Journal of Photogrammetry and Remote Sensing Volume 21, No.3, 2016, pp. 151-161
DOI：10.6574/JPRS.2016.21(3).2
1Associate Professor, Department of Geomatics,National Cheng Kung University Received Date: May 08, 2012
* Corresponding Author, E-mail: tsayjr@mail.ncku.edu.tw Revised Date: Jul. 11, 2016 Accepted Date: Nov. 11, 2016
A DWT-Based New Method for Representation of Image Feature Lines
Jaan-Rong Tsay1*
Abstract
This paper presents a Discrete Wavelet Transform (DWT) based new method for representation of image feature lines. Some tests are done by using the compactly supported orthonormal Haar and Daubechies wavelets, where image feature lines are automatically generated by an automatic edge detection method which utilizes wavelets-derived symmetric gradient operators and wavelets-based zero-crossing approach, and used as test data.
Test results show that feature lines on a 2D image plane can be represented in a flexible manner by the proposed method. The representation can be done as accurately as needed, where the related parameters can be determined automatically. In addition, explicit and user-friendly formulas of DWT and Discrete Wavelet Reconstruction (DWR) are also given in this paper.
Keywords: Discrete Wavelet Transform (DWT), image feature line, Discrete Wavelet Reconstruction (DWR), subband coding, wavelet shrinkage
1. Introduction
In diverse fields of applications and sciences, image feature lines can be utilized for extraction of semantic information such as pattern recognition, optical character recognition (OCR), image understanding, image segmentation, route tracking, etc. Also, multi-scale representation of feature lines can be applied for curve simplification and line generalization as well as linkage of broken image line segments.
Besides multi-scale representation, feature lines on 2D images can also be expressed by other kinds of representation methods such as parametric representation, or the natural equations of a curve. A curve (or a line) in 2D plane or in 3D space can be represented by a vector function 𝑟⃗(t) = x(t)𝑖⃗ + y(t) 𝑗⃗
+ z(t) 𝑘⃗⃗ where x, y, z are Cartesian coordinates. This is called a parametric representation of the curve, t is called the parameter of the representation. The parameter t may be time or something else. Also, a curve (or a line) in 2D plane or in 3D space is uniquely determined (except for its position in space) if we prescribe its curvature (>0) and torsion as continuous functions of arc length s. For this reason, one calls =(s) and =(s) the natural equations of a curve. This also shows why curvature and torsion are basic in the differential geometry of space curves(Kreyszig et al., 2011). In this paper, a new method is proposed for representing diverse forms of feature lines such as fractal, smooth, regular, or
irregular image feature edges. The new method is derived from DWT. It represents either simple or complicated feature line by means of the structure of pixel-by-pixel connection on a digital raster image. It also utilizes the algorithm proposed by Tsay and Liao (2000) for automatic edge detection for a digital image. This algorithm adopts firstly a set of new symmetric gradient operators derived by Tsay (1996) from the wavelet theory and the Daubechies wavelets.
Secondly, a wavelet-based zero-crossing approach is developed and applied for automatic determination of all center edge-lines of one pixel width.
In section 2, explicit formulas for DWT and DWR based on the Daubechies’s Finite Impulse Response (FIR) filters are given and depicted briefly.
Moreover, the idea for representing any types of feature lines is also described. The DWT-based algorithm for representing image feature lines is described in detail in section 3. Section 4 illustrates and analyzes some test results. Some conclusions are drawn in section 5.
2. Mathematical Models
The theory of multi-resolution signal decomposition and detailed mathematical derivations of DWT and DWR was firstly given by Mallat (1989), yet only general DWT and DWR formulas were presented. Frankly, they are not enough user-friendly.
They are still hard to use for most inexperienced wavelet users since, for instance, the start and end
values of multiple variables in the summation formulas for DWT and DWR are not given. Also, the selection of a proper kind of wavelet function is often hard for inexperienced users. In this section, explicit formulas for DWT and DWR based on the Daubechies’s FIR-filters (Daubechies, 1988; 1992) are given and depicted briefly. The detailed mathematical derivations and definitions can be found in (Tsay, 1996).
