Module 5 EMBEDDED WAVELET CODING. Version 2 ECE IIT, Kharagpur

Transcription

1 Module 5 EMBEDDED WAVELET CODING Verson ECE IIT, Kharagur

2 Lesson 5 EBCOT Algorthm Verson ECE IIT, Kharagur

3 Instructonal Objectves At the end of ths lesson, the students should be able to:. Exlan the nadequaces of EZW and SPIHT n wavelet acket encodng.. Defne resoluton scalablty of embedded bt-stream. 3. Defne SNR scalablty of embedded bt-stream. 4. State the basc characterstcs of EBCOT algorthm. 5. Exlan the rate-dstorton otmzaton roblem of code blocks. 6. Exlan how the truncaton onts are selected n EBCOT bt-stream. 7. State the hghlghtng features of block codng algorthm. 8. Defne the sgnfcance of sub-blocks. 9. Exlan the quad-tree structure reresentaton of sub-block sgnfcance. 0. Name the four codng asses n block codng algorthm.. State the role of each of the codng asses n block codng algorthm.. Defne the four codng rmtves used n block codng algorthm. 3. Exlan the formaton of the qualty layers from the embedded code block bt stream. 5.0 Introducton In lesson-3 and lesson-4, we have studed two smlar aroaches to codng of wavelet (n general, subband coeffcents, namely Embedded Zerotree Wavelet (EZW and Set Parttonng n Herarchcal Trees (SPIHT. Both these aroaches use orderng of coeffcents by magntudes for encodng the coeffcents n an embedded bt-stream and exlot the self-smlarty of coeffcents across subbands. The latter aroach s more effcent n terms of codng effcency, as comared to the former, snce t doesn t requre exlct transmsson of orderng nformaton of the coeffcents by effcently arttonng the subsets of coeffcents n satal orentaton trees. However, both EZW and SPIHT can only be aled to dyadc arttonng of coeffcents, n whch only the LL subband at a decomoston level are further analyzed. Ths s regarded as a lmtaton, snce these two algorthms cannot be aled to wavelet ackets n general. Wavelet ackets allow more flexble decomoston of subbands and the subbands at hgher frequences can also be decomosed nto narrower bands. Moreover, these two aroaches only offer SNR scalablty by encodng all the subbands at a gven recson n an teraton of asses (domnant -subordnate asses n EZW and sortng-refnement asses n SPIHT. These algorthms do not offer any resoluton scalablty n the sense that we do not comlete the Verson ECE IIT, Kharagur

4 encodng at a gven resoluton and then do the encodng at the next hgher resoluton and so on. In ths lesson, we are gong to dscuss a more recent aroach to embedded wavelet codng, namely Embedded Block Codng wth Otmzed Truncaton of bt-stream (EBCOT, whch can be aled to wavelet ackets and whch offers both resoluton scalablty and SNR scalablty. Because of ts advantages, the EBCOT algorthm has been acceted ncororated wthn the most recent stll mage comresson standard JPEG-000. In ths lesson, we shall frst defne resoluton and SNR scalabltes and ther combnatons. Ths wll be followed by the basc objectves of EBCOT. The algorthm selects the truncaton onts, that s, where the bt-stream can be termnated based on rate-dstorton (R-D otmzaton. The encodng rmtves and the stes of encodng wll be exlaned later n the lesson. 5. Resoluton Scalablty Before defnng resoluton scalablty, let us consder an examle subband acket decomoston structure, as resented n fg.5.. Fg 5.: Examle subband decomoston structure Verson ECE IIT, Kharagur

5 These subbands are groued nto dfferent resoluton levels L0, L, L, L V where L0 corresonds to the lowest resoluton level and contans only the sngle LL subband (, ndcated n yellow colour. The next resoluton L level s, whch s green n colour and ncludes subbands, 3 and 4. The other resoluton levels are L = { 5,6, 7 } (red, L { 8,9,0,,,} 3 = (whte. Fg 5.: Resoluton scalable bt stream If an embedded bt-stream contans dstnct subsets reresentng each resoluton level, the bt-stream s called resoluton scalable. In the llustratve dagram of fg.5., B l reresents the bt-stream contrbuted by resoluton level l. The embedded bt-stream starts wth the lowest resoluton level corresondng to coarsest reresentaton and rogressvely the hgher resoluton bt-stream follows. 5. SNR Scalablty An embedded bt-stream s sad to be SNR scalable f t contans dstnct subsets Bq such that q B k =0 k together reresents the samles from all the subbands at some qualty (SNR level-q. The best examles that can be cted are the EZW and the SPIHT algorthms. There, what we generate at the end of each domnant-subordnate or sortng-refnement ass s a SNR scalable bt-stream, snce all encoded sgnfcant coeffcents corresond to a bt lane. Fg 5.3: Both resoluton and SNR scalable bt-stream. A bt-stream s sad to be both resoluton and SNR scalable, f t contans dstnct subsets B l, q whch hold the qualty refnements n resoluton level L l. Fg.5.3 llustrates a bt-stream that s both resoluton and SNR scalable. The EBCOT algorthm ncororates both these forms of scalablty. Verson ECE IIT, Kharagur

