaus thiab tsawg zaus. tsuas rov sib sau ua ke cov ntsiab kom tau tus thawj lub teeb liab.

Piv txwv li

Piv txwv peb xav kom hlob cov duab (test duab Lenna). Xav txog cov piv txwv ntawm cov wavelet txia ua lub matrix ntawm pixel brightnesses. Cov high-zaus tivthaiv ntawm daim duab yog lub luag hauj lwm rau displaying zoo kev nthuav dav thiab piav lub suab nrov. Raws li rau lub tsawg zaus, nws muaj cov lus qhia txog qhov zoo ntawm lub ntsej muag thiab du gradients ntawm brightness.

Nta duab ntawm tib neeg xaav muaj xws li hais tias tus tom kawg yog tseem ceeb tivthaiv. Qhov no txhais tau tias thaum compressed ib tug tej yam yog ib feem ntawm lub high-zaus cov ntaub ntawv yuav tsum muab pov tseg. Qhov ntau li ntawd vim hais tias nws muaj tsawg tus nqi thiab yog kho ntau compactly.

Yuav kom nce lub degree ntawm compression yuav tau mus thov ob peb lub sij hawm Haar transformation mus rau ib tug tsawg-zaus cov ntaub ntawv.

Cov kev siv ntawm ob-dimensional arrays

Raws li twb tau hais, cov duab nyob rau hauv lub computer yog nyob rau hauv daim ntawv ntawm ib tug matrix ntawm intensities qhov tseem ceeb ntawm nws cov pixels. Yog li, peb yuav tsum txaus siab rau ib ob-dimensional Haar wavelet txia. Yuav kom siv yog tsim nyog tsuas ua nws seem hloov dua siab tshiab rau txhua leej thiab txhua kem ntawm qhov matrix ntawm lub intensities ntawm pixels nyob rau hauv cov duab.

Qhov tseem ceeb nyob ze rau zero, yuav tau muab pov tseg tsis tseem ceeb kev puas tsuaj rau lub decoded duab. Qhov no tus txheej txheem no hu ua quantization. Thiab nyob rau hauv qhov theem ntawm cov ntaub ntawv no yuav ploj. Los ntawm txoj kev, tus xov tooj ntawm nullable yam tej zaum yuav hloov, li no kho cov neeg kawm ntawv ntawm compression.

Tag nrho cov kauj ruam ua nyob rau hauv hais tias lub matrix yog tau uas muaj loj nyiaj ntawm 0. Nws yuav tsum tau sau ib kab nyob rau hauv ib phau ntawv cov ntaub ntawv thiab hlob tej archiver.

seev suab

Cov rov qab pom nyob rau hauv cov duab hauv qab no algorithm:

• Nws unpacks ib archive;
• siv rov qab Haar txia;
• Lub decoded duab yog hloov dua siab tshiab rau hauv ib lub matrix.

Zoo piv rau JPEG

было сказано, что он основан на ДКП. Thaum xaiv cov algorithm Joint Photographic Kws txawj Group twb hais tias nws yog raws li nyob rau hauv DCT. Qhov no hloov dua siab tshiab yog nqa tawm nyob rau hauv blocks (8 x 8 pixels). Raws li ib tug tshwm sim, yog tias ib tug muaj zog compression rau txo duab yuav appreciable thaiv qauv. Thaum lub sij hawm compression siv wavelets xws li ib tug teeb meem tsis tuaj kawm ntawv. Txawm li cas los, suab nrov yuav tshwm sim lwm yam uas muaj cov tsos ntawm ripples nyob ib ncig ntawm npoo. Nws yog ntseeg hais tias zoo xws li cov artifacts nyob rau nruab nrab tsawg hnov dua "squares" uas tau raug tsim thaum uas siv cov JPEG algorithm.

Tam sim no uas koj paub tias dab tsi wavelets yog dab tsi lawv yog dab tsi thiab kev siv tswv yim rau lawv tau pom nyob rau hauv lub tshav pob ntawm kev ua thiab compressing cov dluab.