Image Sampling
and Quantisation
Introduction to Signal
and Image Processing
Prof. Dr. Philippe Cattin
MIAC, University of Basel
March 29th, 2016
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Contents
Contents
1 Motivation
Introduction and Motivation
3
Sampling Example
4
Quantisation Example
5
2 Sampling
2.1 Tessellation
Tessellation
8
Tessellation Examples by M.C. Escher (1)
9
Tessellation Examples by M.C. Escher (2)
10
Tessellation Basics
11
Tessellation Claim
12
How Many Tessellations Exist with Regular
Polygons?
13
Combinatorial Analysis
14
All Semi-Regular Tessellations
15
All Regular Tessellations
16
Tessellation Rules
17
Advantages of Square Tessellation
18
2.2 A Sampling Model
A Sampling Model
20
The Neighbourhood Function
21
Fourier Transform of the Neighbourhood
Function
22
Filtering with the Neighbourhood Function
23
Sampling of a Continuous 1D Function
24
Sampling of a Continuous 1D Function (2)
25
Sampling of a Discrete 1D Function
26
An Alternative Reasoning for Periodicity in the 27
DFT
Introduction to Signal
and Image
Sampling
of Processing
Two-Dimensional
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Functions
28
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(Images)
Summary Sampling Theorem
29
Aliasing Example 1
30
Aliasing Example 2
31
Aliasing Example 3
32
Remark on the Discrete Fourier Transform
33
Linear, Shift-Invariant Operators
34
Linear, Shift-Invariant Operators (2)
35
Liner, Shift-Invariant Operators (3)
36
Liner, Shift-Invariant Operators (4)
37
3 Quantisation
Quantisation
39
Lloyd-Max Quantisation
40
Quantisation Example
41
Quantisation Example (2)
42
Quantisation Example (3)
43
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Motivation
Introduction and Motivation (3)
In order for computers to process an image, this image
has to be described as a series of numbers, each of finite
precision
This calls for two kinds of discretisation:
Sampling, and
Quantisation
By sampling is meant that the brightness information is only
stored at a discrete number of locations. Quantisation indicates
the discretisation of the brightness levels at these positions.
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Motivation
Sampling Example
(4)
Sampling is the process of measuring the brightness
information only at a discrete number of locations
Fig 4.1: Hight profile of Switzerland
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Fig 4.2: Sampled hight profile
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Motivation
Quantisation Example
(5)
Quantisation is the process of discretising the brightness
at a finite number of positions
Height map with
with
grey values
grey values
with
grey values
with
grey values
Fig 4.3:
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Sampling
Tessellation
Tessellation
(8)
Definition
Tessellations are patterns that cover a plane with
repeating figures so there is no overlapping or empty
spaces
Sampling is best performed following a regular tessellation of
the image:
1. Brightness is integrated over cells of same size
2. Cells should cover the whole image
These cells are usually referred to as picture elements or pixels.
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Tessellation Examples by
M.C. Escher (1)
Tessellation
(9)
Fig 4.4: Sample Escher images
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Tessellation Examples by
M.C. Escher (2)
Tessellation
(10)
Fig 4.5: Sample Escher images
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Tessellation Basics
Tessellation
(11)
Three types of tessellations with polygons exist
1. regular tessellations (using the same regular polygon)
2. semi-regular tessellations (using various regular
polygons)
3. hyperbolic tessellations (they use non-regular polygons)
They are formed by translating, rotating, and reflecting
polygons
Fig 4.6: regular
Fig 4.7: semi-regular
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Fig 4.8: hyperbolic
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Tessellation
Tessellation Claim
(12)
There exist only 11 possible tessellations with regular
polygons that can cover the entire image
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Tessellation
How Many Tessellations
(13)
Exist with Regular Polygons?
Observation 1:
Since the regular polygons in a
tessellation must fill the plane at
each vertex, the interior angle must
be an exact divisor of
Observation 2:
A regular
of
-gon has an internal angle
degrees
Fig 4.9:
Of the regular polygons, only triangles (
), squares (
),
pentagons (
), hexagons (
), octagons (
), decagons (
) and dodecagons (
) can be used for tiling around a
common vertex - again because of the angle value
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Combinatorial Analysis
Tessellation
(14)
A combinatorial analysis of these base polygons produces the
following 14 solutions
Regular
Tessellations
Semi-regular
Tessellations
Semi-regular
Tessellations that
can not be extended
infinitely
4.4.4
6.6.6
3.3.4
3.6.3
3.4.6
3.3.3
4.8.8
3.12
4.6.1
3.4.4
5.5.1
Fig 4.10: Tessellations
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All Semi-Regular
Tessellations
Tessellation
(15)
Eight semi-regular tessellations exist
Snub hexagonal
Trihexagonal
Prismatic trisquare
Truncated
Small
Truncated square
hexagonal
rhombitrihexagonal
Fig 4.11:
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Snub square
Great
rhombitrihexagonal
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Tessellation
All Regular Tessellations
(16)
But only three regular tessellations exist
Triangular tiling
Square tiling
Fig 4.12:
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Hexagonal tiling
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Tessellation Rules
Tessellation
(17)
For practical applications in computer vision the tessellation has
to 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
Regular
Tessellations
4.4.4
6.6.6
Although the hexagonal tessellation offers some substantial
advantages (e.g. no ambiguities in defining connectedness,
closer spatial organisation as found in mammalian retinas), the
square tessellation is the most commonly used.
