A shape-gain approach for vector quantization based on flat tori and - PowerPoint PPT Presentation

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 ∗ 26 de julho de 2018 1 / 29 n

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 ∗ 26 de julho de 2018 2 / 29 n

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 ∗ 26 de julho de 2018 3 / 29 n

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 ∗ 26 de julho de 2018 4 / 29 n

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 ∗ 26 de julho de 2018 4 / 29 n

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 ∗ 26 de julho de 2018 5 / 29 n

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 ∗ 26 de julho de 2018 5 / 29 n

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 ∗ 26 de julho de 2018 6 / 29 n

Vector quantization Q ( x ) ∈ { 1 , 2 , . . . , 2 kR } x ∈ R k x ˆ Encoder Decoder Figure: Vector quantization scheme Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 7 / 29 n

Vector quantization Q ( x ) ∈ { 1 , 2 , . . . , 2 kR } x ∈ R k x ˆ Encoder Decoder Figure: Vector quantization scheme R = 1 x � 2 ] , k log 2 M , D = E [ � x − ˆ (1) Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 7 / 29 n

Vector quantization Q ( x ) ∈ { 1 , 2 , . . . , 2 kR } x ∈ R k x ˆ Encoder Decoder Figure: Vector quantization scheme R = 1 x � 2 ] , k log 2 M , D = E [ � x − ˆ (1) E [ � x � 2 ] SNR = 10 log 10 (2) D E [ � x � 2 ] 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 ∗ 26 de julho de 2018 7 / 29 n

(a) Uniform distribution (b) Normal distribution Figure: Lattice quantizer in R 2 Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 8 / 29 n

(a) Uniform distribution (b) Normal distribution Figure: Lattice quantizer in R 2 Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 9 / 29 n

(a) Uniform distribution (b) Example of quantization using spherical codes. Figure: Lattice quantizer in R 2 Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 10 / 29 n

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 . √ 2 πσ 2 e f ( x 1 , . . . , x k ) ≈ 2 − k (3) k √ � x 2 k ) 2 i ≈ ( σ (4) i =1 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 ∗ 26 de julho de 2018 11 / 29 n

The shape-gain technique Let x ∈ R k 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 x s = g = � x � , x = g s . and then � x � Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 12 / 29 n

The shape-gain technique Let x ∈ R k 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 x s = g = � x � , x = g s . and then � x � 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 ∗ 26 de julho de 2018 12 / 29 n

The shape-gain technique Let x ∈ R k 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 x s = g = � x � , x = g s . and then � x � 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 � = � C g � . � C s � , (5) where C g = 1 , . . . , N g e C s = 1 , . . . , N s . Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 12 / 29 n

The shape-gain technique Let x ∈ R k 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 x s = g = � x � , x = g s . and then � x � 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 � = � C g � . � C s � , (5) where C g = 1 , . . . , N g e C s = 1 , . . . , N s . R = R g + R s Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 12 / 29 n

Shape-gain quantization framework ˆ x = ˆ g ˆ s x = g s points on the spherical code ˆ s s Figure: Example of shape-gain quantization in R 2 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 ∗ 26 de julho de 2018 13 / 29 n

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 ∗ 26 de julho de 2018 14 / 29 n

Spherical Codes A spherical code C ( M , k ) is a set of M points on the surface of the k -dimensional unit sphere S k − 1 ⊂ R k , C ( M , k ) = { x i ; ∈ S k − 1 : � x i � = 1 , 1 ≤ i ≤ M } Z Y L i ˆ S S S L i +1 X Y X (a) Spherical code example (b) Decoding of the shape Fabiano Boaventura ( UNICAMP ) A shape-gain approach for vector quantization based on flat tori and dual lattices A ∗ 26 de julho de 2018 15 / 29 n