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THE GENERALIZED RADIAL HILBERT TRANSFORM AND ITS APPLICATIONS TO

2-D EDGE DETECTION (ANY DIRECTION OR SPECIFIED DIRECTIONS)

Soo-Chang Pei,

Jian-Jiun Ding

Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan, R.0.C

Email address: pei@,cc.ee.ntu.edu.tw

ABSTRACT

It is well-known that the Hilbert transfonn (HLT) is useful for

generating the analytic signal, and saving the bandwidth required

in communication. However, it is known by less people that the

HLT is also a useful tool for edge detection. In this paper, we

introduce the generalized radiant Hilbert transform (GRHLT),

and illustrate how to use it for edge detection. The GRHLT is the

general form of the two-dimensional HLT. Together with some

techniques (such as section dividing and shorter impulse re-

sponse modification), we can use the GRHLT to detect the edges

of images exactly. Using the GRHLT for edge detection has

higher ability of noise immunity than other edge detection algo-

rithms. Besides, we can also use the GRHLT for directional edge

detection, i.e., detect the edges with certain directions.

1. INTRODUCTION

The Hilbert transform (HLT) is defined as [I]:

OM, (g(x))= IFT(H(0). FT(g(x)))

where H(w) = -j sgn(o).

There are some ways to generalize the HLT into the two-

dimensional (2-D) form. The simplest way is combining two I -D

HLT's together:

OHIc(2D)

Y))= OHIt(,) b H ! t ( x ) ( d . 3 Y))J

Recently, in [Z], Davis, McNamara, and Cottrell introduced the

radial Hilbert transform (RHLT):

(1)

(2)

(3)

gH(X,y)=IFT2D[exp(iPB)FT2D(g(x,Y))l

where B =cos-'(dR)=sin?(s/R), R = ( ~ + s * ) " ~ ,

pendent variables in the frequency domain, and Pis any integer.

It is well-known that we can use the HLT to generate the

analytic signal, and save the bandwidth required for a real signal.

In fact, the HLT can also be applied to edge detection [2][3].

Fewer people know this application, but it is very important. The

most serious problem for most of the edge detection algorithms

is that their performance is affected by noise. However, using the

HLT for edge detection can much rcducc the effects of noise.

In this paper, we introduce the generalized radial Hilbert

transform (GRHLT), which is the generalization of the separable

2-D HLT (see (3)) and the RHLT (see (4)). We introduce it in

Sec. 2. Then, we illustrate how to use the GRHLT for 2-D edge

detection and its advantage in Sec. 3. The most important two

advantages are noise immunity and that the ramp edges can be

detected successfully. Using the GRHLT together with the tech-

niques introduced in Sec. 4, we can detect the edges of an image

exactly. Besides, in Sec. 5, we will show that we can also use the

(4)

0, s are the inde-

GRHLT for directional edge detection, i.e., detect the edges with

certain directions.

2. GENERALIZED RADIAL HILBERT TRANS-

FORMS

We define the generalized radial Hilbert transform (GRHLT)

as follows:

gH (& Y ) = IFT~D

where the transfer function H( o,s) is rotational symmetric:

H(w, s) = O(B) when (a,

s) # 0,

/ r) = sin? (s / r) ,

r = dW2+S2, o(@ is any function.

The separable 2-D HLT defined as in (3) is a special case of the

GRHLT where @(8) = 1 when 0 < B < ni2 or z< B < 3ni2,O(@

= -1 when ai2 < B < z or 3d2 < l?< 2z. and Q ( 9 = 0 when B

= Nd2. The RHLT introduced in [2] is a special case of the

GRHLT where

O(@ =exp(iPB).

Besides, many other operations (such as the analytic signals

generating operations, the I-D HLT, and identity operation) are

also the special cases of the GRHLT.

The GRHLT is very general and flexible. Because of its

flexibility, it can solve many problems that can't be solved well

by its special cas& In Secs. 3-5, we show that the GRHLT is a

very powerful tool for edge detection.

When doing the edge detection, we usually use the discrete

counterpart of the GRHLT, i.e., the discrete generalized radial

Hilbert transform:

S H [M,"1=IDFT,D(H[p,91DFT,,(g[m,nl))

(H(0, SIFT20 k ( &

Y)))

(5)

(6)

H(0,O) = 0 ,

where 8 = cos-' (0

(7)

(8)

where p, q, are the discrete independent variables in the fre-

quency domain, and H(p, q] is rotational symmetric:

H[p,q]= @(e)

H[O,O]=O, H[M/Z,O]=O ifMiseven,

H[O, N / 2]= 0

ifNis even, (MxN the size ofg[m, n]),

(9)

B =cos-] (pi.)= sin+(q/ r),

whcn [p, q1 f 10, 01,

r =

Q(4) is any function.

