UTMS 2005–43

November 29, 2005

Detection of irregular points by regularizationin numerical diﬀerentiation and an application
to edge detection
by
X. Q. Wan, Y. B. Wang and M.Yamamoto

Ì

UNIVERSITY OF TOKYOGRADUATE SCHOOL OF MATHEMATICAL SCIENCES
KOMABA, TOKYO, JAPANDETECTION OF IRREGULAR POINTS BY REGULARIZATIONIN NUMERICAL DIFFERENTIATION AND AN APPLICATIONTO EDGE DETECTION
X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO
Abstract. Numerical differentiation is a typical ill-posed problem which canbe treated by the Tikhonov regularization. In this paper, we prove that theL2 -norms of the second order derivatives of the regularized solutions blow up in
any small interval I where the exact solution is not in H 2 (I). One applicationin the image edge detection is presented.

1. Introduction
A numerical differentiation arises in many applications and engineering computations such as the determination of the underground water, the Dupire formulae
in financial mathematics([25]), image edge detection([13]), etc. One of the maindifficulties for this problem is the ill-posedness, which means that the small errorsof measurement may cause huge errors in computed derivatives ([7], [12], [22]).
Therefore one needs regularizing techniques for reasonable computations. Several
numerical algorithms have been proposed for overcoming the instabilities ([5], [7],[8], [9], [10], [11], [12], [16],[19], [23]). It has been shown that the Tikhonov regularization for treating the numerical differentiation problem is one of the effective
methods.
In [24], the authors discussed the numerical differentiation by using the Tikhonov
regularization method. It is shown that, if the exact solution is smooth, then the
Date: November 29, 2005.
Key words and phrases. Ill-posed problem, Numerical differentiation, Tikhonov regularization,Image edge detection.
This work is partially support by NSF of China (No. 10271032, No. 10431030) and theLaboratory of Nonlinear Sciences in Fudan University, Shanghai, China.
The third author was supported partly by Grant 15340027 from the Japan Society for the
Promotion of Science and Grant 17654019 from the Ministry of Education, Cultures, Sports andTechnology.
1

2

X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO

regularized solutions converge to the exact solution, and if the exact solution is
not smooth, then the L2 -norms of the corresponding high order derivatives of theregularized solutions will blow up in the whole interval. The numerical results
indicate that this kind of blow-up happens only near the irregular points. However
to the authors’ knowledge, this property has not been proved up to now.
In this paper we discuss this local property of the regularized solution based on
the work of [24]. We prove that, if the exact solution is not in H 2 (I) for an intervalI, then the norms of the second order derivatives of the regularized solutions in I
blow up. Although we can develop a similar method also in multidimensional cases,
we will discuss only one dimensional case, which is sufficient in many practical cases.
In particular, for the edge detection, we use line-by-line scans to apply the numericaldifferentiation for functions in a single variable (section 4).
This paper is organized as follows: We give the formulation of the problem andshow the theoretic results in section 2. In section 3 we will give numerical examples
to verify this property. An application in image edge detection is presented in
section 4. Some conclusions are given in section 5.

2. Formulation of the problem and theoretical result
Hereforth we set
0

g =

dg
,
dx

00

g =

d2 g
dx2

Suppose that ∆ = {0 = x0 < x1 < · · · < xn = 1} is a uniform grid of [0, 1]. Herexj = nj , j = 0, 1, · · · , n, and we denote h = n1 . Let y = y(x) be a continuousfunction defined on [0,1]. A noisy value of y(x) at point xj is given as yjδ which
satisfies(2.1)|y(xj ) − yjδ | < δ, j = 0, 1, · · · , n,where δ is a given constant called the level of noise in the data. The numericaldifferentiation is then to approximate y 0 (x) from the value of yjδ , j = 0, 1, · · · , n.
Without loss of generality, we assume that there are no errors at the boundary forthe sample data, i. e., y0δ = y(0), ynδ = y(1). Otherwise we can use a new function
Y (x) = y(x) + y0δ − y(0) + (ynδ − y(1) + y(0) − y0δ )x.

