9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythComputer Vision, Lecture 7Professor Hagerhttp://www.cs.jhu.edu/~hager9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythFilter Pyramids• An Exercise:– Suppose I have G(σ) and I perform G(σ) * G(σ) * I• Hint: think about the convolution theorem and the FFT• Suppose I want to subsample images– subsampling reduces the highest frequencies– averaging reduces noise– Can I average and resample and reduce noise while not loosing desirable frequencies?9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythGaussian Pyramid• Algorithm:– 1. Filter with G(σ)– 2. Resample at every other pixel– 3. Repeat• A common use of this is the Laplacian Pyramid9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythLaplacian Pyramid Algorithm• Check F&S9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythLaplacian of Gaussian Pyramid9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythGaussian Pyramid9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythTypes of Edge Operators1. Operators approximating derivatives using differences. • directional: Roberts, Prewitt, DoG, etc.• Rotationally invariant: Laplacian (sum of second derivatives)2. Operators based on the zero crossing of the second derivative (e.g. Canny).3. Operators that attempt to match a specific image profile.9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythFrom Pixels to Edges• Various operators can be used to enhance rapid contrast changes• Detecting these contrast changes involves thresholding to separate noise from signal• Edges are a result of grouping pixels (sometimes called “edgels”) into groups forming continuous curves.Definitions:Edge normal: Unit vector in direction of maximumintensity variationEdge direction: Perpendicular to edge normalEdge position: Image position of pixels of edgeEdge strength: Change in contrast along normal9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythEdge TypesStepRidgeRoof9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythA Simple Edge Detection Algorithm1. Compute image gradient ∇ I(u,v) = [Iu(u,v) Iv(u,v)]’2. Threshold on gradient magnitude: ||∇ I(u,v)|| > τ3. Compute connected components1. maximal sets of pixels whose neighbors exceed threshold4. Remove components smaller than a certain size5. Return resulting points as “edge segments”Problems?9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythCanny Edge Detector• The Plan:– Formulate an optimization problem for detection on 1-D signals– Generalize to 2D signals– Apply thresholding with hysteresis– Apply this operator at various scales• The Assumptions:– Edge enhancement is linear– The edge model is step edges with amplitude A– Noise is additive, white and Gaussian9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythOptimization Criteria• Good Detection: Minimize the probability of false positives and false negatives– Maximize the SNR• Good Localization: Edgels detected should lie as close as possible to the true edge– Maximize 1/distance to edge center which leads to maximizingLOC∫∫−−=∑WWWdttfndttfAfnA)()()(2000∫−=ΛWWdttfnfAfnA)('|)0('|)'(2009/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythOptimization Cont’d• Consider the maximizing the product of both criteria– result is itself a step filter– step filters are noise amplifying!• Additional criterion: single response constraint:– detector should minimize the number of local maxima about and edge (recall what happens with step filter)– RESULT1: localization vs. detection– RESULT2: optimal detector is very close to the first derivative of a Gaussian.)/()()'(1)'()()( wxfxfwherefwfandfwfwww=Λ=ΛΣ=Σ9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythThe Procedure• Enhancement:– compute x and y derivatives using DoG’s.– compute direction and magnitude of gradient (two images)• Nonmaximal Suppression:– Sample along the gradient direction– If given pixel is smaller than neighbor, set it to zero• Hysteresis Thresholding:– Starting from upper left, visit pixels until one exceeds tupper– Follow chains of maxima in edge direction until value drops below tlower– Mark and save all visited values as a connected contour9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythCanny Output9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythLaplacian vs. Canny Comparison9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythSubPixel Precision• It is often hard to exactly localize the maximum of a function• Many algorithms need sub-pixel precision• Thus, it is common to apply a *second derivative* operator locally to locate the edge– note we know the edge direction, so we can compute second directional derivatives!DO IT IN MATLAB GREG!9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythEdges Summary• Filtering is a way of removing noise or suppressing/enhancing frequency content• Typically, we combine some type of image derivative with smoothing• Image gradients are the basic tool in 2D images• Derivative of Gaussian is generally the gradient operator of choice• Canny detector is probably the most widely used algorithm for performing edge detection9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythDetecting Corners• Edges can be thought of as “1D” features• Corners are “2D” features• To make this precise, we need to think about the span of the gradientsC = ∑N∇ I(u) (∇ I(u))’ = R D R’D = diag(λ1,λ2)c = min(λ1,λ2)9/25/2002CS 461, Copyright G.D. Hagerwith slides shamelessly stolen from D. ForsythCorners Algorithm• For every point (u,v)’, compute c for a neighborhood about (u,v)• Threshold all values for which c(u,v) > τ• Sort the resulting values c(u,v)• Read off locations starting with highest values and working downuntil enough locations are found or we run out of locations.9/25/2002CS 461, Copyright G.D.