Due to the DFT and IDFT, the discrete generalized radial Hil-

bert transform has fast algorithm. Its complexity is MN.lag2MN.

(10)

3. USING THE GRHLT FOR EDGE DETECTION

The simplest way for 2-D edge detection is doing the difference

operation. That is, if

0-7803-7663-3/03/$17.00 02003 IEEE

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ICASSP 2003

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(horizontal) Ig[mo,no]-g[m,,no

+Ill >threshold

(vertical) Ig[m,,no]-g[mo +I,n,]l >threshold

we can conclude that the pixel (mo, no) is on the edge. Besides,

there are also some edge detection methods based on the convo-

lution with 3x3 matrix, e. g., the compass gradient mask, Lapla-

cian mask, and statistical mask methods [4]. Their ideas are simi-

lar to the difference operation.

However, most of the above edge detection methods are

highly influenced by noise. We do an experiment in Fig. 1. Fig.

l(a) is the input image, and Fig. I(h) is the image interfered by

the noise. In Fig. I(c), I(d), we use the Laplacian mask method

to detect the edges of Fig. I(a), I(b), respectively. It is appear-

ance that the effect of the Laplacian mask method is much influ-

enced by noise. This is the most important problem for Laplaciau

mask method and most of the existed edee detection aleorithms.

(a) i

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(1 I)

. . .

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0

50

100

-100

0

100

-100

0

100

(C)

-100

(d)

-100

0

0

100

100

-100

0

100

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0

100

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0

100

-100

0

100

Fig. I The experiment for noise immunity. (a): Image. (b): Im-

age +noise. (c)(d): Using the Laplaciau mask to detect the

edges of (a) and (b). (e)(f): Using the GRHLT to detect the

edges of (a) and (b).

However, if we use the GRHLT for edge detection, we can

much reduce the effect of noise.

In (IO), if the transfer function we choose satisfies

@(B) = 4 ( 0 + z)

we can use the GRHLT far 2-D edge detection. This is so be-

cause in this case the impulse response (denoted by h[m, n] =

IDFT(H[o, s ] ) of the GRHLT has the following two properties:

(I) odd symmetric: h[m, n] = -A[-m, -n],

(2) lh[m, n]l has the trend ofbecoming smaller when Iml, In1 grow

larger.

In fact, except for the GRHLT, all the 2-D LTI operations

whose impulse responses satisfy the above hvo constraints can

also be used for edge detection.

for all 8,

('2)

The GRHLT can much reduce the effect of noise because it

has longer impulse response. The impulse response of the differ-

ence method (see (11)) are [-I,

1x2 and 2x1. The impulse responses of the methods based on

3x3 matrix convolution (such as the Laplacian mask method) are

3x3. However, the GRHLT has much longer impulse response,

so it can reduce the effect of noise. In Fig. I(e) and I(f), we do

the GRHLT for Fig. I(a) and I(b), respectively. Fig. l(e) shows

that we can use the GRHLT to detect an image successfully. In

Fig. I(0, we show that even if the original image is interfered by

noise, the performance of edge detection doesn't become worse.

So using the GRHLT for edge detection is noise immunity.

Besides, ifO(S, satisfies

a@)= 50,

then the GRHLT has the property of real-input-real-output. That

is, if the input g[m, n] is real, then the output gH[m, n] of the

GRHLT is also a real function. If we use the RHLT (see (4)), i.e.,

Q(8) = expo@, although it has good performance in edge detec-

tion, for real input, the output may not be real since Q(@ =

expo@ doesn't satisfy (13). We can try to choose Q(@ to satisfy

(12)and(l3).Forexample,wecanchoone0(6'~as:

O(S)= jsgn(c0s(8-z/6))cos~~~(B-z//).

In this case, O(@ satisfy (12) and (13), and the corresponding

GRHLT has the property of real-input-real-output.

The advantages of using the GRHLT for edge detection are:

( I ) Noise immunity.

(2) Ramp edges, which are hard to detect by other edge detection

algorithms, can be detected by the GRHLT.

I] or [-I, 1IT. Their lengths are

(13)

(14)

(3) In the sense of sight, the output has better quality if we use

the GRHLT for edge detection.

The 2"' and 3d advantages can be seen from the comparison of

Fig. l(e) with Fig. I(c).

4. SOME TECHNIQGES TO IMPROVE THE PER-

FORMANCE OF EDGE DETECTION

4.1. Dividing the input into several sections

We like to divide the input image into several sections, and use

each of them as the input of the GRHLT for edge detection. It

has two advantages: (1) The complexity can be reduced. (2) The

performance can be improved.

