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A shape-gain approach for vector quantization based on flat tori and dual lattices A n Fabiano Boaventura de Miranda joint work with Cristiano Torezzan and Sueli Costa Universidade Estadual de Campinas 26 de julho de 2018 Fabiano

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A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

n

Fabiano Boaventura de Miranda

joint work with Cristiano Torezzan and Sueli Costa

Universidade Estadual de Campinas

26 de julho de 2018

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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26 de julho de 2018 1 / 29

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Introduction

In this work we present a vector quantization framework for Gaussian source combining a spherical code in layers of flat tori and the shape-gain technique, using the dual lattice A∗

k.

We focus our attention on the family of dual lattices A∗

k, which

is known to have the thinnest covering radius in dimensions up to 8. We analyze the performance of the lattices A∗

2, A∗ 3 and A∗ 4 to

construct spherical codes for vector quantization, expecting that its covering properties may provide good results for quantization.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Outline

Vector quantization Shape-gain technique Lattices and quantization Spherical Codes in layers of flat tori Proposed vector quantizer Computational results

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The quantization process

Quantization is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The quantization process

Quantization is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set. Underlying challenges:

1 Designing the quantization scheme. 2 Measuring the average distortion. 3 Dealing with the computational cost.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The quantization process - example

Rounding and truncation are simple examples. Supposing sent the information x = 4.75, using an integer closest rounding quantizer, then ˆ x = 5. The representation error is |4.75 − 5| = 0.25.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The quantization process - example

Rounding and truncation are simple examples. Supposing sent the information x = 4.75, using an integer closest rounding quantizer, then ˆ x = 5. The representation error is |4.75 − 5| = 0.25. Digitizing an analog signal

Figure: Example of digitizing an analog signal

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The quantization process - example

Data compression

(a) Original image (b) Compressed image

Figure: Data compression applied to image.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Vector quantization

Encoder Decoder

ˆ x Q(x) ∈ {1, 2, . . . , 2kR} x ∈ Rk

Figure: Vector quantization scheme

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Vector quantization

Encoder Decoder

ˆ x Q(x) ∈ {1, 2, . . . , 2kR} x ∈ Rk

Figure: Vector quantization scheme

R = 1 k log2 M, D = E[x − ˆ x2], (1)

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Vector quantization

Encoder Decoder

ˆ x Q(x) ∈ {1, 2, . . . , 2kR} x ∈ Rk

Figure: Vector quantization scheme

R = 1 k log2 M, D = E[x − ˆ x2], (1) SNR = 10 log10 E[x2] D (2) E[x2] is the average energy of the input vectors in dB.

Gersho, A. Gray, R. M., Vector quantization and signal compression. Boston: Kluwer Academic Publishers, 2001. You, Y., Audio Coding: Theory and Application. New York: Springer, 2010. Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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(a) Uniform distribution (b) Normal distribution

Figure: Lattice quantizer in R2

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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(a) Uniform distribution (b) Normal distribution

Figure: Lattice quantizer in R2

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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(a) Uniform distribution (b) Example

  • f

quantization using spherical codes.

Figure: Lattice quantizer in R2

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Quantizers to Gaussian source

When k → ∞, iid Gaussian random variables tend to be approximately evenly distributed and to lie on the surface of a k-dimensional sphere with radium σ √ k. f (x1, . . . , xk) ≈ 2−k

√ 2πσ2e

(3)

k

  • i=1

x2

i ≈ (σ

√ k)2 (4)

  • J. D. Gibson and K. Sayood, Lattice Quantization, in Advances in Electronics and Electron Physics, P. Hawkes, Ed. New York:

Academic, 1988, vol. 72, pp. 259 − 330 Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The shape-gain technique

Let x ∈ Rk be a iid Gaussian random variable. The vector s is said to be the shape and g is said to be the gain of x, then s = x x and g = x, then x = gs.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The shape-gain technique

Let x ∈ Rk be a iid Gaussian random variable. The vector s is said to be the shape and g is said to be the gain of x, then s = x x and g = x, then x = gs. The scalar g is quantized to be represented by ˆ g and the vector s is independently quantized to be represented by ˆ s.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The shape-gain technique

Let x ∈ Rk be a iid Gaussian random variable. The vector s is said to be the shape and g is said to be the gain of x, then s = x x and g = x, then x = gs. The scalar g is quantized to be represented by ˆ g and the vector s is independently quantized to be represented by ˆ s. The shape-gain codebook is given by C = Cg.Cs, (5) where Cg = 1, . . . , Ng e Cs = 1, . . . , Ns.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The shape-gain technique

