BIM472 Image Processing Image Segmentation

Transcription

1 BIM472 Image Processing Image Segmentation Outline Fundamentals Prewitt filter Roberts cross-gradient filter Sobel filter Laplacian of Gaussian filter Line Detection Hough Transform 2 1

2 Fundamentals Let R represent the entire spatial region occupied by an image. Image segmentation is a process that partitions R into n sub-regions, R 1, R 2,, R n, such that n (a) R = R. i= 1 (b) R is a connected set. i= 1, 2,..., n. i (c) R R =Φ. i i (d) ( R ) = TRUE for i= 1, 2,..., n. i (e) ( R R ) = FALSE for any adjacent regions i R and R. i j j j Note: Union and intersection operations on sets 3 Fundamentals Edge-based Segmentation Region-based Segmentation 4 2

3 Edges contain some of the most useful information in an image. We may use edges to measure the size of objects in an image isolate particular objects from their background recognize or classify objects. An edge may be simply defined as a line of pixels showing an observable difference. 5 In the right hand block, there is a clear difference between the grey values of the second and third columns. Human eye can pick out grey differences of this magnitude easily. Our aim is to develop methods which will enable us to pick out the edges of an image. 6 3

4 7 It appears that the difference tends to emphasize edges, and de-emphasize other components. We can define the difference in three separate ways: 8 4

5 However, an image is a function of two variables, so we can generalize these definitions to include both the x and y values as following where the subscripts in each case indicate the direction of the difference: 9 To see how we might use δ x to determine edges in the x direction, consider the function values around a point (x,y): 10 5

6 To find the filter which returns the value δ x, we just compare the coefficients of the function's values in δ x with their position in the array: This filter thus will find vertical edges in an image and produce a reasonably good result. 11 However, the resulting edges can be a bit jerky. This can be overcomed by smoothing the result in the opposite way by using the filter: 12 6

7 Both filters can be applied at once, using the combined filter: This filter, and its companion for finding horizontal edges: are the Prewitt filters for edge detection. 13 We are now in the position of having to apply two filters to an image. The trick here is to apply each filter independently, and form the output image from a function of the filter values produced. So if px and py are the grey values produced by applying Px and Py to an image, then the output grey value v can be chosen by any of these methods: 14 7

8 MATLAB Implementation: >> ic=imread('circuit.tif'); >> figure; imshow(ic); >> px=[-1 0 1;-1 0 1;-1 0 1]; >> icx=filter2(px, ic); >> figure,imshow(icx / 255) >> py=px'; >> icy=filter2(py, ic); >> figure,imshow(icy / 255) >> edge_p = sqrt(icx.^2 + icy.^2); >> figure,imshow(edge_p / 255); >> edge_t = im2bw(edge_p / 255,0.3); >> figure, imshow(edge_t); 15 Original Vertical Horizontal Combined Binary 16 8

9 MATLAB has also built-in edge detection function MATLAB Implementation (alternatively): >> edge_p=edge(ic,'prewitt'); >> imshow(edge_p); 17 edge function takes care of all the filtering, and of choosing a suitable threshold level; Two results obtained here seem a bit different. This is because the edge function does some extra processing over and above taking the square root of the sum of the squares of the filters. 18 9

10 Slightly different edge finding filters are: Roberts cross-gradient filters: and the Sobel filters: 19 The Sobel filters are similar to the Prewitt filters, in that they apply a smoothing filter in the opposite direction to the central difference filter. In the Sobel filters, the smoothing takes the form: which gives slightly more importance to the central pixel

11 >> ic=imread('circuit.tif'); >> edge_r=edge(ic,'roberts'); >> figure,imshow(edge_r) >> edge_s=edge(ic,'sobel'); >> figure,imshow(edge_s) Original Roberts Sobel 21 Second Differences: The Laplacian Another class of edge-detection method is obtained by considering the difference of differences. These are called as the second differences. To calculate a (central) second difference, take the backward difference of a forward difference: 22 11

12 This can be implemented by the filter: The corresponding filter for second differences in the y direction is: 23 The sum of these two is written as and is implemented by the filter: This is known as a discrete Laplacian

13 The Laplacian (after taking an absolute value, or squaring) gives double edges. It is also extremely sensitive to noise. However, the Laplacian has the advantage of detecting edges in all directions equally well

14 A more appropriate use for the Laplacian is to find the position of edges by locating zero crossings. In general, these are places where the result of the filter changes sign. 27 We define the zero crossings in such a filtered image to be pixels which satisfy either of the following: 1. They have a negative grey value and are next to a pixel whose grey value is positive, 2. They have a value of zero, and are between negative and positive valued pixels

