Lecture: Edge Detection

Transcription

1 CMPUT 299 Winter 2007 Lecture: Edge Detection Irene Cheng

2 Overview. What is a pixel in an image? 2. How does Photoshop, + human assistance, detect an edge in a picture/photograph? 3. Behind Photoshop - How does the computer do it?

3 Pixel as the basic unit An image is processed in memory as bytes (8bits/bytes) Grey scale image [0 255] byte pixel Color image [RGB of values 0-255] 3 bytes pixel

4 (0, 0) X Pixel as the basic unit R G B Y (256, 256)

5 Definition of Edges What are edges in an image? Locations where there is a sudden variation in the grey or color scale. Which image does not contain edges? a b c d e f g h i

6 Definition of Edges Use Photoshop Find Edges tool: a b c d e f g h i

7 Sharpen edges in images Photoshop Demo Sharp Blur High contrast Low contrast Increase contrast and sharpen edges Filter sharper sharpen edges/sharpen more

8 Sharpen edges in images Photoshop Demo Bright Low luminosity Sharpen and image-adjustcurves Discover features Filter sharper sharpen edges/sharpen more

9 Region growing + Edge detection The Magic Wand Tool in Photoshop

10 Can Computer do as well as human? Computer applies edge detection technique based on numeric computation, not based on human perception and cognitive skill

11 Edge detection techniques Noise reduction or elimination Filtering or masking Edge enhancement Edge localization Threshold Examples of Image pixels Concept of filtering An example of Low Pass Filtering: or 0. 9 An example of High Pass Filtering:

12 A Simple Edge Detector - gradient Based on grey scale gradient at a pixel g x ( x, y) f ( x +, y) f ( x, y) x g y ( x, y) f ( x, y + ) f ( x, y ) y Assuming threshold T = 50, a pixel is selected if >= T What is 2 2 x g y g + Answer: ?

13 Gradient detection (continued) Red squares represent selected pixels Detected edge g x g y = Assuming threshold = 50

14 Gradient detection (continued) The two gradients g x and g y computed at each pixel are regarded as the x and y components of a gradient vector, which has gradient magnitude and direction given by: 2 2 = g x g y = g y θ tan g x g + where the orientation θ is measured relative to x axis. Gradient magnitude is sometimes approximated by: g = g x + g y

15 Find edges in images Photoshop Demo Extract R, G, or B Pixels, or luminosity values Stylize find edges Apply Edge Detection

16 Find edges in images Photoshop Demo Find Edges By applying different thresholds T, the results are different: Assign one value v if < T, e.g. v=0 value Assign another value v2 if >= T, e.g. v2=255 value You can switch the resulting values to show a white background increasing threshold Irene Cheng T T2 T3

17 Find edges in images Photoshop Demo Increase contrast Why is there no change? Apply threshold T3

18 Concept Pixel value p Pixel value p2 Compare with threshold T. If T > p => no edge u p p2 p2 p = p (e.g. gradient magnitude) u2 Apply edge detection technique to generate a bigger u value around edges, so that u > T Note: the processing is not in place

19 Apply edge detection kernels. Suppose a nxn (n is an odd number) kernel is used and the centre position of the kernel is denoted by kc. 2. Slide the kernel across the image, one pixel at a time. 3. Compute the new value of each pixel, which is under kc, and repeat for all pixels in the image. 4. Note that the process is not in-place (new values are stored in a new image of same dimension. 5. Computation: For each pixel inside the kernel, multiply the pixel value with the kernel value. Take the sum of the products. 6. (a) The border pixel can be duplicated, or (b) take the mirror values of the border pixels, to fill up the kernel. C Image border R Image (RxC pixels) Kernel (nxn pixels) 6 (a) (b)

20 Convolution operation Based on convolution operations compute weighted averages over a 3x3 neighborhood ), ( ), ( y x f h y x g x x = ), ( ), ( y x f h y x g y y = where the (Prewitt) kernels are: = h x = h y Note: high order convolution kernels like 5x5, etc. can also be used, but they are more computational expensive

21 Convolution operation (continued) h x 0 = 0 h y = 0 0 Assuming threshold = () Is 98 an edge? (2) What are the detected pixels? (3) What will happen if the threshold is 00? 50?

22 Simple Edge Detectors One edge but thick Result from gradient Result from Prewitt kernels Threshold=50 Threshold=00 or 50

23 An Example Gradient magnitudes [Reference ] Apply threshold T: If < T, value If >= T, value 2 (Top Left) Original (Top Right) Apply gradient magnitudes and scale to range (Bottom Left) Apply threshold of 50 to Top Right (Bottom Right) Apply threshold Of 50 to Top Right

24 Scaling grey values onto another range Let O range, O min and O max be the original range, minimum value and maximum value, i.e. O range = O max O min The N range, N min and N max be the original range, minimum value and maximum value, i.e. N range = N max N min To map a pixel P from the original range to P in the new range: P' = P O O min range N range + N min

25 Advantage of a bigger grey value range before processing Threshold 50 Map from 0-00 To Find edges Threshold 28

26 Advantage of a bigger grey value range before thresholding Threshold 28 Map from 0-00 To Find edges Scale To Then apply Threshold 28 Threshold 28

27 Sobel kernels Sobel kernels, which give more weights to onaxis pixels h x = h y = Cross from grey to white +ve value From white to grey -ve value Original image convoluation with hx with hy

28 Noise in an image Problem with edge localization High threshold may suppress meaningful edges Low threshold may include unwanted edges Noise may have high magnitude With noise Without noise x Gradient magnitude Gradient magnitude is very sensitive to noise

29 Blurred edges The detected edge can be rather broad in the case of diffuse edges, resulting in a thick band of pixels instead of a single point of maximum gradient. Grey values Slightly blurred Heavily blurred x

30 Sobel and first-order-derivatives Grey values 255 Sharp edge Blurred edge Sobel kernels give first-order-derivatives that measure the local slope of a surface in the x and y directions.