Let x ∈ Rk be a iid Gaussian random variable. The vector s is said to be the shape and g is said to be the gain of x, then s = x x and g = x, then x = gs. The scalar g is quantized to be represented by ˆ g and the vector s is independently quantized to be represented by ˆ s. The shape-gain codebook is given by C = Cg.Cs, (5) where Cg = 1, . . . , Ng e Cs = 1, . . . , Ns. R = Rg + Rs

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Shape-gain quantization framework

ˆ x = ˆ gˆ s x = gs ˆ s s points on the spherical code

Figure: Example of shape-gain quantization in R2

Hamkins, J., & Zeger, K. Optimal rate allocation for shape-gain Gaussian quantizers. In Proc. IEEE International Symposium on Information Theory (24-29 June 2001), p. 182. Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Objective

Our goal is to designing a shape-gain for vector quantization. It involves: Designing a suitable spherical code Analyze the cost of encoding and decoding

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Spherical Codes

A spherical code C(M, k) is a set of M points on the surface of the k-dimensional unit sphere Sk−1 ⊂ Rk, C(M, k) = {xi; ∈ Sk−1 : xi = 1, 1 ≤ i ≤ M} (a) Spherical code example

Li Li+1 S S ˆ S X Y Z Y X

(b) Decoding of the shape

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Spherical code in layers of flat tori

The construction these code is based in the sphere’s foliation on flat tori. Definition Let c = (c1, c2, . . . , ck) ∈ Sk−1 ⊂ Rk be, ci > 0,

k

  • 1

c2

i = 1 and

y = (y1, y2, . . . , yk) ∈ Rk. Let Φc : Rk → R2k be defined by Φc(y) =

  • c1 cos

y1 c1

  • , c1 sin

y1 c1

  • , . . . , ck cos

yk ck

  • , ck sin

yk ck

  • ,

The image of Φc is the torus Tc, a flat k-dimensional surface on the unit sphere S2k−1. Φc is an embedding of the flat torus Tc, generated by the k-dimensional box Pc =

  • y ∈ Rk; 0 ≤ yi < 2πci
  • ,

1 ≤ i ≤ k.

Torezzan, C.; Costa, S.I.R.; Vaishampayan, V.A., Constructive Spherical Codes on Layers of Flat Tori. Information Theory, IEEE Transactions on , vol.59, no.10, pp.6655,6663, Oct. 2013. Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Example in R4

(i) The set of vectors that lie on surface Sk/2−1 that have only positive coordinates defines a spherical code, which is denoted by C(k/2, MT)+, where MT is the numbers of points in the Sk−1.

c = (c1, c2) Pc 2πc1 2πc2

Figure: Construction of code C(k/2, MT)+

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Example in R4

(ii) Each flattened torus Tc results in a box Pc, that will be filled with a set of points from the lattice A∗

k/2, suitably rescheduled to

achieve a desirable minimum distance.

c = (c1, c2) Pc 2πc1 2πc2

Figure: Allocating of the lattice points into Pc

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Example in R6

Figure: Example of a spherical code C(3, 100)+ ⊂ R3 that represents 100 flat tori in S5 ⊂ R6.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The dual lattice A∗

k A k-dimensional lattice Λ is an infinity, discrete and homogeneous set of points such that Λ =

  • xB | x ∈ Zk

, where B = [b1, b2, . . . , bk] is a matrix with linearly independent rows called generetor matrix of the lattice Λ. The dual latice Λ∗ is defined as the set of all vectors v ∈ Rk such that v, λ ∈ Z, for all λ ∈ Λ. The lattice Ak ⊂ Rk is defined as the set of points in Rk+1, for all k 1, such that the sum of the their coordinates is zero.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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The lattice A∗

k, the dual of Ak, can be written as

A∗

k = k

  • i=0

([i] + Ak) , where [i] = −j

k+1

i ,

  • i

k+1

j and i + j = k + 1 for 0 ≤ i ≤ k. The classes [i] are called cosets of the lattice Ak. A possible generetor matrix for A∗

k is given by

A∗

k =

       1 −1 . . . 1 −1 . . . . . . . . . . . . . . . . . . . . . 1 . . . −1

−k k+1 1 k+1 1 k+1

. . .

1 k+1 1 k+1

       .

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Designing a spherical quantizer on layers of tori

1

Given a R = Rs + Rg, we have a total of N = 2kR points to be distributed between the codebooks Cg and Cs.

