Manual 17548603

Manual 17548603
Technical report CVAP 55, ISRN KTH/NA/P--88/08--SE, 1988. KTH Royal Institute of Technology, Stockholm, Sweden.
Scale-Space for Discrete In1ages
Tony Lindeberg Computer Vision and Associative Pattern Processing Laboratory (CVAP ) Depar tment of Numerical Analysis and Computing Science Royal Institute of Technology S-100 44 Stockholm, Sweden 28 October 1988
1
Abstract
In this paper we address the formulation of a scale-space theory
for discrete images . We denote a one-dimensional kernel a scale-space
kern el if it reduces the number of local extrema and discuss which
discrete kernels are possible scale-space kernels. Unimodality and pos­
itivity properties are shown to hold for such kernels as well as thei r
Fourier transforms. An explicit expression characterizing a ll discrete
scale-space kernels is given.
\Ve propose that there is only one reaso nable way to define a scale­
space family of images L : Z x R+ --. R for a one-dimensional discrete
image f : Z --. R, namely by convolution with the family of discrete
kernels T : Z x R+ --. R
00
L(x; t) =
L
T(n; t)f(x- n)
(xEZ,t2:,0)
n= - oo
where T(n; t) = e-t In(t) and I,. is the modified Bessel function of
order n.
\Vith this representation, comprising a continuous scale par ameter,
we are no longer restricted to specific predetermined levels of scale.
Further, T( n; t) appears naturally in the solution of a discretized
version of the heat equation , both in one and two dimensions.
The family {T(n ; t)} (t 2: 0) is the only one-parameter family
of discrete symmetric shift-invariant kernels satisfying both necessary
scale-space requirements and the semigroup property T(.; s)*T(.; t)
T(.; s + t). Similar arguments appl ied in the continuous case uniquely
lead to the family of gaussian kemels.
The commonly adapted technique with a sampled gaussian pro­
duces undesirable effects. It is shown that scale-space violations might
occur in the family of functions generated by convolution with the sam­
pl ed gaussian kernel. The result exemplifies that properties derived in
the continuous case might be violated after discretization.
A discussion about the numerical im plementation is performed and
an algorithm generating the filter coefficients is supplied.
=
2
1
Introduction
It is well known that objects 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 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 t he 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 importance. Then
a mechanism, which systematically removes finer details or high-frequency
information from an image, is required. This smoothing must be available
at any level of scale.
A method proposed by Witkin [25] and Koenderink, van Doorn [11] is
to embed the original image in a one-parameter family of derived images,
the scale-space, where the parameter t describes the current level of scale
resolution. Let us briefly develop the procedure as it is formulated for one­
dimensional continuous images: Given an image function (a signal) f: R-+­
R a function 1 L: R X R+ - R is defined by L(x; 0) = f(x) and convolution
with the gaussian kernel g : R x R+\ {0} --+- R
L(x; t)
= (g(-; t) * f (·))(x; t ) =
l
oo
(=-oo
1
-.-e-~ / 2 t f(x- Od~
2
.j2i'i
(1)
if t > 0. Equivalently the family can be regarded as defined by the heat
equation
(2)
with appropriate boundary conditions. This family possesses some attrac­
tive properties.
• As the scale p arameter t is increased additional local extrema or ad­
ditional zero crossings caJlnot appear.
• Causality in the sense that L( x; t2) depends exclusively on L(x; tt) if
t2 > tl (tt,t2;::::: 0).
• The blurring is space invariant and does not depend upon the image
values.
1
R+ de notes the set of real non-negative numbers including zero.
3
coarser levels
of scale
increasing t
Figure 1: A scale-space is an ordered set of derived im ages intended to
represent the original image at various levels of scale.
It has been shown by Babaud et.al. [3] that the gaussian function is the
only kernel in a broad class of functions which satisfies adequate scale-space
conditions.
The theory has b een developed and well-established for continuous im­
ages. However , it does not tell us at all about how the implementation
should be performed computationally for real-life i.e. discrete images. In
principle, we feel that there are two approaches possible.
• Apply the results obtained from the continuous scale-space theory by
discretizing the occurring equations. For instance the convolution in­
tegral (1) can be approximated by a sum using customary numerical
methods. Or, the heat equation (2) can be discretized in space with
the ordinary fiv e-point Laplace operator forming a set of coupled or­
dinary differential equations, which can be further discretized in scale.
If the numerical methods are chosen with care we will certainly get
an approximation of the continuous numerical values . Dut we are not
guaranteed that the original scale-space conditions, however formu­
lated , will be preserved .
• Define a genuine discrete theory by postulating suitable axioms.
In this paper we will develop the second item and address the formulation
of a scale-space theory for discrete images. We will mainly consider one­
dimen sional images since, at the moment, the continuous scale-space theory
seems well-analyzed only in the one-dimensional case.
4
With a scale-space we will mean a family of derived images meant to
represent the original image at various levels of scale. Each member of the
family should be associated with a value of a scale parameter intended to
somehow describe the current level of scale. This scale parameter, here
denoted by t, may be either discrete (t E Z+) or continuous (t E R+) and
we will obtain two different types of discrete scale-spaces - discrete images
with a discrete scale parameter and discrete images with a continuous scale
parameter. In both cases we will start from the following basic assumptions:
• All representations should be generated by (linear) convolution of the
original image with a shift-invariant kernel.
• An increasing value of the scale parameter t should correspond to
coarser levels of scale and images with less structure. Particularly,
t = 0 should represent to the original image.
• All images should be real-valued functions : Z -+ R defined on the
same infinite grid, in other words no pyramid representations will be
used.
It will be natural to study linear transformations. A one-dimensional kernel
will be denoted a scale-space kernel if, for all images, the number of lo cal
extrema in the con volved image does not exceed the number of local extrema
in the original image.
In order to get familiar with the consequences of the definition we will
start by illustrating what this scale-space property means. In a straightfor­
ward and constructive manner we will derive some qualitative requirements
of the kernel that are necessarily induced by the a...x.iom. They can be ex­
pressed both in the spatial and the frequency domain. It will be easy to
show that a class of kernels, of which the binomial coefficients are a special
case, possesses scale-space properties.
Later , a complete characterization of a scale-space kernel will be given
using existing results from the theory of total positivity. The characteriza­
tion can b e formulated either in terms of minors of an associated Toeplitz
matrix or as an ex plicit expression for the generating function.
An alternative formulation of the scale-space for continuous functions
will be p erformed based on the scale-space property combined with shift­
invariance and a semi -group requirement posed on the occurring family of
kernels. Using the same argument we will construct the scale-space for
discrete images . A one-parameter family of discrete kernels, which in a sense
5
is the discrete analog of the family of gaussian kernels, will be presented. We
propose that this fan1ily is unique and that it is the canonical one fulfilli ng
reasonable scale-space requirements, including a. continuous scale parameter.
The numerical implementation will be treated and an algorithm gener­
ating the necessary filt er coefficients will be supplied .
vVe end the paper by discussing discrete scale-space properties of some
obvious discretiza.tions of the continuous convolution integral and the heat
equation. It will be shown that convolution with the proposed family of
kernels appears naturally as the solution of a. discretized version of the heat
equation. We will also show that the commonly adapted technique with a.
sampled ga.ussia.n kernel might lead to undesirable effects.
The results presented should h ave implications for image analysis as well
as other disciplines of digital signal processing.
2
Discrete Kernels with Scale-Space Properties
In this section we will consider discrete linear shift-invariant image transfor­
mations expressed on the general form
! out = ]{
00
f ou.t(x) =
L
n=- oo
* f in
(3)
00
L
K(n)fin(x- n) =
K (x - n)fin(n)
(x E Z)
n=-oo
where fin : Z - R is a. discrete image and J( : Z - R is a. discrete
convolution kernel. As a n underlying motivation we imagine fin and !out
as different components of a. multiresolution representation of a. discrete
fun ction. The intention is that the representation at a. coarser scale !out
should be possible to derive from the representation at a. finer level fin using
convolution.
The values K(n) of kernel fun ction]( will sometimes be referred to as
filter coefficients, Cn = K( n). If fin and ]( have finite support we can form
the vectors fin and fout determined by (fi n)n = fin( n) and (fout)n = !out( n ).
Then the convolution transform (3) can be expressed on matrix form:
fout = C fin
(4)
The convolution matrix C is a. matrix with constant values along the
diagonals C i,.i = ci- j (i,j = l..N). Such a matrix is called a Toeplitz
6
matrix. The generating function 2 associated with a kernel ]( with filter
coefficients Cn is
00
2:
<PK(z) =
(5)
CnZn
n =-oo
If we replace z by ei0 we obtain the Fourier spectrum
00
'1/JK(O) =
2:
(6)
CneiniJ
n=-oo
2.1
Basic Requirements of Scale-Space Kernels
By going from a fine scale to a coarser scale in scale-space the high frequency
components of the image should in some sense be suppressed. Formulated in
another way we can say that the convolved image in some sense ought to be
smoother than the input image. If local maxima and minima are regarded as
one measure of smoothness, it is necessary that the number of local extrema
in space does not increase from a fin er to a coarser scale. As will be shown
later, the famil y of functions generated by the gaussian kernel possesses this
quality in the continuous case. In a real implementation one often wants to
trace the local extrema when the blurring proceeds from a finer to a coarser
scale. This problem does for instance involve pract ical complications about
how far extrema can move in space. However, in the present discussion we
confine ourselves to the first requirement and state it as a basic axiom. The
remaining part of this section will be devoted to investigations of how it
limits the class of possible kernels.
Definition 1 We denote a one-dimensional kernel
Z -+ R a scale­
space kernel if for all images fin : Z -+ R the number of local extrema in the
convolved image !out = ](*fin does not exceeil the numbe7' of local extrema
in the original image fin.
J( :
A complication is involved. If either fin or ! out would happen to have a
plateau the question must be raised about how many local extrema the
plateau sh ould be counted as. At this moment we will not go into the
details of those dubious cases. We count a plateau as one local maximum
(minimum) if there are strictly smaller (larger) values bounding it both at
the left and at the right, see Fig 2. A more accurate treatment will be
performed in Section 3.
2
In some llterature this fun ction is referred to as as the characteristic polynomial.
7
(a)
(b)
·f ' . r \
Figure 2: Examples illustrating the definition of local extremum. (a) A local
maximum (generic case). (b) A plateau counted as one local maximum. (c)
A plateau not counted as a local extremum.
