Transform Coding. C.M. Liu Perceptual Signal Processing Lab College of Computer Science National Chiao-Tung University

Transcription

1 Trnsform Codng C.M. Lu Perceptul Sgnl Processng Lb College of Computer Scence Ntonl Cho-Tung Unversty Offce: EC538 (03)

2 Motvtng Exmple

3 Motvtng Exmple: Rotton 3 Consder the (reversble) rotton: θ Ax θ θ 0 x0 cosφ snφ, x, A, sn cos φ θ x φ φ rctn.5 A

4 4 Motvtng Exmple: Trnsformed Sequence

5 Motvtng Exmple: Compresson Step 5 Throw wy the second coordnte For fxed codng, tht s 50% reducton!

6 Motvtng Exmple: Reconstructed Sequence 6 Orgnl Reconstructed

7 Motvtng Exmple: Error Anlyss 7 { xˆ n } reconstructed sequence ˆ θ θ 0,,4, K 0 otherwse N 0 ( x ˆ x ) N 0 ( θ ˆ θ ) Error depends on the mgntude of the θ n members set to zero If mgntude s smll, so s the error I.e., most nformton s the frst element of ech pr

8 Motvtng Exmple: Sttstcl Vew 8 Result: Mx compcton s cheved when the trnsform decorreltes the sequence Prncple Component Method I.e., smple-to-smple correlton s zero.

9 Trnsform Codng 9 Trnsformton Dvde orgnl sequence {x n } nto blocks of sze N Mp ech block nto trnsform sequence {θ n } Usng reversble mppng Quntzton, bsed on Desred verge bt rte Sttstcl propertes of trnsformed sequence Dfferent technques my be used for dfferent subsequences Dstorton Entropy codng Fxed-rte, Huffmn, AC, RLE+AC,

10 The Trnsform 0 Lner forwrd trnsform: θ n N x 0 n, The chrcterstcs of ech element of {θ n } depend on ts poston E.g. odd vs. even elements n motvtng exmple Ths my not be true of {x n } Desgn The vrnce of the trnsform sequence determnes codng scheme N s domn specfc nd s bsed on prctcl consdertons Reconstructon N 0, x θ b n

11 The Trnsform () D trnsform θ Ax x Bθ [ ] [ ] j j j j b,,,, B A I BA AB Θ 0 0,,,,, N N l k j j l k X Seprble D trnsforms Θ 0 0,,,, N N j j k l k X

12 The Trnsform (3) Orthonorml trnsforms T T B Θ B X AXA Θ T A A B ΘA A X T Energy preservton property: 0 0 ) ( N T T T N T T x x x Ax A x Ax Ax θ θ θ

13 Energy Compcton 3 Trnsform codng gn

14 Decomposton Vew of Trnsforms 4 Trnsform rows bss vectors Exmple Frst row: low-pss sgnl Second: hgh-pss sgnl x x θ θ θ θ A 0 0 α α α θ θ

15 Flter Exmple 5 Consder two sequences: low pss : (3, ) hgh pss : (3, -) 3 3

16 Mtrx Vew 6 NN N N N N L M O M M L L A [ ] , N N N N j N j N jn j j jn j j jn j j N j L M O M M L L L M α

17 Mtrx Vew () 7,,0 0, 0,0 α α α α,,0 0 0, 0 0, α θ α θ α θ α θ θ θ θ θ x x x x DC coeffcent AC coeffcents Bss mtrces Decompose nto outer products of the rows.

18 Krhunen-Loéve Trnsform (KLM) 8 A.k.. Hotellng Trnsform Conssts of the egenvectors of the utocorrelton mtrx: [R],j E [X n X n+ -j ] Mnmzes the geometrc mens of the vrnce of the trnsform coeffcents KLM provdes mxml G TC Q: Why do nythng else? Computng KLM s reltvely expensve For (reltvely) sttonry nput, KLM could work For most nput, however, KLM would hve to recomputed/communcted frequently

19 KLM Exmple 9 Egenvlues: Egenvectors: Normlzton: KLM mtrx:

20 Dscrete Cosne Trnsform 0 Dervtve of DFT Better suted for compresson

21 DCT Bss Vectors

22 DCT Bss Mtrces

23 3 DCT Bss for 4x4 Mtrces

24 DFT vs. DCT 4 DFT: DCT:

25 DCT Propertes 5 For Mrkov processes: ρ [ x x ] E n n+ E [ x ] n As ρ gets lrge, DCT pproches KLM compcton In prctce, mny sources re Mrkov DCT s populr choce: JPEG MPEG H.6

26 Dscrete Sne Trnsform (DST) 6 Complmentry propertes to DST: As ρ gets smll, DST pproches KLM compcton

27 Dscrete Wlsh-Hdmrd Trnsform DWHT 7 Hdmrd mtrx of H order N: HH T N I Constructon rules for N k : [] N N N N N H H H H H H, H H 4

