Assignment 3: Edge Detection

Transcription

1 Assignment 3: Edge Detection - EE Affiliate I. INTRODUCTION This assignment looks at different techniques of detecting edges in an image. Edge detection is a fundamental tool in computer vision to analyse objects and to identify correspondence between similar objects. The coursework will look at the multiple stages involved in the Canny detector including Gaussian smoothing, non-maximal suppression and direction aware hysteresis. II. PREWITT AND SOBEL KERNELS Edge detection algorithms commonly use either Prewitt and Sobel kernels. These are applied to the image through convolution. There is a kernel in the x-direction to highlight edges in the x-direction and a kernel to highlight edges in the y-direction (see Fig 2 for comparison). Initially, an algorithm to implement the convolution of each kernel with the image was built (see prewitt.m). This was contrived by scrolling through each row of 9 pixels and taking the dot product with the kernel. Taking the sum of this obtained a weighted value of the central pixel. After further research, it was noticed that this could be achieved more efficiently using the inbuilt conv2 function (see sobel.m). Also, originally, the edges were combined by taking the amplitude from the square root of the squares of the edges in the x and y direction. However, for computation efficiency we can use the following approximation instead: g mag = g 2 x + g 2 y g x + g y (1) The resulting image must be scaled to 255 levels, the standard level of intensity in a grayscale image. Finally, the image is thresholded at a user-defined value. The response of both algorithms are shown in Fig 3 with 90 chosen as the threshold. This was a point seen when not too many high frequency edges were picked up but enough clear edges displayed. There is little perceivable difference between the quality of edge detection using these two algorithms. Using the tic and toc parameters, the benefits of the two adjustments made can be seen. The optimised algorithm is 56 times faster. >> sobel(li,90); Elapsed time is seconds. >> prewitt(li, 90); Elapsed time is seconds. A refinement was implemented using the derivative of a Gaussian to filter the image before applying the edge detection. This has the effect of smoothing the image and removing noise which is prone to detection by the Sobel or Prewitt filters. This makes the algorithm more complex and has additional tunable parameters for the standard deviation and the size of the filter window. The size of the filter window effectively truncates the Gaussian. It is reasonable to do this after 3 standard deviations. The effects of a 0.3 standard deviation can be seen on a lower threshold of 60 in Fig 4. High frequency components are removed in the grass and the brickwork of the lighthouse, leaving more true edges. Finally, the image was edge detected using the in-built Canny algorithm in Matlab. As can be seen in Fig 5, this algorithm is far superior. Edges are detected excellently. The rest of this paper will look at improvements to achieve this level of detection. III. NON-MAXIMAL SUPPRESSION ALGORITHM Non-maximal suppression works by looking at the angles of the edges by taking the arctan of the x-component and y-component. It can be seen that if a pixel value is non-maximal in a line tangential to its edge angle, it may be removed from the edge map. π/2 (i 1, j 1) ±π π/2 if 0 if edge > π/8 && edge < 3π/8 (i + 1, j + 1) FIG. 1: A graphic showing the algorithm behind this implementation of NMS This is implemented by splitting the edge direction into an 8 point compass. If the edge is in the horizontal direction, compare the top and bottom pixels. If the edge is in the vertical direction, compare the left and right direction. If the edge points north west or south east, compare the the top right and bottom left pixels. Alternatively, compare the bottom right and top left pixels. We may do this by looking in a π/8 range and setting the tangential comparison accordingly with a series of if loops. A diagrammatic illustration of this algorithm is shown in Fig 1. The effect of edge thinning is clearly observed when comparing the non-maximal suppression on Sobel, Prewitt, and Sobel with Gauss smoothing to

