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

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1 Module 5 EMBEDDED WAVELET CODING Versio ECE IIT, Kharagpur

2 Lesso 4 SPIHT algorithm Versio ECE IIT, Kharagpur

3 Istructioal Objectives At the ed of this lesso, the studets should be able to:. State the limitatios of EZW algorithm.. Justify the eed for magitude orderig of coefficiets i progressive image trasmissio. 3. Justify the eed for efficiet ecodig of coefficiet sortig i progressive image trasmissio. 4. State the basic objectives of Set Partitioig i Hierarchical Trees (SPIHT) algorithm. 5. Defie spatial orietatio trees. 6. Defie the set partitioig rules. 7. Outlie the steps of SPIHT ecodig ad decodig algorithm. 8. Implemet a wavelet coder ad decoder based o SPIHT. 4.0 Itroductio I lesso-3, we have studied a fast ad efficiet approach for ecodig wavelet coefficiets usig Embedded Zerotree Wavelet (EZW) algorithm. The algorithm has two major stregths. First, the bitstream is embedded ad the coefficiets are ordered i sigificace ad precisio, so that it ca be trucated accordig to the bit-rate requiremets of the chael. Secod, it efficietly utilizes the selfsimilarity betwee the subbads of similar orietatio ad achieves sigificat data reductio. However, we have poited out that the EZW algorithm is ot exactly optimal ad some adjustmets of parameters (like, iitial threshold) may be ecessary to make it optimal with respect to a target bit rate. A sigle embedded file is uable to give the best performace at all target bit rates. Secodly, EZW has some limitatios i efficietly ecodig the isigificat coefficiets. Although it exploits the self-similarity of coefficiets across subbads of same orietatio, it does ot make ay groupig of isigificat coefficiets to improve the codig efficiecy. I this lesso, we are goig to study a modified form of embedded codig of wavelet coefficiets that carries the major stregths of EZW, amely ordered coefficiet trasmissio ad self-similarity across subbads of similar orietatio. I additio, it partitios the set of coefficiets ito subsets of isigificat coefficiets ad idetifies each sigificat coefficiet. The approach, proposed by Said ad Pearlma is kow as Set Partitioig i Hierarchical Trees (SPIHT). At idetical target bit rates, experimetatios have show that SPIHT algorithm has improved performace over EZW, because of its ability to exploit the groupig of Versio ECE IIT, Kharagpur

4 isigificat coefficiets. This lesso will first provide some of the basics of progressive image trasmissio ad the itroduce the cocepts of set partitioig. The ecodig algorithm will be explaied ad the studets should be able to desig ad implemet a SPIHT coder ad decoder. 4, Coefficiet orderig i progressive image trasmissio The hierarchical subbad trasformatio (Wavelet, QMF etc.) may be expressed i a geeral form as c = Ω() s..(4.) where, s is the origial image array, c is the trasformed coefficiet array ad Ω is the uitary hierarchical subbad trasformatio. Both the origial image array ad the coefficiet array have the same dimesios. The ecoder trasmits the coefficiets ad the decoder updates itself accordig to the received bit-stream. From the approximated coefficiet array ĉ, it is possible to recostruct a approximated form of image pˆ through the iverse trasformatio, give by ˆ () cˆ s = Ω...(4.) The mea-squared recostructio error of the image at the decoder is give by D mse s sˆ ( s sˆ ) = = ( N N s, sˆ, ).(4.3) where, N is the total umber of pixels ad s is the pixel itesity at the locatio, (, ). The subbad trasformatio beig lossless, the mea square error will be ivariat to the trasformatio, that is, D mse ( s sˆ ) = D ( ˆ) = ( ˆ mse c c c ), c, N.(4.4 ) where, c is the trasform coefficiet at the locatio (, ),. To start with, the decoder sets cˆ 0 for all coefficiets ad if the ecoder seds the exact value, = of the coefficiet, the mea-square error, give by equatio (4.4) c, ( ) N decreases by c /. This implies that i a embedded bit-stream, the largevalued coefficiets should be trasmitted first as they cotribute more to the, reductio of mea-square error ad better recostructio quality. This justifies why i a embedded bit-stream geeratio, the trasform coefficiets should be Versio ECE IIT, Kharagpur