28 DWH Trnsform 8 Sequency of row: Hlf the number of sgn chnges Dervng trnsform mtrx H from Hdmrd mtrx H N : Normlze H N : multply by Plce rows n sequency order: E.g.: N Performnce Very esy to mplement on constrnt hrdwre Overll, substntlly less compcton thn DCT

29 Codng of Trnsform Coeffcents 9 Bsc observton Dfferent coeffcents crry dfferent mounts of nformton E.g., recll motvtng exmple We should use dfferent quntzton/codng schemes to tke dvntge Two pproches to code ssgnment Optmzton Recursve

30 Optmzton Approch 30 Averge bt rte: R R M M k R k Quntzer nput: θ k k th reconstructon error: r k σ r k α Rk k σθk σ r M α k Rk k σθ k

31 Optmzton Approch () 3 Objectve Fnd R k such tht σ r s mnmzed Phrse s Lgrnge multpler problem: Assume α α k for ll k J α M R M k σ R k θ λ k k M R k R k log M ( α ln σ ) ( α ln σ ) λ M k θ k θ k log R λ

32 Lgrnge Multpler-- Bckgrounds 3 Optmzton Introduce the Lgrnge functon Solve

33 Optmzton Approch (3) 33 R k R + log σ M k θ k ( σ ) θ k M Comments R k wll mnmze σ r : Not gurnteed nteger: Not gurnteed postve: Workrounds: Ignore negtves Unformly reduce R k

34 34 Recursve Algorthm (Zonl Smplng)

35 Threshold Codng 35 Zonl smplng observton Bt lloctons bsed on verge vlue Locl vrton my not be reconstructed properly E.g., edge pxel representton

36 Exmple: Allocton for 8x8 Trnsform for Zonl Codng 36 Threshold codng All coeffcents bove gven threshold re quntzed & coded. Typcl pproch Alwys code frst (DC) coeffcent Threshold code the rest: <coeff, precedng zeroes count> EOB

37 37 Zgzg D Block Trversl

38 JPEG: Intl Processng 38 RGB YUV mppng (lter) 4:: sub-smplng (lter) Assume p-bt encoded mge (For color mges there re three plnes: Y, U, V) Level shftng: X,j X,j p- E.g., Splt pxels nto 8x8 blocks Lst row/column replcted to cheve multple of 8 Added dt s dscrded durng the decodng Apply forwrd DCT

39 Exmple 39 Orgnl mge block Block fter level shftng & DCT

40 JPEG: Quntzton 40 Mdtred quntzton Quntzed vlues referred to s lbels Tble representton E.g.: θ j l j Q j

41 JPEG: Quntzton Exmple 4 θ 00 Q 00 l θ Q00 6 l 00 00

42 JPEG Quntzton Tbles: Nkon D40 4 Source:

43 JPEG: Quntzton () 43 Observtons Usully, only few non-zero elements Quntzton tble effectvely works s threshold operton By vryng quntzton tble we cn vry bt rtes Lower vlues fewer zeroes, less QE hgher rtes/qulty Hgher vlues more zeroes, hgher QE lower rtes/qulty Qulty fctor Vrous scles: 0-4, 0-00 Implemented s quntzton tble multpler

44 Codng 44 DC/AC coeffcents re coded dfferently DCs re dfference-coded from ech other Usng Huffmn ACs re encoded s sequence RLE + Huffmn/AC I.e. DC 0, AC 00, AC 0,, AC 630, DC -DC 0, AC 0, AC,, AC 63 DC codng Unry code for row/ctegory bnry code for column

45 JPEG Procedure 45 Shft by 8

46 JPEG Procedure 46 DCT nd Quntzton

47 JPEG Procedure 47 Huffmn Codng

48 48

49 49 DC Codng Tble

50 50 DC Tble

51 5

52 5 AC Codng Tble

53 Reconstructon 53 Follows the reverse process Decode Huffmn dt Decode DC dfferences Reconstruct quntzed coeffcents Apply the nverse DCT Drop pddng rows/columns (f pplcble) Reverse shftng Interpolte mssng UV components (reverse sub-smplng) YUV RGB

54 Reconstructon Exmple: Coeffcents 54 Orgnl DCT coeffcents Reconstructed DCT coeffcents

55 Reconstructon Exmple: Block Dt 55 Orgnl pxel vlues Reconstructed pxel vlues

56 Imge Exmples bts/pxel 0.4 bts/pxel

57 Imge Exmples () bts/pxel bts/pxel

58 The Modfed DCT (MDCT) 58 Observton Block-bsed trnsforms ntroduce dstorton t block boundres Ide Use overlppng regons to overcome ths effect:

59 MDCT () 59 Applctons Audo: mp3, AAC, Ogg Vorbs Problem Twce s mny coeffcents s smples Wth some mth, ths problem cn be voded

60 References nd Homework 60 Homeworks. Derve the KL trnsform wth length 3 nd summrze the computng steps for the KL mtrx.. In JPEG codng, f the quntzed DC coeffcents of the current block s 30 whle the DC coeffcent n lst block s 36, fnd the coded bnry for the quntzed DC coeffcent bsed on the JPEG stndrd.