Since the GRHLT is implemented by the 2-0 DFT / IDFT, so

its complexity is MN.log,MN where MxN is the size of image. If

we divide the image into 9 sections, then the size of each sec-

tion is near to (M/S)x(N/X) So the complexity becomes

MN

MN

S 2 ~ - l o g 2 ~ = M N l o g 2 ~

S2

S

MN

This is smaller than MN.log2MN, so the complexity is reduced.

We can also write (15) as a function of MoxNo = (M/Sjx(N/S)

where &xN0 is the size of each section:

MN log, M,N,

That is, if the size of sections MoxNo is fixed, the complexity

grows linearly with MxN. Thus, although the edge detection

algorithm using GRHLT seems more complicated than other

edge detection algorithms, if we use the section-division method,

the complexity is in fact O(MN). This is the same as the com-

plexities of other edge detection algorithms.

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Besides, the performance can also be improved if we divide

the input into several sections. This is so because, to conclude

whether a pixel is on the edge, we just require the information

surrounding the pixel. It is unnecessary to use the whole image

as the input of the GRHLT for edge detection. In Fig. 2, we do

some experiments. In Fig. 2(b), we don’t divide the image, and

in Fig. 2(c)(d), we divide the image into 3’ and 9’ sections before

using the GRHLT for edge detection. We can see that, in Fig

2(c)(d), the brightness differences between edee reeions and

non-edge regions are more obvious than that in,Fig. 2(b).

where Z, [m, n] = lg, [m, n] @ A, ,

Ac is a cxc matrix where A,(m, n) = I for all m’s, n’s,

TO is the average value of b [ m , n]l.

In the above, there’re 3 parameters a, b, c. We can adjust them to

achieve better performance. In principle, for a simple image

(such as the fruit image in Fig. 3(a)), we choose larger values of

a and c and smaller value of b. For a complicated image (such as

the Lena image in Fig. 3(b)), we choose smaller value of a and c

and larger value of b. In Fig. 3(c), we choose (a, b, c) as (0.65,

0.8, 15), and in Fig. 3(d), we choose (a, b, c) as (0.5, 0.85,9).

-1 ‘bb

-1 00

(20)

--

0

(a)

0

0

100 100

-100

0

100 -100

0

100

-100 0 100 -100 0

100

Fig. 2 Improving the performance of the GRHLT for edge de-

tection by section-division method. (a) Image. (b) The IC-

sults if we do not separate the input. (c)(d) The results if we

separate the input into 3 ’ sections and 9’ sections.

4.2. Appending tapered borders

When doing edge detection, the borders of an image are usually

misunderstood as the edges. This problem can be overcome by

appending tapered borders to the original image before doing the

edges detection. That is, if the original image is g[m, n] where

1-N, i m, n < N-NI, NI = [N/Z]+l, then we can append the up-

per and lower tapered borders to g[m, n] by:

g[m,nl= (m+B-I+N,)g[l-Nl,

n]/B when 2-AJ-B

g[m,n] = (N+N,+B-m)g[N~N,,~]/B

9 n <I-N,,

when N-N,im< N+N,+B-l

We can use the similar way to append the left and right tapered

borders to g[m, n]. AAer appending tapered borders, the borders

of an image are no longer detected as edges.

4.3. Choosing the adaptive threshold

In Figs. I, 2, we directly show the output oftbe GRHLT. In fact,

we can also choose a threshold. If

(g,[m,,no]( > threshold

where g,,[m, n] is the output of the GRHLT, then we can con-

clude (mo, no) is on an edge. Then, one may ask how to choose

the threshold. The simplest way is choosing the threshold as

some constant. However, it has worse performance for a compli-

cated image. Here, we propose an localized and adaptive method

to choose the threshold function n m , n]:

(17)

(18)

-100 0

100

-100 0

100

(C)

-100

(d)

-100

-50 -50

0 0

50

50

100

100

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0

100

-100 0

100

Fie. 3 Edee detection usinn the GRHLT toeether with the adao-

I

tive threshold funccon defined as in719), (20).

4.4. Shorter impulse response modification

We have stated that the advantage of the GRHLT is that it can

reduce the effects of noise. It is mainly due to the GRHLT has

longer impulse response. However, it has some side effect. That

is, the result is not sharp enough, and sometimes the brightness

of the edges region may not be obviously higher than the bright-

ness of the non-edge regions. This problem can be overcome if

we modify the transfer function of the GRHLT a little. We can

modify the transfer function of the GRHLT in (1 0) as:

H d [ P > q l = @ ( Q ) @ A d ,

where @ is circular convolution, A, is a dxd mamx, Adm, n) = 1

for all m’s, E ’ S , and O(@J is defined the same as previous sec-

tions. If c is larger, the impulse response hd[m, n] = IDFT(H&,

y]) will become shorter. Shorter impulse response makes the

brightness difference between the edge regions and the non-edge

regions become more obvious.