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Advantages of Square
Tessellation
Tessellation
(18)
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
A Sampling Model
(20)
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:
1. Integrate brightness over regions of the pixel size,
2. Read out values only at the pixel positions.
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A Sampling Model
The Neighbourhood
Function
(21)
First a neighbourhood function
has to be
defined, that is 1 inside a region with the shape of a
pixel/cell and 0 outside.
Integrating the incoming intensity
region then yields
over such a
Fig 4.13:
Neighbourhood
function
for square
pixels
(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
.
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A Sampling Model
Fourier Transform of the
Neighbourhood Function
(22)
To gain a deeper understand of the sampling model we need its
Fourier Transform
:
Fig 4.14:
, the
Fourier Transform of
the neighbourhood
(4.3)
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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Filtering with the
Neighbourhood Function
A Sampling Model
(23)
As the Fourier Transform of the neighbourhood function
has negative amplitudes for some frequencies, complete phase
reversals 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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A Sampling Model
Sampling of a Continuous
1D Function
(24)
As the second step after filtering with the neighbourhood
function
we have to select values only at discrete pixel
positions. This is modelled as a multiplication with a 1D or 2D
pattern (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
(4.4)
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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 WhittakerShannon Sampling Theorem
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A Sampling Model
Sampling of a Continuous
1D Function (2)
(25)
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
Sampling of a Discrete 1D
Function
(26)
The preceding example applies to functions of unlimited duration
in the spatial domain. For practical examples only functions
sampled over a finite region are of interest. This situation is
shown graphically below
Consider a real neighourhoodfunction-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
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
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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
in the spatial domain
to
These important practical results establish fundamental
limitations to the treatment of digital functions.
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A Sampling Model
An Alternative Reasoning
for Periodicity in the DFT
(27)
So far, all the results in the Fourier domain have been of a
continuous nature. To obtain a discrete Fourier Transform simply
requires 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 is
covered by
uniformly spaced samples, then
in the
spatial domain and
in the frequency domain. The
latter equation is based on the periodic property of the Fourier
Transform of a sampled function, with period
, as shown
earlier. The Sampling Theorem for discrete signals can thus be
formulated as
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(4.6)
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A Sampling Model
Sampling of
(28)
Two-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
(4.9)
and
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(4.10)
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A Sampling Model
Summary Sampling
Theorem
(29)
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 is
band-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 is
band-limited, then the continuous function
can be
completely reconstructed as long as
and
.
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A Sampling Model
Aliasing Example 1
(30)
The input image contains
regions with clearly different
frequency content. Going from
the centre to boundary, the
frequency increases. It can be
seen that once the Nyquist rate
is higher than the actual
(a) Original pattern
sampling, 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
(c) Sampled
pattern
(b) Sinc size 5
(d) Reconstruction
Fig 4.17 Aliasing example
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A Sampling Model
Aliasing Example 2
(31)
This example shows the
reconstruction of the rolling
pattern for a sampling rate (
) that is well above the
Nyquist rate.
(a) the 128x128 sample rolling
pattern
(a) Original pattern (b) Sinc of size 5
(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
(d) Reconstruction
(c) Sampled
effects)
pattern
Fig 4.18 Aliasing example 2
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Aliasing Example 3
A Sampling Model
(32)
In this example the sampling
rate (
) is below the
Nyquist rate.
(a) the 128x128 sample rolling
pattern
(b) the sinc function for a sampling
rate of
. The grey
(a) Original pattern (b) Sinc size 15
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.
(d) Reconstruction
(c) Sampled
pattern
Fig 4.19 Aliasing example 3
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A Sampling Model
Remark on the Discrete
Fourier Transform
(33)
As already noted,
Sampling in one domain implies
periodicity in the other
If both domains are discretised and thus
should both the original image and its
Fourier Transform be interpreted as periods
of periodic signals.
