Scale-space for discrete signals

TLDR: The proper way to apply the scale-space theory to discrete signals and discrete images is by discretization of the diffusion equation, not the convolution integral.

Abstract: A basic and extensive treatment of discrete aspects of the scale-space theory is presented. A genuinely discrete scale-space theory is developed and its connection to the continuous scale-space theory is explained. Special attention is given to discretization effects, which occur when results from the continuous scale-space theory are to be implemented computationally. The 1D problem is solved completely in an axiomatic manner. For the 2D problem, the author discusses how the 2D discrete scale space should be constructed. The main results are as follows: the proper way to apply the scale-space theory to discrete signals and discrete images is by discretization of the diffusion equation, not the convolution integral; the discrete scale space obtained in this way can be described by convolution with the kernel, which is the discrete analog of the Gaussian kernel, a scale-space implementation based on the sampled Gaussian kernel might lead to undesirable effects and computational problems, especially at fine levels of scale; the 1D discrete smoothing transformations can be characterized exactly and a complete catalogue is given; all finite support 1D discrete smoothing transformations arise from repeated averaging over two adjacent elements (the limit case of such an averaging process is described); and the symmetric 1D discrete smoothing kernels are nonnegative and unimodal, in both the spatial and the frequency domain. >

Full text

Scale-Space for Discrete Signals
Tony Lindeb erg
Computer Vision and Associative Pattern Pro cessing Lab oratory (CVAP)
Royal Institute of Technology
S-100 44 Sto ckholm, Sweden
Abstract
We address the formulation of a scale-space theory for
discrete
signals. In one
dimension it is p ossible to characterize the smo othing transformations completely
and an exhaustive treatment is given, answering the following two main questions:
1. Which linear transformations remove structure in the sense that the number of
lo cal extrema (or zero-crossings) in the output signal do es not exceed the number
of lo cal extrema (or zero-crossings) in the original signal? 2. How should one create
a multi-resolution family of representations with the property that a signal at a
coarser level of scale never contains more structure than a signal at a ner level of
scale?
We prop ose that there is only one reasonable way to dene a scale-space for
1D discrete signals comprising a
continuous
scale parameter, namely by (discrete)
convolution with the family of kernels
T
(
n
;
t
)=
e

t
I
n
(
t
), where
I
n
are the mo died
Bessel functions of integer order. Similar arguments applied in the continuous case
uniquely
lead to the Gaussian kernel.
Some obvious discretizations of the continuous scale-space theory are discussed
in view of the results presented. We show that the kernel
T
(
n
;
t
) arises naturally
in the solution of a discretized version of the diusion equation. The commonly
adapted technique with a sampled Gaussian can lead to undesirable eects since
scale-space violations might o ccur in the corresp onding representation. The result
exemplies the fact that prop erties derived in the continuous case might b e violated
after discretization.
A two-dimensional theory, showing how the scale-space should be constructed
for images, is given based on the requirement that lo cal extrema must not be en-
hanced, when the scale parameter is increased continuously. In the separable case
the resulting scale-space representation can b e calculated by separated convolution
with the kernel
T
(
n
;
t
).
The presented discrete theory has computational advantages compared to a scale-
space implementation based on the sampled Gaussian, for instance concerning the
Laplacian of the Gaussian. The main reason is that the discrete nature of the
implementation has b een taken into account already in the theoretical formulation
of the scale-space representation.
1 Intro duction
It is well-known that ob jects in the world and details in an image exist only over a
limited range of resolution. A classical example is the concept of a branch of a tree which
1

makes sense only on the scale say from a few centimeters to at most a few meters. It is
meaningless to discuss the tree concept at the nanometer or the kilometer level. At those
levels of scale it is more relevant to talk about the molecules, which form the leaves of
the tree, or the forest, in which the tree grows. If one aims at describing the structure of
an image, a multiresolution representation is of crucial imp ortance. Then a mechanism,
which systematically removes ner details or high-frequency information from an image,
is required. This smo othing must be available at any level of scale.
A metho d prop osed by Witkin [23] and Ko enderink, van Do orn [11] is to embed the
original image in a one-parameter family of derived images, the scale-space, where the
parameter
t
describ es the current level of scale resolution. Let us briey develop the
pro cedure as it is formulated for one-dimensional continuous signals: Given a signal
f
:
R
!
R
a function
1
L
:
R

R
+
!
R
is dened by
L
(
x
; 0) =
f
(
x
) and convolution with
the Gaussian kernel
g
:
R

R
+
nf
0
g!
R
L
(
x
;
t
)=
Z
1

=
1
1
p
2
t
e


2
=
2
t
f
(
x


)
d
(1)
if
t>
0. Equivalently the family can b e regarded as dened by the diusion equation
@L
@t
=
1
2
@
2
L
@x
2
(2)
with initial condition
L
(
x
; 0) =
f
(
x
). This family possesses some attractive prop erties.

