VQ widely used in coding speech, image, and video

Transcription

1 at Scalar quantzers are specal cases of vector quantzers (VQ): they are constraned to look at one sample at a tme (memoryless) VQ does not have such constrant better RD perfomance expected Source codng theorems and other results n rate-dstorton theory (due to Shannon) mply that one can always do better (n RD sense) )f vector of samples are coded das unts Note: We can code samples as unts wthout necessarly explotng or knowng nterdependency between the samples VQ wdely used n codng speech, mage, and vdeo

2 at Bref descrpton Some man advantages: explot dependency that may exst wthn an nput vector ablty to generate non-cubc mult-dmensonal parttons of nput whch h provdes better compacton of the nput space ablty to track hgh-order statstcal characterstcs of the nput Some man dsadvantages: encodng complexty and memory requrements ncrease exponentally wth vector sze (under a gven rate) and wth bt-rate Lack of robustness: senstvty to channel nose Conventonal VQ s severely lmted to modest vector and codebook sze Dfferent more robust methods needed

3 at Bref descrpton VQ takes blocks of pxels nstead pxels 1. Dvde mage nto blocks (common sze: 24x24, 16x16, 8x8 ) x = [ ] T 2. Turn block nto a vector 3. Compare x wth best matchng xˆ n codebook Codebook: Table consstng of representatve vectors (reconstructon levels) { r } =1,,M Best matchng: wth respect to a chosen dstance (error) measure. 4. Transmt the ndex k of that best matchng vector: x ˆ = Q ( x ) = r k 5. Recever gets the ndex k and retreves xˆ = r k from ts own stored codebook whch matches transmtter s codebook

4 at Bref descrpton Image block to vector x Search Index Channel look-up xˆ Codebook Codebook Encoder Decoder E : X R N I D : I { r } N I ; r R Exp. Fnd best matchng vector n codebook 00 x =

5 at Motvaton Theorem (from source codng and rate-dstorton theory): As vector sze grows, performance mproves n the ratedstorton sense Practcal constrants: Encodng complexty and memory requrements ncrease exponentally wth vector sze (under a gven bt-rate) and wth bt- rate codebook grows exponentally as a functon of vector sze N and bt-rate r. Other problems: lack of robustness and senstvty to channel nose Conventonal VQ s severely lmted to modest vector and codebook szes Dfferent more robust VQ approaches needed

6 at VQ Desgn In VQ, the N-dmensonal nput set X R N s dvded nto M regons or cells ( quantzaton levels ) V { x X : Q ( x ) = r } ; M = 1 r = th code vector (reconstructon level) Optmal VQ Let d(x,y) = defned dstance measure between x and y Exp.: MSE = E MAE = E Q Q [ N d( x, y) ] ; d ( x, y) = ( x y ) 2 = E ) = 1 N [ d( x, y) ]; d( x, y = x 1 = E ) = 1 y 2

7 at Def: A vector quantzer s sad to be optmal f the expected dstorton D [ d( x, Q( x) )] = d( x Q( x) ) f ( x) d x = E, s mnmzed over all vector quantzers wth M code vectors Two necessary optmalty condtons: 1. The nearest-neghbor condton: For a gven set of code vectors {r } =1,,M, Q(x) must be a nearest-neghbor mappng;.e.: Q ( x) = r ff d( x, r ) d( x, r ); for 1 M

8 at 2. The centrod condton: For a gven set of partton cells {V } =1,,M, each code vector r (1 M) must be chosen so as to mnmze the average dstorton gven a partton cell V ;.e., r s set to be the vector y that mnmzes the condtonal dstorton D ( y ) = E[ d( x, y) x V ] = d( x, y) f ( x) => select r (1 M) such that D x V ( r ) = mn D ( y ) y d x => r centrod of the cell V

9 at Remarks Computaton of a centrod for a partcular cell depends on the dstorton measure d(x,y). Another less mportant necessary condton for optmalty s that for a gven source dstorton, ponts on the boundares between nearest-neghbor cells occur wth zero probabltes. Ths s automatcally satsfed for contnuous-valued nput R.Vs. The nearest-neghbor and centrod condtons hold for scalar quantzers. They are very mportant because they are frequently used as the bass for most of the VQ desgn algorthms.