15 Once we obtain zero crossings, we apply the Laplacian filter. However, this will produce a noisy image because many grey level changes can be interpreted as edges by this method. To eliminate them, we may first smooth the image with a Gaussian filter. This leads to the following sequence of steps for edge detection; the Marr-Hildreth method. 29 Marr-Hildreth method: 1. Smooth the image with a Gaussian filter, 2. Convolve the result with a laplacian, 3. Find the zero crossings. The first two steps can be combined into one, to produce a Laplacian of Gaussian or LoG filter

16 MATLAB Implementation: >> I=imread('circuit.tif'); >> edge(i,'log'); Original LoG 31 Comparison: Original Prewitt LoG Roberts Sobel 32 16

17 Line Detection Edge detection tells us where edges are, but not what they are (geometric descriptions like line, arc, etc.). The next step is to find out if there is any line (or line segment) in the image. To detect a line in an image, we need to see (by finding m and k) if y=m.x+k is satisfied by enough number of points (x,y). 33 Line Detection Recall that, to say (x1,y1), (x2,y2),, (xn,yn) are lying on the same line y=m*x+k, we mean: y1=m.x1+k y2=m.x2+k yn=m.xn+k Given a set of points: (x1,y1),, (xn,yn) The problem is to find (m1,k1),, (mp,kp) such that each pair (mi,ki) determine a line y=mi.x+ki that is contained in the image

18 Line Detection y = m.x + k form is most familiar line equation but cannot handle vertical lines. Another form is better: r = x cos θ + y sin θ 0 r, 0 θ < 2π any r, 0 θ π (don t need to worry about the sign of r) 35 Line Detection Hough Transform Given r and θ, the line equation r=x cos θ + y sin θ determines all points (x,y) that lie on a straightline For each fixed pair (x,y), the equation r=x cos θ + y sin θ determines all points (r,θ) that lie on a curve in the Hough space

19 Line Detection HT take a point (x,y) and maps it to a curve (Hough curve) in the (r,θ) Hough space: 37 Line Detection The pair (r*,θ*) that is common to many Hough curves indicates that the line r*=x cos θ* y sin θ* is in the image How to find the pairs (r,θ) that are common points of a large number of Hough curves? Divide the Hough space into bins and do the counting! 38 19

20 Line Detection Divide Hough space into bins: Accumulate the count in each bin An accumulate matrix H is used. For the figure above, only one entry has count 2; the others are either 0 or Line Detection Initialize accumulator H to all zeros For each edge point (x,y) in the image For θ = 0 to 180 r = x cos θ + y sin θ H(θ, r) = H(θ, r) + 1 end end Find the value(s) of (θ, r) where H(θ, r) is a local maximum The detected line in the image is given by r = x cos θ + y sin θ 40 20

21 Line Detection MATLAB Implementation: I = imread('circuit.tif'); roti = imrotate(i,33,'crop'); BW = edge(roti,'sobel'); [H,T,R] = hough(bw); imshow(h,[],'xdata',t,'ydata',r,'initialmagnification','fit'); xlabel('\theta'), ylabel('\rho'); axis on, axis normal, hold on; P = houghpeaks(h,5,'threshold',ceil(0.3*max(h(:)))); x = T(P(:,2)); y = R(P(:,1)); plot(x,y,'s','color','white'); 41 Line Detection MATLAB Implementation (continued): % Find lines and plot them lines = houghlines(bw,t,r,p,'fillgap',5,'minlength',7); figure, imshow(roti), hold on max_len = 0; for k = 1:length(lines) xy = [lines(k).point1; lines(k).point2]; plot(xy(:,1),xy(:,2),'linewidth',2,'color','green'); % plot beginnings and ends of lines plot(xy(1,1),xy(1,2),'x','linewidth',2,'color','yellow'); plot(xy(2,1),xy(2,2),'x','linewidth',2,'color','red'); % determine the endpoints of the longest line segment len = norm(lines(k).point1 - lines(k).point2); if ( len > max_len) max_len = len; xy_long = xy; end end % highlight the longest line segment plot(xy_long(:,1),xy_long(:,2),'linewidth',2,'color','cyan'); 42 21

22 Line Detection ρ θ 43 Summary Fundamentals Prewitt filter Roberts cross-gradient filter Sobel filter Laplacian of Gaussian filter Line Detection Hough Transform 44 22

23 References Digital Image Processing, 3rd Edition, R. C. Gonzalez & R. E. Woods, Pearson Prentice Hall. Lecture Notes, Frank (Qingzhong) Liu, University of New Mexico Tech. Lecture Notes, Alasdair McAndrew, Victoria University of Technology 45 23