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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
January 6, 2011
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Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
![Page 3: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/3.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images 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
![Page 4: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/4.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images 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
![Page 5: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/5.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images 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
![Page 6: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/6.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images 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
![Page 7: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/7.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images 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 Calculus
Some knowledge of wavelet transforms and their use inimage compressionSome minimal statistics knowledge
![Page 8: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/8.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images 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 compression
Some minimal statistics knowledge
![Page 9: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/9.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images 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
![Page 10: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/10.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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.
![Page 11: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/11.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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.
![Page 12: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/12.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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.
![Page 13: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/13.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
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Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
The thresholding quantization function:
x
qHxL
![Page 15: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/15.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
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Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
4Τ
2
Τ
2 Τ
Τ
Τ 2 Τ
n = 2
![Page 17: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/17.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
8Τ
4Τ
4Τ
2
Τ
2
Τ
2 Τ
Τ
Τ 2 Τ
n = 3
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Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
The quantization function for a given region:
x
qHxL
![Page 19: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/19.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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)
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Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
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Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
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Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
![Page 23: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/23.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
![Page 24: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/24.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
![Page 25: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/25.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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.
![Page 26: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/26.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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.
![Page 27: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/27.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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.
![Page 28: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/28.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
![Page 29: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/29.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
As an example, we apply the algorithm to a grayscale image,with n = 32. The initial bins are chosen at random.
21 iterations, entropy=4.75, PSNR 40.7
![Page 30: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/30.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
As an example, we apply the algorithm to a grayscale image,with n = 32. The initial bins are chosen at random.
28 iterations, entropy=4.56, PSNR 40.0
![Page 31: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/31.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is the quantization function for the second run:
0 50 100 150 200 250x0
50
100
150
200
250
qHxL
![Page 32: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/32.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Quantization applied to “raw” images usually gives bad resultsin areas of gradual gray-value change (sky, water, etc.):
Original image:entropy=7.63
![Page 33: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/33.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Quantization applied to “raw” images usually gives bad resultsin areas of gradual gray-value change (sky, water, etc.):
n = 26:10 iterations, entropy=5.53, PSNR 45.1
![Page 34: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/34.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Quantization applied to “raw” images usually gives bad resultsin areas of gradual gray-value change (sky, water, etc.):
n = 25:18 iterations, entropy=4.35, PSNR 38.0
![Page 35: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/35.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Quantization applied to “raw” images usually gives bad resultsin areas of gradual gray-value change (sky, water, etc.):
n = 24:22 iterations, entropy=3.76, PSNR 34.7
![Page 36: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/36.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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
![Page 37: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/37.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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:
CDF97-transformed image
![Page 38: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/38.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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:
Τ�2=8 Τ=16
Τ=16 2Τ=32
Step sizes for the quantization
![Page 39: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/39.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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:
Quantized transform: entropy=2.75
![Page 40: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/40.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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:
Reconstructed image: PSNR 32.44
![Page 41: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/41.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is the same example with Lloyd-Max quantization:
Original image (-128): entropy=7.74
![Page 42: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/42.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is the same example with Lloyd-Max quantization:
CDF97-transformed image
![Page 43: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/43.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is the same example with Lloyd-Max quantization:
n=24 n=10
n=11 n=6
Number of bins for the quantization
![Page 44: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/44.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is the same example with Lloyd-Max quantization:
Quantized transform: entropy=3.73
![Page 45: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/45.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is the same example with Lloyd-Max quantization:
Reconstructed image: PSNR=33.44
![Page 46: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/46.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Why is the entropy comparatively high after LM-quantization?Here are the histograms of the quantized transformed signals:
Step quantization Lloyd-Max quantization
![Page 47: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/47.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
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.
![Page 48: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/48.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 49: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/49.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 50: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/50.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 51: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/51.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 52: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/52.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 53: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/53.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 54: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/54.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 55: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/55.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 56: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/56.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 57: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/57.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 58: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/58.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 59: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/59.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 60: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/60.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 61: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/61.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 62: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/62.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 63: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/63.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 64: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/64.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 65: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/65.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 66: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/66.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 67: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/67.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
Here is a two-dimensional example:
![Page 68: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/68.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
The End...
Any Questions?
Helmut [email protected]
![Page 69: Lloyd-Max Quantization Schemes - Knausthelmut.knaust.info/presentations/2011/20110106_JMM_DWT.pdf2011/01/06 · Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed](https://reader035.vdocument.in/reader035/viewer/2022070220/61320242dfd10f4dd73a2b86/html5/thumbnails/69.jpg)
Introduction Basic Quantization Lloyd-Max “Raw” Images Transformed Images Generalizations
The End...
Any Questions?
Helmut [email protected]