Flexible Multiple Description Lattice VectorQuantizer with L ≥ 3 Descriptions

Zhouyang Gao and Sorina Dumitrescu, Senior Member IEEEElectrical and Computer Engineering Department

McMaster University, Canadagaoz7@mcmaster.ca; sorina@mail.ece.mcmaster.ca

Abstract

In previous work on multiple description lattice vector quantizers (MDLVQ) with L ≥ 3descriptions, once the central and side lattice codebooks are fixed, the decoding quality is

determined for all numbers k of received descriptions. Therefore, it is not possible to achieve

tradeoffs between the quality of reconstruction for different values of k, 1 ≤ k ≤ L − 1.

This work proposes a flexible MDLVQ capable of overcoming the above drawback. For this, a

different reconstruction method is employed and a heuristic index assignment algorithm, which

uses L − 2 parameters to control the distortions for 2 ≤ k ≤ L − 1, is developed. The second

contribution of this work is a structured index assignment for the case L = 3 and the derivation

of asymptotical expressions of the distortions at high resolution. The proposed index assignment

has a simple mechanism for controlling the tradeoff between the reconstruction quality when

k = 1 and when k = 2, and is able to achieve a wide range of distortion values.

I. INTRODUCTION

A multiple description coder (MDC) consists of L encoders for some L ≥ 2, each encoder

generating a separate description of the signal. Each description is sent to the destination over

a separate channel, which either transmits the whole description correctly or breaks down.

The decoder is able to reconstruct the source to some quality from any subset of received

descriptions, while the fidelity of the reconstruction generally increases with the number of

received descriptions.

An n-dimensional multiple description lattice vector quantizer (MDLVQ) is an MDC consisting

of a so-called central lattice Λc ⊂ Rn, a so-called side lattice Λs, which is a sublattice of Λc,

and an injective mapping α : Λc → ΛLs , which assigns to each central lattice point an L-tuple

of side lattice points. For each λc ∈ Λc, we will denote by αi(λc) the i-th component of the

L-tuple α(λc). The encoder of the MDLVQ quantizes the input vector x = (x1, x2, . . . , xn) to

the closest central lattice point λc and outputs αi(λc) as the i-th description, for 1 ≤ i ≤ L.

When all descriptions λ1, · · · , λL are received at the destination the decoder is able to uniquely

identify the corresponding central lattice point and uses it as the source reconstruction. On the

other hand, when only one description λi is received, the decoder uses it to reconstruct the source.

When all descriptions are lost the decoder outputs the source mean. Notice that when L = 2,

only the above mentioned situations are possible at the decoder.

The MDLVQ framework for L = 2 was introduced and analyzed in [1]. The authors of [1]

proposed a general method for the index assignment design and derived analytical expressions

for the distortions at high resolution, using the squared distance as a distortion measure. While

the aforementioned work addressed the symmetric case, i.e., where the rates of all descriptions

are equal, the extension to the asymmetric case was considered in [2]. In [3], [4], the authors

2014 Data Compression Conference

1068-0314/14 $31.00 © 2014 IEEE

DOI 10.1109/DCC.2014.82

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achieved the tradeoffs between central and side distortions by modifying the encoding rule of [1].

Additionally, an extension to more than two descriptions is briefly mentioned in [3]. A systematic

study of an MDLVQ system for L > 2 descriptions, in the symmetric case, is performed in

[5]. The authors of [5] use the setup described in the previous paragraph and for the additional

situations arisen at the decoder, i.e., when the number of received descriptions is k, 2 ≤ k ≤ L−1,

use the arithmetic average of the received descriptions to reconstruct the source. They propose

an algorithm for optimizing the index assignment and derive asymptotical expressions of the

distortions for different numbers of received descriptions. The authors of [6] propose a simpler

and faster index assignment algorithm and prove its optimality for L = 2 with any N , and for

L > 2 when N → ∞, where N is the index of the side lattice with respect to the central

lattice. They also derive asymptotical expressions of the distortions. The work [7] proposes an

improvement to the index assignment of [6] for finite N . The work [8] uses a simpler method

to analyze the asymptotical performance of the MDLVQ of [6]. A multiple description quantizer

with translated lattice codebooks and the associated optimal index assignment are discussed in

[9]. In [10], the asymmetric MDLVQ with L ≥ 2, which uses the weighted average of the

received descriptions as reconstruction when k < L, is investigated.