DETECTION OF IRREGULAR POINTS BY REGULARIZATION3

It can be easily proved that Y (0) = y0δ , Y (1) = ynδ . Since
¯ ¯
¯ 0
¯
0
¯
¯
¯y (x) − Y (x)¯ = ¯ynδ − y(1) + y(0) − y0δ ¯ ≤ 2δ0

0Therefore the approximation of Y (x) is also an approximation of y (x) (see further[12], [24]).
The numerical differentiation is to find a function f from the data {yjδ }nj=000such that f approximates y . We will solve this problem by the Tikhonov regularization method as in [24]. The following spaces and norms will be used in thispaper:

½
¾
Z 1
2
1/2
L (0, 1) = g | (g (x)dx)
<∞ ,
2

0

n
o
00
H 2 (0, 1) = g | g ∈ L2 (0, 1), g ∈ L2 (0, 1) ,C[0, 1] = {g | g is a continuous function on [0,1]},µZ 1
kgkL2 (0,1) =¶1/2
2|g(x)| dx

,

0

³
´ 12
00
kgkH 2 (0,1) = kgk2L2 (0,1) + kg k2L2 (0,1) ,kgkC[0,1] = max |g(x)|.
x∈[0,1]

Define a cost functional by
n−1

(2.2)

Φ(f ) =00
1X
(f (xi ) − yiδ )2 + αkf k2L2 (0,1)
n i=1

for all f ∈ H 2 (0, 1) with f (0) = y(0), f (1) = y(1), where α is a regularization
parameter.
Then we can prove (e.g. [24]):
Theorem 2.1. Let y ∈ C[0, 1]. There exists a unique minimizer f∗ = f∗ (δ, α, h)of functional (2.2).
The minimizer f∗ is called the regularized solution, and we have the following
error estimate ([24]).

4

X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO

Theorem 2.2. Suppose that y ∈ H 2 (0, 1). Then we have the following error estimation for the regularized solution f∗ (δ, α, h):
r
µ
¶
0
0
00
1
h
δ2
2δ
kf∗ (δ, α, h) − y kL2 (0,1) ≤ 2h + 4α 4 +ky kL2 (0,1) + h
+ 1.
π
α
α4
If we choose α = δ 2 , then
√
√
0
0
00
h
kf∗ (δ, δ 2 , h) − y kL2 (0,1) ≤ (2h + 4 δ + )ky kL2 (0,1) + h + 2 δ.
π
It is practically important to detect subintervals where a state function f is00

not smooth. Thus we are more interested in an interval (a, b) ⊂ (0, 1) where y ∈/L2 (a, b). Therefore, it is necessary to know the behaviour of the regularized solutionf∗ in a subinterval as short as possible. In particular, if (a, b) = (0, 1), then thefollowing theorem is proved in [24].
Theorem 2.3. If y ∈ C[0, 1]\H 2 (0, 1), and we choose the regularization parameterα = δ 2 , then
(2.3)

00

kf∗ (δ, δ 2 , h)kL2 (0,1) −→ ∞,asδ, h → 0.

Remark 2.4. The general study corresponding to Theorem 2.3 is found in [3].
This result indicates only that, if the exact solution is not in H 2 (0, 1), then theL2 -norms of the second order derivatives of the regularized solutions on the wholeinterval [0, 1] blow up. In this paper we will give a localized version in (a, b) ⊂ (0, 1)of this theorem. We want to know if the exact solution is not smooth on a small
00

interval, namely, y ∈ C[0, 1] \ H 2 (a, b), what will happen to kf∗ kL2 (a,b) ?
We state our theoretical result on which our numerical method is based.
Theorem 2.5. Suppose that y ∈ C[0, 1], (a, b) ⊂ (0, 1), and we choose the regularization parameter α = δ 2 . If y ∈/ H 2 (a, b), then the regularized solution f∗ (δ, δ 2 , h)satisfies00

lim kf∗ (δ, δ 2 , h)kL2 (a,b) = ∞.

δ,h→0Remark 2.6. For a piecewise continuous function, the conclusion in Theorem 2.5 isstill true. We can prove it by the same method with some small modifications.
In the proof of this theorem, we will use an interpolation inequality. (e.g., Theorem 4.14 (p.75) in [1]):DETECTION OF IRREGULAR POINTS BY REGULARIZATION

5

Lemma 2.7. Let −∞ ≤ a < b ≤ ∞, 1 ≤ p < ∞ , and 0 < ²0 < ∞, f have them-th order derivative f (m) in (a, b). There exists a constant K > 0 which depends
on ²0 , p, m, and b − a such that for every ², 0 < ² ≤ ²0 , 0 ≤ j < m, we have
Z b
Z b
Z b
−j
1
1
1
(j) p
(m) p
p
p
m−j
(2.4)
(
|f | dt) ≤ K²(|f | dt) + K²(
|f |p dt) p .
aaa