To simplify the definitions, only even number of reference data points are firstly considered in sections 2.1 and 2.2. The related DWT and DWR can also be done for odd number of reference data points by the algorithm given in section 3. Moreover, the main idea for representing image feature lines on a 2D image plane is briefly described also in section 3.
2.1 Explicit formulas for DWT
If discrete reference data Gi on equidistant grid points with coordinates 𝑋𝑖, i = 1,2,…,2n, n N, are given and they have finite square sum or finite energy (Vetterli and Kovacevic, 1995), their smooth and detailed components 𝑆𝑘 ,𝑑𝑘 , k = 1,2,…,n, can be determined respectively as follows (Tsay, 1996;
1998):
2N 1
0 m
1 m 2k m
k h G
s ………..(1)
2N1
0 m
1 m 2k m 1 2N m
k 1 h G
d ……….(2)
where hm , m = 0,1,…,2N-1, are the well-known low-pass filter coefficients of the compactly supported orthonormal wavelets of Daubechies (Daubechies, 1988); and N is the order of Daubechies wavelets. In this paper, the symbols N and Z denote the set of all positive integers and all integers, respectively.
The components 𝑆𝑘 ,𝑑𝑘, k = 1,2,…,n, are also interpreted as a “coded” version of the discrete input signal Gi, i = 1,2,…,2n. The decomposition shown in equations (1) and (2) is often known as subband filtering or subband coding (Kaiser, 1994; Vetterli and Kovacevic, 1995).
2.2 Explicit formula for DWR
Explicit formula for discrete wavelet reconstruction (DWR) is shown in equation (3). It can be derived by synthesizing both smooth and detailed discrete signals 𝑆𝑘 ,𝑑𝑘 , k = 1,2,…,n, where orthonormal bases in a corresponding finer approximation space are utilized. Please refer to Tsay
(1996) for detailed derivations.
m
k m 1 2N m k
m
n
h s 1 h d
G
………….(3)where m = 0,2,…,2N-2 or 1,3,…,2N-1 if n is odd or even respectively, and k = ( n + 1 – m ) / 2.
The original discrete input signal 𝐺𝑖, i = 1,2,…,2n, can be reconstructed perfectly by decoding from the subband codes 𝑆𝑘 ,𝑑𝑘 , k = 1,2,…,n (Kaiser, 1994).
3. Feature Line Modeling Algorithm
Any image feature line can be represented in a similar way as shown in (Davis, 1999) by
x(𝑙𝑖), y(𝑙𝑖),
where x and y are the coordinates of the i-th pixel on an image feature line and 𝑙𝑖 is the path length from the start point of the feature line to the i-th pixel. They represent the coordinates of the i-th pixel of an image feature line. The algorithm presented by Tsay and Liao (2000) outputs also the sequence of pixels that constitute a feature line.
Figure 1 illustrates a DWT-based algorithm for representing an image feature line. In that algorithm, raster image coordinates (𝑐𝑖,𝑟𝑖), i = 1,2,…,2M+ir, of all pixels on a digital image feature line are firstly given, where ir=0 or 1, and (𝑐𝑖,𝑟𝑖) are the column and row coordinates of the i-th pixel on the line. These coordinates are ordered. Adjacent points (pixels) have coordinate indices i and i+1 which differ by 1 from each other. The domain of their coordinate differences 𝑐𝑖, 𝑟𝑖 is the set {-1, 0, 1}. 2M+ir is the number of pixels on the feature line, M , M > 0.
The parameter ir is equal to 0 or 1 and it means that there is even or odd number of pixels on the feature line, respectively.
Except the aforementioned data(𝑐𝑖,𝑟𝑖), a threshold value and number n of multi-resolution levels are also given. The threshold value is used for wavelet shrinkage, please see equation (5).
Moreover, if 2M + ir <2𝑖𝑛, then n is less than or equal to in – 1, in , in > 1.