In this work we propose a more flexible MDLVQ system for L ≥ 3, able to adjust the decoding

quality for various values of k, 1 ≤ k ≤ L − 1. To this aim we use a different reconstruction

method and propose a heuristic index assignment algorithm, which uses L − 2 parameters to

control the reconstruction quality for 2 ≤ k ≤ L− 1. In the second part of this work we propose

a more structured index assignment for the case L = 3 with a simple mechanism to control the

tradeoffs between the distortions when one, respectively two, descriptions are received. We derive

asymptotical expressions for the distortions for k = 1 and k = 2, showing that a wide range of

distortion values can be achieved even if N and R are fixed.

II. PRELIMINARIES

A. Definitions and Notations

Let νc, respectively νs, denote the volume of the fundamental region of the central, respectively

side, lattice. Then N = νs

νc. Note that Λs is assumed to be a clean sublattice of Λc. Let 0 denote

the n-dimensional all-zero vector.

To represent a missing description at the decoder we use the symbol ∗. For instance, for the

case L = 3, if the triple (λ1, λ2, λ3) is transmitted, and the second description is lost, then we

say that the triple (λ1, ∗, λ3) is received. We use L-bit sequences b = b1b2 · · · bL ∈ {0, 1}Lto represent various patterns of received descriptions, where bi = 1 means that description iis received, while bi = 0 means that description i is lost. For λc ∈ Λc and b ∈ {0, 1}L let

us denote by yb(λc) the reconstruction of λc when the pattern of received descriptions is b.

For b ∈ {0, 1}L, let H(b) denote the Hamming weight of b, which equals the number of 1’s

in b. Therefore, H(b) equals the number of received descriptions corresponding to pattern b.

The per symbol squared error is used as a distortion measure. Thus, the distortion of the source

reconstruction corresponding to pattern b is

Db =1

n

∑λc∈Λc

∫Vc(λc)

‖x− yb(λc)‖2f(x)dx,

where ‖y‖ � √< y,y >, with < x,y >�

∑ni=1 xiyi for all x,y ∈ R

n, f(x) =n∏

j=1f (xj) is

the probability density function (pdf) of the source vectors x, and Vc(λc) denotes the Voronoi

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region of λc ∈ Λc, with respect to the central lattice, i.e.

Vc(λc) � {x ∈ Rn : ‖x− λc‖ ≤ ‖x− λ′c‖, ∀λ′c ∈ Λc}.Further, the distortion when k descriptions out of L are received, 1 ≤ k ≤ L, is defined as

DL,k � 1(Lk

) ∑b∈{0,1}L,H(b)=k

Db. (1)

Notice that DL,L equals the distortion of the lattice quantizer with codebook Λc, referred to as

the central distortion and denoted by Dc.

As in prior work we will assume that the index assignment α satisfies the shift-invarianceproperty, i.e., that

αi(λc + u) = αi(λc) + u, ∀λc ∈ Λc, u ∈ Λs, 1 ≤ i ≤ L.

Additionally, since α is shift-invariant it makes sense to assume that the decoder mapping is shift

invariant as well, i.e., yb(λc + u) = yb(λc) + u, for all λc ∈ Λc, u ∈ Λs, and b ∈ {0, 1}L.

As in prior work we will use the high resolution assumption in order to simplify the derivation

of the distortion expression. In other words, we will derive an approximation of the distortion,

which becomes accurate as νc → 0 and νs → 0. Additionally, we will assume that N → ∞.