As a lo cal extremum in a discrete function f is equivalent to a zero­
crossing in its first difference !J.f defined by (LJ.f)(x) = f(x + 1) - f(x) the
definition can be expressed in terms of zero-crossings by replacement of the
string "local extrema" with "zero-crossings" . This result follows since the
difference operator commutes with the convolution operator.
2.2
Results from the Impulse Response
By considering the impulse response it is possible to draw some qualitati ve
conclusions about the properties of a scale-space kernel. Let
1 if
X=
0
f in (X ) = {j (X ) = { 0 otherwise
(7)
Then
fout(x) =(I( *ti)(x)
= K(x)
(8)
o( x) has exactly one local maximum and no zero-crossings. Therefore in
order to be a scale-space kernel J( must not have more than one extremum
and no zero-crossings. Thus,
• All coefficients must have the same sign.
• The coefficient sequence {K(n)}~=-oo must be unimodal3 .
We summarize the results in
3
A real sequence is called unimodal if it is first ascending (descending) and then de­
scending (ascending).
8
•
•
•
•
•
•
•
•
n
Figure 3: The filter coefficient sequence {Cn}~=-oo of a discrete scale-space
kernel must be positive and unimodal.
Proposition 1 Let {en}~=- oo be the coefficients of a discrete kernel with fi­
nite support. Then a necessary condition for the kernel to be a discrete scale­
space kernel is that all Cn have same sign and that the sequence {en}~=-oo
is unimodal.
Without loss of generality we therefore can restrict the rest of the treat­
ment to positive sequences where all Cn 2: 0.
It seems reasonable to require that ]( E it, i.e. that L~ - oo Jcnl is
finite . If f in is bounded and ]( E l 1 then the convolution is well-defined. If,
in addition, fin E l1 then also ! out Eh. Further, this assumption guarantees
that the Fourier transform of the filter coefficient sequence exists. It also
allows u s to normalize the coefficien ts such that L~=-oo en = 1. Particularly
the filter coefficients en must then tend to zero as n goes to infinity.
2 .3 A C lass of D iscrete Scal e-Space Kerne ls with Finite
S u pport
Consider a two-kernel with only two4 non-zero filt er coefficients:
]((
2
)(n)
=
if n = 0
q ifn = 1
{
0 ot herwise
Assume that p 2: 0, q > 0 and p
rpJ((2)(x) = p + qz .
p
+q
(9)
1. The generating function is
4
The superscript within parentheses used in the notation for kernels stands for the
number of non-ze ro filter coefficients.
9
fin(x)
r••,(x)
x-1
_q_
_q_
p-+<j
p-+<j
x+l
X
Figure 4: To convolve an image fin with a two-kernel J( (2 l (n) is equivalent
to to form a weighted average of the sequen ce {f;n(x )}~_ 00 •
It is easy to verify that the number of zero-crossings (local extrema)
in ! out = ]((2 ) * fin cannot exceed the number of zero crossings (local ex­
trema) in fi n · This result follows from the fact that convolution of f in
with 1((2) is equivalent to the formation a weighted average of the sequence
{f;n(x)}~:::::- oo' see Fig 4. The values of the output image can be constructed
geometri cally and will fall on straight lines connecting the values of the in­
put image. The offset along the x- a..x.is is determined by the ratio qf(p + q).
One reali zes that no additional zero-crossings can be introduced by this
transformation .
By repeated application this result we can formulate
Proposition 2 All ker·nels with a generating funct ion on the form
N
cp(z) = c zk I1 (Pi
+ q;z)
(10)
i:::::l
where P·i
> 0 and q; > 0 are discrete scale-space ke rnels.
Proof:
lly direct application of Definition 1 we can state a lemma.
Lemma 1 If two kernels K a. and J(b are scale-space kernels then the com­
posed kernel K a. * J(b is also a scale-space kernel.
Then take an arbitrary image f in and convolve it with a series of two-kernels
2
K f > (i = l.. N), each one having a generating fun ction cpK(2)(z) =Pi + 'JiZ
I
10
(Pi > 0 and qi > 0).
(11)
Ki
2
Let J((N +l) =
) * ](~ ) *... * ]{~) . As linear convolution is an associ ative
operation repeated convolution of fin with the kernels ](?) is equivalent to
2
) is a scale-space kernel J((N+l)
convolution of f in with J((N +I). As each
must also be a scale-space kernel according to the lemma above. Further,
as convolution in the spatial domain is equivalent to multiplication in the
domain of the generating func tions it follows that
2
Jd
(12)
is the generating function of a scale-space kernel. A constant scaling-factor cor a translation IPtranst(z) = zk cannot affect the number of local extrema. Therefore these factors can b e multiplied onto IPJ<{N+t>(z) without changing the the scale-space properties. 0 T he filter coefficien ts ge nerated by (10) can be regarded as a kind of gener­
alized binomial coefficients. T he ordinary binomial coefficients a re obtained, except for a scali ng-factor , as a special case if all Pi and qi are equal. Another formulation of Proposition 2 is also possible. Proposition 3 Let c_m, ... , c_I, eo, c1, ...en be the coefficients of a discrete
kernel with finite suppor·t. Then a sufficient condition for the kernel to be a
scale-space kernel is that all roots of the genemting function
cp(z)
= c_mz - m + ... + c_lz- 1 + eo + c1z + ... + CnZn
(13)
are real and non-positive.
Proof:
Let k = - m, N = n+ m in (10). If all roots of c,o(z) are real and negative
then (10) in Proposition 2 must be the factori zation of ( 13). 0
T he Fourier spectrum of a symmetric sequence on the form (10) has
some interesting properties . The most general generating function of such
a sequence can be written as
N
cp(z) =
C
IT (Pv + qv z)(Pv + qvz- 1 )
v=l
11
(14)
Consider one factor (p11 + q11 z)(p11
t.p 11 (
+ q z-1 ). Its Fourier sp ectrum is
ei0 ) = (Pv + qvei 8 )(Pv + qve-i8 ) = p~ + q~ + 2pvqv cos 0
11
(15)
On the interval [-1r, 1r] this function is non-negative. It assumes its maxi­
mum value (p 11 +qv )2 for() = 0 and its minimum value (Pv- qv) 2 for() = ±1r.
c.p 11 ( ei 8 ) is monotonically increasing on [-1r, 0) and monotonically decreasing
on [0, 1r), i.e. unimodal.
It is easy to show that any finite product of non-negative increasing (de­
creasing) fun ctions is also increasing (decreasing). Subsequently, the Fourier
spectrum c.p( ei8) of a symmetric kernel on the form (10) is non-negative and
unimodal on the interval [-7r, 1r].
In the following three sections we will show that the Fourier spectrum
of any symmetric scale-space kernel must possess these qualities.
2.4 A Restriction on the Sign of the E igenvalues of the
Convolution Matrix
If the convolution matrix has negative eigenvalues then the corresponding
kernel cannot be a scale-space kernel.
Proposition 4 Let c_m , ... ,c_1,Co,C1, ...cn (c_m =f 0, Cn =f 0} be the coef­
ficients of a discrete kernel with finite support. Then a necessary condition
for the kernel to be a discrete scale-space kernel is that no convolution ma­
trix c(N) = banddiagNxN((c_m, .. . ,eo, ... ,en)) of any dimension N > 0 has
a real negative eigenvalue with a correspondi ng real eigenvector.
Co C-1
C1 Co
c(N) =
C_m C-1
C-m Cn
C_m
Cn (16)
C- m
Cn
Cn
eo
C1
c_1
eo
Pm·ticularly if the kern el is symmetric c_i = Ci V i then all eigenvalues must
be real and non-negative.
12
(a)
(b)
(c)
~
,­
_l_
\
Figure 5: (a) The eigenvector V. (b) The components of
indices l..N. (c) The corn ponen ts of ]{ * f in .
c(N) V
having
Proof:
Because of Proposition 2 it is sufficient to study kernels with only non­
negative filter coefficients. Assume that C(N) has a real negative eigenvalue
for some dimension N with corresponding eigenvector v. Let a and f3 be
the indices of the first and the last non-zero components of v. Construct an
extended input image fin such that (see Fig 5):
fin(x)=
{
0
if x <a
Vx
ifa$x${3
0
if X >
(17)
f3
Then the first N components (with indices l..N) of C(N)v and]{ *fin will be
pairwise equal. As v is an eigenvector with a negative eigenvalue it follows
that the components of c(N) v and v have opposite signs. This means that
v, C(N) v and ]( * fin all have the same number of internal zero-crossings
provided that we observe only the components with indices l..N .
The reversal of the components with indices a and f3 guarantee that an
additional zero-crossing will occur. If fin(a) is positive (negative) (K *
f in)(a) will be negative (positive). As all filter coefficients are positive
it is always possible to find at least one positive (negative) component of
]( *fin for a sufficiently small index value. The component with subscript
a- m- n+ 1 does certainly serve our purpose, since it is equal to c_mfin( a).
Consequently, we have found an additional zero-crossing between this posi­
tion and position et. The same argument can be carried out at the right end
point producing another scale-space violation. 0
13
2.5 A Restriction on the Sign of the Fourier Spectrum
A theorem by Toeplitz [23](8] relates the eigenvalues5 of an infinite Toeplitz
matrix C with elements Ci,j = Ci-j to the the values of the generating
function associated with the sequence of filter weights. Assume that <p(z) =
2::~= - oo CnZn is convergent in the ring r < lzl < R, where 0 < r < 1 < R.
Then the eigenvalues of C coincide with the set of complex values that <p(z)
assumes on the unit circle lzl = 1. It allows us to derive an interesting
Corollary from Proposition 4.
Proposition 5 Letc_n , ... ,c_l,co,cl,···cn whereC±n ::/; 0 be the coefficients
of a symmetric d iscrete kernel with finite support. Then a necessary condi­
tion for the kernel to be a discrete scale-space kernel is that the generating
function <p(z) = 2::~- oo CnZn assumes no real negative values on the unit
circle in the complex plane. In other words, the Fourier spectrum must be
non-negative.
Proof:
Let >.~N) denote the smallest eigenvalue of the convolution matrix of
dimension N and let m denote the minimum value6 the generating function
<p(z) assumes on the unit circle in the complex plane. As a consequence
of a theorem by Grenander [8] p65 about the asymptotic distribution of
eigenvalucs of a finite Toeplitz matrix it follows that
lim ),(N)
N -.oo 1
= m
>. (N) > m 1
-
(18)
Assume that <p assumes a strictly negative value on the unit circle. Then
m is strictly negative. As limN-.oo >.~N) = m it follows that for any e > 0
>.iN) must be contained in I= [m, m+ e] when N exceeds some sufficiently
large integer Ne. e is chosen small enough such that m+ e < 0. According
to Proposition 4 the kernel cannot be a scale-space kernel. 0
2.6 A Unimodality Requirement for the Fourier Spectrum
of the Coefficient Sequence
In this section we will show that the Fourier spectrum of a symmetric scale­
space kernel is unimodal on the interval [-rr, rr] . The core of t he proof lies in
5
A is called an eigenvalue of an infinite matrix C if the matrix C-A] has no bounded
inverse. I denotes the unit matrix.