5 ordered accordig to the magitude. We had already see that the EZW algorithm follows this by orderig the sigificat coefficiets i the subordiate pass. The idea of coefficiet orderig ca be exteded to bit-plaes if the coefficiets are raked accordig to their biary represetatios ad the most sigificat bits are trasmitted first. Suppose, we first rak the coefficiets i decreasig order of the miimum umber of bits required for its magitude represetatio, that is, the orderig is doe such that log η ( k ) log cη ( k + ) where, k =,, N c, c η ( k ) represets the coefficiets i the ordered space..(4.5) As a example, let us cosider the followig array of coefficiets before orderig: -3, -9, 6, 5, -57, 8, 38,, -, 4, -7, -6, 5, -7. A arragemet of ordered coefficiets, which obey equatio (4.5), is show i Table-4.. Coefficiet magitude Sig-bit Bit-5 (msb) Bit Bit Bit Bit Bit-0 (lsb) Table-4. Ordered coefficiets accordig to the bit-plae requiremets. I Table-4., we have show the siged-magitude represetatio of the ordered coefficiets alog the colums. It may be oted that although the ordered coefficiets satisfy equatio (4.5), it is ot a strict magitude orderig. For example, the coefficiet with magitude 6 appears before the oe with magitude 7. Both of these require 5-bits for represetatio. To ecode these coefficiets, the bit-stream should carry the orderig iformatio of the coefficiets (i.e., their coordiates) ad also the umber μ which correspods to the umber of coefficiets satisfyig the iequality c +, <. I the give example, we have μ 5 =, μ 4 = 3, μ3 = 4, μ = 3, μ =. While trasmittig the coefficiets, the leadig 0s ad the first s i each colum eed ot be set, as Versio ECE IIT, Kharagpur

6 these are implied with the μ iformatio. The ext set of bits, marked i blue are to be trasmitted sequetially row-wise, from left to right. Thus, progressive trasmissio requires oe sortig pass, where the coefficiets + are ordered accordig to the iequality c <, followed by the, refiemet pass of trasmittig the bits i order of their sigificace. Although the method fulfils the objectives of embedded bit-stream geeratio, the ecoder requires large umber of bits to be trasmitted as orderig iformatio ad this call for better ad efficiet methods for ecodig the orderig iformatio. 4. Basic Objectives of Set Partitioig I set partitioig approach, the orderig iformatio is ot be explicitly trasmitted. Istead, the ecoder ad the decoder follow the same executio path ad if the decoder receives the results of magitude comparisos from the ecoder, it ca recover the orderig iformatio from the executio path. I set partitioig, o explicit sortig of coefficiets is doe. Istead, for a give + value of, the coefficiets are examied if they fall withi c <. If, >, it is sigificat. Otherwise, it is isigificat. A subset T of coefficiets, m c are examied to determie if max c T,, m..(4.6) If this coditio is ot satisfied, the subset is isigificat ad if the coditio is satisfied, the the subset T m is further partitioed to determie isigificat ad sigificat subsets. The sigificat subsets are repetitively partitioed till sigle sigificat coefficiets are idetified. The set partitioig rule is desiged to work i the subbad hierarchy. The objective of the set partitioig algorithm should be such that the subsets expected to be isigificat cotai larger umber of elemets ad the subsets expected to be sigificat should cotai oly oe elemet. Before presetig the set partitioig rule, we defie the spatial orietatio tree, which icludes the hierarchy. 4.3 Spatial Orietatio Tree The spatial orietatio tree, illustrated i fig.4. defies the spatial relatioship betwee the subbads i the form of a pyramid composed of a recursive fourbad split. Each ode of the tree correspods to a pixel ad is idetified by its pixel coordiate. Other tha the leaves, each ode of the tree has four offsprig correspodig to the pixel at the same positio i the ext fier level of the T m Versio ECE IIT, Kharagpur