Thus, we assume that νc is small enough so that the pdf is approximately uniform over each

Voronoi region of λc. This implies that each central lattice point is approximately equal to the

centroid of its Voronoi region. Then [1], [5], [6]

Db ≈ Dc +1

n

∑λc∈Λc

‖λc − yb(λc)‖2∫Vc(λc)

f (x) dx, (2)

where H(b) < L. Denote now by Vs(λs) the set of central lattice points in the Voronoi region

of the side lattice point λs, i.e.,

Vs (λs)Δ=

{λc ∈ Λc : ‖λc − λs‖ ≤

∥∥λc − λ′s∥∥ , ∀λ′s ∈ Λs.

}Relation (2) and the shift-invariance of the index assignment and of the decoding mapping,

together with the assumption that the pdf is uniform over each Voronoi cell of the side lattice,

lead to [1], [5], [6]

Db ≈ Dc +1

nN

∑λc∈Vs(0)

‖λc − yb(λc)‖2.

Using further (1), it follows that

DL,k ≈ Dc +1

nN

1(Lk

) ∑λc∈Vs(0)

∑b∈{0,1}L,H(b)=k

‖λc − yb(λc)‖2. (3)

B. Previous MDLVQ Scheme for L ≥ 3

For every λc ∈ Λc denote μs(λc) =∑L

i=1 αi(λc)L . Additionally, for any L-tuple (λ1, · · · , λL) ∈

ΛLs define its spread as sp(λ1, · · · , λL) =

∑Li=1 ‖λi− 1

L

∑Lj=1 λj‖2 [5]. Then the decoding rule

used in prior work implies that for all 1 ≤ k ≤ L− 1, [5]

DL,k ≈ Dc +1

nN

∑λc∈Vs(0)

‖λc − μs(λc)‖2 +1

nN

L− k

Lk(L− 1)

∑λc∈Vs(0)

sp(α(λc)). (4)

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A key factor in the index assignment design proposed in [6] is the so-called L-fraction lattice

Λs/L, defined as

Λs/LΔ=

1

LΛs =

{λs/L ∈ Rn : λs/L =

k

LGs,k ∈ Zn

},

where Gs is the generator matrix of Λs. Define now the discrete Voronoi cell for each L-fraction

lattice point as [6]

Vs/L(λs/L

) Δ= {λc :

∥∥λc − λs/L

∥∥ ≤ ∥∥λc − λ′s/L∥∥ , ∀λ′s/L ∈ Λs/L}.

It is shown in [6] that, if Λs is clean, then the sets Vs/L(λs/L) are pairwise disjoint. Further, for

each λs/L ∈ Λs/L consider

T(λs/L

)�

⎧⎨⎩(λ1, λ2, ..., λL) ∈ ΛL

s :1

L

∑1≤j≤L

λj = λs/L

⎫⎬⎭ .

It is shown in [6] that the procedure of assigning the L-tuples in T (λs/L) of smallest spread to

the central lattice points in Vs/L(λs/L

), minimizes DL,k for all 1 ≤ k ≤ L− 1, as N →∞.

Since DL,L = Dc, according to [11], one has

DL,L ≈ G(Λc)ν2

nc ,

where G(Λc) denotes the normalized second moment of the lattice Λc. Additionally, the asymp-

totic analysis in [8] leads, for 1 ≤ k ≤ L− 1, to

DL,k ≈ L− k

kL−

L

L−1 (Nνc)2

nG(SLn−n)N2

(L−1)n , (5)

as N → ∞, where G (Sm) is the normalized second moment of a sphere in m dimensions,

m ≥ 1.

III. PROPOSED FLEXIBLE MDLVQ FRAMEWORK FOR L ≥ 3

Notice that in the previous design once Λc and Λs have been chosen the system’s performance

is determined. Interestingly, the index assignment used in evaluating the performance minimizes

the distortion (as N → ∞) for every possible k, 1 ≤ k ≤ L − 1. The fact that is possible to

optimize simultaneously the distortions for all values of k is contrary to our intuition that there

should be tradeoffs between these distortions. The first insight towards solving this conflict is that

the optimality of the index assignment is a result of the particular way the decoder is designed.