6
Due to symmetry of the kernel <,:>(z) assumes only real values on the unit circle. The
minimum value d oes certainly exist since <,:>(ei8 ) is a conti nuous function and the interval
[-rr,rr] is compact.
14
Lemma 3. The remaining technicalities are all aimed at one thing, namely
to avoid problems at the boundaries of the matrix representation of the
convolution transformation.
We start with the an important observation concerning eigenvectors and
eigenvalues. In order to express it clearly we need to distinguish between
different kinds of zero-crossings. A zero-crossing is said to be simple if it
corresponds to one of the sign sequences ( + - ), (- +), (+ 0 -) and (- 0
+). The number of zero-crossings in a vector x will be denoted V(x).
Lemma 2 Let A1 and A2 be real eigenvalues of an x n matrix A and let v1
and v2 be the corresponding eigenvectors. Assume that V(v2) > V(vt) and
that v 1 and v 2 have only simple zero-crossings and no zero end elements.
Then in order· for y = Ax to be a scale-space transformation for all x it is
necessary that IA2I ~ IAtl·
Proof:
Assume that A2 > A1 . The scale-space properties are not affected by a
scaling-factor. Therefore, we can equivalently study B = ~A. For both
eigenvectors we define the the largest and smallest absolute values v(absmin)
and v(absmax) by
v(absmin)
=.
min
jv;j ;
v(absmax)
= . max
t=l..N,
l=l..N, v;;o!'O
lvd
(19)
v;;o!'O
Let x = v2 + a:v 1 where a: is chosen large enough such that V(x) = V(vt)·
This can always be achieved if la:! v~absmin) > v~absmax). Let Iv 1 be the set
of all integers i E [1, n] satisfying i E Iv 1 => (vi )i f. 0. Then V i E Iv 1
the signs of (x)i and (v 1 )i will be equal. As all zero-crossings in v 1 are
simple and no end elements are zero it follows that no additional zero­
crossings can be introduced in x. Nor can any zero-crossings be deleted.
Thus, V(x) = V(vt).
Then consider Bx = ]2 ( a:A 1 v 1 + A2v2) and study
Bkx
=a:(~:) kv + v2
1
(20)
For a fixed value of a: we can always find a sufficiently large value of k such
that V(Bkx) = V(v2 ). In a similar manner to above one verifies that the
condition
la:lj~jk viabsmax) < v~absmin) suffices. Subsequently,
V(Bkx)
>
V(x) which shows that the transformation induced by Bk is not a scale­
space transformation. Therefore, B cannot be a scale-space kernel since at
15
(a)
(b)
Figure 6: (a) Input im age consisting of a low frequency component of high
amplitude and a high frequency component of low amplitude. (b) In the
output image the low fr equency component h as been suppressed while the
high frequency component remains unchanged. As we see, additional zero­
crossings have been introduced.
least one scale-space violation must have occurred in the series of k successive
transform ations. 0
T he essence of this result is that eigenvectors with strong variation must
correspond to small eigenvalues. The next step is to relate the Fourier sp ec­
trum of a kernel to the eigen values and the eigenvectors of the corresponding
convolution matrix. We start by showing that a convolution transformation
equivalently can be described as a multiplication with a circulant1 matrix
provided that both the kernel and th e input image h ave finite support.
Let ]( b e a given symmetric ke rnel with Ci = 0 if li l > n and let
f in be an arbitrary im age with fi n(x) = 0 if lxl > m. Defin e vectors
fin and fout by (fin )i = fin(i) (i = O,±l , ... , ±M) and (fout)i = f out (i)
(i = O,±l, ... , ± Af) for some M > m+ n. Then , the effect of the con­
volution transformation ! out = ]( * fin can equivalently be expressed on
matrix form fout = CTfin > where CT is a Toeplitz matrix with compo­
nents (CT)i ,j = Ci-j (l:,j = O, ±l, ... ,±M). T his follows as the components
with indices O, ±l, .. ., ±M of ! out and fout are equal and fout(x) = 0 when
lxl >M.
However, since th e first and last n components of fin are zero; t he values
of the matrix elements in the upper right and lower left corners do not affect
fout· Therefore, we can replace them with some appropriately selected filter
7
In a circulanL maLrix each row is a circular s hin of Lhe preceding row except for the
firsL row which is a circular shin of Lhe last row.
16 coefficients such that the matrix becomes circulant8 . Hence, the effect of
the convolution transformation can also be written fout = Ccfin • where Cc
is the (2M + 1) X (2M + 1) circulant matrix:
eo
Ct
Ct
eo
Cn
Cn
Ct
Cn
Ct
Cn
Cn
Cc=
Cn
(21)
Cn
Cn
Cn
Cn
Ct
Cn
Cn
eo
Ct
Ct
eo
It is easy to establish the unimodality property for transformations expressed
with circulant matrices since the eigenvalues and eigenvectors can be deter­
mined analytically.
Lemma 3 Let {cJ.L}~=-oo be the filter coefficients of a symmetric discrete
kernel with cJ.L = 0 if IJ.LI > n. For all integers M 2: n it is required that the
transformation given by multiplication with the (2M +1) X (2M +1) symmet­
ric circulant c~n, defined by (CbM))i,j = Cj- j (i,j = o..M) and circulant
extension {21}, should be a scale-space transformation. Then, necessarily
the Fottrier spectrum 1/J( 0) = 2:~=-oo cJ.LeiJ.LB must be unimodal on [-rr, n"].
Proof:
vVe will use the explicit expressions for the eigenvalues >. m and eigenvec­
tors V m of cbM). By verification one shows that
(m= -M..O..M)
(m,k = -M..O..M)
(22 )
(23)
As the kernel is symmetric all eigenvalues except for >. 0 are pairwise degen­
erated P-m = >.m)· Hence, we can choose a new basis of eigenvectors by
8
To express the convolution transformation with a circulant matrix is equivalent to
periodic extension of the input function
17
1/;(0)
0
Figure 7: If the Fourier spectrum is not unimodal on [-1r, 1r], i.e if there
exist (}2 > 01 in [0, 1r] such that 7f;(02) > 7f;(01 ) then the corresponding
transformation cannot be a scale-space transformation.
linear combination of the old ones. Let Vm = -tCY-m - Ym ) (m= -M.. -1)
and Vm = HY-m + Ym) (m= o.. M). Then,
(vm)k
. ( 21r1nk )
=Sill
M
2
+1
21rmk )
(Vm)k = cos ( 2 M
+1
(m= -M.. -1 ,k = -M..O..M)
(24)
(m= O.. M,k = -M..O..M)
We note that the number of zero-crossings in an eigenvector increases as lml
increases. Subsequently, V(vm 2 ) > V(vm 1 ) if lm2l > lm1l·
Introduce the notation 7/J( B) = <p( ei8 ) for the Fourier spectrum. By
comparison of (5) and (22) we see that Am= 7f;( 2~Y~\), i.e. the eigenvalues
of CbM) are given directly by the Fourier spectrum and a larger value of
lml corresponds to a larger absolute value of the argument () to 7/J. Now ,
assume that the Fourier spectrum is not unimodal i.e that there exist some
82 > (} 1 in [0, 1r] such that 7f;(B2) > 7f;(B 1 ), see Fig 7. (Without loss of
generality we can presuppose that 1/J is non-negative on [-1r,1r], because
otherwise , according to Proposition 5, J( cannot be a scale-space kernel.)
Then , as 1/J is a continuous function of() it is possible to find some sufficiently
large integer M such that there exist ()-1 = ;;,~11 and 01 = ;tt~11 satisfying
7f;(02) > 'lj;(0-1 ) for some integers ni2 > ni1 .
To summarize, CbM) has eigenvalues Am2 > Am1 and corresponding
eigenvectors with V(v 1112 ) > V(v1111 ) since ni2 > nit. From (24) we see that
all zero-crossings in the eigenvectors are simple and that no end elements
in them are zero. Consequently, according to Lemma 2 the transformation
given by c2J"1) cannot be a scale-space transformation. ()
18
The last step is to generalize this result to general (non-circulant) con­
volution transformations. Note that Lemma 3 cannot be used without mod­
ification, since the eigenvectors involved are non-zero in the region where
f in is required to be zero. If f in is non-zero in that region the description
with a circulant matrix is no longer equi valent to the description with a
Toeplitz matrix. This means that the superposition of eigenvectors used in
the proof of Lemma 2 is not feasible as input image for the transformation
fout = Crfin· Actually, what Lemma 3 expresses is the desire that if fin is
a periodic fun ction then the number of local extrema in one period should
not increase from a finer to a coarser level of scale.
In view of the last observation we can reformulate Lemma 3 in terms
of p eriodic functions. With the number of local extrema (zero-crossings) in
one period of a periodic function f with period T we mean the number of
local extrema (zero-crossings) in the sequence f(O),J(1), ... , f(T).
Lemma 4 Let ]( be a symmet1·ic kernel with finite support, i.e. c"' = 0 if
IJ.LI > n. A ssume that for all periodic images fin with period larger than
2n + 1 the number of local extrema in one period of the convolved image
]( * fin does not exceed the number of local extrema in one period of the
original image. Then, the Fourier spectrum 1/;(0) = L:::::"=-oo c"' eip.O must be
unimodal on [-1r, 7i).
Proof:
As f in is periodic !out = J( *fin will also be periodic with the same period.
Subsequently, it suffices to study one p eriod of each function. We only need
to create a counter-example. Therefore, we can restrict the analysis to the
case where the period T is odd, i.e. T = 2M + 1 for some integer M.
Define vectors fin and fout by (fin)i
fi n(i) (i
0,±1, ...,±M) and
(fout)i = f out(i) (i = 0, ±1 , ... , ±M). Then, the effect of the convolution
transformation ! out = ](*fin can equivalently b e expressed on matrix form
fout = Ccfi 0 , where Cc is the (2M + 1) X (2M + 1) circulant m atrix with
components (Cc )i,j = Ci-j ( i, j = 0, 1, .. . , M) and circulant extension (21).
