Digital Image Processing

Transcription

1 Dgtal Iage Processg Chater 08 Iage Coresso Dec. 30, 00 Istructor:Lh-Je Kau( 高立人 ) Deartet of Electroc Egeerg Natoal Tae Uversty of Techology /35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Huffa Codg /35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

2 Huffa Code Huffa Code: Varable Legth Code Theore: X, dscrete rado varable wth f (X). Let C be ay refx code for X ad C Huff be a Huffa code, the l( CHuff ) l( C) ad H ( X ) l( C ) H ( X ) + Huff Wth odoubt The worst case 00//3 3/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Etroy The frst-order etroy ( 熵 ) s defed by, H P( x) LogP( x) ( S) P( x) Log x x P( x) Where x s the alhabet/sybol the source S. The ut of etroy:bts/sale A less lkely outcoe wll brg ore forato 00//3 4/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

3 Exale of Huffa Code x P Ste: Ste: //3 5/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Exale of Huffa Code Ste3: 0.6 Ste4: //3 6/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

4 Ste5: 0 0 Exale of Huffa Code C: l( C) *0.4 + *0. + 3*0. + 4*0.+ 4*0.. Sgle-sde Growg Huffa Tree bts/sybol H ( X ) 0.4Log + 0.Log + 0.Log Log + 0.Log bts/sybol //3 7/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Exale of Huffa Code Aother soluto of the sae exale. Ste : Ste : Rearrage the order //3 8/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

5 Exale of Huffa Code Ste 3: Ste 4: Rearrage the order //3 9/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Exale of Huffa Code Ste 5: Rearrage the order aga Ste 6: //3 0/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

6 Exale of Huffa Code Ste 7: Rearrage the order aga Ste 8: //3 /35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Ste9: Exale of Huffa Code.0 0 l( C) * *0.+ 3*0.+ *0. + *0.. bts/sybol C: Mu Codeword Legth Huffa Tree 00//3 /35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

7 Note: Exale of Huffa Code Though the above codes have the sae average bt rates, C has saller varace tha C Better to be stored eory Thought the 4-bt codeword legth seldo occurs, the storage should have the eory wdth elarged case ths does take lace. Therefore, C s ot as good as C. 00//3 3/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology A Short Suary For u codeword legth varace: Cobe rob. wth shorter legth Sort rob. each ste, large to sall, fro left to rght Move cobed rob. as close to the left as ossble, ad cobe rob. o the rght.( 已經 cobe 過的必然造成較長的長度 ) 00//3 4/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

8 Rearks for Huffa Code Reark: Percetage creasg H ( X ) l( CHuff ) H ( X ) + l( CHuff ) H ( X ) Percetage creasg H ( X ) Eg. x 3 H ( X ) 0.86 P l( CHuffa ) *0.8 + * * 0.0. l( CHuff ) H ( X ) P.I. 47% H ( X ) //3 5/35 Pretty hgh bts/sybol Though.bt/sybol, stll far aart fro the etroy Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Rearks for Huffa Code Reark: Logest codeword legth (Worst case) Alhabet sze > logest codeword legth s - Eg. Alhabet 5 > logest codeword legth //3 6/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

9 Ercal PMF For Huffa trees X: dscrete rado varable Alhabet H: {,,, } N : Legth of the sequece to be coded : uber of occurrece of sybol k N k Ercal f Nk : k, k,, K, N Bary tree ca be obtaed fro Nk stead of k 00//3 7/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology A exale of usg N k x N k k N //3 8/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

10 Storage Requreet Proble: How to costruct the header such that the header sze s otzed? The codeword legth for the worst case s (-), where s the uber of sybols. Therefore, the sace requred for the storage of the code s x(-) //3 9/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Two-Pass Huffa Ecodg Two-ass Frst ass for k Secod ass for ecodg Trassso of k or codewords are requred k ot udated for varyg statstcs. 00//3 0/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

11 Basc Iforato Theory Theore: For a sybol source, S, wth sybols, the etroy of the source satsfes H ( S) log. Proof: Suose the robabltes of the sybols are,,,, resectvely, ad. Defe the etroy of the source to be H log. 00//3 /35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology H log Basc Iforato Theory log [log That s H Log 0. log + log ] log ( log ) log log 0 Moreover, we see that a uforly sread source results a larger etroy. 00//3 /35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

12 Soethg About Log Fucto Logx x 00//3 3/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Kraft Iequalty Gve a sybol source s wth alhabets a, a,, a. Suose the a refx code s costructed for the source wth codeword legth l, l,, l, resectvely. The the code satsfes the followg Kraft Iequalty Pf: Let lax l axl ax lax l l ax l 00//3 4/35 lax l l 黃色為夭折 ( 未長出的部份 ) Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

13 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology 00//3 5/35 Codg Legth of a Prefx Code Gve a sybol source S wth alhabets a, a,, a. Suose the a refx code s costructed for the source wth codeword legth l, l,, l, resectvely. The the average codg legth L satsfes the followg Iequalty Pf: Let L S H ) ( 0 ) ( log log log ) ( l l l l l L S H Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology 00//3 6/35 For Shao-Fao codg, the average codg legth s bouded the rage. ) ( ) ( S H Log Log l L < ) ( ) ( S H L S H + < 又 For Shao-Fao codg P Log l P Log + < Codg Legth of a Prefx Code

14 Vector Quatzato X x, x,..., ) C C, C,..., C ) ( x6 j ( j j j6 4 4 d ( X, C ) L j X 0 6 ( x C j ) C 6 M O M X L 5 Orgal Iage C M 56 00//3 7/35 C 55 codebook( 編碼簿 ) Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology JPEG Stll Iage Codg Stadard Please see Lecture-JPEG for detals. 8/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

15 Vdeo Codg 在 MPEG 中, 先將視訊影像分成三類, 分別為 I P 和 B 影像 I 影像用傳統的 JPEG 壓縮即可 ;P 可利用前面的 I 影像, 透過移動匹配 (Block Matchg) 和補償 (Coesato) 來壓縮, 夾在 I 和 P 之間的 B 影像之區塊就由 I 和 P 所匹配到的區塊內插而成 I: Itra frae ( 以真實的 Frae 直接進行壓縮編碼 ) P: Predcted frae ( 以先前編碼過的 I frae 或 P frae 進行移動估測編碼 ) B: B-drectoal frae ( 以前或後之 I frae 或 P frae 進行編碼 ) 00//3 9/35 I B B B P Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Block Matchg 在前一張參考影像中找到某一區塊使得這找到的區塊和目前區塊最匹配 通常是採用在前張影像中先訂出一個搜尋視窗, 在這搜尋視窗內包含許多與目前區塊相同大小的正方形區塊 Curret Frae Prevous Frae 00//3 30/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

16 Moto Coesated Codg 00//3 3/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Mea Absolute Dfferece(MAD) 目前區塊, 算出利用初始移動向量所得的區塊位置, 然後和對應於參考影像中的區塊計算兩者的 MAD: c B c B C ' ' MAD( B, B ) c r N N x y B r ' ' B ( x, y) B ( x, y) c r B B c Full-Search Method Logarthc Search Method ( 每次搜尋都縮小尚未搜尋範圍的一半 ) Three-Ste Search Method (Three-Ste Search Algorth) 00//3 3/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

17 Fully Search 00//3 33/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology Logarthc Search( 對數搜尋 ) 00//3 34/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology

18 Three-Ste Search( 三步搜尋 ) 00//3 35/35 Lh-Je Kau Multeda Coucato Grou Natoal Tae Uv. of Techology