Example: Canny Edge DetectionMore: facts and figuresAnswer..Choice of Some comparisonsCannyMarr-Hildreth vs. CannyImage PyramidsGaussian PyramidLaplacian PyramidsLaplacian PyramidReconstructionSpliningSplining (Blending)Example images from GTechOutlineMatching with Invariant FeaturesExample: Build a PanoramaHow do we build panorama?Matching with FeaturesSlide 21Slide 22Slide 23Slide 24More motivation…Corner DetectionCorner Detection: Analyzing Gradient CovarianceContentsHarris Detector: MathematicsSlide 30Slide 31Slide 32Slide 33Slide 34Harris DetectorHarris Detector: WorkflowSlide 37Slide 38Slide 39Slide 40Example: Gradient CovariancesExample: Corner Detection (for camera calibration)Example: Corner DetectionHarris Detector: SummarySlide 45Tracking: compression of video informationOne More..Harris Detector: Some PropertiesSlide 49Slide 50Slide 51Slide 52We want to:Models of Image ChangeSlide 55Rotation Invariant DetectionSlide 57Scale Invariant DetectionSlide 59Slide 60Slide 61Slide 62Slide 63Scale Invariant DetectorsHarris LaplacianHarris-LaplacianSlide 67Scale Invariant Detection: SummaryComputer Vision : CISC 4/689Example: Canny Edge Detection(Matlab automatically set thresholds)Computer Vision : CISC 4/689More: facts and figures•The convolution of two Gaussians with variances {1}2 and {2}2 is {1}2+{2}2. This is same as consecutive smoothing with the two corresponding SD’s. •Thus, generic formula is: i{i}2•Problem: A discrete appx. to a 1D Gaussian can be obtained by sampling g(x). In practice, samples are taken uniformly until the truncated values at the tails of the distribution are less than 1/1000 of the peak value.a) For =1, show that the filter is 7 pixels wide.Computer Vision : CISC 4/689Answer..•Lets pick (n+1) pixels from the center of kernel(including center). This way, total kernel size is 2n+1, n pixels on either side of origin. Exp(-{(n+1)2}/{22}) < 1/1000So, n > 3.7 -1n must be the nearest integer to 3.7 -0.5For =1, n=3, 2n+1=7.Filter coefficients can be obtained as {-3,-2,-1,0,1,2,3}Computer Vision : CISC 4/689Choice of •The choice of  depends on the scale at which the image is to be displayed.•Small values bring out edges at a fine scale, vice-versa.•Noise is another factor to look into the selection, along with computational costComputer Vision : CISC 4/689Some comparisonsZero-crossings easy to find than thresholdComputer Vision : CISC 4/689Canny•Many implementations of the Canny edge detector approximate this process by first convolving the image with a Gaussian to smooth the signal, and then looking for maxima in the first partial derivatives of the resulting signal (using masks similar to the Sobel masks). •Thus we can convolve the image with 4 masks, looking for horizontal, vertical and diagonal edges. The direction producing the largest result at each pixel point is marked. •Record the convolution result and the direction of the edge at each pixel.Computer Vision : CISC 4/689Marr-Hildreth vs. Canny•Laplacian is isotropic, computationally efficient: single convolution, look for zero-crossing. (one way to explain zero-crossing is, if first derivative can be looked at as a function, its maximum will be its derivative=0).•Canny being a directional operator (derivative in 4 or 3 directions), more costly, esp. due to hysterisis.•Two derivatives -> more sensitive to noiseComputer Vision : CISC 4/689Image Pyramids•Observation: Fine-grained template matching expensive over a full image –Idea: Represent image at smaller scales, allowing efficient coarse- to-fine search•Downsampling: Cut width, height in half at each iteration: from Forsyth & PonceComputer Vision : CISC 4/689Gaussian Pyramid•Let the base (the finest resolution) of an n-level Gaussian pyramid be defined as P0 = I. Then the ith level is reduced from the level below it by: •Upsampling S"(I): Double size of image, interpolate missing pixels courtesy of WolframGaussian pyramidComputer Vision : CISC 4/689Laplacian Pyramids•The tip (the coarsest resolution) of an n-level Laplacian pyramid is the same as the Gaussian pyramid at that level: Ln(I) = Pn(I)•The ith level is expanded from the level above according to Li(I) = Pi(I) ¡ S"(Pi+1(I))•Synthesizing the original image: Get I back by summing upsampled Laplacian pyramid levelsComputer Vision : CISC 4/689Laplacian Pyramid•The differences of images at successive levels of the Gaussian pyramid define the Laplacian pyramid. To calculate a difference, the image at a higher level in the pyramid must be increased in size by a factor of four prior to subtraction. This computes the pyramid. •The original image may be reconstructed from the Laplacian pyramid by reversing the previous steps. This interpolates and adds the images at successive levels of the pyramid beginning with the lowest level. •Laplacian is largely uncorrelated, and so may be represented pixel by pixel with many fewer bits than Gaussian. courtesy of WolframComputer Vision : CISC 4/689ReconstructionComputer Vision : CISC 4/689Splining•Build Laplacian pyramids LA and LB for A & B images•Build a Gaussian pyramid GR from selected region R•Form a combined pyramid LS from LA and LB using nodes of GR as weights:LS(I,j) = GR(I,j)*LA(I,j)+(1-GR(I,j))*LB(I,j)Collapse the LS pyramid to get the final blended imageComputer Vision : CISC 4/689Splining (Blending)•Splining two images simply requires: 1) generating a Laplacian pyramid for each image, 2) generating a Gaussian pyramid for the bitmask indicating how the two images should be merged, 3) merging each Laplacian level of the two images using the bitmask from the corresponding Gaussian level, and 4) collapsing the resulting Laplacian pyramid. •i.e. GS = Gaussian pyramid of bitmask LA = Laplacian pyramid of image "A" LB = Laplacian pyramid of image "B" therefore, "Lout = (GS)LA + (1-GS)LB"Computer Vision : CISC 4/689Example images from GTechImage-1 bit-mask image-2Direct addition splining bad bit-mask choiceComputer Vision : CISC 4/689Outline•Corner detection•RANSACComputer Vision : CISC 4/689Matching with Invariant FeaturesDarya Frolova, Denis SimakovThe Weizmann Institute of ScienceMarch 2004Computer Vision : CISC 4/689Example: Build a PanoramaM. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003Computer Vision : CISC 4/689How do we build