Keywords:Digital Image, Edge Center Line, Gradient Operator, Wavelets, Zero-Crossing
Abstract Firstly, this paper proposes a set of new symmetric gradient operators. They were derived from the wavelet theory and some well-known compactly supported orthonormal wavelets. Some tests are done using digital images. They conclude that such operators are really better edge-detectors e.g. than the Sobel ones. Secondly, a new approach for automatic determination of edge center lines in a digital image is presented. It integrates these new wavelet-based gradient operators with the general zero-crossing principle. Test results show apparently that it can automatically determine all center edge-lines of one pixel width.
1. Introduction
At present, edge features are often used in different kinds of image-based techniques, e.g. photogrammetry, remote sensing, computer vision, image understanding and so on. Based on those edge features, geometrical, physical-radiometric and semantic data and information as well can be extracted semi- or even full-automatically by different algorithms, where digital or digitized images are the main input data. In the field of image processing, there are already many different kinds of operators and methods for detecting three basic types of discontinuities: points, lines, and edges. For detail, please see e.g. (Gonzalez and Woods, 1993). In this paper, an alternative new method for automatic edge detection in an image is presented.
The section 2 describes briefly the mathematical derivation and presents a set of new gradient operators. Some tests and analyses are also given there. In section 3, a new approach for automatic determination of edge center lines is given. Some preliminary test results are analyzed in section 4. Conclusions are made in section 5.
2. A Set Of New Gradient Operators
2.1 Brief Depiction on Mathematical Derivations
If a signal f exists in a so-called Hilbert space, it can be represented in a linear form based on orthonormal bases. For instance, the space of square-summable sequences and the space of square-integrable functions depicted in (Vetterli and Kovacevic, 1995) are two examples of the so-called Hilbert spaces.
The compactly supported Daubechies wavelets proposed in (Daubechies, 1988) are examples of orthonormal bases.
The theory for multiresolution signal decomposition was firstly presented in (Mallat, 1989).
It was derived based on those signals f in a Hilbert space.
The signal component
A
jf of f in a resolution space
V
j is so defined that it has the property of minimal error energy, i.e.
ò
(A
jf(t)-f(t))
2 dt is minimal.
Herewith, all coefficients in the linear representation are defined.
A
jf can be further decomposed into two orthogonal components
A
j-1f and
D
j-1f. They exist in two V
j 's subsets.
One is a coarser resolution space V
j-1 than V
j .
The other one is the orthogonal complement of
V
j-1 in V
j.
It is a so-called wavelet space and denoted by O
j-1.
On all equidistant dyadic grid points stated in (Rioul and Duhamel, 1992; Jawerth and Sweldens, 1994), the decomposition derives the basic mathematical formulas of image decomposition shown in (Mallat, 1989). (Tsay, 1996) gave a more user-friendly formulas for image decomposition and image reconstruction. If the first derivatives of those wavelet bases used in that decomposition exist, a new gradient operator can be derived from the derivative values on all dyadic grid points. In case that Daubechies wavelets are used, these operators are defined as follows (Tsay, 1998):
where
bm is the m-th element of the operator;
h
k is the k-th low-pass filter coefficient of the Daubechies wavelet;
; N is the order of Daubechies wavelets;
f and
y are the father and mother wavelet of Daubechies, respectively. Table 1 shows the new gradient operators derived from the Daubechies wavelets of order N = 3 (1) 10. These new gradient operators are to be named as "Daubechies operators" in this paper.
2.2 New Symmetrical Gradient Operators
Apparently, they are symmetrical gradient operators. Moreover, it is verified that the operators determined by the asymmetric Daubechies wavelets are the same with the ones determined by the least asymmetric Daubechies wavelets of the same order N.
Table 1. New gradient operators derived from the Daubechies wavelets of order N = 3 (1) 10,
where
b0=0
and the other
bm
-coefficients are equal to zero (|m|>2N+1) or they are insignificant coefficients with
|
bm|<10
-6 ( |m|
£2N+1).
| |
m |
bm=-b-m |
|
|
m |
bm=-b-m |
| N=3 |
1
2
3
4 |
0.745205
-0.145205
0.014612
0.000342 |
N=7 |
1
2
3
4
5
6
7
8 |
0.868744
-0.282965
0.090189
-0.022687
0.003881
-0.000337
-0.000004
0.000002 |
| N=4 |
1
2
3
4
5
6 |
0.793010
-0.191999
0.033580
-0.002224
-0.000172
0.000001 |
N=8 |
1
2
3
4
5
6
7 |
0.883446
-0.303259
0.106364
-0.031290
0.006958
-0.001032
0.000077 |
| N=5 |
1
2
3
4
5
6 |
0.825906
-0.228820
0.053353
-0.007461
0.000239
0.000054
|
N=9 |
1
2
3
4
5
6
7
8 |
0.895316
-0.320312
0.120954
-0.039953
0.010617
-0.002103
0.000278
-0.000020 |
| N=6 |
1
2
3
4
5
6
7 |
0.850137
-0.258553
0.072441
-0.014546
0.001589
-0.000004
-0.000012 |
N=10 |
1
2
3
4
5
6
7
8
9 |
0.905071
-0.334784
0.134055
-0.048427
0.014669
-0.003526
0.000631
-0.000077
0.000005 |