6 5.3 Basc Characterstcs of EBCOT algorthm Before we descrbe EBCOT algorthm s detal, we lst some of ts basc characterstcs: It offers both resoluton and SNR scalablty. It has a random access attrbute. Gven any regon of nterest and a wavelet transform wth fnte suort kernel, t s ossble to dentfy the regons wthn each subband. Each subband s arttoned nto small non-overlang block of samles, known as code blocks. EBCOT generates an embedded bt-stream for each code block. The bt-stream assocated wth each code block may be truncated to any of a collecton of rate-dstorton otmzed truncaton onts. A code block s encoded by a two-ter codng structure, as shown n fg.5.4. In the frst stage, a small amount of summary nformaton s collected durng the generaton of each code block s embedded btstream. Based on ths summary nformaton, the truncaton onts for each code block are determned, usng whch the qualty layers are formed n the second stage. Fg 5.4: Two-ter codng structure of EBCOT. The EBCOT algorthm works on a layered bt-stream concet, where each layer corresonds to a qualty and a collecton of these layers form a ratedstorton otmzed mage. A sngle bt-stream offers a range of scalablty sngle layer ossesses resoluton scalablty, few layers target at some secfc bt-rate of nterest and a large number of layers offer SNR and resoluton scalablty wth random access attrbutes. 5.4 Rate-Dstorton Otmzaton Suose that a truncaton ont n s selected for a code-block B. The rate and the dstorton for the code-block assocated wth ths truncaton ont are gven B Verson ECE IIT, Kharagur

7 n by R and D n resectvely. The dstorton metrc used s usually the mean square error and s addtve over all the code blocks. Thus, the overall dstorton n the mage s gven by n D = D... (5. whch s to be mnmzed subject to the constrant on the maxmum bt-rate gven by max R, n R max R = R (5. The Rate-Dstorton (R-D otmzaton roblem requres mnmzaton of D + λr, where λ s a Lagrange multler. If we can fnd a value of λ such that the n selected from dfferent code blocks acheve R =, we truncaton onts { } can say that λ R max { n λ } s the otmal set of truncaton onts. Snce the truncaton onts are dscrete, t s not ossble to exactly make R = Rmax. In ractce, the code blocks are small n sze and there are many truncaton onts. It s suffcent to determne the smallest value of λ such that R R. We have searate mnmzaton roblem for each code block max. To determne λ n the truncaton ont, whch mnmzes ( n D + λr for all values of λ, we select a set N n of feasble truncaton onts, enumerated by j < j < whose rate- jk jk jk dstorton sloes S = ΔD / ΔR (where and ΔR jk = R jk R jk λ λ B are strctly decreasng. Then, the otmal selecton of λ jk truncaton onts s gven by n = max jk N S > λ. Ths unque set may be determned through a rocess of convex hull analyss. 5.5 Block Codng Algorthm { } ΔD jk = D jk D The block codng algorthm generates searate embedded bt-stream for every code block. Some hghlghtng features of the block codng algorthm are: Uses the concet of fractonal bt lane, n whch every bt-lane s encoded n multle numbers of asses. Uses context-senstve arthmetc codng. Code blocks are further subdvded nto sub-blocks whose sgnfcance are effcently encoded ror to samle by samle encodng. jk Verson ECE IIT, Kharagur

8 Before resentng the algorthm, we ntroduce you to the symbols used n the encodng rocess: s ( k,k : -D sequence of subband samles belongng to the code block B. For the LL, LH and HH subbands, kand k reresent the horzontal and the vertcal ostons resectvely. The HL subband s transosed, so that kand k reresent the vertcal and the horzontal ostons resectvely. χ k,k : The sgn of the subband samle s (. Hence, χ ( k k {, } υ ( ( k,k ( k, k k,k : Quantzed magntude of the subband samle, gven by s = ( k, k υ δ β where, B δ β belongs. ( k, k, s the quantzaton ste-sze for the subband to whch the code-block υ : The th bt-lane of ( max sgnfcant bt and s the most sgnfcant bt. υ k,k. = 0 corresonds to the least σ ( k,k : A bnary state-varable, ndcatng the sgnfcance ma. Its entry s ntalzed to zero, but s set to one, when the relevant samle s frst non-zero btlane k k s encountered. υ (, = 5.5. Sub-blocks and ther sgnfcance: Every code-block s further subdvded nto sub-blocks, each of whch s tycally max of the sze 6 x 6. For each bt-lane 0, the sgnfcance of the subblocks are reresented n a quad-tree structure, where the sub-blocks belong to 0 the leaf nodes of the quad-tree B ( j, j ( j, j are the coordnates of the subblock and 0 s the level of the leaf node. The nodes at the level-t n the tree are formed from the nodes at the level- (t- as t t B j, j = B j + z, j + z where, 0 t T ( ( z, z { 0,} T where, T s the level corresondng to the root of the tree, gven by ( 0,0 B, whch ndcates the entre code-block. The quad-tree data structure for the subblocks s llustrated n fg.5.5. Verson ECE IIT, Kharagur

