Abstract The problem of designing a multiple description vector quantizer with lattice codebook \Lambda is considered. A general solution is given to a labeling problem which plays a crucial role in the design of such quantizers. Numerical performance resu

4PRELIMINARIES

andy=(y1,y2,...,yL)isgivenby x,y =(1/L)

L

9

xiyiandthecorrespondingnorm

i=1

is x = x,x 1/2,i.e.,theinnerproductandnormaredimension-normalized).Thecorre-¯0,d¯1andd¯2.Weassumethatanentropycoderspondingaveragedistortionsaredenotedbyd

isusedinordertotransmitthelabeledvectorsataratearbitrarilyclosetotheentropy,i.e.,Ri=H(αi(Q(X)))/L,i=1,2,whereH(U),theentropyoftherandomvariableUtakingval-

uesinalphabetUwithprobabilitydistributionPisgivenbyH(U)= u∈UP(u)logP(u).

¯0subjecttod¯1≤Ds,Theproblemistodesignthelabelingfunctionαsoastominimized¯2≤Ds,andRi≤R,i=1,2,forspeci edvaluesoftherateRanddistortionDs.d

def , λ ∈Λ}.V(λ)={x: x λ ≤ x λ

TheVoronoi(ornearestneighbor)regionV(λ)ofapointλ∈Λisde nedtobe

(5)

Witheachsublatticepointλ′∈Λ′,weassociateadiscreteVoronoiset(withNelements)

V0(λ′)={λ∈Λ: λ λ′ ≤ λ λ′′ , λ′′∈Λ′}.

def

(6)

In(5)and(6)ties(i.e.,pointsforwhichequalityholdsinthede ningcondition)arebrokeninsomeprearrangedmanner.Theexistenceofsublatticesforwhichnotiesoccurisdiscussedin[8].TheVoronoiregionV(0)andthediscreteVoronoiregionV0(0)arebothillustratedinFig.3.

→Weregardthelabelforλasadirectededge e=(α1(λ),α2(λ))ofthegraphwithvertexsetΛ′.Thecorrespondingunorderedpaire={α1(λ),α2(λ)}willbereferredtoastheundirectededgeorundirectedlabelassociatedwithλ.Theessentialdi erencebetweenadirectededgeorlabel(λ′a,λ′b)andtheundirectededge{λ′a,λ′b}isthatforthedirectededgethereisanimplicitassociationbetweenedgecomponentandchannel(λ′aissenton

channel1andλ′bissentonchannel2)whereasfortheundirectededgenoassociationisimplied.Graphically,anedgeconnectingtwosublatticepointsλ′aandλ′b,withanarrowpointingfromλ′atoλ′b,indicatesthatλ′aissentonchannel1andλ′bissentonchannel2,or