Assume that th e Fourier spectrum is not unimoda l on [-7r,7r). Hence,
according to Lemma 3 it is possible to find some sufficiently large M such
that the transformation given by multiplication with
is not a scale­
space tra nsformation.
T here is one difference between the non-p eriodic a nd the periodic case
that must b e noted. Namely t hat the number of local extrema in one pe­
riod of a periodic function includes "wrap-around". However, the proof of
Lemma 3 can b e carried out in this case as well. All we have to do is to
=
=
eft>
19
(b)
, ~A~
'\['r'\)V
n
..
w
•
T
(2n+l)T
\~t\~
'\}'f '\} ~
n
Figure 8: Construction of the image fin from the periodic fun ction /per ·
replace the definition of V(x) and remove the requirement "no zero end ele­
ments from Lemma 2. Consequently, if the Fourier spectrum is not unimodal
the number of local extrema in one period will increase. 0
From this result the generalization to general convolution transforma­
tions can be obtained by construction of a counter-example.
Proposition 6 Let {c'"'}~=-oo be the filter coefficients of a symmetric dis­
crete kernel ]( with finite support. Then a necessary condition for the kP.r­
nel to be a discrete scale-space kernel is that the Fourier spectrum tj;(O) =
L~=-oocl-'ei 0 ~-' is tmimodal on [-rr,rr].
Proof:
Assume that]( is a symmetric discrete kernel with non-unimodal Fourier
spectrum and finite support, c'"' = 0 if IJ.LI > n.
Then, according to Lemma 4 it is possible to find a periodic function
/per of some period T such that the number of local extrema in one period
of the convolved image exceeds the number of local extrema in one period
of the original image.
Let I2n+t be an interval with 2n + 1 consecutive periods and construct a
new function fin, which is equal to /per on l2n+t and at then closest points
at each boundary, see Fig 8 . At all other points f in should be zero. Due
to the construction of fin and the finiteness of ]( it follows that ]( * fin
and ](*/per will be equal on hn+l· Thus, provided that we only count the
points in I2n+t we have introduced at least 2n + 1 additional zero-crossings.
Outside hn+t we might expect to find more zero-crossings in ](*fin·
The support region of J( *fin is general larger than the support region of fin·
20
However, fi n cannot have more than a total of 2n additional zero-crossings
since fin is non-zero only at 2n points outside h n+l· Consequently, ]( * fin
contains at least one zero-crossing more than fin, which shows that]( cannot
be a scale-space kernel. 0
2.7
A Result for Kernels with Three Non-Zero Elements
For a three-kernel ]((3) with exactly three non-zero consecutive elements
c_ 1 > 0, eo > 0 and c1 > 0 it is possible to determine the eigenvalues of the
convolution matrix and the roots of the characteristic equation analytically.
It is easy to verify that the eigenvalues >.,_. of the convolution matrix
are all real and equal to
fl'lr
>.JJ. =eo- 2Jc_tcl cos( - N )
+1
(IL
= l..N)
(26)
and that the roots of generating function <p[((3)(z) = c_ 1 z- 1 +eo+ c1 z are
Z1 2
'
=
-eo±
Jc6- 4c_l c1
2Ct
(27)
From (26) we deduce that the eigenvalues of a convolution matrix of arbi­
trary dimension are all non-negative if and only if eo~ 2jc_ 1c1 . (27) says
that the roots of the characteristic equation are real a nd non -positive if and
only if c6 ~ 4c_ 1 c1 . Thus, the necessary condition in Proposition 4 and the
sufficient condition in Proposition 3 both lead to the condition c6 ~ 4c_ 1 c1
and we obtain a complete classification for all possible values of c_ 1 , c0 and
c1 . We conclude that a three-kernel with positive elements is a scale-space
kernel if and only if it can b e written as the convolution of two two-kernels
with positive elements.
At this moment one could ask one-self if the result can be generalized to
hold for kernels with arbitrary numbers of non-zero filter coefficients. I.e. if
all discrete scale-space kernels with finite support have a generating function
on the form (10). This question will be answered in the next section.
21
3
Classification of Discrete Scale-Space Kernels
Until now we have postulated an axiom in terms of local extrema or equiv­
alently zero-crossings and investigated some of its consequences for image
transformations expressed as linear convolution with a shift-invariant kernel.
We have seen t h at the sequence of filter coefficients must be positive and
unimodal a nd that its sum should be convergent. For symmetric kernels the
Fourier spectrum must be positive and unimodal on (-1r, 1r].
In this section we will perform a complete characterization of the scale­
space kernels. We have studied the literature and seen that several interest­
ing results can be derived from the theory of total positivity. Sign-regular
kernels and Polya frequency sequences turn out to be important concepts as
they characterize the interesting kernels completely. The proofs of the im­
porta nt theorems are sometimes of a rather complicated nature for a reader
with an engineering background. We will not burden the presentation with
them but confine ourselves to references.
The pioneer in the subject of va riation-diminishing transforms was I.J.
Sch oenberg. He studied t he subject in a series of papers from 1930 to 1953 .
Later the theory of total positivity has been covered in a monumental mono­
graph by Karlin [1 3]. A recent paper by Ando [2] reviews the field using
skew-symmetric vector products and Schur complements of matrices as ma­
jor tools. T he questions issued in the pap er constitute a new application to
these not too well-known but very powerful results.
3.1
Definitions
We will firs t consider general linear transformations of discrete images where
t he kernel does not need to be shift-invariant .
00
f out(x)
=
L
K(x, y )f in(Y)
(x
E Z)
(28)
y=-oo
Two notions of sign changes in vectors will be used. Let x = (xbx2, ... , xn)
be a vector of n real numbers. We denote by v-(x) the number of sign
cl1 anges obtained in the seque nce x 1,x2, ...,xn if all zero terms are deleted
and by v+(x ) t he maximum number of sign ch anges possible in the sequence
XI, x2, ... , Xn if each zero value is allowed to be replaced by either + 1 or -1.
We use a special convention saying that the number of sign changes in the
null vector is - 1.
22
The interesting sequences and kernels will defined in terms of minors.
Given a kernel ]( : X x Y ~ R we form minors of arbitrary order r by
selections of x 1 < x2 < ... < Xr from X and of YI < Y2 < ... < Yr from Y.
The determinant of t he resulting matrix with components {K(xi, Yi )}i,i=l..r
will be called "a minor of order r" and denoted by
K(x1,Y2)
K(x2, Y2)
K(xi. Y1)
K(x2, Yl)
]( ( X1, X2, ... ,Xr )
YI. Y2, ... , Yr
K(xl,Yr)
K(x2,Yr)
(29)
K(xr, Yl)
K(xn Yr)
K(xr, Y2)
The single most important concept is sign-regularity.
Y ~ R is sign-regular (SRoo) if all its
r-order minors have same sign for every order r from 1 through oo, i.e. if
there exists a sequence of constants e 1, e2, ... each +1 or -1 such that
Definition 2 We say that]{: X
er]( (
for all choices of X1
X
X2, ... ,Xr
Y1, Y2, ... , Yr
X1,
< X2 < ... <
)
~O
Xr from X and Y1
(30)
< Y2 < ... < Yr
from Y.
In other words sign-regularity means that it is impossible to find two minors
of same order having opposite signs. If strict inequality holds for all r then
](is said to be strictly sign-regular (SSR 00 ).
3.2 Characterization Theorem for General Linear Trans­
formations with Scale-Space Properties
General linear transformations having scale-space properties can be fully
characterized in terms of sign-regularity.
X m real matrix with n ~ m. Then the linear
map A from Rm to Rn diminishes vm·iations in sign in the sense that
Theorem 1 Let A be an n
for all x E Rm x
-::/=
0
(31)
if and only if A is strictly sign-regular {SSR 00 ).
The original proof of this powerful theorem, forming the foundation of the
theory for variation-diminishing transforms, can be found in Schoenberg [22].
Ando [2] p201 derives it using skew-symmetric vector products.
Another formulation is possible [2] p202 if A is known to be of full rank.
23
Theorem 2 Let A be an n X m real matrix of rank m. Then
(32)
holds for all x E Rm (x =P 0) if and only if A is sign-regular (SR=)·
We note that the condition (32) is equivalent to the formulation we expressed
in Definition 1. Consequently, sign-regularity and full rank are the necessary
and sufficient conditions for a kernel to be a potential scale-space kernel.
3.3 Characterization Theorem for Convolution Transfor­
mations with Scale-Space Properties
A narrower class of kernels is obtained if all minors are required to be non­
negative.
Definition 3 A kernel]( : X X Y --t R is said to be totally positive (T P=)
if all its minors are nonnegative; i.e. if
Xt,X2, .•• ,Xp
Y1, Y2, ... , Yp
Xl
<
X2
< ... < Xp;
Yl
<
)
~ 0
< ... < Yp;
Y2
(33)
p
= 1, 2, ... , 00
An important case ofTP 00 -sequences appears if the discrete kernel is shift­
invariant i.e. if K(x , y) can be written as k(x- y) = Cx -y·
Definition 4 We say that {en} ~= -= is a P6lya frequ ency sequence if any
minor of the infinite Toeplitz matrix
C=
eo
c1
c- 1
eo
c-2
c_ l
C2
C1
Co
(34)
is nonnegative.
The importance of the Polya frequency sequences becomes apparent when
we require tha t the generating function con verges, which for instance holds
if the sum of the filter coefficients is convergent.
24
Definition 5 A P 6lya f requency sequence {cn }~=- oo having a generating
function <p(z) = 2:::~=-oo CnZn which converges in an annulus r < lzl < R
(0 < r < 1 < R) such that <p(z) =J 0 is called a normalized P6lya frequency
sequence.
According to a theorem by Schoenberg [21] p363 sign-regularity com­
bined with the Toeplitz struct ure implies total positivity. Subsequently,
Theorem 3 The convolut ion transformation
00
f out(x) =
L
cn f in(x- n)
n= - oo
is variation-diminishing i.e.
holds for all fin if and only if the sequence of filte r coefficients
a normalized P6lya frequency sequence.
{cn } ~= - oo
is
In other words, each shift-invariant discrete scale-space kernel corresponds
to a normalized Polya fr·e.quency sequence.
There exists a remarkably explicit characterization theorem for the gen­
erating function of a P F00-sequ ence. It has been proved in several steps b y
Edrei and Schoenberg, see [22] or [13].
Theorem 4 An infinite sequence { cn } ~=-oo is a P6lya frequency sequence
if and only if its generating fu nction <p( z) = 2:::~=-oo CnZn is of the form
(35)
(X)
c
> 0; k; E Z
q_b q1 , a;,{3i,/i, 6; ~ 0 ;
L (a;
+ {3; + li + 6;) < oo
i=l
T he sequence {en } ~=-= is normalized if a nd only if it in addition holds that
{3; < 1 and li = 1, see [13] p423.
