4 -1Properties of Laplacian•As a second order derivative, the Laplacian typically is unacceptably sensitive to noise.•Moreover, the Laplacian produces double edges and is unable to detect edge direction.•Sometimes it is used to detect whether a pixel is on the dark or light side of an edge.• A more general use of the Laplacian is finding the location of edges using its zero-crossing property.•This concept is based on convolving an image with the Laplacian of a 2-D Gaussian function (Marr and Hildreth 1980).Cross Section of LOGLaplacian of Gaussian Intensity functionLOG Detector •Convolving with the Marr-Hildreth kernel blurs theimage with blurring proportional to .•Although this property has value in terms of noise reduction, the usefulness lies in the zero-crossings.•Edge detection by gradient operators tends to work well when there are sharp intensity transitions and low noise.•Zero-crossings offer an alternative in cases where edges are blurred or when high noise content ispresent.Example of LOGOriginal Convolvingwith LOG Binarized Zero CrossingsEdge Linking and BoundaryDetection•Edge detectors detect intensity discontinuities and should yield pixels lying only on the boundarybetween regions.•In practice, the set of pixels yielded seldom characterizes the boundary because of noise, and breaks in the boundary due to non-uniformillumination, focus etc.•Typically edge detection is followed by linking and other operations to assemble edge pixels intomeaningful boundaries.•Many techniques are used.Local Processing•Link all points that are similar in a, say, 3x3 or 5x5 neighbourhood surrounding a given point (x,y)•Similarity is established in terms of strength and direction of gradient function•That is, similarity is established for pixels at (x,y) and (x’,y’) with thresholds T and A, ifwhere ∝is the gradient angleA yx yxTyxfyxf≤-≤∇-∇|)','(),(||)','(),(|ααExample of Edge LinkingOriginal GxG y Vertical and Horizontal Edge LinkingCanny Edge Detector (and Linker)•Canny, J., A Computational Approach To EdgeDetection, IEEE Trans. Pattern Analysis and MachineIntelligence, 8(6):679–698, 1986.–Apply Gaussian filter to smooth the image in order to remove the noise–Find the intensity gradients of the image–Apply non-maximum suppression to get rid of spurious response to edge detection (ridge following)–Apply double threshold to determine potential edges–Track edge by hysteresis: Finalize the detection of edges by suppressing all the other edges that are weak and notconnected to strong edges.Canny Edge Detector Often the preferred methodGlobal Processing via the HoughTransform•Here we consider linking points by determining whether they lie on a specified curve or line•This processing considers global relationships between pixels •Suppose for n points in an image, we want to find subsets of these points that lie on straight lines•One possible solution is to first find all lines determined by every pair of points and then find subsets of points associated witheach line•Involves n(n-1)/2~n2lines and (n)(n(n-1))/2~n3comparisons •prohibitively expensive in all but trivial situations!Hough Transform •Hough (1962) proposed an alternative referred to as the Hough transform •Consider a point (x i ,y i ) and the general equation of a straight line y i = a x i +b •Infinitely many lines pass through (x i ,y i ) but they all satisfy y i = a x i +b for varying values of a and b •However, expressing this equation as b = -x i a +y i and considering the (a,b) parameter plane yields the equation of a single line for a fixed pair (x i ,y i ) •Further a second point (x j ,yj ) also has a lineassociated with it and the two lines intersect at (a’,b’)XY and Parameter Planesyx a b(x i,y i)(x j,y j)a’b’b = -xia +yib = -xia +yiComments•One problem with using ab space is that the slope and the intercept approach infinity as the line approaches the vertical •A solution is to use the following representation for a line •Now points yield sinusoidal curves in parameter space •Method can be extended to detect circles and other shapes rather than lines.ρθθ=+sin cos y xHough Transform ExampleOriginalInfraredImageEdge DetectedHough Transform Projected Back onto Image。