1 Two-dimensional wavelet transform modulus maxima and image multi-scale edge extraction 1.1 Wavelet transform modulus maxima and multi-scale edge detection The multi-scale modulus maximal concept of digital image is established below, between the wavelet modulus maxima and the image edge point. Correspondence relationship.
Let the two-dimensional smoothing function (u, v) (such as Gaussian function or cubic spline wavelet) satisfy (u,v)0, R2(u,v)dudv=1,limu,v(u,v)=0, remember s(u,v)=1s2us, vs, for any f(u,v)L2(R2), (fs)(u,v) denotes the smoothed image of f(u,v), where s>0 For smooth scales. Define two two-dimensional wavelets from (u,v) as 1(u,v)=(u,v)u, denoted as 1s(u,v)=1s21us, vs;2(u,v)=(u, v)v, recorded as 2s(u,v)=1s2us, vs.
Then the two-dimensional wavelet transform of f(u,v) on the scale s includes two components W1f(s, u, v)=R2f(x, y)1s1x-us, x-vsdxdy=f1s(u,v), W2f(s, u, v) = R2f(x, y)1s2x-us, x-vsdxdy=f2s(u, v).
Where: ks(u,v)=1s2ks(-u,-v);k=1,2.
According to the two-dimensional binary wavelet transform (take s=2j), the sufficient condition for fully and stably expressing f is easy to prove that W1f(x, u, v)W2f(s, u, v)=sf1s(u,v)f2s( u, v)=su(fs)(u,v)v(fs)(u,v)=s(fs)(u,v).
Thus, the modulus of the gradient vector (fs)(u,v) of (fs)(u,v) is proportional to the modulus of the following wavelet transform: Mf(s,u,v)=W1f(s,u,v)2 W2f(s, u, v) 2. The angle between the gradient direction and the horizontal u (phase angle or angle) is Af(s, u, v)=arctanW2f(s, u, v)W1f(s, u, v ).
Under the scale s, if the modulo Mf(s, u, v) is at the point (u1, v1) along the direction of (u, v) = (u1, v1) lf (u, v1), when l is sufficiently small, the local is obtained. The maximum value, then the point (u1, v1) is an edge point of (fs)(u,v), which is a sudden point of f(u,v), and the direction of the boundary is perpendicular to nj(u1,v1) . This shows that the edge point of the image can be determined by detecting the modulus maxima of the two-dimensional wavelet transform. Since wavelet transform provides edge information of an image on various scales, it is called a multi-scale edge. Connecting the edge points at any scale along each boundary direction forms a modulus max curve along the boundary at that scale.
1.2 Multi-scale edge extraction of digital image Assume that image D has NN pixels, ie D={dn, mn, m=0, 1, 2,, N-1}. The specific processing and methods are as follows: (1) Under the scale s=2j, the two-dimensional wavelet transform W1f(2j,n,m), W2f(2j,n,m), n,m=0,1,,N-1 of the digital image D is calculated, where 1jJ= The number of scales of log2N. decomposition can be determined as needed.
(2) Calculate the tangent of the modulus value and phase angle Af(2j, n, m) of each point. For each pixel (n, m), calculate: Mf(2j, n, m) = W1f(2j, n, m)2 W2f(2j, n, m)2; tanAf(2j, n, m)= W2f(2j, n, m) W1f(2j, n, m).
(3) Find the boundary point. Determining the threshold T>0, for n, m=0,1,2,,N-1, if Mf(2j,n,m)T, and Mf(2j,n,m) obtain a local maximum, ie ( n, m) is the modulus maximum point, then (n, m) is a boundary point. From the structure of the digital image, it is known that there are only 8 adjacent points around each pixel, and they divide a plane into 8 sectors, which are sequentially -225 (, 225 (, 675 (, 1125 (, 1575 ( , -1575 (, -1 215 (, - 675 (, - 225 (two-two, sequentially represented by 0, 1, 2, 3, 4, 5, 6, 7). The average line direction of each sector is used in the middle. The black arrow indicates that the discrete gradient direction is indicated. Therefore, only 8 directions at each point can be used as the gradient direction. Considering the symmetry of the gradient direction, only the gradient directions of the four sectors 0, 1, 2, and 3 are considered.
Use CodeAf(2j,n,m) to mark the interval number in which tanAf(2j,n,m) falls, and assign Mf(2j,n,m) to the sector CodeAf(2j,n,m) in the gradient direction. The modulus values ​​of two adjacent pixels can be compared to determine whether Mf(2j, n, m) obtains local maxima; (4) connect boundary points on each scale to form a maximum curve along each boundary at each scale. . The boundary points in the image generally form a curve, which is usually the boundary of some important structures. By connecting the maximum points of the wavelet modes, a maximal curve along the boundary is formed. In the discrete case, the maximal curve is formed by concatenating two adjacent boundary points (n, m) and (n, m) (n, m) in the image discrete sampling grid, where (n, m) perpendicular to the gradient direction corresponding to the sector CodeAf(2j, n, m).