For kernels with finite support q_ 1 , qt, f3i and /i must be zero and the
infinite product must be replaced with a finite one. Thus the class of ker­
nels we arrived at in Section 2.3 (Proposition 2) is exactly t he set of P F00
sequences having finite support, i.e the kernels wit h a generating function
25
on the form (10) are the only finite scale-space kernels. An immediate con­
sequence of this characterization is that all finite scale-space kernels can
be decomposed into kernels with two real strictly positive consecutive filter
coefficients.
The representation (35) can sometimes be very convenient for further
analysis. For example, starting from (35) it is almost trivial to show that the
Fourier spectrum of a symmetric P F 00 -sequence, i.e. a general symmetric
scale-space kernel, is unimodal and non-negative on the interval [-7r,7r].
Due to the symmetry q_1 = q1, a 11 = 011 and f3v = 'Yv· As a first step
one replaces z with iB and shows that each one of the factors e(q- 1 z-t+qtz),
(1 +a 11 z)(1 +011 z- 1 ) and (1- /3 11 z )(1- -y11 z- 1 ) is a non-negative and unimodal
function of 8 on [-71" , 7r]. The rest is left as an exercise for the reader.
As we see, the theory of total positivity provides us with a complete
classification as well as a simple and powerful representation of the inter­
esting kernels. The key result (Theorem 3) from this discussion is worth
restatement:
Proposition 7 A discrete kernel]( : Z - R is a scale-space kernel if and
only if the corresponding sequence of filter coefficients {K(n)}~=-oo is a
normalized Polya frequency sequence.
4
Axiomatic Construction of the Discrete Scale­
Space
In last section we studied linear transformations expressed as convolutions
and concluded that a discrete kernel is a scale-space kernel if and only if the
sequence of filter coefficients {K(n)}~=-oo is a normalized P6lya frequency
sequence. \Vith this result in mind , an apparent way to get a multireso­
lution representation of an image f is to define a set of discrete functions
Li (i = 0.. n) 'vhere Lo = f and each coarser level is calculated by con vo­
lution from the previous one Li = J(ir-i- 1 * Li-1 (i = l..n). The kernels
J(i +-i- l should be approp riately selected scale-space kernels corresponding
to suitable amounts of blurring. The scale-space condition for each kernel
guarantees that the number of extrema at a coarser level (larger value of i)
does not exceed the number of local extrema at a finer level. From empir­
ical results [16] it is known that not more than approximately seven levels
of scale are relevant in human perception of typical real-life images. This
provides a good motivation for this so-called sampled scale-space with a dis­
26
crete scale parameter. However, in general the locations of those levels are
not known a priori.
The goal in this section is to tie together scale-space kernels correspond­
ing to different degree of smoothing in a systematic manner such that a
continuous resolution parameter can be introduced. The concept of a con­
tinuous scale parameter is of considerable importance since, we will no longer
be locked to fixed discrete levels of scale. It allows us to defocus images with
arbitrary amount of blurring, which will certainly make it easier to locate
and trace events in scale-space. Of course, it is impracticable to generate
the representations at all levels of scale in a real implementation. However,
the important idea is that any representation can be calculated if desired.
We will not consider the question about how to choose a suitable set of
scale levels in a practical case. Imagine for instance that we want to trace
events, like local extrema or convex and concave regions, as the blurring
proceeds 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. It is certainly a non-trivial problem to make an appropriate
selection of these levels. It seems very reasonable that the sampling rate
along the scale direction should depend on the behaviour of the image as a
function of the space coordinates. If the representation at some level of scale
is relatively smooth we should be able to allow a larger scale step than if it
were strongly varying. We will thus be lead to methods that automatically
regulate the scale step, based on the interconnection between the appearance
of the family of images as a function of the space coordinate and as a function
of the scale parameter. The point with a scale-space with a continuous scale
parameter is that it provides a theoretical framework for the development
of such algorithms. We do not need to select the scale levels a priori, but
can leave the decision open to the actual situation.
As an illustrative background we start by deriving the continuous scale­
space for one-dimensional im ages starting from the variation-diminishing
property. Then, the discrete scale-space will be constructed with a. similar
approach.
4.1 A Continuous Formulation which Leads to the Gaus­
sian Kernel
Koenderink, van Doorn [11] derive the heat equation, or equivalently the
gaussian kernel, from three assumptions- causality, homogeneity and isotropy
- using differential geometry. Another way to reach to the same conclu­
27
sion is to postulate that the scale-space for a continuous image should be
constructed by convolution with a one-parameter family of kernels . The
parameter should, of course, describe resolution such that a higher param­
eter value corresponds to a more blurred representation of the original im­
age. A representation at a coarser scale t2 should be possible to calculate
from the rep resentation at a fin er scale t 1 (t 2 > t 1 ) by convolution with a
kernel from the family. This can b e formulated as a semi-group property
T (·; s) * T(·; t) = T(·; s + t). We require each kernel to be shift-invariant
and to possess the same variation diminishing property as a discrete scale­
space kernel i.e. we require that the number of local extrema9 in a convolved
image must not exceed the number of local extrema in the original image.
The latter two conditions lead us to study the Polya frequency fun ctions10 ,
the continuous correspondence to the P6lya frequency sequences. As in the
discrete case this class of function s does precisely consist of continuous shift­
invariant scale-space kernels, see [9] p83-84. A theorem by Karlin [13] p354
shows that these conditions uniquely define the gaussian family of kernels.
Theorem 5 Let kt(x): X X R+--+ R (0 < t < oo) denote a one-parameter
family of P6lya frequency functions integrable on the real axis and fulfilling
the semi-group property
(36)
Suppose also that kt( x) is Borel-measurable as a function oft. Then, nec­
essarily
- oo
< x < oo; t > 0 6 E R
(37)
Consequently, this theorem provides both a new formulation of the one­
dimensional scale-space theory for continuous images as well as a further
justification of the selection of the gaussian kernel as the canonical contin­
uous scale-space kernel. The assumption of Borel-measurability means no
9
In the continuous case the va riation-diminish.i ng p roperty is normally expressed in
terms of zero-crossings. Thus, this formulation is valid only if the differentiation oper­
ator commutes with the convolution operator. In problems occur we prefer to base the
discussion on zero-crossings instead.
10
A continuous function k : R - R is said to be a Polya frequency function if the
function f( : R x R - R defined by K(x , y) = k(x- y) is totally positive i.e. if all
minors on the form (29) are non-negative for arbitrary selections of Xt < X2 < ... < Xr
and Yt < Y2 < ... < Yr from R.
28
important restriction. It is well known that all continuous functions are
Borel-measurable.
In this one-dimensional case we based the foundation on the following
fundamental assumptions:
• As we go from a finer to a coarser level of scale (increasing t) the
number of local extrema must not increase.
• The smoothing method must be independent of both position and the
image values (linear shift-invariant filtering).
• The scale-space family of functions should be constructed from a one­
parameter family of kernels possessing a semi-group property.
The last condition makes it possible to calculate the representation L(·; t2)
at a coarser level t 2 from the representation L(·; ti) at a finer level t 1
(t 2 > ti) using convolution with a kernel from the one-parameter family. In
summary,
L(·; t2) ={definition}= T(·; t2) * f ={semi-group}=
(38)
(T(·; t2- tt) * T(·; t1)) * f = {associativity} = T(·; t2- ti) * (T( ·; tl) *f)= {definition}= T(· ; t2- t 1 )
4.2
* L(·;
t 1) The Discrete Analog of the Gaussian Kernel
When we construct a scale-space for discrete images we proceed in the same
way as in last section. Let us be a little more formal this time. Given
any one-dimensional discrete image f : Z - R we define a derived one­
parameter family of discrete images L : Z x R+ - R. It is this entire
family of functions regarded as one unit that we denote as the scale-space.
The parameter t is meant to somehow describe the resolution at the current
level. We start at the finest resolution level available and let L(x; 0) = J(x)
(x E Z) . The last two assumptions mentioned in the end of last section lead
us to postulate that the family should be generated by convolution with a
one-parameter family of di screte kernels T: Z x R+ - R,
00
L(x; t) =
L
T(n; t)f(x- n)
(x E Z, t > 0)
(39)
n =-oo
and that the family must possess the semi-group property T ( ·; t ) *T(-; s) =
T(· ; t + s). The firs t two assumptions from last section combined wit h the
29
complete characterization of shift-invariant scale-space kernels in Proposi­
tion 7 leads to the requirement that each kernel in the family must be
equivalent to a normalized P F 00 -sequence. A discrete version of Theorem 5,
also by Karlin [13] p450, says that under these conditions only one class of
discrete functions is possible.
Theorem 6 The only semi-group of normalized P6lya frequency sequences
has a genera ting function on the form
(40)
Another reasonable requirement is that the kernel in no way should prefer
left to right or the other way round. Thus, we only consider symmetric
kernels where c_i = Ci V i. Then the generating function must necessarily
satisfy cp( z- 1 ) = cp( z) which in our case leads to q_ 1 = q1 . For simplicity we
let q_ 1 = q1 = ~. This means no serious restriction since it only affects the
scaling of the scale parameter t. Then,
'Pt(z)
= ef(z-l+z} =
00
L
In(t)zn
( 41)
n=-oo
As filter coefficients In we recognize the modified Bessel functions of integer
order n, which are the solutions to the differential equation:
d2w
t2 dt2
dw
+ tdt- (t2 + n2)w =
0
(42)
These functions with real arguments are except for an alternating sign se­
quence equal to the ordinary Bessel functions l n of integer order with purely
imaginary arguments. They can also be expressed as an infinite sum. In
this context i denotes the imaginary unit.
n
.
ln(t) = L n(t) = ( -1) ln(zt) =
oo
L
k=O
(tj2)n+2k
k' (
k)' (n 2:: 0, t > 0)
• n +
·
(43)
It has been pointed out by Norman [17] that this kernel is the proper
di screte analog of the gaussian kernel. We obtain a normalized kernel if we
let T : Z x R+ - R be d efined by
(44)
This is easily understood by setting z to 1 in the generating function ef (z - l+z} =
L~= -oo ln(t)zn . Then it follows that L~=-ooln(t) =et, which means that
30
L~=-oo
T(n; t) = 1. The semi-group p roperty is trivially preserved after
normalization.
For the special case t = 0 it holds that
1 if n = 0
In(O) = Ln(O) = h(n) = {
tl
.