31 The Laplacian The second-order-derivatives ( 2 f ) measure the rate at which the slope of the grey scale surface changes in the x and y directions, and therefore can be used for edge localization. Let represent dark to bright grey. On the 2 darker side, f is decreasing (-ve) and on the brighter side, 2 is increasing (+ve). f

32 st & 2 nd order derivatives Grey values Sobel gives first derivatives (increase when approaching the border and decrease when leaving the border.) Second order derivative measures the rate at which the slope of the grey level changes with distance It changes sign at the centre of the edge, so the edge can be localized at the zero crossing. band Gradient pixel at the zero crossing

33 Another example of 2 nd order derivatives pixel grey values gradient values rate of slope change Assuming threshold is 20 grey values ve decreasing -> increasing -> +ve pixel (x-direction) gradient values pixel (x-direction)

34 The Laplacian (continued) The Laplacian of an image f combines the second-order-derivatives: f f f = x y The Laplacian is seldom used on its own because it is sensitive to noise. It is often used as part of the Laplacian of Gaussian (LoG) filter. a Gaussian filter to blur the image a Laplacian to enhance edges localization is done by finding zero crossings

35 [reference ] Radially-symmetric two-dimensional Gaussian with r 2 = x 2 + y 2 : = exp ) ( σ r r h The Laplacian is: = exp σ σ σ r r h LoG The standard deviation value σ acts as a threshold value

36 The Canny Edge Detector There is a trade-off between noise reduction and edge localization. Improved noise reduction is at the expense of good localization. The Canny detector provides the best compromise between the two.. Smoothing step: a Gaussian low pass filter 2. Enhancement step: calculate the gradient vector at each pixel of the smoothed image Gaussian separability: compute x and y gradients with onedimensional kernels 3.

37 The Canny Edge Detector (continued) 3. Localization step: non-maximal suppression Thins the wide ridges around local maxima in gradient magnitude down to only one pixel wide [reference ]

38 The Canny Edge Detector (continued) Algorithm: Non-maximal suppression in the Canny edge detector Create an output image, g s, with the same dimensions as g For all pixel coordinates, x and y, do Approximate θ(x, y) by θˆ, one of the angles 0, 45, 90, 35 if g(x, y)<g at neighbour in direction θˆ or g(x, y)<g at neighbour in direction θˆ +80 then g s (x, y) = 0 else end for end if g s (x, y) = g(x, y)

39 Hysteresis thresholding Separate into gradient and θ images. On the gradient image, apply hysteresis thresholding. Assuming T high has been applied, expand the neighborhood using T low. Some techniques use 8 neighbors, and some use neighbors along a line normal to the gradient orientation at the edge pixel only [reference ]

40 Rank or order statistic filtering Non-linear In image processing it is usually necessary to perform high degree of noise reductionin an image before performing higher-level processing steps, such as edge detection. A non-linear digital filtering technique is often used to remove noise from images or other signals.

41 Rank or order statistic filtering (continued) Compile a list of grey scales in the neighborhood of a given pixel, sort them in ascending order and select a value as the new value Median filter; any structure that occupies less than half of the filter s neighborhood will be eliminated Minimum filter and Maximum filter Range filter: output the difference between the maximum and minimum

42 Rank or order statistic filtering (continued) Hybrids of linear and non-linear filters, e.g. α-trimmed mean filter: sorts the neighborhood into ascending order, discards a number of them and outputs the mean of the remaining (α is the number of values removed, in the range [0, (n 2 )/2] from each end of the list) when α = 0: mean filter when α = (n 2 )/2: median filter n 2 2α 2 n α f i i= α +

43 Median filter Non-linear technique Consider all pixels inside the filter. Compile a list of grey values and sort them in ascending order. select a value as the new value Median filter; any structure that occupies less than half of the filter s neighborhood will be eliminated Have the advantage of non kernel-based; no problem to filter a smaller neighborhood at the corners or sides of the image The shape of the filter applied at the corner can give different results, e.g. square vs. cross-shaped

44 Rank filtering examples What is the filtered value of the centre pixel after applying a:. median filter 2 2. mean filter 3 3. maximum filter minimum filter 5 5. range filter trim filter trim filter trim filter 2

45 Median filtering example The square and the cross are both median filters p P (a) (b) What is the filtered value of P in (a)? What is the filtered value of P in (b)?

46 References. Digital Image Processing a practical introduction using Java Nick Efford, Pearson Education 2000.