Proof of Theorem 2.5: Henceforth for simplicity, we setf∗ (δ, n)(x) = f∗ (δ, δ 2 , h)(x)where we recall h = n1 .
Assume contrarily that the conclusion of the theorem is not correct. This meansthat there exist two sequences {δk }, {hk }, k = 1, 2, · · · , such thatlim δk = lim hk = 0

k→∞

k→∞

and
(2.5)

00

kf∗ (δk , nk )kL2 (a,b) ≤ M,

k = 1, 2, · · ·where M is a positive constant and nk = h1k .
By y ∈ C[0, 1], we can take a sequence of functions ym ∈ H 2 (0, 1) satisfying
ym (0) = y(0),ym (1) = y(1),

and
(2.6)

kym − ykC[0,1] ≤1,m

sup kym kL2 (a,b) < ∞.
m

In fact, ym can be constructed for example by suitable interpolated polynomials.
Next we will prove
(2.7)

00

sup kym kL2 (a,b) = ∞.
m

00

Assume contrarily that supm kym kL2 (a,b) < ∞. Then from the definition of thenorm, we know that supm kym kH 2 (a,b) < ∞. By the reflexiveness of H 2 (a, b), there
exists a subsequence ymk ∈ H 2 (a, b) and ye ∈ H 2 (a, b) so that ymk → ye weakly in
H 2 (a, b). On the other hand, we see from (2.6) that ym → y strongly in L2 (a, b),so ye = y and thus we have y ∈ H 2 (a, b). This is a contradiction. Thus the proof of(2.7) is complete.

6

X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO

Moreover, for each k ∈ N , we set
00`(k) = min{j ∈ N ; δj kyk kL2 (0,1) < 1}.
Such an `(k) exists uniquely because δj → 0 as j → ∞. By (2.7), we note thatlimk→∞ `(k) = ∞. For simplicity we denote δ`(k) and n`(k) again by δk and nk
respectively. Hence00δk kyk k2L2 (0,1) < 1.

(2.8)

We set f∗k = f∗ (δk , nk ). Since Φ(f∗k ) ≤ Φ(yk ), we have
(2.9)

nk −1 ³´21 X
f∗k (xj ) − yjδk ≤ Φ(f∗k )nk j=1nk −1
00
1 X≤ Φ(yk ) =(yk (xj ) − yjδk )2 + δk2 kyk k2L2 (0,1)
nk j=1≤

nk −1 ³´
00
2 X
(yk (xj ) − y(xj ))2 + (y(xj ) − yjδk )2 + δk (δk kyk k2L2 (0,1) )nk j=1≤2+ 2δk2 + δk .
k2

At the last inequality, we used (2.8) and1,k|yk (xj ) − y(xj )| ≤1 ≤ j ≤ nk − 1,by (2.6).
Suppose that xi0 −1 ≤ a < xi0 and xj0 < b ≤ xj0 +1 . Then by (2.9), we have
´2
1 X³
f∗k (xj ) − yjδknk j=i≤nk −1 ³´21 X
f∗k (xj ) − yjδknk j=1≤2+ 2δk2 + δk .
k2

j0

0

Moreover|yjδk |2 = (yjδk − y(xj ) + y(xj ))2≤ 2((yjδk − y(xj ))2 + y(xj )2 )≤ 2(δk2 + kyk2L∞ (0,1) ).

DETECTION OF IRREGULAR POINTS BY REGULARIZATION

7

Therefore by (2.1) and (2.9)
j0
j0
´
1 X 2
2 X³
f∗k (xj ) ≤(f∗k (xj ) − yjδk )2 + |yjδk |2
nk j=ink j=i(2.10)

0≤0

4+ 4δk2 + 2δk + 4(δk2 + kyk2L∞ (0,1) ) ≤ A.
k2

Here A > 0 is a constant which is independent of k ∈ N . Moreover, denote
ηi0 = a,ηi =xi−1 + xi, i = i0 + 1, · · · , j0 ,2

ηj0 +1 = b.