2

Use the Lloyd-Max algorithm to design the gain codebook, containing Ng = 2kRg points.

3

Select an appropriated spherical code in dimension k/2 with positive coordinates, C(k/2, MT)+ to define the tori and distribute Ns = 2kRs lattice points of A∗

k/2 proportionality to the volume of each torus, such that

MT

j=1 Mj ≤ 2kRs.

4

Apply the mapping Φc given in equation (6) to embed the tori in the surface of the unit sphere Sk−1 ⊂ Rk. The spherical code CT(k, M) will be given by CT(k, M) =

  • c∈C(k/2,MT )+

Φ(YTc)

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Encoding and decoding process for the proposed quantizer

Sk−1 ⊂ Rk

x s = x x w ˆ s z a

Λ ⊂ R

k 2

Tcξ Pcξ

ˆ s

Figure: Vector quantization

process Given a random vector x ∈ Rk.

  • 1. Compute g = x and s = x

g .

  • 2. Use the gain codebook to quantize g to

ˆ g.

  • 3. Find the index ξ that identifies the closest

torus Tcξ from s.

  • 4. Project the coordinate of s within of the

box Pcξ and find the closest lattice point a in A∗

k/2.

  • 5. Calculate ˆ

s = Φcξ(a).

  • 6. Get the vector quantized ˆ

x = ˆ gˆ s.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Computational results

(a) Comparison of some C(k, MT) for rate R = 3. (b) Performance of the proposed vec- tor quantizer.

Figure: Computational results of the simulation

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Computational results

Table: Comparison of the vector quantizer CT(k, M) with the results showed in [3]

rate/samples Quantizer 1 2 3 4 5 6 Shannon bound 6.02 12.04 18.06 24.08 30.10 36.12 QV using CT(k, M) 4.454 10.058 16.168 22.058 27.938 33.346 QV using WΛ24 2.44 11.02 17.36 23.33 29.29 35.27 Lloyd-Max QE 4.40 9.30 14.62 20.22 26.02 37.81 Uniform QE 4.40 9.25 14.27 19.38 24.57 29.83

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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Summarizing

We introduce a shape-gain vector quantization based on spherical codes in layers of flat tori. We explore the lattices A∗

k in each layer.

The computational complexity is dominated by finding the closest lattice point in A∗

k, that is of order O(n2 log n)

Good computational results for low rates.

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References

Torezzan, C.; Costa, S.I.R.; Vaishampayan, V.A., Spherical codes on torus layers, 2009 IEEE International Symposium on Information Theory, Seoul, 2009, pp. 2033-2037. Torezzan, C.; Costa, S.I.R.; Vaishampayan, V.A., Constructive Spherical Codes on Layers of Flat Tori. Information Theory, IEEE Transactions on , vol.59, no.10, pp.6655,6663, Oct. 2013. Hankins, J. Design and Analysis of Spherical Codes. PhD thesis, University

  • f Illinois, 1996.

Hamkins, J. Zeger, K.. Gaussian Source Coding With Spherical Codes. IEEE

  • Trans. Inf. Theory, vol. 48, pp. 2980-2989, Nov. 2002.
  • J. H. Conway, N. J. A. Sloane, and E. Bannai. Sphere-Packings, Lattices,

and Groups. Springer-Verlag New York,1987. Inc., New York, NY, USA. Gersho, A. Gray, R. M., Vector quantization and signal compression. Boston: Kluwer Academic Publishers, 2001.

Fabiano Boaventura (UNICAMP) A shape-gain approach for vector quantization based on flat tori and dual lattices A∗

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References

  • J. H. Conway and N. J. A. Sloane. Soft decoding techniques for codes and

lattices, including the Golay code and the Leech lattice. IEEE Trans. on lnformation Theory, IT-32:41-50, 1986.

  • J. D. Gibson and K. Sayood, Lattice Quantization, in Advances in

Electronics and Electron Physics, P. Hawkes, Ed. New York: Academic, 1988, vol. 72, pp. 259 − 330

  • G. D. Forney Jr. A bounded-distance decoding algorithm for the Leech

lattice, with generalizations. IEEE Trans. on lnformation Theory, IT-35:906-909, July 1989.

  • Y. Be’ery, B. Shahar, e J. Snyders. Fast decoding of the Leech lattice. IEEE

Selected Areas on Communications, 7:959-967, August 1989. Shannon, C. E. Probability of error for optimal codes in a Gaussian channel, in The Bell System Technical Journal, vol. 38, no. 3, pp. 611-656, May 1959.

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OBRIGADO!!

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