Edge operators are based on estimating derivatives.While first derivatives show approximately where the edges are, zero crossings of second derivatives were shown to be better.Ignoring that entirely, Canny developed his own edge detector that everyone uses now.After finding good edges, we have to group them into lines, circles, curves, etc. to use further.The Hough transform for circles works well, but for lines the performance can be poor. The Burns operator or some tracking operators (old ORT pkg) work better.
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1.Edge Detection EE/CSE 576 Linda Shapiro
2.Edge Detection EE/CSE 576 Linda Shapiro
3.Edges are caused by a variety of factors. depth discontinuity surface color discontinuity illumination discontinuity surface normal discontinuity Origin of edges
4.Characterizing edges An edge is a place of rapid change in the image intensity function image intensity function (along horizontal scanline) first derivative edges correspond to extrema of derivative 4
5.The gradient of an image: The gradient points in the direction of most rapid change in intensity Image gradient 5
6.The discrete gradient How can we differentiate a digital image F[ x,y ]? Option 1: reconstruct a continuous image, then take gradient Option 2: take discrete derivative ( “ finite difference ” ) 6
7.The gradient direction is given by: H ow does this relate to the direction of the edge? The edge strength is given by the gradient magnitude Simple image gradient How would you implement this as a filter? 7 perpendicular or various simplifications 0 -1 1
8.Sobel operator -1 0 1 -2 0 2 -1 0 1 -1 -2 -1 0 0 0 1 2 1 Magnitude: Orientation: In practice, it is common to use: What’s the C/C++ function? Use atan2 Who was Sobel?
9.Sobel operator Original Orientation Magnitude
10.Effects of noise Consider a single row or column of the image Plotting intensity as a function of position gives a signal Where is the edge? 10
11.Effects of noise Difference filters respond strongly to noise Image noise results in pixels that look very different from their neighbors Generally, the larger the noise the stronger the response What can we do about it? Source: D. Forsyth 11
12.Where is the edge? Solution: smooth first Look for peaks in 12
13.Differentiation is convolution, and convolution is associative: This saves us one operation: Derivative theorem of convolution f Source: S. Seitz How can we find (local) maxima of a function? 13 We don’t do that.
14.Remember: Derivative of Gaussian filter x -direction y -direction 14
15.Laplacian of Gaussian Consider Laplacian of Gaussian operator Where is the edge? Zero-crossings of bottom graph 15
16.2D edge detection filters is the Laplacian operator: Laplacian of Gaussian Gaussian derivative of Gaussian 16
17.Edge detection by subtraction original 17
18.Edge detection by subtraction smoothed (5x5 Gaussian) 18
19.Edge detection by subtraction smoothed – original (scaled by 4, offset +128) 19
20.Using the LoG Function (Laplacian of Gaussian) The LoG function will be Zero far away from the edge Positive on one side Negative on the other side Zero just at the edge It has simple digital mask implementation(s) So it can be used as an edge operator BUT, THERE’S SOMETHING BETTER 20
21.Canny edge detector This is probably the most widely used edge detector in computer vision J. Canny, A Computational Approach To Edge Detection , IEEE Trans. Pattern Analysis and Machine Intelligence, 8:679-714, 1986. Source: L. Fei-Fei 21
22.The Canny edge detector original image (Lena) 22 Note: I hate the Lena images.