The discrete Fourier Transform is
therefore not the Fourier Transform
of the image as such, but rather of
the periodic signal created by
repeating the image data both
horizontally and vertically
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Periodically repeated image
Flipped images
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A Sampling Model
Linear, Shift-Invariant
Operators
(34)
Convolution theory is not only important in image acquisition but
plays an important role at several other occasions. To fully
benefit from the convolution theorem a little bit more
background theory is required. In fact, it will be explained that
Every linear, shift-invariant operation can be expressed
as a convolution and vice versa.
Definition:
Consider a 2D system
that
produces 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 shiftinvariant if
the output
is
produced when the input is
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Fig 4.20: Linear system
Fig 4.21: Shift-invariant system
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A Sampling Model
Linear, Shift-Invariant
Operators (2)
(35)
Suppose a process, e.g. camera with lens system, can be
modeled as a linear, shift-invariant operation . As we have
seen, any image can be considered as a sum of point sources
(Dirac impulses).
The output of
for a single
point source is called Point
spread function (PSF) of
which we denote as
.
Fig 4.22: Point spread function
Knowledge of the PSF can be used to determine the
output for
Assuming shift-invariance implies that the output to such a Dirac
pulse is always the same irrespective of its position. In terms of
image acquisition, we assume that the light comming from a
point source will be distributed over the image following a fixed
spatial pattern. The projection of such a point will therefore
always be blurred in the same way independent of its position in
the image.
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A Sampling Model
Liner, Shift-Invariant
Operators (3)
Let us consider an input picture
linear combination of point sources
(36)
. It can be written as a
(4.11)
For the linear and shift-invariant operation
we obtain
(4.12)
The linear, shift-invariant operation
has led to a convolution
operation. This is true in general and every LSI operation can be
written as a convolution and vice versa.
A simple variable substitution shows that the above expression
can also be written as
(4.13)
so that
(4.14)
i.e. convolution is commutative (convolution is also associative).
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A Sampling Model
Liner, Shift-Invariant
Operators (4)
(37)
Suppose we would like to process an image by first convolving
with , followed by a convolution with , thus
(4.15)
the global operation can therefore be interpreted as applying a
single (generally larger) filter
.
The reverse analysis might be useful too, i.e. if a filter
(separable) can be decomposed as a convolution of two simpler
filter efficiency can be increased by applying the smaller filters
sequentially.
Example
The Figures on the right show a 2D
Gauss kernel
and a 1D Gauss
kernel
of size
and
respectively.
Fig 4.23: 2D Gauss kernel
It can be easily shown numerically
that the kernel
can be separated
into two 1D kernels
and
thus
(4.16)
Fig 4.24: 1D Gauss kernel
Convolving the image sequentially
with the 1D kernels is computationally
more efficient than convolving the
entire image with the 2D kernel.
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Quantisation
Quantisation
(39)
The subjective image quality depends on (1)
the number of samples
and (2) the
number of grey-values . Figure 4.26 shows
this relation.
The key point of interest is, that
isopreference curves tend to become more
vertical as the detail in the image increases
→ images with large amount of detail require
fewer grey levels.
Fig 4.26: Isopreference
curves for the three
sample images
Fig 4.25: (a) Low detail face image, (b) Cameraman
with mid detail, and (c) crowd with high detail content
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Quantisation
Lloyd-Max Quantisation
(40)
In the Introduction of this Lecture we
have already shortly explained the
effect of using more or less
quantisation levels. This part is
concerned with the optimal placement
of these quantisation levels
Suppose we create
intervals in the
range of possible intensities, defined
by the decision levels
.
Fig 4.27: Principle of the
Lloyd-Max quantiser
We therefore assign to all intensities
in the interval
the new grey
level . The mean-square
quantisation error between the input
and output of the quantiser for a given
choice of boundaries and output
levels is thus
(4.17)
where
is the probability density function for the input
sample value.
For a given number
of output levels, we would like to
determine the output levels and interval boundaries that
minimise . The partial derivatives of with respect to and
must thus vanish:
(4.18)
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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 optimal
quantised 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 can be
disputed. The main problem is, that Lloyd-Max quantisation does
not take local image structure or interpretation into account.
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Quantisation
Quantisation Example
Original image with 256
grey values
32 equally spaced grey
values
(41)
32 Lloyd-max quantised
grey values
Fig 4.28: Quantisation example with 32 grey values
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Quantisation
Quantisation Example (2)
Original image with 256
grey values
16 equally spaced grey
values
(42)
16 Lloyd-max quantised
grey values
Fig 4.29: Quantisation example with 16 grey values
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Quantisation
Quantisation Example (3)
Original image with 256
grey values
8 equally spaced grey
values
(43)
8 Lloyd-max quantised
grey values
Fig 4.30: Quantisation example with 8 grey values
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