2 the pre-suppressed image in Figs 6, 7 & 8. IV. HYSTERESIS THRESHOLDING In image processing, hysteresis compares two images to build an intermediate image. The function takes two binary images that have been thresholded at different levels. The higher threshold has a smaller population of white pixels. The values in the higher threshold are more likely to be real edges. In hysteresis, if a value in the larger population is connected to a point in the smaller population, we can assume it is a real edge and add it to hysteresis image. A simple way to do this is to use the bwselect(bw1,c,r,n) function. This returns a binary image by observing objects in the image BW1 that overlap pixels at row R, column C. An object is defined by the parameter N. With N = 8, the algorithm looks for 8-connected neighbours set to 1. The effects of hysteresis can be seen in Figs 9, 10 & 11. The hysteresis is not particularly intelligent and chooses to add large patches of high frequency information based on connectivity. Later, a better edge aware technique will be implemented. V. FAST FOURIER TRANSFORM The discrete fast Fourier transform function fft2() converts a 2 dimensional signal into the frequency spectrum. This is implemented in MATLAB using the Cooley- Tukey algorithm [1] which reduces computational operations for an N-length vector from O(N 2 ) needed in a standard discrete Fourier transform (DFT) to O(N log(n)) by expressing the DFT recursively in terms of smaller DFTs. Once converted into the domain, the image must be scaled by the magnitude in order to display a range of frequencies. The magnitude of the signal shows how much of a frequency is present in the image. The phase specifies where this component exists in the image. Small thin edges are displayed as high frequency components and large edges are seen as lower frequency components. A log scale was taken for the lighthouse image. Observing a difficult spectrum such as seen in Fig 12 provides little useful information but the spectral form can be useful for processing as is seen in the next section. A simpler image shows its components more clearly in Fig 13. VI. HIGH PASS AND LOW PASS FILTER To apply a high pass filter, low frequencies must be attenuated and the high frequencies left untouched. Likewise, with a low pass filter, the high frequencies are attenuated. This is easy to achieve in the frequency domain where the components of frequency are known. An ideal filter will have an infinite roll-off. Each value can be multiplied by 1 or 0 dependent on whether it is passed or attenuated. Rather than using a box filter as suggested by the script, is more natural to use a circular filter. The radius of the filter defines the cut off frequency. For a low pass, only the values within the radius of the circle will be kept. For the high pass, it is the values outside of the radius which are kept. An algorithm was built to draw a circle based on the input cut-off frequency radius by the user. This uses the equation of a circle r = ± x 2 + y 2 defining the vectors x and y to cover all quadrants, positive and negative, to make the full circle. From Fig 14, it is noted most of the spectral data is contained in the low frequencies. This is intuitive for this is where the most defined edges in the image lie. Removing a small diameter circle from the image loses most of the information. VII. EDGE DIRECTION AWARE HYSTERESIS ALGORITHM A new algorithm was written to use edge orientation in hysteresis. The high threshold is considered the base image. If pixels in the lower threshold are in the same edge direction as adjacent ones in the higher threshold, they are included in the output image. This uses the same pi/8 window condition explained for non-maximal suppression. The effects of this algorithm can be seen in Fig 15. This form of hysteresis is much more subtle. Notice the roof of the shed is more joined up after hysteresis compared to the high threshold from 3 to 6 pixels on the right and 4 to 10 on the left. Large areas of random connected pixels have not been added. Turning down the direction tolerance, i.e. having a larger window than 9-point connectivity, would improve on these results. VIII. SCALE SPACE PYRAMID First, a hysteresis is performed on two images thresholded at 50 levels but with the first σ = 0.5 and the second σ = 5. The larger standard deviation removes more of the high frequency components. The original hysteresis function must be used as these two images will have different edge directions. The result of this can be seen in Fig 16. Some of the intermediate frequencies are absent. A scale space pyramid iteratively applies Gaussians on the image with the effect of downsampling. A full pyramid would have many different layers. Each layer is defined by the standard deviation required to downsample the image to half its original size. The standard rule is to use σ 2 = t where t is the scaling of the original image. This algorithm will not attempt an accurate pyramid but merely look at the addition of several different standard deviations to find frequency components. A ten level pyramid is shown in Fig 17. It must be noted that changing the range of standard deviations and the steps between them as well as the initial threshold level changes edge detection results dramatically. The top image has a 2 of 12

3 threshold of 20, starting at 0.4 standard deviations and increasing in 0.2 increments. The bottom has a threshold of 50, starting at 0.5 standard deviation and increasing in increments of IX. CONCLUSIONS In conclusion, several stages of edge detection were explored and algorithms produced. Different kernels were compared. The benefits of non-maximal suppression and hysteresis were discussed. Further advanced forms of hysteresis were attempted using edge direction and gaussian pyramids. The techniques are very much user-defined based on thresholding and standard deviation values. For each image, these parameters can be tweaked to get a good response. A more robust technique would bypass these human parameters. [1] J. Cooley and J. Tukey, An algorithm for the machine calculation of complex Fourier series, Mathematics of computation, of 12

4 FIG. 2: Here shows the edges detected using the Sobel kernels for the x-direction and the y-direction. FIG. 3: Here are the Prewitt and Sobel algorithms applied to the image with a 90 level threshold test FIG. 4: Here is the effect of pre-blurring on the image before edge detection. Notice high frequency components are no longer present 4 of 12

5 FIG. 5: The in-built canny detector is superior at detecting edges FIG. 6: Non maximal suppression on the Sobel edge detection 5 of 12

6 FIG. 7: Non maximal suppression on the Prewittt edge detection FIG. 8: Non maximal suppression on the Sobel edge detection with Gaussian blurring 6 of 12

7 FIG. 9: Hysteresis on the Sobel edge detection FIG. 10: Hysteresis on the Prewitt edge detection FIG. 11: Hysteresis on the Sobel edge detection with Gaussian blurring 7 of 12

8 FIG. 12: The FFT of the lighthouse image FIG. 13: FFT of a box 8 of 12

9 FIG. 14: Applying high pass and low pass filters to an image 9 of 12

10 FIG. 15: The effect of edge aware hysteresis in comparison to basic connectivity hysteresis 10 of 12

11 FIG. 16: Using two different standard deviation values to form the basis of hysteresis 11 of 12

12 FIG. 17: Here are the results on a ten level pyramid. 12 of 12