0 o 1erw1se
(n 2: 0)
Thus, the convolution expression (39) is valid for t = 0 as well.
We summarize this very important conclusion in the following proposi­
tion.
Prop o s ition 8 Given any one-dimensional image f : Z --+ R let L : Z X
R+ --+ R be a one-pammeter family of func tions defined by L(x; 0) = f(x)
(x E Z) and L(x; t) = L~= -oo T(n; t)f(x- n) {x E Z, t > 0), where
T : Z X R+ --+ R is a one-pammeter family of symmetric functions satisfying
the semi-group property T(·; s)*T( ·; t) = T( ·; s+t) and the normalization
cr·iterion L~=-oo T(n; t) = 1. For all images f it is required that if t2 > t1
then the number of local extrema in L(x; t2) must not exceed the number of
local extr·ema in L(x; t1). Then necessarily, T(n; t) = e-atln(at) for some
non-negative real a, where I n are the modified Bessel f unctions of integer
order.
Conseq uently, these continuous and discrete results provide us with explicit
controlled methods to preserve structure in the spatial domain as we let the
original image erode by blurring it to coarser levels of resolution through
scale-space.
5
Numerical Imple me ntation of the Discrete Scale­
Space
According to the definition of the scale-space for discrete images the repre­
sentation of an image f at a scale-level t is given by,
00
L(x; t) =
I:
T(n; t) f (x - n)
n=-oo
(x E Z, t > 0)
( 45)
where T(n; t) = e-tin(t). When this transformation is to be implemented
computationally a few numerical problems must be considered:
• The infinite convolution sum must be replaced with a finite one.
31
• Normally, the modified Bessel functions are not available as standard
library routines. Therefore, we must design an algorithm to generate
the required filter coefficients T( n; t) for a given value oft.
• A realistic image is finite, but a finite approximation of (45) might
need additional values.
In this section we will discuss the first two items. We will not go into the
complications11 , which arise from finite images. Instead we assume that
f is defined for all those integers, where image values are required for our
algori tluns.
5.1
Truncation of the Infinite Convolution Sum
A reasonable approach to approximate (45) is to truncate the infinite sum
for some sufficiently large value of N,
N
L(x; t) ~
L
T(n; t)f(x- n)
(x E Z, t > 0)
( 46)
n=-N
chosen such that the absolute error in L due to truncation does not exceed
a given error limit C:trunc· If we assume that f is bounded (lf(x)l ~M) we
get the sufficient condition
00
2M
L
T(n; t) :S
C:trunc
(47)
n=N+1
Formally, if f is not defined for all integers the convolution {45) is undefined. If we
instead use the finite approximation ( 46) the values of L become undefined as soon as a
part of the convolution mask reaches outside the given image. Thus, the size of a blurred
image will diminish when we go to coarser levels of scale. However , in general the filter
coefficients in the outer regions of the kernel are relatively small compared to the central
ones, which means that the value of L should not be very much affected if only a minor
part of the convolution mask stretches outside the given image. With this motivation in
mind it is possible that we could get a better result by extending f in some way. Several
ad hoc extension methods have been invented, but they seem to suffer from the intrinsic
disadvantage that the quality of result depends very much upon the image.
It is also possible to use a genuine finite approach. A main issue, which then has to be
answered, is how the finiteness should be treated i.e. if a finite image should be regarded
as a fixed-size photograph or a window to the infinite real world.
In some sense the problems due to infinite images seem somewhat artificial. One way
to avoid them could be to use an alternative camera geometry, where the image values are
mapped onto a sphere/circle instead of an infinite plane/line.
11
32
In order to estimate N as a function of t and eiXfc we need a simpler
expression for the remainder. In [15] Section A.2 it is shown that
(48)
where
oo
SN+ 1 =
1
""
+~
'YN+1
n
_
'YN+1
fli==l (N + 1 + i) - l + N + 2 + (N
2
'YN+1
+ 2)(N + 3)
+...
( 49 )
and
I+~
t
IN+ 1 =
e \lh;~
2 . _(_1_+-~V-N+_\_1)-N=-:-+-:-1 '
1
aN+1 = --;===::=====
2
(Ztt) + (N~1f
(50)
If fS'.t~ < 1 the sum ( 49) can be estimated by a comparison with a geometric
series. Then,
(51)
Otherwise it can be calculated from the recurrence relation
n+l
Sn+1 = - - (Sn- 1)
'YN+l
So=
e'YN+I
(52)
Starting from these expressions it is not difficult to write an algorithm , which
for a given value oft returns a sufficiently large value of N. The criterion is
not sharp 12 and will in general overestimate N. However , once a sufficient
number of filter coefficients has been calculated, it is easy to determine
how many that are actually needed from the condition L,~==- N T(n; t) 2:
1-
!wL=
2M ·
12
We do not claim that the estim ation given by (47) and (48) is optimal in any way.
However, it is sufficient for our purpose of implementation.
33
5.2
Calculation of the Filter Coefficients
Given N, the next step is to calculate T( n; t) (n = - N .. N). It is not
particularly efficient to use the series expression ( 43) since the sum converges
slowly for large values oft. Instead we make use of recurrence equation.
2n
l n- I(t)- ln+I(t) = -tl n(t)
(53)
This relation is always unstable for upward recurrence , but stable in the
reverse direction. We use Miller's algorithm [18] p142 and start the recur­
rence with an arbitrary seed IN.tart = 1 and I N.tart+ I = 0 for a sufficiently
large start index Nstart· As n decreases the iterates obtained from (53) will
successively approach the correct solution. One can show (see [18] p175,711)
that if d significant digits are required in IN it is enough to start at
Nstart = 2(N + ch/N)
(54)
The sequence of iterates can be normalized if ! 0 ( t) is calculated by a separate
routine. A sketch of an algorithm generating the filter coefficients is given
in Appendix A.2. Tables with coefficient values for a few values of t are
supplied in Appendix A.l.
5 .3
Asymptotic expressions for T(n; t)
We close this section with some asymptotic expressions for the filter coeffi­
cients.
1
T(n; t) ~ J7C;
(55)
(n < < t)
y 27rt
T(n; t)
6
~
-e-t1
n.
(t)
-
2
n
(n
> > t)
(56)
Discrete Scale-Space Properties of Some Nu­
merical Approximations of the Continuous Scale­
Space Theory
In this section we will consider some numerical approximations, which are
close at hand for the convolution integral (1) and the heat equation (2).
Using the results derived in previous sections we will investigate if the oc­
curring transformations possess scale-space properties in the discrete sense.
34
The aim is to analyze the previously commonly adapted approach where the
filter coefficients are set to sampled values of the gaussian kernel. We show
that some undesired effects occur, mainly due to the fact that the semi­
group property does not hold after discretization. We also show that the
transformation obtained by convolution with the presented discrete analog
of the gaussian kernel is equivalent to the solution of a discretized version
of the heat equation. This provides another motivation for the selection of
T as the canonical discrete scale-space kernel.
The rendering is of necessity somewhat technical and can be skipped by
the hasty reader.
6.1
The Rectangle Rule of Integration
Maybe the most obvious way to approximate the convolution integral
L(x;t) =
l
oo
1
2
r;c:;e-f. / 2tf(x- Od~
f.=-oo V 27rt
(x E R,t > 0)
(57)
numerically is to first truncate the interval of integration at sufficiently large
values and then apply the rectangle rule of integration to the remaining part.
L(x;t) ~
f
N+J.2
jf.=-(N+~)
g(~;t)f(x- Odf. (x E R ,t > 0)
Preferably N is chosen as an integer. If it is known that
(58)
f is bounded
(lf(x)l :S M) we can choseN such that the error13 in L(x; t) due to trun­
cation of the integral does not exceed a given error limit E:trunc ·
r-(N+~)
}f.=-oo
I
r oo
I
g(~; t)f(x- Od~ + }f.=N+~ g(~; t)f(x- ~)d~ :S E:trunc
(59)
This inequality does certainly hold if 2 *M* erf( -(N + !)f,fi) :Se:, where
erf denotes the error function 14 . The rectangle rule of integration then gives
13
Approximating Lex; t) numerically raises another question, which we however will
make no attempt to answer. Given that ftn is bounded ll(x)l $ M and afflicted with
an error not exceeding t:in and that we accept errors of magnitude smaller that t:out in
l out· What are then the requirements of the kernel if we demand that the transformation
l ou t = K * hn must b e a scale-space transformation modulo local extrema within the error
margins.
2
14 erf(x) = 1
f:r:
e-~ f 2 d{
~ J~=- 00
35
the following approximation formula
N
L(x; t) ~
L
g(n; t)J;n(x- n) (x E Z, t > 0)
(6 0)
n=-N
which we recognize as a discrete convolution. The filter coefficients
g(i; t) are values of the gaussian kernel sampled at the node points.
Ci
= g(i;
t)
= -..[2ii
-1 e - •2/2 t
1
Ci
(61)
From the definitions of PF00 -functions and PF00 -sequences it is clear that
uniform sampling of a P F 00 -function will produce a P F 00 -sequence. Sub­
sequently, if N goes to infinity the approximating transformation (60) pos­
sesses scale-space properties.
However, the semi-group property will not hold exactly after discretiza­
tion. Therefore, if we calculate the representation at a level t 2 > 0 from
the original image we will not obtain the same result as we get if L(·; t 2 ) is
calculated via an intermediate level t 1 (0 < t 1 < t2) applying the approxi­
mation formula (60) in two steps. Of course, we can use the formula (60)
to generate the discrete representations at all levels of scale directly from
the original image. But then we are not guaranteed that new local extrema
cannot be created from a level t 1 f. 0 to a coarser level t2. In Appendix B.1
we show that the transformation between these levels does in general not
possess scale-space properties, i.e. the representation L(x; t 2 ) might have
more local extrema than L(x; t 1 ) although t2 > t1.
Another minor disadvantage with this approximation appears for small
values of t (t ~< 1). Then the coefficient sequence is dominated by a
very sharp peak at eo . Even thought the integral of the continuous kernel is
normalized to one the central peak can drive the sum of the filter coefficients
to a value substantially greater than one. In some way the sampled gaussian
kernel appears as having a smaller sigma-value than it should, see Fig 9.
A renormalization does of course not solve the problem since the mutual
relation between the filter coefficients remains unchanged.