10 at Remarks (contnued) From above optmalty condtons: For gven reconstructon levels (code vectors) {r } =1, M, the quantzaton levels are defned n terms of regons wth centrod r such that V ={ x : d(x,r ) d(x,r ) } ; {1,,M} If MSE, d(x,r ) = x-r 2 and V ={ x : x-r 2 x-r 2 }

11 at Remarks (contnued) Smple algorthm for performng VQ: 1. For each nput x, compute dstances d(x,r ) ; {1,,M} 2. Choose such that d(x,r(, ) mn 1 M d(x,r(, ) (Choose level correspondng to closest centrod)

12 at Smple VQ Algorthm (contnued) If more than one quantzaton level possble, use some predefned rule to make decson or smply make an arbtrary decson Computatonal requrements: f M code vectors (.e., codebook has M entres), we have to compute M dstances and make M-1 1 comparsons for each nput sample x Ths and the memory requred to store the centrods put a lmt on the practcal sze of the codebooks For gven quantzaton levels {V } =1, M 1, the optmal reconstructon levels n the mean-square sense (.e., d(x,r ) = x-r 2 ) are r = V V x f f ( x) ( x) dx dx

13 at Remarks (contnued) Equatons V ={ x : d(x,r ) d(x,r ) } and r V = V x f f ( x ) ( x) dx dx can be used teratvely to desgn codebooks (fnd {r }) for vector quantzers that are optmal n the mse sense most popular and classcal VQ desgn technque s the Generalzed Lloyd Algorthm (GLA), also known as the LBG (Lnde, Buzo and Gray) algorthm (1981) another wdely used algorthm: Parwse Nearest Neghbor (PNN) by Equtz (1989). Sgnfcantly reduces computaton and no need for ntal codebook, comparable reconstructed mages qualty.

14 at Generalzed Lloyd (LBG) algorthm based on the optmalty condtons mentoned earler most popular (although not best), wdely used for comparson wth other codebook desgn methods adaptaton of the k-means clusterng algorthm Basc steps: Step1: Start wth a tranng set of vectors (get a large quantty of representatve vectors: tran on one set, test wth others) Step 2: Start wth an ntal codebook of sze M (selected from tranng set); example: randomly selected vectors from tranng set Step 3: Vector Quantze each tranng vector usng current codebook (cluster tranng data)

15 at = codevectors = tranng vectors M = 4 Step 4: Use centrod of clusters as the updated d codebook centrod = mean of cluster for mse and for a statonary and ergodc nput snce tme/space averages replace statstcal averages centrod = center of mass 5. Repeat from Step 3 untl dstorton between old and new codebook s smaller than a selected small threshold

16 at Remarks on LBG At each teraton, the LBG algorthm constructs {r } and {V } satsfyng x f x d x V ={ x : d(x,r ) d(x,r ) } and LBG guaranteed to converge and fnds a locally optmal quantzer for a tranng set (may not be locally optmal for the nput x). r = V V f ( ) ( x) Fnal resultng codebook depends d on ntal t choce algorthm nfluenced by choce of ntal codebook (cluster centers), and by the choce and geometrcal propertes of tranng data. dx

17 at Remarks on LBG (contnued) Local optmal desgn for fxed number of levels M. In codng, VQ usually used n conuncton wth entropy codng lmt the entropy of quantzed sgnal rather than number of quantzaton levels l n the desgn process entropy-constraned VQ (EC-VQ)

18 at Intalzaton n LBG Most mportant ssue snce t can sgnfcantly affect the performance of desgned codebook Several codebook ntalzaton methods proposed. Popular ones: 1. Random selecton from tranng set 2. Bnary splttng for LBG codebook desgn» uses fxed perturbatons of the current code vectors (centrods) to create more code vectors: twce as many at each step

19 at Basc steps of Bnary Splttng for LBG: 1. Step 1: Start wth the centrod of the tranng set 2. Step 2: Perturb the current centrod(s) (usually 2 opposte drectons f sze doubles at each teraton) 3. Step 3: VQ all the tranng vectors, and take centrods of the new resultng clusters 4. Step 4: Repeat Step 2 untl we get the desred numbe of centrods (codevectors) for the ntal t codebook 5. Do LBG Advantage: can reduce search complexty by usng Tree search hvq nstead of exhaustve search hvq