23.The Canny edge detector norm of the gradient 23
24.The Canny edge detector thresholding 24
25.Get Orientation at Each Pixel theta = atan2 (- gy , gx ) 25
26.The Canny edge detector 26
27.The Canny edge detector thinning (non-maximum suppression) 27
28.Non-maximum suppression Check if pixel is local maximum along gradient direction Picture from Prem K Kalra 28
29.Compute Gradients (DoG) X-Derivative of Gaussian Y-Derivative of Gaussian Gradient Magnitude 29
30.Canny Edges 30
31.Canny on Kidney 31
32.Canny Characteristics The Canny operator gives single-pixel-wide images with good continuation between adjacent pixels It is the most widely used edge operator today; no one has done better since it came out in the late 80s. Many implementations are available. It is very sensitive to its parameters, which need to be adjusted for different application domains. 32
33.Effect of ( Gaussian kernel spread/size) Canny with Canny with original The choice of depends on desired behavior large detects large scale edges small detects fine features 33
34.An edge is not a line... How can we detect lines ? 34
35.Finding lines in an image Option 1: Search for the line at every possible position/orientation What is the cost of this operation? Option 2: Use a voting scheme: Hough transform 35
36.Connection between image (x,y) and Hough (m,b) spaces A line in the image corresponds to a point in Hough space To go from image space to Hough space: given a set of points (x,y), find all (m,b) such that y = mx + b x y m b m 0 b 0 image space Hough space Finding lines in an image 36
37.Hough transform algorithm Typically use a different parameterization d is the perpendicular distance from the line to the origin is the angle of this perpendicular with the horizontal. 37
38.Hough transform algorithm Basic Hough transform algorithm Initialize H[d, ]=0 for each edge point I[ x,y ] in the image for = 0 to 180 H[d, ] += 1 Find the value(s) of (d, ) where H[d, ] is maximum The detected line in the image is given by What’s the running time (measured in # votes)? 38 How big is the array H? Do we need to try all θ ? d Array H
39.Example 0 0 0 100 100 0 0 0 100 100 0 0 0 100 100 100 100 100 100 100 100 100 100 100 100 - - 0 0 - - - 0 0 - 90 90 40 20 - 90 90 90 40 - - - - - - - - 3 3 - - - 3 3 - 3 3 3 3 - 3 3 3 3 - - - - - - 360 . 6 3 0 - - - - - - - - - - - - - - - - - - - - - 4 - 1 - 2 - 5 - - - - - - - 0 10 20 30 40 …90 360 . 6 3 0 - - - - - - - - - - - - - - - - - - - - - * - * - * - * - - - - - - - (1,3)(1,4)(2,3)(2,4) (3,1) (3,2) (4,1) (4,2) (4,3) gray-tone image DQ THETAQ Accumulator H PTLIST distance angle 39
40.Chalmers University of Technology 40
41.Chalmers University of Technology 41
42.How do you extract the line segments from the accumulators? pick the bin of H with highest value V while V > value_threshold { order the corresponding pointlist from PTLIST merge in high gradient neighbors within 10 degrees create line segment from final point list zero out that bin of H pick the bin of H with highest value V } 42
43.Line segments from Hough Transform 43
44.Extensions Extension 1: Use the image gradient same for each edge point I[ x,y ] in the image compute unique (d, ) based on image gradient at ( x,y ) H[d, ] += 1 same same What’s the running time measured in votes? Extension 2 give more votes for stronger edges Extension 3 change the sampling of (d, ) to give more/less resolution Extension 4 The same procedure can be used with circles , squares, or any other shape, How? Extension 5; the Burns procedure . Uses only angle, two different quantifications, and connected components with votes for larger one. 44
45.A Nice Hough Variant The Burns Line Finder 1. Compute gradient magnitude and direction at each pixel. 2. For high gradient magnitude points, assign direction labels to two symbolic images for two different quantizations. 3. Find connected components of each symbolic image. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Each pixel belongs to 2 components, one for each symbolic image. Each pixel votes for its longer component. Each component receives a count of pixels who voted for it. The components that receive majority support are selected. -22.5 +22.5 0 45 45
46.Example 46 Quantization 1 Quantization 2 Quantization 1 leads to 2 yellow components and 2 green. Quantization 2 leads to 1 BIG red component. All the pixels on the line vote for their Quantization 2 component. It becomes the basis for the line. 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 -22.5 +22.5 0
47.Burns Example 1 47
48.Burns Example 2 48
49.Hough Transform for Finding Circles Equations: r = r0 + d sin c = c0 - d cos r, c, d are parameters Main idea: The gradient vector at an edge pixel points to the center of the circle. *(r,c) d 49
50.Why it works Filled Circle: Outer points of circle have gradient direction pointing to center. Circular Ring: Outer points gradient towards center. Inner points gradient away from center. The points in the away direction don’t accumulate in one bin! 50
51.51 Procedure to Accumulate Circles Set accumulator array A to all zero. Set point list array PTLIST to all NIL. For each pixel (R,C) in the image { For each possible value of D { - compute gradient magnitude GMAG - if GMAG > gradient_threshold { . Compute THETA(R,C,D) . R0 := R - D*sin(THETA) . C0 := C + D*cos(THETA ) . increment A(R0,C0,D) . update PTLIST(R0,C0,D) }}
52.52
53.Finding lung nodules (Kimme & Ballard) 53
54.Finale Edge operators are based on estimating derivatives. While first derivatives show approximately where the edges are, zero crossings of second derivatives were shown to be better. Ignoring that entirely, Canny developed his own edge detector that everyone uses now. After finding good edges, we have to group them into lines, circles, curves, etc. to use further. The Hough transform for circles works well, but for lines the performance can be poor. The Burns operator or some tracking operators (old ORT pkg ) work better. 54