After the above-mentioned data are given, multi-resolution decomposition is to be done. For each resolution level il = n,n-1,…,1, both discrete 𝑐𝑖
and 𝑟𝑖 signals are decomposed by means of discrete wavelet transform (DWT), respectively. In this paper, DWT is done by the compactly supported Haar and Daubechies wavelets (Tsay, 1998) as follows:
1 N 2
0
1 2 1 N 2 1
N 2
0
1 2 1 2 1
N 2
0
1 2 1 N 2
0
1 2
h 1
h 1 h
h
m
m k m m
k m
m k m N m k
m
m k m k
m
m k m k
r dr
c dc
r sr
c sc
………..(4)
where (sck,dck), k = 1,2,…,M, are the approximation and detail components of the discrete ci signal, respectively; (srk,drk), k = 1,2,…,M, are the approximation and detail components of the discrete ri signal, respectively; N = order of the Daubechies wavelet used; N=1 denotes also the family of Haar wavelets; hm, m = 0,1,…,2N-1, are the normalized low-pass filter coefficients of Daubechies wavelet;
i.e.∑2N−1hm
m=0 = 1; periodical 𝑐𝑖 and 𝑟𝑖 signals are assumed to appear outside the distribution area of data points (𝑐𝑖, 𝑟𝑖), i = 1,2,…,2M+ir,i.e. 𝑐𝑖=𝑐𝑖+2𝑚𝑀
and ri=ri+2mM ri=, m .
Data Input: (𝐶𝑖,𝑟𝑖), i = 1,2,…, 2M+ir, and n Multi-resolution Decomposition:
For resolution level il = n,n-1,…,1
DWT for the discrete 𝐶𝑖 and 𝑟𝑖 signal respectively
Wavelet shrinkage for the decomposed 𝑑𝑐𝑘 and 𝑑𝑟𝑘 signals Multi-resolution Reconstruction:
For resolution level il = 1,2,…,n
inverse DWT to reconstruct synthesized 𝐶𝑖 and 𝑟𝑖 signals permute the reconstructed elements
Output the synthesized image feature line Error analysis
Figure 1 A DWT-based algorithm for representing an image feature line
c
1c
2c
3c
4c
5c
6c
7 (level 0) decomposition
sc
1dc
1sc
2dc
2sc
3dc
3c
7 permutation
sc
1sc
2sc
3dc
1dc
2dc
3c
7 (level 1) decomposition
Sc
1Dc
1sc
3dc
1dc
2dc
3c
7 ( level 2)Figure 2 An example of multi-resolution wavelet decomposition, where the number of s-components on the level 0, 1, and 2 is odd
If there is odd number of approximation components on a resolution level, the last smooth element is directly coded without any further decomposition process, see an example shown in Figure 2.
The decomposed detailed components, namely the discrete 𝑑𝑐𝑘 and 𝑑𝑟𝑘 signals, are also known as the so-called wavelet coefficients. Individual wavelet coefficients 𝑑𝑐𝑘 and 𝑑𝑟𝑘 are then shrunk. For instance, a method for shrinking individual wavelet coefficient is shown as follows:
>
, 0
k k
k
k k
dc for dc
c d
dc for
c d
……….(5)
Please refer to e.g. (Horgan, 1998; Vidakovic and Mueller, 1991) for other thresholding schemes.
After wavelet shrinkage is completed, the signal components and the shrunk detail components 𝑑𝑐𝑘′
and 𝑑𝑟𝑘′, k, are then available for the next process, namely multi-resolution reconstruction.
On each resolution level il = 1,2,…,n, inverse DWT is done to reconstruct a synthesized discrete
c
i andr
i signal, respectively. The reconstruction formulas are
m
k m m k
m i
m
k m m k
m i
r d sr
r
c d sc
c
1 N 2
1 N 2
h 1 h
2
h 1 h
2
………(6)
where(𝑐𝑖′,𝑟𝑖′
), i = 1,2,…,2M, are the raster
coordinates of all pixels on a reconstructed digital feature line image; m = 0,2,…,2N-2 if i is odd;
m = 1,3,…,2N-1 if i is even; k = ( i + 1 – m ) / 2 ( or m = i – 2k + 1).
On each resolution level, the reconstructed signal is permuted immediately after the reconstruction is completed. Figure 3 shows an example of the entire processes.
Finally, the synthesized (reconstructed) image feature line is generated. Some adjacent pixels may be projected onto an almost completely straight line segment because of insignificant detailed components.