The second insight is that the decoding rule is natural for the side decoders, i.e., when k = 1,

while for 2 ≤ k ≤ L− 1, it seems to rather be dictated by convenience. These observations lead

to the conclusion that distortion DL,1 is, indeed, the smallest possible, but there should be room

for further decreasing DL,k, for 2 ≤ k ≤ L− 1, if DL,1 is allowed to increase. Nevertheless, for

this to be possible, the decoding mapping for 2 ≤ k ≤ L− 1 has to be changed.

Therefore, in order to introduce more flexibility into the system we will consider a different

decoding mapping for 2 ≤ k ≤ L − 1, as follows. For any L-tuple (ξ1, ξ2, · · · , ξL) ∈(Λs ∪

{∗})L

that has at least one component equal to ∗ and at least two other components in Λs, we

compute the reconstruction value as the arithmetic average of the central lattice points in the set

α−1(ξ1, ξ2, · · · , ξL), where

α−1(ξ1, ξ2, · · · , ξL) � {λc ∈ Λc : αi(λc) = ξi for all i, 1 ≤ i ≤ L, with ξi ∈ Λs}.

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Notice that this decoding rule is optimal assuming that the pdf is uniform over the set α−1(ξ1, ξ2, · · · , ξL).Further, in order to control the performance when k, 2 ≤ k ≤ L − 1, descriptions are re-

ceived, , we introduce L − 2 parameters: δk, 2 ≤ k ≤ L − 1, satisfying the inequalities:

δ2 ≥ δ3 ≥ · · · ≥ δL−1 ≥ 0 and impose the following condition.

Condition A. For any k, 2 ≤ k ≤ L − 1, and any L-tuple (ξ1, ξ2, · · · , ξL) ∈(Λs ∪ {∗}

)L

having exactly L − k components equal to ∗, one must have ‖λc − λ′c‖ ≤ δk for any λc, λ′c ∈

α−1(ξ1, ξ2, · · · , ξL).It is easy to see that Condition A guarantees that

DL,k � Dc + δ2k, ∀k, 2 ≤ k ≤ L− 1.

Finally, when designing the index assignment α we attempt to minimize DL,1 while ensuring

that Condition A is satisfied. Note that the decoding rule for k = 1 is the same as in previous

work, therefore the value of DL,1 can be still computed using equation (4) for k = 1, leading to

DL,1 ≈ Dc +1

nN

∑λc∈Vs(0)

‖λc − μs(λc)‖2 +1

nNL

∑λc∈Vs(0)

sp(α(λc)). (6)

The proposed index assignment algorithm first selects Ln L-fraction lattice points λs/L ∈ Λs/L

situated in the Voronoi region of 0 with respect to the side lattice Λs, such that the difference of

any two such points is not in Λs. Let F denote this set and C = ∪λs/L∈FVs/L(λs/L). Notice that

it is sufficient to specify the index assignment for the central lattice points in C and then extend

it to Λc via shifting. Guided by (6), for each λs/L ∈ F we assign the central lattice points in

Vs/L(λs/L) to L-tuples in T (λs/L), to minimize the first sum in (6). However, we can no longer

select the L-tuples of smallest spread to be assigned as in [6], since this would lead to violations

of Condition A. Having in mind the need to keep the last sum in (6) as small as possible we

proceed in a greedy manner as described next.

The algorithm maintains a list T of candidate triples to be assigned. The assignment is built

up gradually such that at every moment Condition A to be satisfied for the assignment obtained

by extending via shifting the partial assignment built so far. The set T is initialized to T =∪λs/L∈FT (λs/L). At each iteration the L-tuple of smallest spread from the set T is selected

as the current candidate to be assigned. Let (λ1, · · · , λL) denote this L-tuple. Next the value

λs/L =∑

i=1Lλi

L is determined and for each λc in Vs/L(λs/L) it is tested whether assigning the

L-tuple to λc preserves Condition A for the assignment extended by shifting. Specifically, it is

checked whether λc satisfies Condition B, stated next.