7 pyramid of same orietatio, as show by arrows i the diagram. The oly exceptioal case is the LL subbad existig at the highest level of pyramid. Pixels i this subbad form the root ad groups of adjacet x pixels are composed. Other tha oe of the pixels (marked as * ) out of these four, all remaiig three pixels have their four offsprig i the HL, LH ad HH subbads of the same scale, as show. Oe out of the four is obviously left out sice oly three subbads exist for determiig the descedats. Fig.4. Spatial Orietatio Tree The spatial orietatio tree discussed as above has close resemblace with the hierarchical tree structure that was used to examie the zerotrees ad the zerotree roots i EZW algorithm. However, there is a major differece too. I the hierarchical tree of EZW, every LL subbad pixel at the highest level has three offsprig at the HL, LH ad HH subbads, whereas the offsprig relatioship of LL subbad pixels i the spatial orietatio tree is as what has bee already discussed i the last paragraph. 4.4 Set Partitioig Rules We first defie the followig sets before presetig the set partitio rules: O (, ) : The offsprig set. It cotais the coordiates of the pixels, which are offsprig of the ode (, ). Click oce o fig.4.. Oe of the pixels starts blikig. It is this ode s offsprig which we are goig to determie. Click o the figure oce more. The four pixels which blik together are the offsprig of the ode ad all of them belog to the offsprig set. ( D, ) : The descedats set. It cotais all the coordiates of the pixels which are descedats of the ode (, ). Click oce more o fig.4.. All the pixels which are descedats of the ode (, ) start blikig. All these belog to the descedat set. L (, ) : It is the differece set of D (, ) ad O (, ). It therefore, other tha the offsprig. cotais the descedats of the ode ( ), Click o fig.4. oce more. Now oly the elemets of the set start blikig. L (, ) H: This set icludes the coordiates of all spatial orietatio tree roots, which belog to the highest level of pyramid, that is, the LL subbad. Based o the above set defiitios, the set partitioig rules are preseted below: Rule-: The iitial partitio cotais D ( ) for each ( ) H,,. Versio ECE IIT, Kharagpur

8 ( ( Rule-: If, is foud sigificat, partitio it ito L, plus four sigle elemet sets with ( i, j ) O (, ). Rule-3: If L (, ) is foud sigificat, partitio it ito four sets of D ( i, j), i, j O. where ( ) ( ) D ) ), 4.5 SPIHT Ecodig ad Decodig The SPIHT algorithm applies the set partitioig rules, as defied above o the subbad coefficiets. The algorithm is idetical for both ecoder ad decoder ad o explicit trasmissio of orderig iformatio, as eeded i other progressive trasmissio algorithms for embedded codig, are ecessary. This makes the algorithm more codig efficiet as compared to its predecessors. Both the ecoder ad decoder maitai ad cotiuously update the followig three lists, viz. List of Isigificat Pixels (LIP) List of Sigificat Pixels (LSP) List of Isigificat Sets (LIS) I all lists, each etry is idetified by a coordiate (, ). I LIP ad LSP, the etry represets idividual pixels, whereas i LIS, the etry represets either the set D (, ) or the set L(, ). As a iitializatio step, the umber () of magitude refiemet passes that will be ecessary is determied from the maximum magitude of the coefficiets. Iitially, all pixels are treated as isigificat. The iitializatio is followed by three major passes the sortig pass, the magitude refiemet pass ad the quatizatio step update pass which are iteratively repeated i this order till the least sigificat refiemet bits are trasmitted. Durig the sortig pass, the pixels i the LIP, which were isigificat till the previous pass, are tested ad those that become sigificat are moved to the LSP. Similarly, the sets i LIS are examied i order for sigificace ad those which are foud to be sigificat are removed from the list ad partitioed. The ew subsets with more tha oe elemet are added to the LIS ad the sigle pixels are added to LIP or the LSP, depedig upo their sigificace. Durig the magitude refiemet pass, the pixels i the LSP are ecoded for th most sigificat bit. The ecodig algorithm ca be summarized as follows: Step-: Iitializatio: = log max Output ( ( ){ }), c, Set the LSP ={ } Versio ECE IIT, Kharagpur

