lloyd-max quantization schemes -...

Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations

Lloyd-Max Quantization Schemes

Helmut KnaustDepartment of Mathematical Sciences

The University of Texas at El PasoEl Paso TX 79968-0514

[email protected]

January 6, 2011


1 Introduction

2 Basic Quantization Schemes

3 Lloyd-Max Quantization Setup

4 Lloyd-Max Quantization for “Raw” Images

5 Lloyd-Max Quantization for Transformed Images

6 Generalizations


Learning Outcomes and Prerequisites

Learning Outcomes:

Students will reflect on the role of quantization in imagecompression.Students will improve their programming skills.

Prerequisites:Multi-variable CalculusSome knowledge of wavelet transforms and their use inimage compressionSome minimal statistics knowledge



Learning Outcomes:Students will reflect on the role of quantization in imagecompression.

Students will improve their programming skills.Prerequisites:

Multi-variable CalculusSome knowledge of wavelet transforms and their use inimage compressionSome minimal statistics knowledge



Learning Outcomes:Students will reflect on the role of quantization in imagecompression.Students will improve their programming skills.





Prerequisites:

Multi-variable CalculusSome knowledge of wavelet transforms and their use inimage compressionSome minimal statistics knowledge




Prerequisites:Multi-variable Calculus

Some knowledge of wavelet transforms and their use inimage compressionSome minimal statistics knowledge




Prerequisites:Multi-variable CalculusSome knowledge of wavelet transforms and their use inimage compression

Some minimal statistics knowledge


Quantization

Quantization reduces ranges of values in a signal to asingle value, thereby reducing entropy.

Quantization is an integral part of lossy compressionalgorithms.

Quantization is usually employed after transformation.


Basic Quantization Scheme: Thresholding

The most basic quantization technique is Thresholding:

Given a signal ~x = (xi) and a single threshold σ, we replacevalues as follows:

q(xi) =

{0 if |xi | ≤ σxi if |xi | > σ


The thresholding quantization function:

x

qHxL


The JPEG2000 Quantization Scheme

The Lossy JPEG2000 Quantization Scheme also has onefixed parameter, τ . After wavelet transformation, a “step”quantization

q(xi) = sgn(xi) · σ ·⌊|xi |σ

⌋is applied to each region with a parameter σ determined asfollows:

Τ

2 Τ

Τ 2 Τ

n = 1






Τ

4Τ

2

Τ

2 Τ

Τ

Τ 2 Τ

n = 2






Τ

8Τ

4Τ

4Τ

2

Τ

2

Τ

2 Τ

Τ

Τ 2 Τ

n = 3


The quantization function for a given region:

x

qHxL


For a fixed signal ~x and a fixed positive integer n, Lloyd-Maxquantization uses two sets of parameters:

Bin-boundaries ~L = (L1,L2, . . . ,Ln+1), withmin~x = L1 < L2 < · · · < Ln < Ln+1 = 1 + max~x ,and replacement values ~p = (p1,p2, . . . ,pn).

The quantization function replaces the x-values in the bin[Lj ,Lj+1) by the value pj :

q(xi) = pj , when xi ∈ [Lj ,Lj+1)


The goal is to choose ~L and ~p to minimize the resultingquantization error

E(~L, ~p) =m∑

i=1

|xi − q(xi)|2

This is a classical Calculus problem. Rewriting we obtain:

E(~L, ~p) =n∑

j=1

∑xi∈[Lj ,Lj+1)

(xi − pj)2


E(~L, ~p) =n∑

j=1

∑xi∈[Lj ,Lj+1)

(xi − pj)2

Since minima will occur only if all partial derivatives are equal to0, the following conditions need to be satisfied:

∂ E∂pj

=∑

xi∈[Lj ,Lj+1)

2(xi − pj) = 0⇔ pj =

∑xi∈[Lj ,Lj+1)

xi

#{i | xi ∈ [Lj ,Lj+1)}(1)

∂ E∂Lj

= 0 ⇔ Lj =12(pj−1 + pj) (2)

p j-1 L j p jxi


These equations can usually not be solved explicitly; instead atwo-step procedure is used repeatedly until a fixed point hasbeen (nearly?) reached:

Equations (1) are used to update the values ~p:

pnewj = ave

{xi | xi ∈ [Lj ,Lj+1)

}Equations (2) then yield new values for ~L:

Lnewj =

12(pnew

j−1 + pnewj ) , j = 2, . . . ,n

The values for L1 and Ln+1 are left unchanged.




pnewj = ave

{xi | xi ∈ [Lj ,Lj+1)

}

Equations (2) then yield new values for ~L:

Lnewj =

12(pnew

j−1 + pnewj ) , j = 2, . . . ,n





pnewj = ave

{xi | xi ∈ [Lj ,Lj+1)

}Equations (2) then yield new values for ~L:

Lnewj =

12(pnew

j−1 + pnewj ) , j = 2, . . . ,n



As an example, we apply the algorithm to a grayscale image,with n = 32. The initial bins are chosen at random.

Original image, entropy=7.65



21 iterations, entropy=4.75, PSNR 40.7



28 iterations, entropy=4.56, PSNR 40.0


Here is the quantization function for the second run:

0 50 100 150 200 250x0

50

100

150

200

250

qHxL


Quantization applied to “raw” images usually gives bad resultsin areas of gradual gray-value change (sky, water, etc.):

Original image:entropy=7.63



n = 26:10 iterations, entropy=5.53, PSNR 45.1


We now compare step quantization to Lloyd-Max quantizationfor a transformed image:

We use the CDF97 wavelet transform once.For the step quantization we use τ = 16:

Original image (-128): entropy=7.74




CDF97-transformed image




Τ�2=8 Τ=16

Τ=16 2Τ=32

Step sizes for the quantization




Quantized transform: entropy=2.75




Reconstructed image: PSNR 32.44


Here is the same example with Lloyd-Max quantization:

Original image (-128): entropy=7.74



CDF97-transformed image



n=24 n=10

n=11 n=6

Number of bins for the quantization



Quantized transform: entropy=3.73



Reconstructed image: PSNR=33.44


Why is the entropy comparatively high after LM-quantization?Here are the histograms of the quantized transformed signals:

Step quantization Lloyd-Max quantization


Lloyd-Max quantization can be generalized to higherdimensions.

The bin boundary conditions in Equations (2) now becomeVoronoi cell conditions.The update step given by Equations (1) still remains: Thenew replacement values are the average of the signalvalues in the Voronoi cell.


Here is a two-dimensional example:


The End...

Any Questions?

Helmut [email protected]

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