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Image Sampling &Quantisation
Biomedical ImageAnalysis
Prof. Dr. Philippe Cattin
MIAC, University of Basel
March 1st/7th/8th, 2016
March 1st/7th/8th, 2016Biomedical Image Analysis
1 of 47 21.02.2016 22:35
Contents
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Contents
1 Motivation
Introduction and Motivation
Sampling Example
Quantisation Example
2 Sampling
2.1 Tessellation
Tessellation
Tessellation Examples by M.C. Escher (1)
Tessellation Examples by M.C. Escher (2)
Tessellation Basics
Tessellation Claim
How Many Tessellations Exist with RegularPolygons?
Combinatorial Analysis
All Semi-Regular Tessellations
All Regular Tessellations
Tessellation Rules
Advantages of Square Tessellation
2.2 A Sampling Model
A Sampling Model
The Neighbourhood Function
Fourier Transform of the NeighbourhoodFunction
Filtering with the Neighbourhood Function
Sampling of a Continuous 1D FunctionMarch 1st/7th/8th, 2016Biomedical Image Analysis
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Sampling of a Continuous 1D Function (2)
Sampling of a Discrete 1D Function
An Alternative Reasoning for Periodicity in theDFT
Sampling of Two-Dimensional Functions(Images)
Summary Sampling Theorem
Aliasing Example 1
Aliasing Example 2
Aliasing Example 3
Remark on the Discrete Fourier Transform
Linear, Shift-Invariant Operators
Linear, Shift-Invariant Operators (2)
Liner, Shift-Invariant Operators (3)
Liner, Shift-Invariant Operators (4)
3 Quantisation
Quantisation
Lloyd-Max Quantisation
Quantisation Example
Quantisation Example (2)
Quantisation Example (3)
March 1st/7th/8th, 2016Biomedical Image Analysis
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Motivation
March 1st/7th/8th, 2016Biomedical Image Analysis
(3)Introduction and Motivation
In order for computers to process an image, this imagehas to be described as a series of numbers, each of finiteprecision
This calls for two kinds of discretisation:
Sampling, and
Quantisation
By sampling is meant that the brightness information is onlystored at a discrete number of locations. Quantisation indicatesthe discretisation of the brightness levels at these positions.
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Motivation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Sampling Example
Sampling is the process of measuring the brightnessinformation only at a discrete number of locations
Fig 4.1: Hight profile of Switzerland Fig 4.2: Sampled hight profile
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Motivation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Quantisation Example
Quantisation is the process of discretising the brightnessat a finite number of positions
Height map with grey values with grey values
with grey values with grey values
Fig 4.3:
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Sampling
Tessellation
March 1st/7th/8th, 2016Biomedical Image Analysis
(8)Tessellation
Definition
Tessellations are patterns that cover a plane withrepeating figures so there is no overlapping or emptyspaces
Sampling is best performed following a regular tessellation ofthe image:
Brightness is integrated over cells of same size1.
Cells should cover the whole image2.
These cells are usually referred to as picture elements or pixels.
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Tessellation Examples byM.C. Escher (1)
Fig 4.4: Sample Escher images
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Tessellation Examples byM.C. Escher (2)
Fig 4.5: Sample Escher images
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Tessellation Basics
Three types of tessellations with polygons exist
regular tessellations (using the same regular polygon)1.
semi-regular tessellations (using various regular
polygons)
2.
hyperbolic tessellations (they use non-regular polygons)3.
They are formed by translating, rotating, and reflecting
polygons
Fig 4.6: regular Fig 4.7: semi-regular Fig 4.8: hyperbolic
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Tessellation Claim
There exist only 11 possible tessellations with regularpolygons that can cover the entire image
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
How Many TessellationsExist with Regular Polygons?
Observation 1:
Since the regular polygons in atessellation must fill the plane ateach vertex, the interior angle mustbe an exact divisor of
Observation 2:
A regular -gon has an internal angle
of degrees Fig 4.9:
Of the regular polygons, only triangles ( ), squares ( ),pentagons ( ), hexagons ( ), octagons ( ), decagons (
) and dodecagons ( ) can be used for tiling around acommon vertex - again because of the angle value
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Combinatorial Analysis
A combinatorial analysis of these base polygons produces thefollowing 14 solutions
RegularTessellations
4.4.46.6.6
Semi-regularTessellations
3.3.43.6.33.4.63.3.34.8.83.12.12.gi4.6.1
Semi-regularTessellations thatcan not be extendedinfinitely 3.4.45.5.1
Fig 4.10: Tessellations
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
All Semi-RegularTessellations
Eight semi-regular tessellations exist
Snub hexagonal Trihexagonal Prismatic trisquare Snub square
Small
rhombitrihexagonalTruncated square
Truncated
hexagonal
Great
rhombitrihexagonal
Fig 4.11:
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
All Regular Tessellations
But only three regular tessellations exist
Triangular tiling Square tiling Hexagonal tiling
Fig 4.12:
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Tessellation Rules
For practical applications in computer vision the tessellation hasto adhere to the following rules
The tessellation must tile an infinite area with no gaps or
overlapping
Each vertex must look the same
The tiles must all be the same regular polygon
This leaves us with the following three regular tessellations
RegularTessellations 4.4.4
6.6.6Although the hexagonal tessellation offers some substantialadvantages (e.g. no ambiguities in defining connectedness,closer spatial organisation as found in mammalian retinas), thesquare tessellation is the most commonly used.