The porous algorithm of the 1.3B spline binary wavelet refers to the construction of the B-spline wavelet, and the value of the B-spline wavelet filter is calculated. The analytical filter value of the third-order B-spline wavelet is h[-1]/2=0125, h[1]/2=0375, g(0)=-05, h[0]/2=0375, h[-2]/2=0125,g[1]=05.
Since the filter is simple, the calculation speed is much faster than the Canny algorithm, which is one of the advantages of wavelet transform edge detection.
The traditional Canny edge detector is the first derivative of the Gaussian function, but it does not have the multi-scale analysis function of the image. The image edge information obtained is often messy and the image processing effect is not good.
Since the scaling function and wavelet of the dyadic wavelet transform are designed by the filters h, g, h, g, the fast algorithm for wavelet decomposition and reconstruction can also be realized by it. In this paper, the porous algorithm (A) trousAlgorithm) is used to realize the wavelet decomposition. It can find the wavelet transform value of the sampling point at each resolution and realize multi-scale detection of image edges.
Let the original input signal be a two-dimensional discrete signal with limited energy or image a0={a0n,m}n,mZ, with a sampling interval of 1, and set (x,y)=(x)(y), correspondingly, 2j(x, y)=122jx2jy2j, then there is a two-dimensional function f(x, y)L2(R2) such that (f)(n,m)=a0n,m, for any j0, remember ajn,m=2j (f2j)(n,m), for j0, on integer grid points (n,m), the dyadic wavelet coefficients are given by: dj,1n,m=W1f(2j,n,m)=2j (f12j) (n, m) dj, 1n, m = W2f (2j, n, m) = 2j (f12j) (n, m).
Then for any scale 2j>1, discrete signal sequence {d1,1,d1,2}, {d2,1,d2,2},,{dj,1,dj,2}, aj is called discrete two-dimensional a0 Binary wavelet transform.
2 The experimental research image will be affected by the noise of the camera during the acquisition process. The wavelet mode greatly suppresses the noise while performing multi-scale detection on the edge, and achieves the dual requirements of suppressing noise and accurately positioning the edge. According to the wavelet modulus maximal theory and its porous algorithm, the image edge detection is implemented in Matlab software. Combined with the Canny processing function in Matlab software, the Canny algorithm is used to detect the edge of the image. The specific image processing results and experimental analysis are as follows.
2.1 Comparison of gear contour extraction effects The following are the gear images taken by means of CCD (Charge Coupled Devices) and the edge detection by Canny algorithm.
In the Matlab software, the wavelet contour maximal multi-scale edge detection is realized, and the gear contour graph obtained when the scale j=1, 2, 3, 4 is obtained.
The non-gear contour maximal curve appears in two places in the above-mentioned gear contour pattern, which is caused by the background during the shooting process and the pouring port of the plastic gear.
2.2 Performance analysis of edge detection Experiments show that the edge extraction characteristics of cubic B-spline binary wavelet are better than Canny algorithm, especially when the scale is j=3.
From the analysis of the relationship between the modulus maxima of the wavelet and the boundary point, to the programming of the third-order B-spline binary wavelet modulus and its multi-hole algorithm, the theoretical framework of the Canny algorithm and the wavelet algorithm is compared, and the Matlab is combined with the two algorithms. The realization effect can be seen as follows: (1) The Canny operator selects the Gaussian function as the gear contour map extracted by the smoothing function, and the wavelet modulus maximal edge extraction selects the cubic B-spline function as the gear contour map extracted by the smoothing function.
(2) Description: The derivative of the B-spline function can construct a binary wavelet, and there is a fast algorithm for multi-scale edge extraction. This fast algorithm has multi-scale image edge detection. For the Canny operator, it is not easy to give a fast and effective multi-scale algorithm, which has limitations when used.
(3) From the above figure of the gear contour extraction effect, it can be seen that the edge details of each scale are different when the multi-scale edge detection is performed, and the higher the scale, the less detail, which is multi-scale. Advantage. When j=4, the contour shape is deformed due to excessive rounding off the detail portion, resulting in distortion, which is not conducive to accurate measurement of the gear parameters. It can be seen that the modulus maximal edge detection of B-spline wavelet is best at j=3.
3 Conclusions The comparison between the modulus maximal edge detection algorithm based on spline wavelet and the traditional Canny algorithm is compared with the traditional Canny algorithm. The modulus maximal edge detection algorithm is better than the Canny algorithm. When the smooth scale j=3, the effect is obtained. Better, a clear gear profile is obtained. The multi-scale and multi-resolution edge detection algorithm based on spline wavelet has stronger ability to obtain different levels of edge information, and solves the problems of poor performance and low recognition value of traditional edge detection algorithms. Technical support is provided for high-precision detection of gear parameters.
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