Condition B. Given λc ∈ Λc, we say that λc satisfies Condition B if and only if for every

λ′c ∈ C assigned so far and every k, 2 ≤ k ≤ L− 1, such that there are sk ∈ Λs and k different

positions i1, i2, · · · , ik ∈ {1, · · · , L} with the property that αij (λ′c) = αij (λc) + sk, 1 ≤ j ≤ k,

the inequality ‖λ′c − sk − λc‖ ≤ δk holds.

If a point λc satisfying Condition B is found then the L-tuple is assigned to it and removed

from the list T . Otherwise the L-tuple is simply removed from the list T without being assigned.

When all central lattice points from the set Vs/L(λs/L), for some λs/L, are assigned, all L-tuples

in T (λs/L) ∩ T are removed from T . Finally, the algorithm stops when T becomes empty.

We point out that when δk = ∞ for all 2 ≤ k ≤ L − 1, the algorithm produces an index

assignment as in [6].

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0 10 20 30 400

500

1000

1500

2000

2500

D3,2

D3,1

Fig. 1. The value of D3,1 versus D3,2 for the hexagonal lattice A2 with N = 307 and various values of δ2.

TABLE Iδ2 δ3 d1 d2 d320 ∞ 2914.68 26.93 7.1825 ∞ 1569.43 40.07 9.9630 ∞ 801.68 52.43 14.6935 ∞ 228.43 66.50 21.06∞ ∞ 198.68 64.78 22.47

TABLE IIδ2 δ3 d1 d2 d320 10 3025.06 25.88 3.1525 10 2736.06 36.20 3.0930 10 1974.56 49.68 3.2135 10 1356.56 67.51 3.7740 10 888.81 89.68 4.25∞ 10 391.68 114.53 5.71

TABLE IIIδ2 δ3 d1 d2 d320 0 5256.18 28.14 0.0730 0 3659.56 48.20 0.0740 0 2282.93 88.79 0.0750 0 1077.43 161.79 0.0760 0 684.31 187.83 0.07∞ 0 666.93 189.28 0.07

TABLE IVδ2 δ3 d1 d2 d3∞ 0 666.93 189.28 0.07∞ 5 530.18 150.20 0.90∞ 10 391.68 114.53 5.71∞ 15 262.18 78.42 16.24∞ ∞ 198.68 64.78 22.47

TABLE Vδ2 δ3 d1 d2 d330 0 3659.56 48.20 0.0730 5 2820.56 50.50 0.4930 10 1974.56 49.68 3.2130 15 1089.31 49.99 9.2130 ∞ 801.68 52.43 14.69

TABLE VIδ2 δ3 d1 d2 d320 0 5256.18 28.14 0.0720 4 5037.68 23.97 0.2520 8 3218.31 25.57 1.8420 10 3025.06 25.88 3.1520 ∞ 2914.68 26.93 7.18

IV. EXPERIMENTAL RESULTS

In this section, we assess empirically the performance of the proposed index assignment

algorithm for L = 3 and L = 4. In both cases n = 2 and the central lattice Λc is the hexagonal

lattice A2, i.e., the lattice generated by the vectors (1, 0) and (−1/2,√3/2).

For L = 3 we consider a clean sublattice Λs of Λc with index N = 307. We use various values

for the parameter δ2, starting at 0 and incrementing by 2τ up to 24τ , then incrementing by 10τ ,

where τ is the squared distance between two neighboring central lattice points, i.e., τ = 1. In

our experiments incrementing δ2 beyond 24τ did not produce any change in performance.