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Tessellation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Advantages of SquareTessellation
They directly support operations in the Cartesian coordinate
frame
Most algorithms (FFT, Image pyramids) are based on square
tessellations
The resolution is often a power of 2: e.g. 16x16, 32x32,
..., 256x256, 512x512
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A Sampling Model
March 1st/7th/8th, 2016Biomedical Image Analysis
(20)A Sampling Model
As we have seen,
The intensity value attributed to a pixel corresponds to the
integration of the incoming irradiance over a cell of the
tessellation
The cells are only located at discrete locations
The sampling process can thus be modeled in a 2-step scheme:
Integrate brightness over regions of the pixel size,1.
Read out values only at the pixel positions.2.
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
The NeighbourhoodFunction
First a neighbourhood function has to be
defined, that is 1 inside a region with the shape of apixel/cell and 0 outside.
Integrating the incoming intensity over such a
region then yields
(4.1)
rewriting this expression as
(4.2)
we recognise it as the convolution of with
which can also be written as
. Since is symmetric we can
equally well write .
Fig 4.13:
Neighbourhood
function
for square
pixels
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Fourier Transform of theNeighbourhood Function
To gain a deeper understand of the sampling model we need itsFourier Transform :
(4.3)
Fig 4.14: , the
Fourier Transform of
the neighbourhood
function (notice
the negative values)
Because is real and even its Fourier Transform is too
→ the neighbourhood filter will not change the phase but only
their amplitude.
Since becomes negative for some some
frequencies undergo a complete phase reversal (shift over -
see next slide).
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Filtering with theNeighbourhood Function
As the Fourier Transform of the neighbourhood function
has negative amplitudes for some frequencies, complete phasereversals can be observed for higher frequencies:
Fig 4.15: Star pattern that increases its
frequency towards the centre
Fig 4.16: Complete phase reversals
occur at higher frequencies
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Sampling of a Continuous1D Function
As the second step after filtering with the neighbourhoodfunction we have to select values only at discrete pixel
positions. This is modelled as a multiplication with a 1D or 2Dpattern (train) of Dirac impulses at these discrete positions.
Consider the real neighbourhood function
filtered
Suppose its Fourier Transform is band
limited and thus vanishes outside the
interval
To obtain a sampled version of simply
involves multiplying it by a sampling
function , which consists of a train of
Dirac impulses apart
Its Fourier Transform is also a train of
Dirac impulses with a distance inversely
proportional to , namely apart
By the convolution theorem multiplication
in the image domain is equivalent to
convolution in the frequency domain
The transform is periodic, with period
, and the individual repetitions of
can overlap → aliasing!!!
The centre of the overlap occurs at
To avoid these problems, the sampling
interval has to be selected so that
, or
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(4.4)
Once the individual are separated a
multiplication with the window function
yields a completely isolated
The inverse Fourier Transform then yields
the original continuous function
Complete recovery of a band-limited
function that satisfies the above
inequality is known as the Whittaker-
Shannon Sampling Theorem
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Sampling of a Continuous1D Function (2)
All the frequency domain information of a band-limited
function is contained in the interval
If the Whittaker-Shannon Sampling Theorem or Nyquist
Sampling Theorem
(4.5)
is not satisfied, the transform in this interval is corrupted by
contributions from adjacent periods. This phenomenon is
frequently referred to as aliasing.
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A Sampling Model
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Sampling of a Discrete 1DFunction
The preceding example applies to functions of unlimited durationin the spatial domain. For practical examples only functionssampled over a finite region are of interest. This situation isshown graphically below
Consider a real neighourhood-
function-filtered function
Suppose its Fourier Transform is
band limited and thus vanishes
outside the interval
The sampling function fulfils
the Whittaker-Shannon Theorem
As the Whittaker-Shannon
Sampling Theorem (aka Nyquist
Criterion) is fulfilled, the are
well separated and no aliasing is
present
The Sampling Window
and its Fourier Transform
has Frequency components
that extend to infinity
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Because has frequency
components that extend to infinity,
the convolution of these functions
introduces a distortion in the
frequency domain representation
of a function that has been
sampled and limited to a finite
region by
These considerations lead to the important conclusion that
No function of finite duration can be band limited
Conversely,
A function that is band limited must extend from to in the spatial domain
These important practical results establish fundamentallimitations to the treatment of digital functions.