In Fig. 4, we do some experiments. In Fig. 4(a)(c), we show

the slicing (along x-axis) of the impulse response of the original

GRHLT and the modified GRHLT (d = 9). It can be seen that the

impulse response of the modified GRHLT becomes shorter.

Then, in Fig. 4(b)(d), we show the transform results of the origi-

nal GRHLT and modified GRHLT, respectively. The brightness

difference between the edge and the non-edge regions of the

result of the modified GRHLT is more obvious.

Thus, using the shorter impulse response GRHLT for edge

detection can obtain better performance. However, the ability of

noise immunity is reduced. There is a tradeoff between the two

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goals: (I)

for noise immunitv.

better performance of edge detection, (2) higher ability

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100

O b . . . . - __-..

-0.5

-1 0

0

10

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100

(d)

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50

100

-0.5 1-

-1 n

Fig. 4 Experiments for using the shorter impulse response modi-

fied GRHLT for edge detection. (a)(b) The impulse response

(x-axis slicing) and the transform results of the original

GRHLT. (c)(d) The impulse response and he transform re-

sults of the shorter impulse response modified GRHLT.

0

i n

..

-inn

n

inn

..

5. DIRECTIONAL EDGE DETECTION

In Secs. 3,4, we detect the edges whatever the direction it has. In

fact, if the transfer function of the GRHLT is chosen properly,

we can detect the edges with certain directions. For example, If

we want to detect the edges with the direction in the region of [#,,

4 1 where 4-4, < n, then, in (IO), we can choose

@(@ = -j when 4, i B S 4,

@(@ = 0 othenvise.

@(@ = j when # , - d B <4-n,

Then we do the GRHLT. If the output r&m,n)l > threshold, then

we can conclude that (m,

n) is an an edge, and the direction of

the edge is in the range of [+,, 41. In Fig. 5, we give some ex-

ample. We use a circle (Fig. 8(a)) and the Lena image (Fig. 8(b))

as the input. In Fig. 5(c), we plot the imaginary part of the trans-

fer function &, q] = O(@, where a(@

we choose (I = d3 and 4 = 2d3. Then we do the GRHLT for

the two inputs, and use the method described in subsection 4.3 to

choose the adaptive threshold. We plot the transform results in

Fig. 8(e) and 8(g). The edges found in Fig. 8(e) are just two arcs,

and the angle ranges of the two arcs are [d3, 2d3] and [-m’3,

d3], respectively. In Fig. 5(g), we also find the edges with direc-

tion in the range of [d3,2d3] successfully.

Besides, ifwe want to detect the edges in the region of [#,, 4 1

and [4, 4 at the same time, we can choose O(@ as:

when (, i B i h, h 5 B i d4,

when #,-x< . 9 5 4-z, h-n< B 2 k-n,

@(@ = j

O(@ = 0

otherwise.

is defined as (22). Here,

For example, in Fig. 8(d), we choose

choose [4, rj4] = [2d3, 5461. In Fig. 5(f) and 5(h), we plot the

transfer results of the GRHLT (with threshold) with this transfer

function for the circle image and Lena image. We detect the

edges with the direction in the ranges of [d6, ”31 and [2d3,

8d61 successfully.

Thus, using the GRHLT for directional edge detection is very

flexible and convenient.

4l = [d6.

“31, and

(22)

O(B) = -j

(23)

(b)

-1 00

(a)

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0

100 100

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a

100

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0

100

4 O I / - j \ [

4 0 ; ~

i i / o

-40-20 0 20 40 -40-20 0

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100

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100

(hl

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(9)

.loo

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0

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Fig. 5 Experiments for directional edge detection. (a)(b): The

inputs. (c)(d): Transfer functions a(@.

form results if we use (c) as the transfer function. (f)(h) The

transform results if we use (d) as the transfer function.

(e)(g): The trans-

6. CONCLUSION

We have introduced the generalized radiant Hilbert transform

(GRHLT), and illustrated how to use it for 2-D edge detection.

Using the GRHLT for edge detection is very flexible, and can

much reduce the effect of noise. We can even use the GRHLT

for directional edge detection. Thus, except for communication,

the GRHLT is also an important tool for image processing.

7 . REFERENCES

[I] S. C. Hahn, Hilbert Transforms in Signal Processing, Boston:

Anect House, 1996.

[2] J. A. Davis, D.

E. McNamara, and Don M. Cottrell, “Image

processing with the radial Hilbert transform theory and experi-

ments”, Opt. Lett., vol. 28, p. 99-101, Jan. 2000.

[3] A. W. Lohmann, D.

Mendlovic, and Z. Zalevsky, “Fractional

Hilbert transform”, Opt. Lett., vol. 21, p 281-283, Feb. 1996.

[4] Pratt, William K, Digitd image processing, 3‘d Ed., New

York: Wiley, 2001.

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