Wavelet-based Multi-scale Edge Detection

In Section 4.4, we discussed the basic concepts of wavelet theory, and its important properties: 1) applicable to stationary and non-stationary signals; 2) simultaneously equipped with resolution capability for time-frequency location; 3) equipped with frameworks of multi-resolution, so resolution can be adjusted according to problem complexities (Daubechies 1992; Burrus et al. 1998). A new edge filter based on a wavelet transform has been proposed by some researchers such as Mallar and Hwang (1992), as well as Mallat and Zhong (1992). They considered most wavelets to be derivations of smoothing functions (such as the quadratic spline function and the cubic one) that can compose very interesting edge detectors. Consequently, the idea of wavelet-based

multi-scale image edge detection was derived..

In (Mallat and Zhong 1992), Mallat and Zhong combined the properties of a wavelet transform and a gradient method to form a multi-scale edge detector. In their work, the first derivative of a cubic spline function was applied to detect the local extreme values of a wavelet transform as edge points. Here, we simply describe the principle of the multi-scale detection of edges. We consider a smoothing function φ(x) whose integral is equal to 1 and that converges to 0 at infinity. (A typical example of such smoothing function is the Gaussian function). In order to obtain smoothed versions of a signal at different scales, we introduce the smoothing function φs(x) at scale s:

( )

1 (5.1)

Multi-scale edges are defined as the modulus maxima points of the first derivative of smoothed versions of the signal. They are inflexion points of the filtered signal. If the smoothing function is differentiable, the first derivative of a function f, smoothed at scale s can be expressed by:

where * denotes the convolution product.

This concept of multi-scale edge detection can be also expressed in terms of wavelet transform. Here, we introduce the wavelet function ψ(x) and the wavelet function )ψ_s(x at scale s:

Then

where Wf denotes the wavelet transform of function f, with respect to the wavelet ψ(x), at the scale s and the position x. Equation (5.4) proves that multi-scale edges can be detected by means of a wavelet transform. Based on our computation for Wf (s, x), our task is now to detect local maxima of the modulus of the wavelet transform. It should be noted that this calculation can be made for any value of the scale parameter s.

Furthermore, in order to obtain superior results for image feature extraction, a edge detector combines an optimization process and a wavelet transform is proposed (Hsieh 1996). It uses Canny’s three criteria of good detection, good localization, and low spurious response as a guide to derive a wavelet that is successful for edge detection.

Canny believed that the three criteria could be used to evaluate the performance of an edge detector, and thus, we have converted these conceptual criteria into practical rules.

Good detection for our purposes will mean detection of a high signal-to-noise-ratio.

Good localization criterion is equivalent to obtaining a low mean-squared deviation value of the detected edges. The filter should delete spurious edges. A detailed presentation of the three criteria can be found in (Canny 1986).

Based on the above, a wavelet-based edge filter is derived using an optimization process. This allows us to obtain the low-pass filter {L(n)} and high-pass filter {H(n)}, respectively. A detailed procedure for deriving the coefficients of the edge filter is described (Hsieh 1996). Once we have the coefficients {L(n)} and {H(n)}, it is easy to implement the discrete wavelet transform. Since a digital image is a 2D array, let us first briefly introduce the algorithm to implement the 2D discrete wavelet transform.

Let )S20 f(x,y be the finest resolution and S21f(x,y)be the coarser resolution of f(x, y). Here, we assume L₂j(n) and H₂j(n) denote the discrete filters obtained by putting 12^j − zeros between all two consecutive coefficients of L(n) and H(n), respectively. The algorithm of the 2D discrete wavelet transform (DWT) can be illustrated as follows (Hsieh 1996):

; convolutions with respect to m and n, respectively, of an image I(m, n) using the 1-D filters a(m) and b(n).

Next, we will use W₂¹j f(x,y) and W₂²j f(x,y)to calculate related information about the detected edge points from an image. An important feature can be found, that is, the local maximum in the position of the horizontal detail image W₂¹j f(x,y) and the vertical detail image W₂²j f(x,y) corresponds to the image edge point in the horizontal and vertical directions. At scale 2^j, the modulus of the gradient vector of the input image f(x, y) is defined as follows:

)

and the orientation of the gradient vector is:

(

^{( , ) ( , )}

)

^(5.9) larger than a threshold and it is thus classified as an edge point. Using this method, we can detect image edge points at scale 2^j in any direction. At the corresponding locations, using the magnitudes of M₂j f(x,y) , the edge strengths can be determined.

Furthermore, based on the values ofA₂j f(x,y), the directions of the detected edges are determined. As above, the scale 2^j can be varied. In general, when scale increases, the degree of the image’s smoothness is strengthened, the resolution capability is weakened, and image edges are coarser. On the other hand, when scale decreases, the image is less smooth, resolution capability is greater, and image edges will appear finely. Thus, the method satisfactorily inhibits noise. This procedure is called “multi-scale” edge detection.

In this thesis, the main focus of tunnel excavation face image processing is extracting geological features, namely the images’ edges. We adopted the multi-scale analysis of wavelet transforms to perform tunnel image edge detection. A wavelet-based multi-scale edge detection framework is depicted in Fig. 5.2 and experimental results are shown in Fig. 5.3. One can observe the geological features at both the fine resolution and the coarse resolution. Additionally, at the coarser resolution, there is less noise. As stated above, noises caused by the uneven surfaces of excavation faces, boring, or light deficiency generally appear as irregular bands. They could be eliminated by applying the multi-scale concept at coarser resolutions. Along with the detected edges, the weak planes remain in the image. The findings thus indicate that the application of a wavelet-based multi-scale edge detector is indeed superior to a Sobel edge detector.

Tunnel Excavation Face Images

Image Standardization

Extract High-pass signals at each decomposition scale

Low-Pass Signals

High-Pass Signals

Smoothed Image

Loop

Gradient Calculation

Optimal Threshold Selection

Geological Feature Extraction From Excavation Face Images

Wavelet Decomposition

Fig. 5.2 Framework of wavelet-based multi-scale edge detection

(a)

(b)

(c)

Fig. 5.3 Experimental results of wavelet-based edge detection with various resolutions.

在文檔中 隧道開挖面地質特徵自動化擷取技術之探討與開發 (頁 60-67)