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
An Alternative Reasoningfor Periodicity in the DFT
So far, all the results in the Fourier domain have been of acontinuous nature. To obtain a discrete Fourier Transform simplyrequires to sample it with a train of Dirac impulses that are units apart.
Consider the signals and
as the results of the operation
sequence on the previous slide
To sample we multiply it with
a train of Dirac impulses that
are units apart
The inverse Fourier Transform of
yields , an other train of
Dirac impulses with inversely
spaced pulses
The graph shows the
result of sampling
As the equivalent of a
multiplication in the Fourier
domain is a convolution in the
spatial domain, it yields a periodic
function, with period
If samples of and are taken and the spacings
between samples are selected so that a period in each domain iscovered by uniformly spaced samples, then in thespatial domain and in the frequency domain. The
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latter equation is based on the periodic property of the FourierTransform of a sampled function, with period , as shown
earlier. The Sampling Theorem for discrete signals can thus beformulated as
(4.6)
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Sampling ofTwo-Dimensional Functions(Images)
The preceding sampling concepts (after some
modifications in notation) are directly applicable to
2D functions
The sampling process for these functions can be
formulated making use of a 2D train of Dirac
impulses
For a function , where and are
continuous, a sampled function is obtained by
forming the product . The equivalent
operation in the Frequency domain is the
convolution of and , where is
a train of Dirac impulses with separation and
. If is band limited it might look like
shown on the right
Let and represent the widths in and
direction that completely enclose the band-limited
function
No aliasing is present if and
The 2D sampling theorem can thus be formulated as
(4.7)
and
(4.8)
A periodicity analysis similar to the discrete 1D case
shown previously would yield a 2D Sampling Theorem
of
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(4.9)
and
(4.10)
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Summary SamplingTheorem
The One-Dimensional Sampling Theorem states that
If the Fourier Transform of a function is zero for all
Frequencies beyond , i.e. the Fourier Transform isband-limited, then the continuous function can be
completely reconstructed as long as .
The Two-Dimensional Sampling Theorem states that
If the Fourier Transform of a function is zero for
all Frequencies beyond , i.e. the Fourier Transform isband-limited, then the continuous function can be
completely reconstructed as long as and
.
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Aliasing Example 1
The input image containsregions with clearly differentfrequency content. Going fromthe centre to boundary, thefrequency increases. It can beseen that once the Nyquist rateis higher than the actualsampling, aliasing occurs.
(a) the 256x256 sample pattern
(b) the sinc function for a sampling
rate of (grey is zero,
brighter is positive, and darker is
negative)
(c) the original pattern is sampled
with
(d) the reconstructed pattern. In
regions where the Nyquist rate is
higher strong aliasing artefacts are
present
(a) Original pattern (b) Sinc size 5
(c) Sampled
pattern
(d) Reconstruction
Fig 4.17 Aliasing example
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Aliasing Example 2
This example shows thereconstruction of the rollingpattern for a sampling rate (
) that is well above theNyquist rate.
(a) the 128x128 sample rolling
pattern
(b) the sinc function for a sampling
rate of . The grey
background is zero, brighter is
positive, and darker is negative
(c) the original pattern is sampled
with
(d) the reconstructed rolling
pattern. The reconstruction is
perfect (except for boundary
effects)
(a) Original pattern (b) Sinc of size 5
(c) Sampled
pattern
(d) Reconstruction
Fig 4.18 Aliasing example 2
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Aliasing Example 3
In this example the samplingrate ( ) is below theNyquist rate.
(a) the 128x128 sample rolling
pattern
(b) the sinc function for a sampling
rate of . The grey
background is zero, brighter is
positive, and darker is negative
(c) the original pattern is sampled
with
(d) the reconstructed rolling
pattern is no longer valid. It is
interesting that not only the
frequency changed, but even the
orientation of the pattern.
(a) Original pattern (b) Sinc size 15
(c) Sampled
pattern
(d) Reconstruction
Fig 4.19 Aliasing example 3
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Remark on the DiscreteFourier Transform
As already noted,
Sampling in one domain impliesperiodicity in the other
If both domains are discretised and thusshould both the original image and itsFourier Transform be interpreted as periodsof periodic signals.
The discrete Fourier Transform istherefore not the Fourier Transformof the image as such, but rather ofthe periodic signal created byrepeating the image data bothhorizontally and vertically
Periodically repeated image
Flipped images
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Linear, Shift-InvariantOperators
Convolution theory is not only important in image acquisition butplays an important role at several other occasions. To fullybenefit from the convolution theorem a little bit morebackground theory is required. In fact, it will be explained that
Every linear, shift-invariant operation can be expressedas a convolution and vice versa.