Let P(x,y) and Q(x,y) be two of such adjacent pixels and 1(x1,y1) and 2(x2,y2) be the two end points of the line segment. For error analysis, one computes the vertical distance |d| of the pixel P(x,y) from the line passing through the two pixels 1(x1,y1) and 2(x2,y2) using the following formulas:
A = |x2*y1+x*y2+x1*y-x1*y2-x2*y-x*y1| / 2 12̅̅̅̅=
x1x2
2 y1y2
2 ……….(7)|d| = 2 * A / 12 where A = area of the triangle P12 (Wolf and
Brinker, 1994); 12 = distance from the point 1 to the point 2.Moreover, it is defined as shown in Figure 4 that d > 0, if the point P(x,y) is on the right side of the vector12⃗⃗⃗⃗ . d < 0, if the point Q(x,y) is on the left side of the vector12⃗⃗⃗⃗ .
In the following text, the number d is also known as offset.
Sc
1Dc
1sc
3dc
1dc
2dc
3c
7 ( level 2) reconstruction
sc
1sc
2sc
3dc
1dc
2dc
3c
7 (level 1) permutation
sc
1dc
1sc
2dc
2sc
3dc
3c
7 reconstruction
c
1c
2c
3c
4c
5c
6c
7 (level 0)Figure 3 An example of multi-resolution wavelet reconstruction; an inverse process of the multi-resolution decomposition shown in Figure 2
(a) d > 0 (b) d < 0
Figure 4 Definition of a offset d of a pixel P(x,y) or Q(x,y) from the line vector12⃗⃗⃗⃗ . (a) d > 0, if P(x,y) is on the right side of 12⃗⃗⃗⃗ (b) d < 0, if Q(x,y) is on the left side of 12⃗⃗⃗⃗
4. Test Results and Analyses
Figure 5 shows an image of the entrance to the Castle Garden “Eingang zum Schlogarten” in Darmstadt, Germany. The color image has the size of 322 x 236 pixels. It is firstly converted into a B&W image. Its edges are then automatically detected by the Daubechies gradient operators, by the wavelet-based zero-crossing approach without/with edge linking, respectively, where asymmetric Daubechies wavelet of order N=3 is utilized. The wavelet-based zero-crossing method and the Daubechies gradient operators are depicted in detail in Tsay and Liao (2000). In the wavelet-based zero-crossing approach with edge linking operations, altogether 6988 white pixels in Figure 5(c) are detected as new black pixels on edges, since they satisfy the following conditions:
(if G’ > s0 and G” decreases from positive to negative numbers)
or
(if G’ < s0 and G” increases from negative to positive numbers)………(8)
where s0 is a positive real number which is a first derivative threshold evaluated from the input image. G’ and G” are the first and second derivatives respectively determined by the gradient operators derived by Tsay (1996) using the Daubechies wavelets. s0=30 is automatically determined and used in this case. Until now, all test results show that the edge detection is done very well and correctly if s0 is equal to 0.7-1 times the root mean square values of first derivatives.
To test the proposed DWT-based representation method, an edge feature line shown in Figure 6 is selected. The edge image has the size of 306 x 220 pixels. The edge is composed of 1448 black pixels.
Figure 7 shows four edges represented by the DWT method based on the Haar wavelets, where an allowable maximal offset D of 3, 10, 30, and 50 pixels is given for each case, respectively. Apparently, the DWT method represents a more simple trend line, if a larger D-value is adopted. The larger the D-value is, the larger the offsets become. Some exact statistics are shown in Table 1. It indicates the following statistics in the represented image feature line function by DWT based on the Haar wavelets:
D = a given allowable maximal offset;
n = number of significant subband codes 𝑠𝑐𝑘, 𝑠𝑟𝑘, 𝑑𝑐𝑘′, 𝑑𝑟𝑘′, k, on all resolution levels in column- and row-coordinates;
nrp = number of reconstructed pixels; the ratio of nrp to the number (1448) of pixels on the original line is shown in percentage in the parenthesis;
= threshold for wavelet shrinkage;
min(d), max(d), and rms(d) = minimal, maximal, and root-mean-squared values of offsets, respectively.
Table 1 shows also clearly that less number of significant subband codes is used to represent a trend curve, if a larger D-value is given. The afore-mentioned notations D ~ rms(d) are also adopted in Table 2 and Table 3.