Figure 1 plots D3,1 versus D3,2 for the specified values of δ2. The rightmost point in the plot

is achieved for the largest δ2. Its corresponding pair of distortions (D3,2, D3,1) is the same as in

[6]. We see that as δ2 decreases towards 0, the value of D3,2 decreases towards Dc, while the

value of D3,1 increases.

For L = 4 we consider N = 703. The results are presented in Tables I-VI. In these tables dkrepresents the value of D4,k, for 1 ≤ k ≤ 3. From the simulation results, we can see that we can

achieve the tradeoffs between d1, d2 and d3 by varying δ2 and δ3.

V. STRUCTURED INDEX ASSIGNMENT FOR L = 3

The index assignment proposed in Section III lacks structure and therefore it is difficult to

analyze theoretically. In this section we propose a structured index assignment for a flexible

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MDLVQ in the case L = 3 and derive its asymptotical performance. Our results show that the

proposed scheme can achieve a wide range of values for D3,1 and D3,2 when Λc and Λs are

fixed. We first develop an index assignment ensuring D3,2 = Dc and then proceed to the case

D3,2 > Dc.

A. Case D3,2 = Dc

Consider the set A0 consisting of the N side lattice points that are closest to 0. Further, let

T0 denote the set of triples (λ,0,−λ) with λ ∈ A0. The central lattice points in Vs(0) are

assigned to the triples in T0 in an one-to-one manner. Further, the index assignment is extended

via shifting to all points in Λc. Thus, the central lattice points in Vs(v), for v ∈ Λs are assigned

triples of the form (λ+ v, v,−λ+ v). It is easy to see that any two components of any assigned

triple (λ1, λ2, λ3) uniquely identify the corresponding central lattice point λc = α−1(λ1, λ2, λ3).Therefore, one has

D3,2 = Dc ≈ G(Λc)ν2

nc .

In order to proceed with the derivation of D3,1 we need the following result, which follows from

[1].

Proposition. Let Λ′ be a lattice in Rn with ν ′ denoting the volume of its fundamental region,

and let N0 be a positive integer. Denote by S(N0,Λ′) the sum of the squared distances to 0 of

the N0 lattice points that are closest to 0. Then one has

S(N0,Λ′) = N

n+2

n

0 (ν ′)2

nnG(Sn)(1 + o(1)), (7)

as N0 →∞.

Using (4) with k = 1, one obtains

D3,1 ≈ Dc +1

nN

∑λc∈Vs(0)

‖λc‖2 +2

3nN

∑λ∈A0

‖λ‖2. (8)

We will first prove that

Dc +1

nN

∑λc∈Vs(0)

‖λc‖2 ≈ G(Λs)(Nνc)2

n . (9)

Let Ds denote the distortion of a quantizer having Λs as a codebook. Then, according to [11],

as νs approaches 0, one has

Ds ≈∫Vs(0)

‖x‖2dxnNνc

= G(Λs)(Nνc)2

n ,

where Vs(0) denotes the Voronoi region of 0 with respect to the side lattice. The above relation

further implies that

Ds ≈∑

λc∈Vs(0)

∫Vc(λc)

‖x− λc + λc‖2dxnNνc

≈N

∫Vc(0)

‖x‖2dxnNνc

+

∑λc∈Vs(0)

‖λc‖2

nN,

which proves (9) since Dc ≈∫Vc(0)

‖x‖2dxnνc

[11].

Further, notice that∑

λ∈A0

‖λ‖2 = S(N,Λs) and based on (7), (8) and (9), one obtains the

following

D3,1 ≈ G(Λs)(Nνc)2

n +2

3G (Sn)N

2

n (Nνc)2

n ≈ 2

3G (Sn)N

2

n (Nνc)2

n , (10)

where the last relation holds because as N →∞ the second term becomes dominant.