Definition:
Consider a 2D system thatproduces output and
when given inputs
and respectively.
The system is called linear if
the output
is
produced when the input is
The system is called shift-invariant if
the output is
produced when the input is
Fig 4.20: Linear system
Fig 4.21: Shift-invariant system
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Linear, Shift-InvariantOperators (2)
Suppose a process, e.g. camera with lens system, can bemodeled as a linear, shift-invariant operation . As we haveseen, any image can be considered as a sum of point sources(Dirac impulses).
The output of for a singlepoint source is called Pointspread function (PSF) of which we denote as .
Fig 4.22: Point spread function
Knowledge of the PSF can be used to determine theoutput for
Assuming shift-invariance implies that the output to such a Diracpulse is always the same irrespective of its position. In terms ofimage acquisition, we assume that the light comming from apoint source will be distributed over the image following a fixedspatial pattern. The projection of such a point will thereforealways be blurred in the same way independent of its position inthe image.
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Liner, Shift-InvariantOperators (3)
Let us consider an input picture . It can be written as a
linear combination of point sources
(4.11)
For the linear and shift-invariant operation we obtain
(4.12)
The linear, shift-invariant operation has led to a convolutionoperation. This is true in general and every LSI operation can bewritten as a convolution and vice versa.
A simple variable substitution shows that the above expressioncan also be written as
(4.13)
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so that
(4.14)
i.e. convolution is commutative (convolution is also associative).
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Liner, Shift-InvariantOperators (4)
Suppose we would like to process an image by first convolvingwith , followed by a convolution with , thus
(4.15)
the global operation can therefore be interpreted as applying asingle (generally larger) filter .
The reverse analysis might be useful too, i.e. if a filter(separable) can be decomposed as a convolution of two simplerfilter efficiency can be increased by applying the smaller filterssequentially.
Example
The Figures on the right show a 2DGauss kernel and a 1D Gausskernel of size and respectively.
It can be easily shown numericallythat the kernel can be separatedinto two 1D kernels and
thus
(4.16)
Convolving the image sequentiallywith the 1D kernels is computationallymore efficient than convolving theentire image with the 2D kernel.
Fig 4.23: 2D Gauss kernel
Fig 4.24: 1D Gauss kernel
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Quantisation
March 1st/7th/8th, 2016Biomedical Image Analysis
(39)Quantisation
The subjective image quality depends on (1)the number of samples and (2) thenumber of grey-values . Figure 4.26 showsthis relation.
The key point of interest is, thatisopreference curves tend to become morevertical as the detail in the image increases→ images with large amount of detail requirefewer grey levels.
Fig 4.25: (a) Low detail face image, (b) Cameraman
with mid detail, and (c) crowd with high detail content
Fig 4.26: Isopreference
curves for the three
sample images
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Quantisation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Lloyd-Max Quantisation
In the Introduction of this Lecture wehave already shortly explained theeffect of using more or lessquantisation levels. This part isconcerned with the optimal placementof these quantisation levels
Suppose we create intervals in therange of possible intensities, definedby the decision levels
.
We therefore assign to all intensitiesin the interval the new grey
level . The mean-square
quantisation error between the inputand output of the quantiser for a givenchoice of boundaries and outputlevels is thus
Fig 4.27: Principle of the
Lloyd-Max quantiser
(4.17)
where is the probability density function for the input
sample value.
For a given number of output levels, we would like todetermine the output levels and interval boundaries that
minimise . The partial derivatives of with respect to and
must thus vanish:
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(4.18)
For not equal to zero we obtain the Lloyd-Max Quantiser
Equations
(4.19)
We see that
the decision levels are located halfway between the output
levels
whilst each is the centroid of the portion of between
and
If the sample values occur equally frequently, the optimalquantised will spread the values and uniformly, and the
Lloyd-Max Quantiser Equations can be simplified to
(4.20)
As can be seen from the following examples, improvement canbe disputed. The main problem is, that Lloyd-Max quantisationdoes not take local image structure or interpretation intoaccount.
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Quantisation
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Quantisation Example
Original image with 256
grey values
32 equally spaced grey
values
32 Lloyd-max quantised
grey values
Fig 4.28: Quantisation example with 32 grey values
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Quantisation Example (2)
Original image with 256
grey values
16 equally spaced grey
values
16 Lloyd-max quantised
grey values
Fig 4.29: Quantisation example with 16 grey values
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Prof. Dr. Philippe Cattin: Image Sampling & Quantisation
Quantisation Example (3)
Original image with 256
grey values
8 equally spaced grey
values
8 Lloyd-max quantised
grey values
Fig 4.30: Quantisation example with 8 grey values
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