By the mean value theorem, we can choose ξj ∈ (ηj , ηj+1 ), andkf∗k k2L2 (a,b)

Z ηj0 +1

=

ηi0

2
f∗k
(x)dx =

j0 Z ηj+1
X
j=i0

ηj

j0

2
f∗k
(x)dx ≤2 X 2
f (ξj )
nk j=i ∗
0

j0
X

j0
X

0

0

=

2
2
2
(f 2 (ξj ) − f∗k(xj )) +f 2 (xj )nk j=i ∗knk j=i ∗k

≤

Z b¯
j0
¯
2
2 X 2
¯ 0
¯
f (xj ).
¯2f∗k (x)f∗k (x)¯ dx +nk a
nk j=i ∗k0Here we used

≤

¯
¯
¯ ¯
¯X
¯
¯ ¯X
j0 Z ξj
¯ j0
¯
¯
¯
d
2
2
2
¯
¯
¯
¯
(f
(x))
dx
(f
(ξ
)
−
f
(x
))
=
∗k
∗k j
∗k j ¯
¯
¯
¯
¯j=i0
¯
¯ ¯j=i0 xj dx
j0 Z ξj
X
xj

j=i0

0

2|f∗k (x)f∗k (x)|dx ≤Z ηj0 +1
ηi0

0

2|f∗k (x)f∗k (x)|dx.

Hence (2.10) and the Schwarz inequality yield
kf∗k k2L2 (a,b) ≤(2.11)

04
kf∗k kL2 (a,b) kf∗k kL2 (a,b) + 2A.
nk

Choosing parameters as p = 2, m = 2, j = 1, ² = 1 in Lemma 2.7, we have0

(2.12)

00

kf∗k kL2 (a,b) ≤ K(kf∗k kL2 (a,b) + kf∗k kL2 (a,b) ).

Therefore, (2.11) yields
(1 −

00
4K
4K
)kf∗k k2L2 (a,b) ≤kf∗k kL2 (a,b) kf∗k kL2 (a,b) + 2A.
nknk

Substituting
00

kf∗k kL2 (a,b) kf∗k kL2 (a,b) ≤and choosing k0 ∈ N large, we have

1 00
1kf∗k k2L2 (a,b) + kf∗k k2L2 (a,b)
2
2

8X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO√
00
kf∗k kL2 (a,b) ≤ K1 kf∗k kL2 (a,b) + K1 A
for all k ≥ k0 . Here the constant K1 > 0 is independent of k.
By using this inequality and (2.5), we havesup kf∗k kL2 (a,b) < ∞.

k≥k0

Therefore, by (2.12) we havesup kf∗k kH 2 (a,b) < ∞,kk ≥ k0 .

Since H 2 (a, b) is reflexive, there exist a subsequence f∗k , which is denoted by
the same letter, and fe ∈ H 2 (a, b) such thatf∗k → fe weakly in

H 2 (a, b).

Since the embedding from H 2 (a, b) to C[a, b] is compact, we see thatf∗k → fe strongly in C[a, b].
That is,lim kf∗k − fekC[a,b] = 0.

(2.13)k→∞

By the definition of the integral, for any ² > 0, there exists k = k(²) ∈ N suchthat by (2.1) and (2.9) we obtainky − fek2L2 (a,b)
j0

≤

1 X
(y(xj ) − fe(xj ))2 + ²
nk j=i
0

≤

j0
X

3
{(y(xj ) − yjδk )2 + (yjδk − f∗k (xj ))2 + (f∗k (xj ) − fe(xj ))2 } + ²
nk j=i0≤ 3δk2 +6+ 6δk2 + 3δk + 3kf∗k − fekC[a,b] + ²
k2

for any k ≥ k1 (²).
Hence letting k → ∞, we haveky − fekL2 (a,b) ≤ ².

DETECTION OF IRREGULAR POINTS BY REGULARIZATION

9

Since ² is arbitrary, we haveky − fek2L2 (a,b) = 0,
that is,y(x) = fe(x)for almost all x ∈ [a, b].