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B. Case D3,2 > Dc

Let μ > 0 and let U denote the set of side lattice points in the n dimensional sphere of radius

μ, centered at 0. Let A denote the set of the closest � N|U|� side lattice points to 0. Note that

|U| denotes the cardinality of the set U . Assume that 1 < |U| ≤ N . Then the triples assigned

to central lattice points in Vs(0) are from the set T � {(λ, u,−λ) : λ ∈ A, u ∈ U}. To

perform the assignment the set Vs(0) is first partitioned into |U| subsets Lu, u ∈ U , such that

� N|U|� ≤ |Lu| ≤ � N

|U|� and Lu contains at most one point λc for which −λc is not in Lu. A

moment of thought reveals that such a partition is possible since Vs(0) is symmetric with respect

to 0. The latter condition ensures that

1

N

∑λc∈Lu

λc ≈ 0, (11)

as N →∞. Finally, for each u ∈ U the central lattice points in each Lu are assigned to triples

of the form (λ, u,−λ), where λ ∈ A. The index assignment is extended to Λc using shifting.

Let us first derive D3,2. To simplify the analysis we assume the following suboptimal decoding

method. For every (λ1, λ2, λ3) ∈ Λ3s, γ(λ1, λ2, ∗) � λ2, γ(∗, λ2, λ3) � λ2 and γ(λ1, ∗, λ3) �

λ1+λ3

2 , where γ denotes the decoding mapping. It is easy to see that this decoder is shift invariant.

Then, according to (3), the following holds

D3,2 ≈ Dc +1

3nN

∑λc∈Vs(0)

(‖λc − y110(λc)‖2 + ‖λc − y011(λc)‖2 + ‖λc − y101(λc)‖2

). (12)

Using the fact that y101(λc) = 0 for all λc ∈ Vs(0) and relation (9) it follows that∑λc∈Vs(0)

‖λc − y101(λc)‖2 =∑

λc∈Vs(0)

‖λc‖2 ≈ nN(G(Λs)(Nνc)

2

n −Dc

), (13)

Further, one gets

1N

∑λc∈Vs(0)

‖λc − y110(λc)‖2 ≈ 1N

∑u∈U

∑λc∈Lu

‖λc − u‖2

≈ 1N

∑λc∈Vs(0)

‖λc‖2 + 1N |A|

∑u∈U

‖u‖2 = n(G(Λs)(Nνc)

2

n −Dc

)+ 1

N |A|S(|U|,Λs), (14)

where the second relation in the above sequence is obtained based on (11). A similar reasoning

leads to

1

N

∑λc∈Vs(0)

‖λc − y011(λc)‖2 = n(G(Λs)(Nνc)

2

n −Dc

)+

1

N|A|S(|U|,Λs). (15)

Relations (12)-(15) imply that

D3,2 ≈ G(Λs)(Nνc)2

n +2

3nN|A|S(|U|,Λs).

Using further (7) and the fact that |A||U| ≈ N when N is large enough, one gets

D3,2 ≈ (Nνc)2

n

(G(Λs) +

2

3G(Sn)|U|

2

n

). (16)

Let us now evaluate D3,1. According to (4) we have

D3,1 ≈ Dc +1

nN

∑λc∈Vs(0)

‖λc − μs (λc)‖2 +1

3nN

∑λc∈Vs(0)

3∑j=1

‖αj (λc)− μs (λc)‖2, (17)

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Then

1

N

∑λc∈Vs(0)

‖λc − μs (λc)‖2 =1

N

∑u∈U

∑λc∈Lu

‖λc −u

3‖2

≈ 1

N

∑λc∈Vs(0)

‖λc‖2 +1

9N|A|

∑u∈U

‖u‖2 = n(ds −Dc) +1

9N|A|S(|U|,Λs)

= n(G(Λs)(Nνc)

2

n −Dc

)+

1

9(Nνc)

2

nnG(Sn)|U|2

n . (18)

where the second relation follows from (11) and the last from (7). Let us evaluate now the last

sum in (17).