Haar wavelets are the special family of Daubechies wavelets with order N=1, and they are the simplest orthogonal wavelets so far. They enable fast computation e.g. for DWT and DWR, but their results are not smooth due to the inherent characteristics of step functions of Haar wavelets. In contrast against Haar wavelets, the other higher order Daubechies wavelets (N>1) need higher computational complexity, but their results are smoother. Figure 8 and 9 show four edges represented by DWT based on the asymmetric Daubechies wavelet of order N=4, and the least asymmetric Daubechies wavelet of order N=10, respectively. As shown clearly in Figure 10, in contrast against the Haar wavelets, the asymmetric
Daubechies wavelets of order N=4 are smoother basis and the least asymmetric Daubechies wavelets of order N=10 are the most smooth one. These two sets of test results show that the DWT method based on
these two kinds of Daubechies wavelets represent evidently a smoother line than the one by the Haar wavelets. Exact statistics are also shown in Table 2 and 3, respectively.
(a) original image (b) Daubechies operator (N=3)
(c) zero-crossing without edge linking (d) zero-crossing with edge linking
Figure 5 (a) an image of the entrance to the Castle Garden “Eingang zum Schlogarten” in Darmstadt, Germany, and its edges automatically detected (b) by the Daubechies gradient operator, (c) by the wavelet-based zero-crossing approach without edge linking, and (d) by the wavelet-based zero-crossing approach with the edge linking operations, where asymmetric Daubechies wavelet of order N=3 is adopted
Figure 6 An edge image with the size of 306 x 220 pixels is adopted as test data; There are 1448 black pixels on the edge
(a) D = 3 pixels (b) D = 10 pixels
(c) D = 30 pixels (d) D = 50 pixels
Figure 7 an original edge (solid line) and an edge represented by DWT based on the Haar wavelets (dashed line), where an allowable maximal offset D is given
(a)D = 3 pixels (b) D = 10 pixels
(c) D = 30 pixels (d) D = 50 pixels
Figure 8 An original edge (solid line) and an edge represented by DWT based on the asymmetric Daubechies wavelet of order N=4 (dashed line), where an allowable maximal offset D is given
Table 1. some statistics of the image feature line represented by DWT based on the Haar wavelets.
D(pixels) n nrp (pixels) d = offset (pixels)
min(d) max(d) rms(d)
3 530 359(24.8%) 1.1 -2.81 2.38 0.81
10 156 122( 8.4%) 3.8 -10.00 9.97 3.29
30 43 31( 2.1%) 11.3 -22.25 25.33 10.13
50 24 17( 1.2%) 18.0 -30.62 39.86 15.04
Table 2 some statistics of the image feature line represented by DWT based on the asymmetric Daubechies wavelet of order N=4
D(pixels) n nrp (pixels) d = offset (pixels)
min(d) max(d) rms(d)
3 227 1304(90.1%) 0.9 -2.70 2.54 0.74
10 74 1179(81.4%) 3.3 -9.34 9.20 2.87
30 25 1025(70.8%) 12.8 -23.47 23.23 8.34
50 14 927(64.0%) 22.1 -47.75 45.95 20.76
(a) D = 3 pixels (b) D = 10 pixels
(c) D = 30 pixels (d) D = 50 pixels
Figure 9 An original edge (solid line) and an image feature line represented by DWT based on the least asymmetric Daubechies wavelet of order N=10 (dashed line), where an allowable maximal offset D is given
Table 3 some statistics of the image feature line represented by DWT based on the least asymmetric Daubechies wavelet of order N=10.
D(pixels) n nrp (pixels) d = offset (pixels)
min(d) max(d) rms(d)
3 256 1342(92.7%) 0.7 -2.03 2.20 0.62
10 45 1160(80.1%) 4.1 -9.62 9.21 3.70
30 17 913(63.1%) 15.5 -28.70 23.58 11.53
50 16 951(65.7%) 17.0 -33.06 33.67 12.45
(a) Haar father and mother wavelets (left), (right)
(b) asymmetric Daubechies father (left) and mother (right) wavelet of order N=4
(c) least asymmetric Daubechies father (left) and mother (right) wavelet of N=10 Figure 10 three wavelet bases used in the tests
In the case of Haar wavelets, the number of reconstructed pixels is 17~359 (1.2~24.8% of 1448 pixels). It is significantly much less than the ones in both Daubechies cases that are 913~1342 (63.1~92.7% of 1448 pixels). Yet, the number of significant subband codes shows no significant difference in three cases. The number of significant subband codes in the Haar case is 24~530. The one in both Daubechies cases is 14~256 that is about half of the one in the Haar case. It means that both Daubechies cases represent a line using less significant subband codes, but they produce more feature points to represent the line. The reason is very clear. Figure 10 shows the three wavelet-bases used in these tests. The Haar wavelets are piecewise linear step functions. They construct also piecewise linear step functions between any two adjacent feature points that refer to the significant subband codes.