9 Set the LIP = Step-: Sortig pass: {( ) } { }, H ad LIS= D( ) ( ) H,,, Step-.: For each etry i the LIP, output the sigificace ( if sigificat, 0 if ot sigificat). If foud sigificat, remove it from the LIP ad add to the LSP. Step-.: For each etry i the LIS, output the sigificace. If foud sigificat, output its sig. Perform the set partitioig usig the rule- or rule-3, depedig upo whether it is the D (, ) set or the L (, ) set. Accordig to the sigificace, update the LIS, LIP ad LSP. Step-3: Refiemet pass: For each etry i the LSP, except those which are added durig the sortig pass with the same, output the th most sigificat bit. Step-4: Quatizatio-step update pass: I this pass, is decremeted by ad the steps-, 3 ad 4 are repeated util = 0. The decoder steps are exactly idetical. Oly the output from the ecoder will be replaced by the iput to the decoder. As with ay other codig method, the efficiecy of the algorithm ca be improved by etropy codig its output, but at the expese of very high codig / decodig time. We ca explai this algorithm, by cosiderig a example 8x8 array, which is coverted ito two levels of subbad decompositio ad the coefficiet array is show i fig.4.. First, we perform the iitializatio o this array. The maximum magitude of the coefficiet is 63 ad thus, = log 63 = 5. This value will be sed as a output from the ecoder. The iitial LIP will cotai the etire LL subbad coefficiets LIP = 0,0, 0,,,0,, coordiates ad are give by {( ) ( ) ( ) ( )} The iitial LIS will cotai the descedat subsets of LL subbad coefficiets, i.e., LIS = D 0,, D,0, D, { ( ) ( ) ( )} Note that this list does ot iclude D( 0,0) as the ode (,0) descedats. Also, the iitial LSP list is empty, that is, = { } 0 does ot have ay LSP. Now, let us follow the first sortig pass. The coefficiets will be examied from the LIP ad the first elemet to cosider is (0, 0). This is sigificat, sice, Versio ECE IIT, Kharagpur

10 6 ( 0,0) 5 c <. The sigificace (a bit value of ) ad its correspodig sig (i this case, 0, sice it is positive) will be set as output. Now, we move the coordiate etry (0,0) from the LIP to the LSP. The other LIP coefficiets will be examied i the order. The studets may please verify that the remaiig sigificace bit outputs will be: {,0,0}. Also, the sig of the other sigificat coefficiet will be output as (sice c ( 0,) = 34, i.e. egative). At the ed of this LIP sortig, the status of the LIP ad the LIS will be LIP = LSP = {(,0 ), (, )} {( 0,0),( 0,) } Next, we follow the LIS sortig. First D( 0,) is examied. This is sigificat ad hece this sigificace status (=) will be first set as output. The set D( 0,) eeds partitioig. So, we examie the offsprig of D ( 0,), which are (0,), (0,3), (,) ad (,3). We output their sigificace ad sig (if sigificat) as before. The coefficiets (0,) is sigificat. So, it is added to the LSP. All remaiig oes are isigificat ad added to the LIP. Further, the set L( 0, ) is added at the ed of the LIS ad D ( 0, ) is removed. The studet may ow carry out these steps for the other etries i the LIS. The refiemet pass oly cosiders the etries i the LSP ad their ecodig for the curret value of is very straightforward. Versio ECE IIT, Kharagpur