October 2, 2014Computer Vision Lecture 8: Edge Detection I 1 Edge Detection

October 2, 2014Computer Vision Lecture 8: Edge Detection I 2 Why Edge Detection? How can an algorithm extract relevant information from an image that is enables the algorithm to recognize objects? The most important information for the interpretation of an image (for both technical and biological systems) is the contour of objects. The most important information for the interpretation of an image (for both technical and biological systems) is the contour of objects. Contours are indicated by abrupt changes in brightness. Contours are indicated by abrupt changes in brightness. We can use edge detection filters to extract contour information from an image. We can use edge detection filters to extract contour information from an image.

October 2, 2014Computer Vision Lecture 8: Edge Detection I 3 Types of Edges One-dimensional profiles of different edge types

October 2, 2014Computer Vision Lecture 8: Edge Detection I 4 Types of Edges One-dimensional profile of actual edges

October 2, 2014Computer Vision Lecture 8: Edge Detection I 5 Edge Detection First we need some definitions: An edge point is a point in an image with coordinates [i, j] at the location of a significant local intensity change. An edge fragment corresponds to the i and j coordinates of an edge and the edge orientation , which may be the gradient angle. An edge detector is an algorithm that produces a set of edges (edge points or edge fragments) from an image.

October 2, 2014Computer Vision Lecture 8: Edge Detection I 6 Edge Detection A contour is a list of edges or the mathematical curve that models the list of edges. Edge linking is the process of forming an ordered list of edges from an unordered list. By convention, edges are ordered by traversal in a clockwise direction. Edge following is the process of searching the (filtered) image to determine contours.

October 2, 2014Computer Vision Lecture 8: Edge Detection I 7Gradient 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

October 2, 2014Computer Vision Lecture 8: Edge Detection I 8Gradient 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.

October 2, 2014Computer Vision Lecture 8: Edge Detection I 9Gradient In order to compute G i and G j 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).

October 2, 2014Computer Vision Lecture 8: Edge Detection I 10 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

October 2, 2014Computer Vision Lecture 8: Edge Detection I 11 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):

October 2, 2014Computer Vision Lecture 8: Edge Detection I 12 Gradient vs. Edge Orientation Here, the gradient orientation is horizontal (pointing to the right) and the edge orientation is vertical. Note: Edge and gradient orientation are perpendicular to each other:

October 2, 2014Computer Vision Lecture 8: Edge Detection I 13 Sobel Filters Calculating the magnitude of the brightness gradient with a Sobel filter. Left: original image; right: filtered image.

October 2, 2014Computer Vision Lecture 8: Edge Detection I 14 Sobel Filters and Thresholding

October 2, 2014Computer Vision Lecture 8: Edge Detection I 15 Sobel Filters Note that smoothing the image before applying a Sobel filter typically gives better results. Even thresholding the Sobel filtered image cannot usually create precise, i.e., 1-pixel wide, edges. Idea: The pixels with local maxima of the intensity gradient should be more precise edge indicators. We can use second-derivative methods such as Laplacian Filters for finding these maxima.

October 2, 2014Computer Vision Lecture 8: Edge Detection I 16 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:

October 2, 2014Computer Vision Lecture 8: Edge Detection I 17 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.