9 B j, j for a bt-lane s defned as follows. If any of the quantzed magntudes υ ( k, k for ( k, k B ( j, j, the subblock B ( j, j s defned as sgnfcant for the bt-lane. The sgnfcance of a sub-block s roagated to the nodes at the hgher levels n the quad-tree, whch means that the node B t ( j, j s sgnfcant, only f at least one of ts descendant sub-blocks s sgnfcant. The sgnfcance of a quad-tree node at level-t for btlane s ndcated byσ ( B t ( j, j. The sgnfcance of the quad-tree datastructure s encoded by arthmetc codng. If a quad-tree node s nsgnfcant, the sgnfcance of ts descendants need not be encoded. Also, f a quad-tree node s sgnfcant n the revous bt-lane that s, (+, t wll reman sgnfcant n all the bt-lanes from onwards. The sgnfcance of a sub-block ( The encodng of the sub-block sgnfcance, sub-blocks contanng one or more sgnfcant samles are dentfed and ths leads to an effcent codng, snce all Verson ECE IIT, Kharagur

10 other sub-blocks whch are nsgnfcant, are by-assed n the remanng codng hases for the bt-lane Codng Passes: The embedded bt-stream for each bt-lane belongng to a code block s generated n four dfferent asses n the followng order, namely (a Forward ( sgnfcance roagaton ass P ( P ( P (, (c Magntude refnement ass 3 and (d Normalzaton ass quad-tree sgnfcance code normalzaton ass lane- are gnored untl wth the most sgnfcant bt-lane 0 bt-lane. P 4 S, (b Reverse sgnfcance roagaton ass P 4. The for the th bt-lane s sent before the, so that the coeffcents that become sgnfcant n bt P 4. The generaton of the embedded bt stream starts max and roceeds u to the least sgnfcant Fg 5.6: Aerance of codng asses n EBCOT bt-stream. Fg.5.6 llustrates the bt-stream generaton order. The codng asses are descrbed below: (a Forward sgnfcance roagaton ass ( P : In ths ass, the coeffcents n a code block are vsted n scan lne. Those whch are nsgnfcant tll the revous bt-lane, that s υ (, + k k and have a referred neghborhood are coded and the rest are sked. For LL, LH and HL subbands (note that the HL subband coeffcents are already transosed, as descrbed earler., the samle s ( k,k s sad to have a referred neghborhood, f at least one of ts horzontal neghbors s sgnfcant, that s σ ( k ±, k =. The HH subband s samle s ( k,k s sad to have a referred neghborhood, f at least one of ts four dagonal neghbors are sgnfcant, that s, σ k ±, k =. To each such samle, ( ± one of the two codng rmtves, namely Zero Code (ZC or Run Length Code (RLC (to be descrbed shortly s aled to ndcate f the samle frst becomes sgnfcant n the current bt-lane. If so, another codng rmtve, Sgn Code (SC s nvoked to encode the sgn of the coeffcent. Ths ass s referred to as sgnfcance roagaton, snce, the coeffcents Verson ECE IIT, Kharagur

11 already found to be sgnfcant serve as seeds and roagate ther sgnfcance n the drecton of ther scan. ( (b Reverse sgnfcance roagaton ass P : Ths codng ass s dentcal to P excet for the order of scannng the coeffcents whch s reverse. Also, the concet of referred neghborhood s extended to eghtconnected neghbors of the samle. ( P (c Magntude refnement ass 3 : In ths ass, the coeffcents whch are already found to be sgnfcant are encoded usng the Magntude Refnement (MR rmtve (to be descrbed shortly. ( (d Normalzaton ass P 4 : All coeffcents whch were sked n the earler three asses are encoded n ths ass. Ths ncludes coeffcents whch are nsgnfcant tll the revous bt-lane and do not have any referred neghborhood. To encode such coeffcents, we make use of RLC and SC codng rmtves. Although the orgnal EBCOT algorthm roosed by Taubman uses the four asses lsted above, n the JPEG-000 mage codng standard, the four asses are smlfed to three by usng only one sgnfcance ass, nstead of the forward and the reverse and usng eght neghbors for referred drecton Codng Prmtves: To encode the code block coeffcents, one of the followng four codng rmtves are used: Zero Codng (ZC: Ths rmtve s used n the sgnfcance roagaton asses to encode those coeffcents whch are nsgnfcant tll the last btlane, have a referred neghborhood and do not form a run of nsgnfcant samles. The coeffcent under consderaton.e. s ( k,k s encoded usng the context of ts neghbors n terms of sgnfcance. The sgnfcance of the neghbors of s ( k,k are groued nto three categores: o Horzontal: Gven by h ( k k ( k, k. 0 h {,} ( k z, = σ k, so that +, z o Vertcal: Gven by v ( k k that ( k, k 0 v. {,} ( k k z, = σ, so, + z Verson ECE IIT, Kharagur