However, both Daubechies cases define an inconstant function between any two adjacent points. There appear more feature points, namely more pixels are produced.
5. Conclusion
1. This paper proposes a new method for representing image feature lines. It is derived from Discrete Wavelet Transform (DWT) and represents image feature lines in the type of a digital raster image, where all edges are automatically extracted by the
algorithm proposed by Tsay and Liao (2000) for automatic edge detection for a digital image.
2. Test results show that image feature lines on a 2D image plane can be represented in a flexible manner by the DWT scheme. The representation can be done as accurately as needed, where the related parameters (e.g. wavelet bases, threshold for wavelet shrinkage, number of resolution levels) can be determined automatically by some given constraints, e.g. allowable maximal offset, minimal number of significant subband codes.
3. All test results show that the edge detection is done very well and correctly if s0 is equal to 0.7-1 times the root mean square values of first derivatives.
4. In principle, the mathematical models and algorithms proposed in this paper can also be extended to represent a line or a curve in a 3D or multi-dimensional space. They are suitable for representing image feature lines in the type of a multi-dimensional raster image. They are to be further studied.
5. In the Douglas-Peucker algorithm for line simplification, the points between two endpoints of a line are eliminated, if their perpendicular distances to the line don’t exceed a given threshold (Monmonier, 1982). Similarly, the new method proposed in this paper can also eliminate some insignificant points, if necessary, and represent an approximation of a line.
Thus, the method should and might possess the potential application on line simplification or even
also on line generalization. They will and should be further studied in the future.
6. Besides the above-mentioned DWT-based representation method, curve modeling can also be done by the wavelet series. The first method, namely the DWT-based representation method, is available for representing a digital feature line image. It has the discrete raster data type. Yet, the second one, namely the wavelet series, is suitable to describe a continuous line or curve function. For example, digital vector line data can be it’s input data. The second method will also be further studied.
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Daubechies, I., 1992, Ten Lectures on Wavelets.
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Davis, T.J., 1999, Fast Decomposition of Digital Curves into Polygons Using the Haar Transform.
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Tsay, J.R., and Liao, Y.C., 2000. A New Algorithm for Automatic Edge Detection in a Digital Image, Festschrift anlaesslich der Vollendung des 65. Lebensjahres von Prof. Dr. Ing.
Bernhard P. Wrobel, Heft 10, ISBN 3-9804926-9-9, Schriftenreihe Fachrichtung Geodaesie, Fachbereich Bauingeneurwesen und Geodaesie, November 2000, Technical University Darmstadt, Germany , pp. 223-236.
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1國立成功大學測量及空間資訊學系 副教授 收到日期：民國 101 年 05 月 08 日
*通訊作者, E-mail: tsayjr@mail.ncku.edu.tw 修改日期：民國 105 年 07 月 11 日 接受日期：民國 105 年 11 月 11 日
運用離散小波轉換表達影像特徵線的新方法
蔡展榮
1*摘 要
本篇論文提出一個新方法來表達一般的影像特徵線，它是從離散小波轉換演化設計而得。本文使用 有限承載正則哈爾和多布吉斯小波來進行實驗測試，其中，影像特徵線是採用小波推導出的對稱梯度運 算元並搭配「過零法(zero-crossing)」發展出的自動化影像邊緣線偵測法求定而得。測試結果顯示，使用 此方法可以靈巧地表達影像特徵線，而且依據所需的精度要求來準確表達影像特徵線，當中的計算參數 可以自動決定。此外，簡明具體的離散小波轉換公式和離散小波重建公式也一併於文中載明。