∑λc∈Vs(0)

∑3j=1 ‖αj(λc)− μs (λc)‖2 ≈

∑u∈U

∑λ∈A

(‖λ− u

3‖2 + ‖2u3 ‖2 + ‖λ+ u3‖2

)

=∑u∈U

∑λ∈A

(2‖λ‖2 + 2

3‖u‖2)= 2|U| ∑

λ∈A‖λ‖2 + 2

3 |A|∑u∈U

‖u‖2

= 2|U|S(|A|,Λs) +23 |A|S(|U|,Λs) = N(Nνc)

2

nnG(Sn)(2|A| 2n + 2

3 |U|2

n

), (19)

where the last equality follows from (7). Finally, (17)-(19) lead to

D3,1 ≈ (Nνc)2

n

(G(Λs) +G(Sn)

(23|A| 2n +

1

3|U| 2n

)).

When N → ∞ the last term in the expression inside the outer pair of parentheses dominates,

therefore we may write

D3,1 ≈ G(Sn)(Nνc)2

n

(23|A| 2n +

1

3|U| 2n

). (20)

Assume now that 1 ≤ |U| ≤√N . Then |U| = N

α

2 for some α ∈ (0, 1). Replacing in (16), one

obtains

D3,2 ≈ (Nνc)2

n

(G(Λs) +

2

3G(Sn)N

α

n

)≈ 2

3G(Sn)(Nνc)

2

nNα

n , (21)

where the last relation is obtained by keeping only the dominant term, as N → ∞. Further,

replacing |U| by Nα

2 and |A| by N1−α

2 in (20), one gets

D3,1 ≈ G(Sn)(Nνc)2

n

(23N

2

n(1−α

2) +

1

3N

α

n

)≈ 2

3G(Sn)(Nνc)

2

nN2

n(1−α

2). (22)

Notice that as α varies in the interval (0, 1), the product D3,1D3,2 is constant and

D3,1D3,2 ≈4

9(G(Sn))

2(Nνc)4

nN2

n . (23)

On the other hand, according to (5), the distortion product for the MDLVQ in prior work [10]

[6] [8]

D3,1D3,2 ≈ 1

27(G(S2n))

2(Nνc)4

nN2

n . (24)

It can be seen from (23) and (24) that for each scheme D3,1D3,2 = O(N6

n ). However, the previous

scheme can only achieve one distortion pair with the ratio D3,1/D3,2 = 4, while the proposed

scheme can achieve a wide range of pairs with ratios D3,1/D3,2 = N2

n(1−α), for all α ∈ (0, 1).

On the other hand, the distortion product for the proposed scheme is 12 times higher than for

the previous framework. This fact motivates the search for better structured index assignments,

261

problem which we will address in future work. An idea towards this is to consider subsets Lu

of decreasing sizes as ‖u‖2 increases.

It is interesting to compare the performances of the proposed index assignment for D3,2 > Dc

when α→ 0 versus the index assignment achieving D3,2 = Dc from subsection V-A. Relations

(21) and (20) imply that for the former index assignment, one has

limα→0

D3,2 ≈2

3G(Sn)(Nνc)

2

n , limα→0

D3,1 ≈2

3G(Sn)(Nνc)

2

nN2

n .

Comparing with (7) and (10) we see that the value of D3,1 is the same, while there is a big gap

between the values of D3,2. Designing a better index assignment for D3,2 > Dc, to close this

gap, is another objective for our future research efforts.

VI. CONCLUSION

This work proposes a flexible MDLVQ. A different reconstruction method is adopted and a

heuristic index assignment algorithm, which uses L − 2 parameters to control the distortions

for 2 ≤ k ≤ L − 1, is developed. Additionally, a structured index assignment, amenable to

theoretical analysis, is proposed for the case L = 3. By deriving the asymptotical expressions

of the distortions at high resolution, we show that a wide range of values can be achieved for

the distortions when k = 1 and k = 2. Future research efforts will be directed to improve the

performance of the proposed structured index assignment and to generalize it to higher values of

L.

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