Since fe ∈ H 2 (a, b), this implies that y ∈ H 2 (a, b), which contradicts the assumption y ∈/ H 2 (a, b). Thus the proof is complete.
3. Numerical Example
In this section we make two numerical tests on the basis of Theorem 2.5.
First we will consider a continuous function but with irregular points. Let



2x,
0 ≤ x < 0.4y(x) = −3x + 2,
0.4 ≤ x < 0.6−5(x − 1)2 + 1.
0.6 ≤ x < 1which is shown as Figure 1. This function has two irregular points: 0.4 and 0.6.
1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1: y(x)i
We set δ = 0.001, n = 200 that is, xi = 200, 0 ≤ i ≤ 200, and the regularized

solution f∗ can be obtained by the algorithm in [24].
We set00

F (xi ) = kf∗ (δ, δ 2 ,

1 2
)k 2
n L (xi−1 ,xi )and F (xi ) is shown as Figure 2.
Choose the different parameters δ = 0.0001, n = 400, then F (xi ) is shown as
Figure 3.

10

X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO

By Theorem 2.5, we know that this value will grow up near the irregular points.
From Figure 2 and Figure 3 we can see that F (xi ) is large near 0.4 and 0.6. The
numerical results are consistent with the theoretic results. Thus by our method,
we can reconstruct the irregular points.
30

150

25

20

100

15

10

50

5

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0

Figure 2: F (xi ),δ = 0.001, n = 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 3: F (xi ),δ = 0.0001, n = 400

Next we will show that our result still works for a discontinuous function, although Theorem 2.5 does not cover discontinuous functions. Lety(x) =




1,






x − 1,0 ≤ x < 0.2
0.2 ≤ x < 0.5−x + 1,−0.5.

0.5 ≤ x < 0.7
0.7 ≤ x < 1which is shown as Figure 4. This function has three discontinuous points: 0.2, 0.5
and 0.7.
1

0.8

0.6

0.4

0.2

0

−0.2

−0.4

−0.6

−0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

Figure 4: y(x)

0.7

0.8

0.9

1

DETECTION OF IRREGULAR POINTS BY REGULARIZATION

11

The numerical results are shown in Figure 5 and Figure 6.
4

1800
2.5

x 10

1600

1400

2

1200
1.5

1000

800
1

600

400
0.5

200

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

0

Figure 5: F (xi ),δ = 0.0005, n = 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 6: F (xi ),δ = 0.0001, n = 400

There is a difference in the reconstruction from the first example of a continuousfunction, and the numerical results show extra two peaks around the true discontinuous points. However our numerical method can still localize discontinuous points.

4. Application
In this section we will give one application of the numerical differentiation in theimage edge detection. In the image processing, the reconstruction of a good image
relies heavily on how to locate accurately the discontinuous points of an unknown
source function. This depends also on how to reconstruct the derivatives of thesource function from scattered measured data. This is called the edge detection
problem which can become very complicated if the source function is non-smooth.
At present, several numerical methods for solving the edge detection problem havebeen proposed. These include the logarithmic image processing method ([4], [17]),
which originated from the theory of functional analysis; a gradient operator method
in which the ratio of the local gradient magnitude is used to compute the local mean
intensity value of the source function ([13], [21]); and the genetic approach ([2], [18]).
Most of the existing edge detection techniques are based on finding the maxima of
the first-order derivative or the zero-crossing of the second order derivatives of thesource function and see also [6],[14],[15],[20].
Here we will use a new method for edge detection which is based on Theorem
2.5. Numerical results show that our method is effective. Here we give the outlineof our method:

12

X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO(1) Consider a line-by-line scan of an image with size N1 ×N2 where N1 and N2
are positive integers. For simplicity we only consider gray level pictures.
The two examples presented here are all 8-bit gray level pictures, whichmeans the colour value will vary from 0 to 28 − 1 = 255.
(2) The colour values of an digital image are noisy observation data at discrete
points. Here we denote the colour value of a pixel at (i, j) as c(i, j).
j−1(3) Let M = 28 and define yiδ (xj ) = c(i,j)
M , where xj = N2 −1 , j = 1, 2, · · · , N2 .

Then 0 ≤ yiδ (xj ) < 1 and the value of the grid length h in Theorem 2.5 is1
N2 −1 .

(4) The numerical differentiation algorithm developed in [24] is then applied tofind the approximation of the real colour function yi (x),0 ≤ x ≤ 1 fromthe noisy values yiδ (xj ) on each line. The approximation is denoted by f∗i .
(5) The result of Theorem 2.5 is then applied to locate the discontinuous pointsof the function yi (x). In image processing, the edge of an image is usually
defined as the area where a rapid change of magnitude occurs. This rapidchange is then detected by the magnitude of the H 2 norm of the minimizerf∗i . A large value can be set as the threshold for detecting the edge of theimage.
(6) It usually takes two different directions of line-by-line scans to obtain an
acceptable approximation result. Note that the directions are not required
to be perpendicular.
(7) The pictures are taken by a camera and we will add some random noise to
test our method.