6.2
Integrated Values of the Gaussian Kernel
One a d hoc way to avoid that the central coefficient c0 becomes too dominant
for small t-values is to use integrated values of the gaussian over the pixel
36
-1
1
2
3
Figure 9: The sampled gaussian kernel is donilnated by the central coefficient
for small values oft. The peak value is greater than one when t is smaller
than 1/2rr.
support region instead of sampled values . One can for example let
j'+2
_1_ e-e2
i-~ yl2it
.
ci
=
I
/2t
d~
(62)
This choke of filter coefficients is equivalent to the continuous formulation
(57) if we let the continuous image f be a piecewise constant function,
which is equal to the discrete pixel value over each pixel support region.
One can show, [15] Section 6.2, that the kernel given by (62) is a scale-space
kernel. This means that the transformation from original image (t = 0) to
an arbitrary level of scale (t 1 > 0) is always a scale-space transformation.
However, we cannot expect a semigroup property to hold exactly and will
probably arrive at similar scale-space problems as with the sampled gaussian
kernel.
6.3
Discretizing the Heat Equation in Space
Consider the one-dimensional heat equation
(63)
with initial condition L(x ; 0) = f(x) for some given image f. We discretize
it in space at an equidistant grid Xi= i (i E Z) with the usual approximation
fx¥(xi) ~ ~(Li+1 - 2Li + Li- 1) = {h = 1} = Li+l - 2Li + Li-1· Li
37
does of course denote L(xi; t). This leads to a system of coupled ordinary
differential equations
(64)
with initial conditions Li( 0) = J( i). These equations can be further dis­
cretized in scale. Using Eulers explicit method with uniform step length D.t
(tk = ktlt; *(tk) ~ lt(Li,k+l- Li,k)) we get the recursion formula,
(65)
which we recognize as a discrete convolution in space. The kernel Kstep
has the filter coefficients c_ 1 = ~t, eo = 1 - D.t and c1 = ~t and the
generating function IPstep(z) = ~t z- 1 + (1 - D.t) + ~t z. From the scale­
space requirement for three-kernels
2: 4c_ ICI derived in Section 2.7 it
follows that the transformation (65) possesses scale-space properties if and
only if
c5
D.t
1
<
- ­2
(66)
It is not too difficu lt to get the analytical solution to the system of
scale-continuous equations (64). Assume that we want to calculate the so­
lution for a fixed t-value. We can use the discretization (65) with n steps
in the scale-direction such that the step length D.t = tfn satisfies (66) .
As each iteration step consists of a linear convolution the final solution
achieved can equivalently be obtained by convolution with the composed
kernel K composed = *~I Kstep· Let us derive an asymptotic expression for its
generating function.
As the kernel describing one iteration is a symmetric scale-space kernel it
can be decomposed into two-kernels with non-negative coefficients such that
IPstep(z) = (p + qz-1 )(p + qz). By identification of the two representations
of cp8 tep(z) we get the equation system
D.t
pq = -
2
;
(67)
One verifies that the only non-negative solutions are
p
= J1- tlt +2v'1- 26,t = 1- -D.t
+ O( ut2)
2
A
38
(68)
q=
1 - /J.t - .J1 - 2/J.t
/J.t
=2
2
V
+ 0( ut 2)
A
and the solution corresponding to interchanged values of p and q. The
generating function of the composed kernel then becomes,
IPcomposed,n( Z)
= (lPstep( Z) t = (p + qzt(P + qz- 1 )n
( 69)
which can be written as
lPcomposed n( Z) = (1- _!_(1- z- 1 ) + 0( ...!:_2 )t(l- _!_(1- Z) + 0( ...!:_2 ))n (70)
'
2n
n
2n
n
after substitution of -k for !J.t in the Taylor expansions for p and q. Since
limn--.. 00 (1 + ~ )n = e01 if limn--.. 00 Cl'n =a it follows that
(71)
We recognize the generating function of the family of discrete kernels we
arrived at when we constructed the discrete scale-space in Section 4.2. e-t
is the normalization factor. Consequently, the transformation obtained by
convolution with the discrete analog of the gaussian is equivalent15 to the
analytical solution of the system of equations obtained by discretizing the
heat equation on a fixed equidistant grid in space. In other words (39) is
the solution of (64). This is not surprising bearing Theorem 6 in mind.
7
7.1
Summary and Discussion
The One-Dimensional Case
The aim of this paper has been to investigate the discrete aspects of the
one-dimensional scale-space theory. We have studied linear shift-invariant
transformations and stated a requirement on kernels saying that the num­
ber of local extrema in a convolved image must not exceed the number of
local extrema in the original image. As an immediate consequence we saw
that the coefficient sequence must be non-negative and unimodal. For sym­
metric kernels the same requirements hold for the Fourier spectrum. We
showed that the interesting kernels could be completely classified in terms
15
The observant reader notes that the conclusion is valid only if the discretization (65)
converges to the solution of the continuous equations (64) when b.t _. 0. This does for
instance hold if f E l 1 or f E l2.
39
of sign-regularity - all shift-invariant scale-space kernels are equivalent to
normalized P6lya frequency sequences. The generating function of such a
sequence/kernel possesses a very simple characterization.
Then we introduced a continuous scale parameter and showed that the
only reasonable way to define a scale-space for discrete images is by convolu­
tion with the one-parameter family of kernels T(n; t) = e - ti n(t), where In
arc the modified Bessel functions of integer order. Similar arguments applied
in the continuous case uniquely lead to the gaussian kernel. The kernel T
does also have the attractive property that it is equivalent to the limit case
of a certain di scretization of the heat equation. The idea of a continuous
scale parameter even for discrete images is of considerable importance, since
it p ermits arbitrary degrees of smoothing, i.e. we are no longer restricted
to specific predetermined levels of scale. Due to the semi-group property
the scale-space condition holds between any two levels of representation.
\Ve showed that the commonly used technique, where the "scale-space" is
constructed by convolution with the sampled gaussian kernel, might lead to
undesirable effects, since in general the transformation from an arbitrary fine
level to a randomly selected coarser level is not a scale-space transformation.
An important point with t his result is that it constitutes an example of
a property, which has been derived in the continuous case, but does not hold
after di scn~tization.
The treatment concentrates on one-dimensional images defmed for all in­
tegers. \Ve have not gone into the complications that occur at the boundary
if the image function is defined only for a subset of the integers. However,
there is one thing we want to emphasize if the proposed method is applied
to a finite image, using some extension method. If one wants the semi-group
property to hold exactly (except for rounding errors) it is necessary that all
reprcsen tations at all levels are generated di1·eclly from the original extended
image using the approximation (46). If several small steps are taken then
the intermediate representations must be truncated, which means that we
might get a different result, since the semi-group property no longer holds
exactly.
Of course, a genuine finite approach is also possible. In this presentation
we have chosen not to develop the subject, since the associated problems
are somehow artificial and difficult to handl e in a consistent manner. There
is no getting away from the fact that all finite images have boundaries and
that problems arise if one tries to analyze objects near them. Dy necessity,
the peripherical image values of a smoothed finit e image will be less reliable
than the central ones. However, in a practical case additional data can often
40
be acquired simply by moving the camera such that image values become
available in a sufficiently large neighbourhood of the object of interest. For
some very simple cases it might be enough do an ad hoc extension. But this
requires a priori information about the scene.
One way to avoid both the infiniteness and the boundary problems
could be to use an alternative camera geometry, where the image values
are mapped onto a sphere/circle instead of aJl inftnite plane/line. The idea
seem s rather natural from a biological point of view. Then, the ordinary pla­
nar camera geometry would appear as an approximate description for foveal
vision, i.e. small solid angles in the central field of vision. This approach is
very closely related to projective geometry.
7.2
Extension to Two Dimensions
The extension to two dimensions is not obvious since it is possible to show
that there does not exist any non-trivial kernel on R 2 or Z 2 having the
property that for all images the number of local extrema in the convolved
image does not exceed the number of local extrema in the original image.
Pizer and Lifshitz [14] present an illuminating counter-example:
Imagine a two-dimensional image function consisting of two hills, one of
them somewhat higher than the other one. Assume that they are smooth
wide rather bell-shaped surfaces situated some distance apart clearly sepa­
rated by a deep valley running between them. Connect the two tops by a
narrow sloping ridge without any local extrema. Then the top of the lower
hill is no longer a local ma...·dmum. Let this configuration be the input im­
age. When the heat equation is applied to the geometry the ridge will erode
much faster than t he hills. After a while it has eroded so much that the
lower hill appears as a local maximum again. Thus, a new local extremum
has b een created.
The same argument can be carried out in the discrete case. Of course,
we have to consider connectedness when we define what we mean by local
extrema. But this question is only offorma.l nature. Given an arbitrary non­
tri vial convolution kernel it is always possible to create a counter-example.
The width of the ridge can be set to one pixel. ·w hen the convolution
kernel is applied to the image the ridge will erode much faster than the hills
provided that the hills are sufficiently wide and that the valley is chosen
deep and wide enough. Therefore, it is not clea.r what we should mean
with a scale-space property in two space dimensions. In one dimension
several formulations arc equivalent. Does there for example exist a natural
41
/
0
/
Figure 10: New local extrema can be created by the heat equation in the
two-dimensional case
formulation su ch that a variation-diminishing property holds in some weaker
sense ? Or, can we express an axiom, which in one dimension is equivalent
to t he formulation in terms of local extrema, and directly generali zes to
higher dimensions. It would be very interesting to get the answers to these
questions both in the continuous and the discrete case.
Is it true t hat the discrete an alog of the gaussian kernel used as a sep­
arated kernel is the natural discrete kernel in two dimensions ? If one due
to computational consider ations wants to use separable discrete kernels, one
could, of course, heuristically argue that the kernel should at least have a
good performance in one dimension. Another indication in that direction
can b e taken from the t wo-d imensional heat equ ation discretized on a fixe d
equidi stant grid in space, such th at the con volu tion kernel , describing one
iteration along the scale direction , is separ able in space. In Appendix D.2
we show that as the step length along th e scale-axis goes to zero the solution
of the discrete approximation approaches the result we get if we apply the
one-dimensional discrete analog of th e gaussian kernel as a separated kernel
- first along one coordi nate direction and t he along the ot her. Subsequently,
convolution with the presented discrete analog of the gaussian kernel de­
42 scribes the solution of the system of equations which appears if the heat
equation discretized in space but solved analytically in time. This result
hold s both in one and two dimensions.
8
A cknowle dgme nts
I would like to thank P rof. Jan-Olof Ek1undh for a lot of valuable advice.
Hi s guidance and support and his way to always find time for discussions
is very much appreciated. I also wish to thank Dr. Stefan Carlsson for
stimulating discussions, which provided a large source of inspiration to thi s
work.
