September 26, 2013Computer Vision Lecture 8: Edge Detection II 1Gradient In the one-dimensional case, a step edge corresponds to a local peak in the first derivative of the intensity function. In the two-dimensional case, we analyze the gradient instead of the first derivative. Just like the first derivative, the gradient measures the change in a function. For two-dimensional functions it is defined as
September 26, 2013Computer Vision Lecture 8: Edge Detection II 2Gradient Gradients of two-dimensional functions: The two-dimensional function in the left diagram is represented by contour lines in the right diagram, where arrows indicate the gradient of the function at different locations. Obviously, the gradient is always pointing in the direction of the steepest increase of the function.
September 26, 2013Computer Vision Lecture 8: Edge Detection II 3Gradient In order to compute G x and G y in an image F at position [i, j], we need to consider the discrete case and get: G i = F[i+1, j] – F[i, j] G j = F[i, j+1] – F[i, j] This can be done with convolution filters: Gj =Gj =Gj =Gj = Gi =Gi =Gi =Gi = To be precise in the assignment of gradients to pixels and to reduce noise, we usually apply 3 3 filters instead (next slide).
September 26, 2013Computer Vision Lecture 8: Edge Detection II 4 Sobel Filters Sobel filters are the most common variant of edge detection filters.Sobel filters are the most common variant of edge detection filters. Two small convolution filters are used successively:Two small convolution filters are used successively: SjSjSjSj SiSiSiSi
September 26, 2013Computer Vision Lecture 8: Edge Detection II 5 Sobel Filters Sobel filters yield two interesting pieces of information: The magnitude of the gradient (local change in brightness): The magnitude of the gradient (local change in brightness): The angle of the gradient (tells us about the orientation of an edge): The angle of the gradient (tells us about the orientation of an edge):
September 26, 2013Computer Vision Lecture 8: Edge Detection II 6 Sobel Filters Calculating the magnitude of the brightness gradient with a Sobel filter. Left: original image; right: filtered image.
September 26, 2013Computer Vision Lecture 8: Edge Detection II 7 Laplacian Filters Let us take a look at the one-dimensional case:Let us take a look at the one-dimensional case: A change in brightness:A change in brightness: Its first derivative:Its first derivative: Its second derivative:Its second derivative:
September 26, 2013Computer Vision Lecture 8: Edge Detection II 8 Laplacian Filters Idea: Smooth the image, Smooth the image, compute the second derivative of the (2D) image, compute the second derivative of the (2D) image, Find the pixels where the brightness function “crosses” 0 and mark them. Find the pixels where the brightness function “crosses” 0 and mark them. We can actually devise convolution filters that carry out the smoothing and the computation of the second derivative.
September 26, 2013Computer Vision Lecture 8: Edge Detection II 9 Laplacian Filters
September 26, 2013Computer Vision Lecture 8: Edge Detection II 10 Laplacian Filters Since we want the computation centered at [i,j]: Obviously, each function can be computed using a 1, -2, 1 convolution filter in horizontal or vertical direction. Since 2 is defined as the sum of these two functions, we can use a single 3 3 (or larger) convolution filter for its computation (next slide).
September 26, 2013Computer Vision Lecture 8: Edge Detection II 11 Laplacian Filters Continuous variant: Discrete variants (applied after smoothing):
September 26, 2013Computer Vision Lecture 8: Edge Detection II 12 Laplacian Filters Let us apply a Laplacian filter to the same image:
September 26, 2013Computer Vision Lecture 8: Edge Detection II 13 Laplacian Filters 3 3 Laplacian 5 5 Laplacian zero detection 7 7 Laplacian
September 26, 2013Computer Vision Lecture 8: Edge Detection II 14 Gaussian Edge Detection As you know, one of the problems in edge detection is the noise in the input image. Noise creates lots of local intensity gradients that can trigger unwanted responses by edge detection filters. We can reduce noise through (Gaussian) smoothing, but there is a trade-off: Stronger smoothing will remove more of the noise, but will also add uncertainty to the location of edges. For example, if two parallel edges are close to each other and we use a strong (i.e., large) Gaussian filter, these edges may be merged into one.
September 26, 2013Computer Vision Lecture 8: Edge Detection II 15 Canny Edge Detector The Canny edge detector is a good approximation of the optimal operator, i.e., the one that maximizes the product of signal-to-noise ratio and localization. Basically, the Canny edge detector is the first derivative of a Gaussian function. Let I[i, j] be our input image. We apply our usual Gaussian filter with standard deviation to this image to create a smoothed image S[i, j]: S[i, j] = G[i, j; ] * I[i, j]
September 26, 2013Computer Vision Lecture 8: Edge Detection II 16 Canny Edge Detector In the next step, we take the smoothed array S[i, j] and compute its gradient. We apply 2 2 filters to approximate the vertical derivative P[i, j] and the horizontal derivative Q[i, j]: P[i, j] (S[i+1, j] – S[i, j] + S[i+1, j+1] – S[i, j+1])/2 Q[i, j] (S[i, j+1] – S[i, j] + S[i+1, j+1] – S[i+1, j])/2 As you see, we average over the 2 2 square so that the point for which we compute the gradient is the same for the horizontal and vertical components (the center of the 2 2 square).
September 26, 2013Computer Vision Lecture 8: Edge Detection II 17 Canny Edge Detector P and Q can be computed using 2 2 convolution filters: P: Q: We use the upper left element as the reference point of the filter. That means when we place the filter on the image, the location of the upper left element indicates where to enter the resulting value into the new image.
September 26, 2013Computer Vision Lecture 8: Edge Detection II 18 Canny Edge Detector Once we have computed P[i, j] and Q[i, j], we can also compute the magnitude and orientation of the gradient vector at position [i, j]: This is exactly the information that we want – it tells us where edges are, how significant they are, and what their orientation is. However, this information is still very noisy.
September 26, 2013Computer Vision Lecture 8: Edge Detection II 19 Canny Edge Detector The first problem is that edges in images are not usually indicated by perfect step edges in the brightness function. Brightness transitions usually have a certain slope that extends across many pixels. As a consequence, edges typically lead to wide lines of high gradient magnitude. However, we would like to find thin lines that most precisely describe the boundary between two objects. This can be achieved with the nonmaxima suppression technique.
September 26, 2013Computer Vision Lecture 8: Edge Detection II 20 Canny Edge Detector We first assign a sector [i, j] to each pixel according to its associated gradient orientation [i, j]. There are four sectors with numbers 0, …, 3: 270 90 225 180 135 45 0000 315
September 26, 2013Computer Vision Lecture 8: Edge Detection II 21 Canny Edge Detector Then we iterate through the array M[i, j] and compare each element to two of its 8-neighbors. Which two neighbors we choose depends on the value of [i, j]. In the following diagram, the numbers in the neighboring squares of [i, j] indicate the value of [i, j] for which these neighbors are chosen: 103 2[i,j]2 301
September 26, 2013Computer Vision Lecture 8: Edge Detection II 22 Canny Edge Detector If the value of M[i, j] is less than either of the M-values in these two neighboring positions, set E[i, j] = 0. Otherwise, set E[i, j] = M[i, j]. The resulting array E[i, j] is of the same size as M[i, j] and contains values greater than zero only at local maxima of gradient magnitude (measured in local gradient direction). Notice that an edge in an image always induces an intensity gradient that is perpendicular to the orientation of the edge. Thus, this nonmaxima suppression technique thins the detected edges to a width of one pixel.