The original pictures and pictures with 1%, 5%, 10%, 20% and 30% random
noises are displayed left and the computed edges are given right respectively. Table
4.1 gives the parameters and the computational times for these two images. It canbe observed from both the figures and the table that the numerical differentiation
algorithm successfully reconstructs the edges of the images in a short period of CPU
time.

DETECTION OF IRREGULAR POINTS BY REGULARIZATION

Figure 7a: without noise

Figure 7b: with 1% random noise

Figure 7c: with 5% random noise

13

14

X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO

Figure 7d: with 10% random noise

Figure 7e: with 20% random noise

Figure 7f: with 30% random noiseDETECTION OF IRREGULAR POINTS BY REGULARIZATION

Figure 8a: without noise

Figure 8b: with 1% random noise

Figure 8c: with 5% random noise

15

16

X. Q. WAN, Y. B. WANG, AND M. YAMAMOTO

Figure 8d: with 10% random noise

Figure 8e: with 20% random noise

Figure 8f: with 30% random noiseDETECTION OF IRREGULAR POINTS BY REGULARIZATION

17

Table 4.1: Parameters and CPU time for the computations
size of the image

αtime used for computation

1360 × 1236

10−12

218 seconds

2048 × 1536

−12548 seconds

Figure 7
Figure 8

10*The computations were performed on a HP DX200 P4 2800MHz with 512M RAM.

5. Conclusion
In this paper, near irregular points we established the behaviour of the Tikhononv
regularized solutions in the numerical differentiation. We proved that, if the exact
solution has irregular points, then the norms of the second order derivatives of theregularized solutions in any small interval which contains the irregular points, willblow up. Unlike the result of [24], we proved that the blow-up will take place near
the irregular points. Our numerical example and the application to edge detection
show that irregular points can be located very easily by using our method.

References
1. R. A. Adams, Sobolev Spaces. Pure and Applied Mathematics, Vol. 65, Academic Press, New
York-London, 1975.
2. J. Canny, A computational approach to edge detection, IEEE Trans. Patt. Anal. Mach. Intell.,Vol. PAMI-8 (1986), pp. 679-698.
3. J. Cheng and M. Yamamoto, Regularity of the exact solution and the boundedness of the
Tikhonov regularized solution. Preprint 2001.
4. G. Deng and J. C. Pinoli, Differentiation-based edge detection using the logarithmic image
processing model, Journal of Mathematical Imaging and Vision, Vol. 8 (1998), pp. 161-180.
5. H. W. Engl, M. Hanke and A. Neubauer, Regularization of Inverse Problems. Kluwer Academic
Publishers, Dordrecht Hardbound, 1996.
6. S. Godbole and A. Amin, Mathematical morphology for edge and overlap detection for medical
images, Real-Time Imaging 1 (1995), pp. 191-201.
7. C. W. Groetsch, The Theory of Tikhonov Regularization for Fredholm Equations of the First
Kind. Research Notes in Mathematics, 105. Pitman (Advanced Publishing Program), Boston,1984.
8. C. W. Groetsch, Differentiation of approximately specified functions. Amer. Math. Monthly
98 (1991), pp. 847-850.
9. C. W. Groetsch, Lanczos’ generalized derivative. Amer. Math. Montyly 105 (1998), No. 4, pp.
320-326.