12 o Dagonal: Gven by d ( k, k ( k, k 4 0 d {,} ( k + z k z σ, + z, z =, so that ( The label assocated wth the ZC rmtve deends on the values of h k,k, ( k,k and d ( k, k. These are quantzed to nne dstnct codng contexts. v Run-length Codng (RLC: Ths rmtve s used n lace of the ZC rmtve, when each of the followng condtons hold good: o Four consecutve samles are all nsgnfcant,.e. σ k + z k = 0 for 0 z ( 3, o All these samles have nsgnfcant neghbors,.e. h k + z k = v k + z, k = d k + z, k for 0 z ( ( ( 3, o The samles must resde wthn the same code block. o Horzontal ndex of the frst of these four samles, k must be even. When a grou of four samles satsfy the above condtons, a sngle symbol s used to dentfy whether any of the four samles become sgnfcant n the current bt lane. Sgn Codng (SC: Ths rmtve s used only once for each samle, when a revously nsgnfcant samle s found to be sgnfcant durng a ZC or RLC oeraton. It s observed that the sgn bts χ ( k,k of adjacent samles tend to be correlated and that s why, the label assgned to the SC rmtve takes nto consderaton the contexts of h ( k,k, v ( k,k and χ ( k,k. Magntude Refnement (MR: Ths rmtve s used to encode the subband samles n a bt-lane, whch are already sgnfcant from the revous ass. A new state varable ~ σ (, k k s ntroduced whch makes a transton from 0 to after the MR rmtve s frst aled to s ( k,k. The bt υ ( k, k s coded wth contexts that deend uon h ( k,k and v ( k,k. 5.6 Formaton of Qualty Layers We now focus our attenton to the second stage of the two-ter codng structure shown n fg.5.4, whch accets the embedded bt-stream and other summary nformaton from each of the code blocks to form the qualty layers. As dscussed Verson ECE IIT, Kharagur

13 B n secton-5.4, each code block has a collecton of truncaton onts { n q, q =,, } corresondng to the dfferent qualty layers. The embedded bt stream for each code block, generated from the frst stage of the codng engne s comosed of a collecton of qualty layers Qq, where q =,, are the ndces of the qualty layers n the ncreasng order of qualty. The frst bytes n the embedded bt-stream of s comosed of the qualty layers Q to Q, n whch B q q q n n L = R R the layer q makes an ncremental contrbuton 0 from the code block B. The code blocks may make emty contrbutons to some of the qualty layers. For each code block, followng quanttes are sent as summary nformaton: (a the ndex of the qualty layer to whch the code block makes the frst nonemty contrbuton, (b the ncremental contrbuton to each qualty max layer and (c the value of. These quanttes are avalable for all the code blocks from the frst stage. Two quanttes show sgnfcant amount of nter code max block redundances. These are and the ndex q of the qualty layer to whch the code block B makes ts frst contrbuton. These nter code block redundances are exloted by usng a searate embedded quad-tree code wthn each subband. The second-stage of the codng engne, utlze the summary nformaton from the code blocks rather than each samle to comose the qualty layers. The qualty layer formaton s llustrated n fg.5.7. The bt-stream from each code block, truncated nto secfc truncaton onts corresondng to the qualty layers are arranged n ncreasng order of q. To comose the bt-stream corresondng to each qualty layer, we have to roceed along the code blocks n ther scannng order and add ther contrbutons to the qualty layer. Q q q L q n R q Verson ECE IIT, Kharagur

14 5.7 Summary and Conclusons In ths lesson, we have resented EBCOT algorthm, whch offers resoluton scalablty, and SNR scalablty wthn fnely embedded bt-streams for relatvely small blocks of subband samles. The EBCOT algorthm forms the bass for the latest stll mage comresson standard, JPEG-000. We shall dscuss about the devatons of the JPEG-000 algorthm from the orgnal EBCOT n lesson-7. Verson ECE IIT, Kharagur