This paper describes research conducted at the Computer Vision and As­
sociative Pattern Processing Laboratory (CVAP ), Royal Institute of Tech­
nology, Stockholm, Sweden. The support from the National Swedish Board
for Technical Development is gratefully acknowledged.
43
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[21] Schoenberg I.J. (1948) "Some Analytical Aspects of the Problem of
Smoothing", Courant Anniversary Volume, Studies and Essays, New
York, pp351-370.
[22] Schoenberg I.J. (1953) "On Smoothing Operations and their Generating
Functions", Bull. A mer. Math. Soc., 59, pp199-230.
[23] Toeplitz 0. (1911) "Zur Theorie der quadratischen und bilinearen
F01·men von unendlichvielen Veranderlichen. I. Teil: Theorie der 1­
formen", Mathemalisch e Annalen, 70 , pp351-376.
[24] Whittaker E.T., Watson G.N. (1952) Mod em Analysis, Cambridge Uni­
versity Press.
[25] Witkin A.P. (1983) "Scale-Space Filtering", Pmc. 7th International
Joint Confe7'ence on A1'lificial Intelligence, pp1019-1022.
45
A
Numerical Implementation of the Discrete Scale­
Space (Continued)
A.l
Tabulated Values of the Discrete Analog of the Gaus­
sian Kernel
For reference purpose we supply some kernel values for t = 1 and t = 10.
They have been generated by the algorithm presented in Section 5. The
kernel size was determined from ( 47) and ( 48) with Etrun c = 0.25 and M =
255. The estimation yielded N = 5 and N = 13 while the correct values
were N = 4 and N = 11.
±n
T(n; 1)
0
1
2
3
4
4.657596e-01
2.079104e-01
4.993877e-02
8.155307e-03
1.006930e-03
±n
T(n; 10)
0
1
2
3
4
5
6
7
8
9
10
11
1.278333e-01
1.212626e-01
1.035808e-01
7 .983036e-02
5.568258e-02
3.528429e-02
2.039829e-02
1.080634e- 02
5.269408e-03
2.375292e-03
9.938819e-04
3.875283e-04
A .2 Algorithm Sketch Filter Coefficient Generation
Given: t = t =Scale parameter value
d = d = Number of significant digits in the filter coefficients
epsilon = t:trun c = Maximum truncation error in the ou tput image pixel values
M= M = Upper bound for the input image pixels values ihn(x)i:::::; M
maxreal Maximum real number allowed in hardware
=
• Determine an upper bound for N either from the estimation (48)-(52) or a
pre-calculated table.
Nestimated = estimate_N(t, epsilon/(2•M));
• Determine a start index for the iterations.
Nstart = integer(2•(Nestimated + d*sqrt(Nestimated));
46
• Initial values.
T(Nstart) : = 1;
T(Nstart+1) := 0;
• Calculate iterates from the recursion formula.
for n : = Nstart step -1 until 1 do begin T(n-1) := T(n+1) + 2•n•T(n)/t; if T(n-1) > maxreal then for i := n-1 step 1 until Nstart do T(i) /= maxreal; end for; • Renormalize the iterates
renorm := T_O(t)/T(O); for n := 0 step 1 until Nstart do T(i)
renorm; •=
where T_O is a routine which calculates T0 (t) , see [18] p711.
real procedure T_O(t); real t; begin real z; if (t < 0) then error; else if (t < 3.75) then begin z := t•t/(3.75•3.75);
T_O .- exp(-t)•(1.0 + z•(3.5156229 + z*(3.0899424 +
z•(1.2067492 + z•(0.2659732 +
z•(0.0360768 + z•0.0045813))))));
end;
else begin
z := 3 . 75/t;
T_O . - 1.0/sqrt(t)•(0.39894228 + z•(0.01328592 +
z•(0.00225319 + z•(-0.00157565 +
z•(0.00916281 + z•(-0.02057706 +
z•(0.02635537 + z•(-0.01647633 +
z*0.00392377))))))));
end if-else ; end procedure; • Determine how ma ny coefficients that must be used in the discrete mask in
order to obtai n the desired accuracy.
47
N := 0;
sum := T(O); while (sum < 1-epsilon) do begin N += 1;
sum+= 2*T(N); end while; • Now the filter coefficients are given by
Cn
= T(n) (n = -N.. O..N).
B Numerical Approximations to the Continuous
Scale-Space Theory (Continued)
B.l Scale-Space Violations for the Sampled Gaussian Ker­
n el
Assume that we construct the "scale-space" for a discrete image by convolu­
tion with the sampled gaussian kernel, i.e. given a discrete image f : Z -+ R
we define the family of functions L : Z x R+ -+ R by L( x; 0) = f( x) (x E Z)
and
L(x; t) =
L
g(n ; t)f(x - n)
(x E Z,t > 0)
(72)
n=-oo
where
(nE Z, t > 0)
(73)
Earlier we have shown that for all images f the number of local extrema in
L(x; t) (t > 0) does not exceed the number of local extrema in f . In this
section we will show that this scale-space property does not hold between
two arbitrary levels.
We will need an expression for the generating function for the discrete
kernel corresponding to the sampled gaussian. For simplicity we let qt =
1
e-21. One can show [21] that
where
~
Ct = y 2rrt
fr (1 - qzn)
(75)
n=l
Let t1 and t2 be two levels (t2 > t 1 > 0) of the representation (72) and
let <pin be the generating function of the input image. Then the generating
48
functions of l(x; t 1 ) and l(x; t2) are
(76)
Let l.?diff descri be t he transformation from l(x; t 1 ) to l(x; t2) . Thus,
<pL 2 (z)
= 'PdiJJ(z) 'PL 1 (z) (77)
Combination of (76), (77) and (75) gives
. (z)- <pL2(z )- Ct2
'Pdt f f
- r £ (z) t1
IT~=O(l + q~2m+lz)(1 + qz2m+lz-l)
(1 + n2n+l
z)(1 + q2n+l
z-1)
n=O
"lh
(]
c . noo
(f) -
1
(78)
According to the complete characterization of scale-space kernels it follows
that the corresponding kernel is a scale-space kernel if and only if (78) can
be written on the form (35) . Then, for each factor (1 + fJl1n+l z±1 ) in the
denominator there must exist a corresponding factor in th e numerator (1 +
fJZ2m+lz±1 ), i .e for each n there must exist an m such that
(79)
1
Insertion of
requirement
fJt;
e- '217 and red uction gives the necessary and sufficient
2m
= -t2 (2n + 1) tl
(80)
1
It is clear that this relation cannot hold for all n E Z if t 1 and t 2 are chosen
ar bitrarily. The transformation from l(x; t 1 ) to l(x; t2) (t 2 > t 1) is a
is an odd integer.
scale-space transformation if and only if the ratio
i7
B .2 Discretizing the Two-Dime nsiona l H eat Equation
Space
111
We will derive the solution of the two-dimensional heat equation
(81)
discretized on an equidistant grid in space. The initial condition L(x, y; 0) =
f(x, y) is given .
Let x; = i, Yi = j and L;,j = L(x;, Yi> t ) (h = 1). There are two common
discrete approximations to t he two-dimensional Laplace operator {ir + ~
49
(
1
1 -4
1
1) c /2 -2 1/2 )
P~2 )
a
-4a- 2{3
a
[3 /2
fJ/ 2
( p2
1/2
1/2
a b..t / 2
Pt.t/4
ab..t/2 1 - (2a + [J) D.t ab..t/2
{JD.t/4
ab..t/2
[J D.t /4
( pt.t/4
)
Figure 11: Computational molecules approximating the two-dimension al
Laplaceoperator (a) h 2 \lg (b) h 2 \l~ (c) h 2 (a\l g+{3 \l~) . (d) Computational
symbol for the kernel K x,y corresponding to one iteration in scale.
namely the five-point operator \lg and the cross-operator \l~ (see Fig lla
and Fig llb). Since we are interested in a separable kernel we choose the
linear combination ~ + ~ ~ a\lg + {3 \l~ where a+ [3 = 1, a :2: 0 and
[3 :2: 0 (Fig llc) . We then get a system of coupled ordinary differential
equations,
a
a
L i,j -1 + -(2a + [J)L;,j + Li,J+1 +
2
2
[3
a
[3
4 Li,j-1 + 2Li ,i + 4_Li ,i+I
We want to calculate the solution of this system for a given value of t.
Vle di scretize in scale using n equidistant points tk = kD.t where D.t =
~ - Let L ;,j,k denote L ;,j(tk) · The derivative 8 ~;-z (tk) is approximated by
~ t (L;,j,k+I- L ;,j,k) according to Eulers explicit method . Then,
{JD.t
L;,j,k+I = - - L i -l ,j- 1
4
aD.t
- - L i,j-1
2
L ;- l,j
[J D.t
+ - 4-
L i -l ,J+l
ab.t
+ (1- (2a + [J) D.t)L;,j + - 2-
[J D.t L ·.
4
a D.t
+ - 2-
t,J - l
+
aD.t L ·.
2
t ,J
+
L i,j+I
+
(83)
+
[J D.t L ·.
4
t,J+ l
The associated computational molecule is shown in Fig lld. Vve want to
investigate if the corresponding kernel J( x,y is separable for some choice of a
50
and {3 . Let Kx,y = K x * Ky where K x and K y correspond to convolutions in
the x- and y-directions respectively. Due to symmetry t he filter coefficien ts
of ](x and Ky must consist of the same set of values. Denote them by 1,
1 - 21 and 1 (~, > 0) . Then these relations must hold:
2
~,
{JD.t
- ·
-
4
'
(1 - 21) 2 = 1- (2a + {J) t::.t
(84)
D.t
2
(85)
The only non-n egative solution is
Q
= 1- D.t;
{3
= D.t ;
1= -
Subsequently, the kernel is se parable if and only if we choose a = 1- t::.t and
{3 = D.t. Then the generating function <.?x,y(z,w) = L~ - oo L~ -ooci,jZiwi
of the kernel K x,y can be written as IPx,y(z,w) = IPx(z)cpy(w) where <.?x(z) =
~t z- 1 + (1- t::.t ) + ~t z 1 and cpy('w) = ~tw- 1 + (1- t::.t ) + ~tw 1 .
The composed kern el K composed,n describing the fi nal solution after n
steps has a generating fun ction
After calculations similar to t hose performed in Section 6.3 it follow s t hat
(87)
I.e. the solution of the two-dimensional heat equation discretized in space
is obtained by convolut ion wit h the di screte analog of the gaussian kernel
applied first in one coordinate direction and then in the other.
51
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