As the scale parameter
t
is increased additional local extrema or additional zero
crossings cannot app ear.

Causality in the sense that
L
(
x
;
t
2
) depends exclusively on
L
(
x
;
t
1
) if
t
2
> t
1
(
t
1
;t
2

0).

The blurring is shift invariant and do es not dep end up on the image values.
It has b een shown by Babaud et.al. [3] that the Gaussian function is the only kernel
in a broad class of functions which satises adequate scale-space conditions.
The scale-space theory has b een developed and well-established for continuous signals
and images. However, it do es not tell us at all ab out how the implementation should be
p erformed computationally for real-life i.e. discrete signals and images. In principle, we
feel that there are two approaches p ossible.

Apply the results obtained from the continuous scale-space theory by discretizing
the o ccurring equations. For instance the convolution integral (1) can be approxi-
mated by a sum using customary numerical metho ds. Or, the diusion equation (2)
can be discretized in space with the ordinary ve-point Laplace operator forming
a set of coupled ordinary dierential equations, which can be further discretized in
scale. If the numerical metho ds are chosen with care we will certainly get reasonable
approximations to the continuous numerical values. But we are not guaranteed that
the original scale-space conditions, however formulated, will b e preserved.

Dene a genuine discrete theory by p ostulating suitable axioms.
1
R
+
denotes the set of real non-negativenumbers.
2

The goal in this paper is to develop the second item and to address the formulation of
a scale-space theory for discrete images. We will start with a one-dimensional signal
analysis. In this case it is p ossible to characterize exactly which kernels can be regarded
as smo othing kernels and a complete and exhaustive treatment will b e given. One among
many questions which are answered is the following: If one p erforms rep eated averaging,
do es one then get scale-space b ehaviour? We will also present a family of kernels, which
are the discrete analog of the Gaussian family of kernels. The set of arguments, which in
the discrete case uniquely leads to this family of kernels do in the continuous case uniquely
lead to the Gaussian family of kernels.
The structure of the two-dimensional problem is more complex, since it is diÆcult
to formulate what should be meant by preservation of structure in this case. However,
arguing that lo cal extrema must not be enhanced when the scale parameter is increased
continuously, we will give an answer to how the scale-space for two-dimensional discrete
images should b e calculated. In the separable case it reduces to separated convolution with
the presented one-dimensional discrete analog of the Gaussian kernel. The representation
obtained in this way has computational advantages compared to the commonly adapted
approach, where the scale-space is based on dierent versions of the sampled Gaussian
kernel. One of many spin-o pro ducts which come up naturally is a well-conditioned
and eÆcient metho d to calculate (a discrete analog of ) the Laplacian of the Gaussian.
It is well-known that the implementation of the Laplacian of the Gaussian has lead to
computational problems [8].
The theory develop ed in this paper do es also have the attractive prop erty that it is
linked to the continuous theory through a discretized version of the diusion equation.
This means that continuous results may be transferred to the discrete implementation
provided that the discretization is done correctly. However, the importantpoint with the
scale-space concept outlined here is that the prop erties we want from a scale-space hold
not only in the ideal theory but also in the discretization
2
, since the discrete nature of
the problem has b een taken into account already in the theoretical formulation of the
scale-space representation. Therefore, we believe that the suggested way to implement
the scale-space theory really describes the prop er way to do it.
The presentation is organized as follows: In Section 2 we dene what we mean by a
scale-space representation and a one-dimensional discrete scale-space kernel. Then in a
straightforward and constructive manner Section 3 illustrates some qualitative prop erties
that must be p ossessed by scale-space kernels. A complete characterization as well as an
explicit expression for the generating function of all discrete scale-space kernels are given in
Section 4. Section 5 develops the concept of a discrete scale-space with a
continuous
scale
parameter. The formulation is equivalent to the previous scale-space formulation, which
in the continuous case leads to the Gaussian kernel. The numerical implementation of
this scale-space is treated in Section 6. Section 7 discusses discrete scale-space prop erties
of some obvious discretizations of the convolution integral and the diusion equation.
Section 8 describes some problems which o ccur due to the more complicated top ology
in two dimensions. In Section 9 we develop the scale-space for two-dimensional discrete
2
In a practical implementation we are of course faced with rounding and truncation errors due to nite
precision. But the idea with this approach is that we hop e to improve our algorithms by including at least
the discretization eects already in the theory. In ordinary numerical analysis for simulation of physical
phenomena it is almost always p ossible reduce these eects by increasing the density of mesh points, if
the current grid is not ne enough to give a prescrib ed accuracy in the result. However, in computer
vision we are often locked to some xed maximal resolution, beyond which additional image data are not
available.
3

