25feb: Edge detection

The theme of this week is edge detection. First I experiment with edge detection in noisy images, then I detail a direction selective edge detector.

Activity 1: Edge detection in noisy images

Here I add Gaussian noise to a lena image and study the effect of noise on edge detection. As we know any edge detection uses some kind of high pass filtering at its core. Therefore if we try to detect edges of a noisy image, the high frequency noise component gets amplified and edge detection algorithms give poor performance, due to detection of spurious edges. A simple way to correct that would be to apply a low pass filter before we feed the image to the edge detection algorithm.

In the images below we see the original image, its spectrum, and its edges. Then if we compare it to the noisy image, we see that the spectrum looks more "whitened" that is in the original image the high frequency components (towards the corners) were darker, but in the noisy image the corners look brighter, due to the introduction of high frequency noise. Also the edges detected in the noisy image look worse than the original edges. Note that all spectrums displayed are centered and show as log of magnitude.

Next I used a Gaussian low pass filter of size 5 and variance 1 and then I passed the processed image to a Canny edge detector. The results are below. We can clearly see that the high frequency portions of the spectrum look darkened after the low pass filtering operation. Also the edges detected are much less noisy and comparable to the original edges detected.

Finally I tried median filtering with window size 5 (same as the Gaussian filter). Median fitering is good for removing salt and pepper type noise, but not Gaussian noise. In the images below we see that it removes too much details and the edges detected after median filtering is even less than the edges detected in the original image

Here is the MATLAB code for this

function filtering_edgedetect()

close all;

img = rgb2gray(imread('lena.jpg'));

figure; imshow(img); dispfft(img);

noisylena = imnoise(img, 'gaussian'); %add gaussian noise to the image

figure; imshow(noisylena); dispfft(noisylena);

im0 = edge(img, 'canny');

figure; imshow(im0); %edge detection on original image

im1 = edge(noisylena, 'canny'); %perform edge detection on noisy image

figure; imshow(im1);

filter2 = fspecial('gaussian', 5, 1);

im2 = imfilter(noisylena, filter2); %perform LPF

figure; imshow(im2); dispfft(im2);

im3 = edge(im2, 'canny'); %perform edge detection on LPF image

figure; imshow(im3);

im4 = medfilt2(noisylena, [5 5]);

figure; imshow(im4); dispfft(im4);

im5 = edge(im4, 'canny'); %perform edge detection on median filtered image

figure; imshow(im5);

end

%function to display fft spectrum

function dispfft(h)

temp = log(abs(fftshift(fft2(h))));

figure; imshow(temp/max(max(temp)));

end

Activity 2: Direction selective edge detection

In this section I will attempt to develop a rough direction sensitive edge detection code in MATLAB. The basic idea is first I use Prewitt operators to get horizontal and vertical edges, then estimate the gradient at each point. Then if the direction of gradient matches the desired direction in which we are looking for edges and if magnitude of gradient is above a certain threshold, we select that point as a valid edge along the desired direction.

The table below summarizes the Prewitt filter.

function directionaledge()

close all;

img = rgb2gray(imread('stripes1.jpg'));

figure; imshow(img);

filt1 = [1 1 1 ; 0 0 0 ; -1 -1 -1]; %horizontal edges

filt2 = filt1'; %vertical edges

%apply the vertical and horizontal filters using convolution

im1 = conv2(double(img), double(filt1));

figure; imshow(normalize(im1));

im2 = conv2(double(img), double(filt2));

figure; imshow(normalize(im2));

%calculate magnitude and direction of the estimated gradient

sz = size(img); mag = zeros(sz); angle = zeros(sz);

for i = 1:sz(1)

for j = 1:sz(2)

mag(i,j) = sqrt(double(im1(i,j)^2 + im2(i,j)^2));

angle(i,j) = atan(double(im2(i,j)/im1(i,j)));

end

end

figure; imshow(normalize(mag));

ang = pi/4 ; %set desired angle here

angleedge = zeros(sz);

mag = normalize(mag);

for i = 1:sz(1)

for j = 1:sz(2)

%if the magnitude is strong enough and if the direction is correct

if (mag(i,j) > 0.2 && abs(angle(i,j) - ang) < 0.15)

angleedge(i,j) = mag(i,j);

end

end

end

figure; imshow(normalize(angleedge));

end

function normimg = normalize(img)

normimg = img - min(min(img)); %gets rid of negative values

normimg = normimg/max(max(normimg)); %scales in the range 0 to 1

end

Let us first examine this image and see the direction selective edge detection in action

In the above image there are 4 images that are passed through direction sensitive edge filters (0^o, 90^o, 45^o, -45^o and 20^o).

Figure 1

The original image has stripes at about 20^o. The horizontal Prewitt operator gives larger values than the vertical Prewitt operator because the angle is close to the horizontal. The magnitude clearly shows the diagonal lines. Now consider the direction selective edge detector results. The 0^o detector shows no edge, except at the top edge (which is circled). The 90^o detector likewise shows no edges except an artifact at the left edge. The 45^o detector shows a faint edge (marked in green) as 45 is quite close to the 20^o edges present in the image. The -45^o detector is almost wholly empty. But the 20^o detector shows the strongest edges, like the one present in the image.

Figure 2

In this geometric image the edges in the different directions are detected by appropriate filters. Notice for the 45^o detector, some edges (marked in red) are brighter than others (marked in green). Thats because in the original image, some white 45o lines are embedded in black background (hence they give a stronger and brighter edge), while others are embedded in grey background (hence they give a weaker and darker edge). Thats because the white line has more difference from a black background than from a grey one. Note that the -45^o detector returns edges of same strength as all the lines in that direction are embedded in the same gray shade.

Figure 3

Another geometric figure showing correct edges detected by appropriate filters.

Figure 4

There are strong horizontal stripes in the upper half of the dress, which is well captured by the 0^o filter, while the 45^o stripes in the lower part are captured selectively by the 45^o filter. A few vertical edges in the border of the woman are captured by the 90^o filter.