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10. C. W. Groetsch, Optimal order of accuracy in Vasin’s method for differentiation of noisy
functions. Journal of Optimization Theory and Applications 74 (1992), No. 2, pp. 373-378.
11. C. W. Groetsch and O. Scherzer, The optimal order of convergence for stable evaluation ofdifferential operators. Electronic Journal of Differential Equations 1993 (1993), No. 4, pp.
1-10.
12. M. Hanke and O. Scherzer, Inverse problems light: numerical differentiation. Amer. Math.
Monthly 108 (2001), No. 6, pp. 512-521.
13. R. P. Johnson, Contrast-based edge detection, Pattern Recogn., Vol. 23 (1990), pp. 311-318.
14. D. Marr and E. Hildreth, Theory of edge detection, Proc. R. Soc. London B207 (1980), pp.
187-217.
15. R. Mehrotra and S. M. Zhan, A computational approach to zero-crossing-based twodimensional edge detection, Graphical Models and Image Processing, Vol 58, No, 1, January(1996), pp. 1-17.
16. D. A. Murio, Automatic numerical differentiation by discrete mollification, Compputers andMathematics with Applications 13 (1987), pp. 381-386.
17. J. C. Pinoli, A contrast definition for logarithmic images in the continuous setting, Acta
Stereol., Vol. 10 (1991), pp. 85-96.
18. W. K. Pratt, Digital Image Processing, 2nd edition, John Wiley, New York, 1991.
19. A. G. Ramm and A. B. Smirnova, On stable numerical differentiation. Math. Comp. 70 (2001),
pp. 1131-1153.
20. L. G. Roberts, Machine perception of three-dimentional solids, Optical and Electro-Optical
Information Processing, MIT Press, Cambridge, MA, 1965.
21. G. S. Robinson, Edge detection by compass gradient masks, Comput. Graph. Image Proc.
Vol. 6 (1977), pp. 492-501.
22. A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-posed Problems. Winston and Sons, Washington, 1977.
23. V. V. Vasin, The stable evaluation of a derivative in the space C(−∞, ∞). USSR Computational Mathematics and Mathematical Physics 13 (1973), pp. 16-24.
24. Y. B. Wang, X. Z. Jia and J. Cheng, A numerical differentiation method and its application
to reconstruction of discontinuity. Inverse Problems 18 (2002) No. 6, pp. 1461-1476.
25. B. B. Yin and Y. Z. Ye, Recovering the Local volatility in Black-Scholes model by numericaldifferentiation. Preprint 2005.

DETECTION OF IRREGULAR POINTS BY REGULARIZATION

19

School of Mathematical Sciences, Fudan University, Shanghai 200433, CHINA
E-mail address: xqwan@fudan.edu.cn
School of Mathematical Sciences, Fudan University, Shanghai 200433, CHINA
E-mail address: ybwang@fudan.edu.cn
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba,
Tokyo, 153-8914 Japan
E-mail address: myama@ms.u-tokyo.ac.jp

Preprint Series, Graduate School of Mathematical Sciences, The University of Tokyo
UTMS
2005–33 Taro Asuke: Inﬁnitesimal derivative of the Bott class and the Schwarzian
derivatives.
2005–34 Dan Tiba, Gengsheng Wang and Masahiro Yamamoto: Applications of convexity in some identiﬁcation problems.
2005–35 X. Z. Jia and Y. B. Wang: A boundary integral method for solving inverse heat
conduction problem.
2005–36 Ganghua Yuan and Masahiro Yamamoto: Lipschitz stability in inverse problems for a Kirchhoﬀ plate equation.
2005–37 Ken-ichi Yoshikawa: Discriminant of certain K3 surfaces.
2005–38 Noriaki Umeda: Existence and nonexistence of global solutions of a weakly
coupled system of reaction-diﬀusion equations.
2005–39 Yuji Morimoto: Homogeneous coherent value measures and their limits undermultiperiod collective risk processes.
2005–40 X. Q. Wan, Y. B. Wang, and M. Yamamoto: The local property of the regularized solutions in numerical diﬀerentiation.
2005–41 Yoshihiro Sawano: lq -valued extension of the fractional maximal operators fornon-doubling measures via potential operators .
2005–42 Yuuki Tadokoro: The pointed harmonic volumes of hyperelliptic curves with
Weierstrass base points.
2005–43 X. Q. Wan, Y. B. Wang and M. Yamamoto: Detection of irregular points byregularization in numerical diﬀerentiation and an application to edge detection.

The Graduate School of Mathematical Sciences was established in the University ofTokyo in April, 1992. Formerly there were two departments of mathematics in the University of Tokyo: one in the Faculty of Science and the other in the College of Arts andSciences. All faculty members of these two departments have moved to the new graduate school, as well as several members of the Department of Pure and Applied Sciencesin the College of Arts and Sciences. In January, 1993, the preprint series of the former
two departments of mathematics were uniﬁed as the Preprint Series of the Graduate
School of Mathematical Sciences, The University of Tokyo. For the information about
the preprint series, please write to the preprint series oﬃce.
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