Figure 1: A scale-space is an ordered set of derived signals/images intended to represent
the original signal/image at various levels of scale.
images. Here we also compare the discrete scale-space representation with the commonly
used approach, where the scale-space implementation is based on various versions of the
sampled Gaussian kernel. Finally, Section 10 gives a brief summary of the main results.
The results presented should have implications for image analysis as well as other
disciplines of digital signal processing.
2 Scale-Space Axioms
With a scale-space we mean a family of derived signals meant to represent the original
signal at various levels of scale. Each member of the family should be asso ciated with a
value of a scale parameter intended to somehow describ e the current level of scale. This
scale parameter, here denoted by
t
,may b e either discrete (
t
2
Z
+
)orcontinuous (
t
2
R
+
)
and we obtain two dierenttyp es of discrete scale-spaces - discrete signals with a discrete
scale parameter and discrete signals with a continuous scale parameter. However, in b oth
cases we start from the following basic assumptions:

All representations should b e generated by (linear) convolution of the original image
with a shift-invariantkernel.

An increasing value of the scale parameter
t
should corresp ond to coarser levels of
scale and signals with less structure. Particularly,
t
= 0 should represent to the
original signal.

All signals should be real-valued functions :
Z
!
R
dened on the same innite
grid; in other words no pyramid representations will b e used.
The essential requirement is that a signal at a coarser level of scale should contain less
structure than a signal at a ner level of scale. If one regards the numb er of lo cal extrema
as one measure of smo othness it is thus necessary that the number of lo cal extrema in space
do es not increase as we go from a ner to a coarser level of scale. It can be shown that
the family of functions generated by convolution with the Gaussian kernel p ossesses this
prop erty in the continuous case. We state it as the basic axiom for our one-dimensional
analysis and dene:
4

Figure 2: (a) Input signal. (b) Convolved with (
1
3
;
1
3
;
1
3
). (c) Convolved with (
1
2
;
1
2
). (d)
Convolved with (
1
4
;
1
2
;
1
4
).
Denition 1
A one-dimensional discrete kernel
K
:
Z
!
R
is denoted a scale-space
kernel if for al l signals
f
in
:
Z
!
R
the number of local extrema in the convolved signal
f
out
=
K

f
in
does not exceed the number of local extrema in the original signal.
An imp ortant observation to note is that this denition equivalently can be expressed in
terms of zero-crossings just by replacing the string \lo cal extrema" with \zero-crossings".
The result follows from the facts that a lo cal extremum in a discrete function
f
is equivalent
to a zero-crossing in its rst dierence 
f
, dened by (
f
)(
x
) =
f
(
x
+1)

f
(
x
), and
that the dierence op erator commutes with the convolution op erator.
However, the stated denition has further consequences. It means that the number of
lo cal extrema (zero-crossings) in any
n
:th order dierence of the convolved image cannot
b e larger than the number of lo cal extrema (zero-crossings) in the
n
:th order dierence of
the original image. Actually,theresult can be generalized to arbitrary linear op erators.
Prop osition 1
Let
K
:
Z
!
R
be a discrete scale-space kernel and
L
a linear operator
(from the spaceofreal-valued discrete functions to itself ), which commutes with
K
. Then
for any
f
:
Z
!
R
(such that the involved quantities exist) the number of local extrema in
L
(
K

f
)
cannot exceed the number of local extrema in
L
(
f
)
.
Pro of:
Let
g
=
L
(
f
). As
K
is a scale-space kernel the number of lo cal extrema in
K

g
cannot be larger than the number of lo cal extrema in
g
. Since,
K
and
L
commute
K

g
=
K
L
(
f
)=
L
(
K

f
)and the result follows.
}
This shows that not only the function, but also all its \derivatives" will become
smo other. Accordingly, convolution with a discrete scale-space kernel can really be re-
garded as a smo othing op eration.
To realize that the number of lo cal extrema or zero-crossings can increase even in a
rather uncomplicated situation consider the input signal
f
in
(
x
)=
8
>
<
>
:

3 if
n
=0
2 if
n
=

1
0 otherwise
(3)
an convolve it with the kernels (
1
3
;
1
3
;
1
3
), (
1
2
;
1
2
) and (
1
4
;
1
2
;
1
4
). The results are shown
in Fig. 2 (b), (c) and (d) resp ectively. As we see, b oth the number of lo cal extrema and
the number of zero-crossings have increased for the rst kernel, but not for the two latter
ones. Thus, an op erator which naively can be apprehended as a smo othing op erator,
might actually give a less smo oth result. Further, it can really matter if one averages over
three instead of two points and how the averaging is p erformed.
5

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Scale-space for discrete signals" ?

In one dimension it is possible to characterize the smoothing transformations completely and an exhaustive treatment is given, answering the following two main questions: 1. Which linear transformations remove structure in the sense that the number of local extrema ( or zero-crossings ) in the output signal does not exceed the number of local extrema ( or zero-crossings ) in the original signal ? The authors propose that there is only one reasonable way to de ne a scale-space for 1D discrete signals comprising a continuous scale parameter, namely by ( discrete ) convolution with the family of kernels T ( n ; t ) = e In ( t ), where In are the modi ed Bessel functions of integer order. Some obvious discretizations of the continuous scale-space theory are discussed in view of the results presented. The authors show that the kernel T ( n ; t ) arises naturally in the solution of a discretized version of the di usion equation. 

Q2. What is the generating function of a kernel on the form (6)?

All kernels with the generating function 'K(z) = P1 n= 1K(n)z n on the form'K(z) = C z k NY i=1 (pi + qiz) (7)where pi > 0 and qi > 0 are discrete scale-space kernels. 

Q3. What is the commonly used technique to implement the scale-space theory for discrete signals?

A commonly adapted technique to implement the scale-space theory for discrete signals has been to discretize the convolution integral (1) using the rectangle rule of integration. 

Q4. What is the natural way to apply the scalespace theory to discrete signals?

The natural way to apply the scalespace theory to discrete signals is apparently by discretization of the di usion equation, not the convolution integral. 

Q5. What is the essential requirement for a signal at a coarser level of scale?

The essential requirement is that a signal at a coarser level of scale should contain less structure than a signal at a ner level of scale. 

Q6. How can the authors analyze the behaviour in scale-space?

In order to analyze the behaviour in scale-space, the continuum of multiresolution representations must be sampled at some levels of scale. 

Q7. What is the idea of a continuous scale parameter even for discrete signals?

The idea of a continuous scale parameter even for discrete signals is of considerable importance, since it permits arbitrary degrees of smoothing, i.e. the authors are no longer restricted to speci c predetermined levels of scale. 

Q8. How can the scale-space representation be calculated?

In the separable case the scale-space representation can be calculated by separated convolution with the one-dimensional discrete analog of the Gaussian kernel. 

Q9. What is the generating function for the Bessel functions of integer order?

For simplicity, let a = b = 2 , and the authors get the generating function for the modi ed Bessel functions of integer order, see [1] (9.6.33).'t(z) = e t 2 (z 1+z) = 1X n= 1 In( t)z n (21)The authors obtain a normalized kernel if the authors let T : Z R+ ! 

Q10. What is the condition about suppression of local extrema?

This shows that, combined with the requirements about a continuous scale parameter and semi-group structure, the condition about suppression of local extrema is in one dimension equivalent to the condition about decreasing number of local extrema. 

Q11. How do the authors apply scale-space theory to two-dimensional discrete images?

The proper way to apply the scale-space theory to two-dimensional discrete images is apparently by discretization of the di usion equation. 

Q12. How do the authors convert the one-dimensional scalespace theory to discrete images?

The authors have also shown that the only reasonable way to convert the one-dimensional scalespace theory from continuous images to discrete images is by discretization of the di usion equation. 

Q13. What is the nal formula for smoothing?

As mentioned earlier, this form on the smoothing formula corresponds to the requirements about linear shift-invariant smoothing and a continuous scale parameter.