WAVELET EDGE DETECTION BASED ON A

小波变换

An improved method of image edge detection based on wavelet transform

WAVELET EDGE DETECTION BASED ON A`TROUS

A. theory of wavelet edge detection algorithm

Owing to the character of wavelet transform, its modulus maximum point is corresponding to the singular point of signal. So we can get image’s edge points by solving the modulus maximum points. For wavelet transform exist in every scale, so wavelet transform at each scale provides certain edge information, it is called multiscale edge[6]. Supposingθ (x, y) is a properly smooth 2-D signal, and satisfies equation

(6):

From the definition of gradient, at fixed scale s, if the module Ms f( x, y) gets a maximum value point in the direction of As f ( x, y) , it shows that the point (x, y) is the singular point of f * θ s(x, y); videlicet it is the singular point of f ( x, y) . Hence, the process of acquire edge points is transformed into a process of evaluate the module maximum points of wavelet transform.

B. edge detection algorithm based on a`trous

Via interpolating approximating of limited filter, a`trous do undecimated discrete wavelet transform.

Relative to other wavelet algorithms, it has three characters below[7]:

1) The requirement of space and time of calculation is rational obviously, and it’s easy to programmed realization.

2) It is isotropic on 2-D, the process of transform can be realized by filtering.

3) It is benefit to the acquirement of image’s minutia characters without sampling and

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interpolating. The concret description of a`trous is that: supposing the original image data is C (x ) , the data after filtering with scale function (x) is C (x) , so the equation W (x) =

C (x) C(x)means the information gap between image data at two different scale, namely minutia signal (wavelet plane). Actually, a`trous wavelet transform decompose input image data into several minutia signals and one background signal. The image’s minutia characters is concentrated on wavelet plane, the original image is the superposition of minutia signals and background signal. Supposing image I contains N×N pixels, it needs decompose image I at J scale, includes J = log N +1, scale s = 2 j (1≤ j ≤ J ) . Realization steps as follows:

1) Transform image to double type;

2) Calculate the size of image to ensure decompositionscale J;

3) Design a loop structure, do a`trous wavelet transform in row and column respectively, then get the magnitude and phase of wavelet coefficient;

4) Find out the modulus maximum point and mark down the position of the singular point.

5) Chaining the adjacent singular points, wipe off these undesirable points, then obtain the edge image correspond to scale s.

IV. SIMULATION AND ANALYSIS

Select the ‘cameraman’ image with Gaussian noises to be the original image. Do edge detection using traditional edge detection algorithms and a`trous method, list the edge detection results at j=1, 2, 3. The simulation result is show in Figure 2

Figure2. the effect pictures of traditional algorithms and the improved method

A. analysis of traditional edge detection operator

Robert operator: Using local differential operators to find

the edge, it has high positioning accuracy, but it is easy to lose part of the edge and do not have the ability to suppress noises.

Sobel and Prewitt operator: Both of them to do weighted smoothing of the image firstly, and then do differential operation. Therefore, they have a certain ability of noise suppression, but can not completely rule out the false test results appear in the edge.

Prone to appear multi-pixel edge, making the detection accuracy decreased Log operator: Firstly do image filtering with a Gaussian function, and then do Laplacian transform with filtered image, consider the point correspond to zero is the boundary

point. It may smooth out the original sharp edge while suppressing noise. It is necessary

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to select the appropriate variance parameter σ of Gaussian function to get the best image processing effect.

Canny operator: The edge detection operator is derived based on optimization

ideological[8]. First with the Gaussian filter to smooth the image, and then calculate gradient magnitude and direction of the filtered image, and then apply non-maxima suppression method to the gradient magnitude to obtain the edge detection image. This operator uses a dual-threshold method to detect and connect edge, this method is less susceptible to noise and is ability to detect weak edge, but the continuity of the boundary is worse than Log operator. From the simulation result, all of these traditional operators have the problem of error detection or missing detection. The detection accuracy can not achieve requirements really or technically.

B. analysis of the new method

The new edge detection method get the detection results at different scales by making the best of the scale

characteristic of wavelet transform. At j=1, the new method can detect the edge details effectively. In addition, it has good accuracy. It forms the outline of image edge as the j increased. We can choose suitable edge detection result within all results at different scale according to actual needs. The partial enlargement picture of detection result at j=1 is shown in Figure 3.

Figure3. the partial enlargement picture of detection result at j=1

In this picture, the edge details are reserved completely, and it has a high-precision positioning.