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3-D Computer Vision CSc83020 / Ioannis Stamos Revisit filtering (Gaussian and Median) Introduction to edge detection 3-D Computater Vision CSc 83020

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3-D Computer Vision CSc83020 / Ioannis Stamos Linear Filters Given an image In(x,y) generate a new image Out(x,y): For each pixel (x,y) Out(x,y) is a linear combination of pixels in the neighborhood of In(x,y) This algorithm is Linear in input intensity Shift invariant

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3-D Computer Vision CSc83020 / Ioannis Stamos Discrete Convolution This is the discrete analogue of convolution The pattern of weights is called the “kernel” of the filter Will be useful in smoothing, edge detection

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3-D Computer Vision CSc83020 / Ioannis Stamos Computing Convolutions What happens near edges of image? Ignore (Out is smaller than In) Pad with zeros (edges get dark) Replicate edge pixels Wrap around Reflect Change filter

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3-D Computer Vision CSc83020 / Ioannis Stamos Example: Smoothing Original: Mandrill Smoothed with Gaussian kernel

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3-D Computer Vision CSc83020 / Ioannis Stamos Gaussian Filters One-dimensional Gaussian Two-dimensional Gaussian

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3-D Computer Vision CSc83020 / Ioannis Stamos Gaussian Filters

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3-D Computer Vision CSc83020 / Ioannis Stamos Gaussian Filters

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3-D Computer Vision CSc83020 / Ioannis Stamos Gaussian Filters Gaussians are used because: Smooth Decay to zero rapidly Simple analytic formula Limit of applying multiple filters is Gaussian (Central limit theorem) Separable: G 2 (x,y) = G 1 (x) G 1 (y)

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3-D Computer Vision CSc83020 / Ioannis Stamos Size of the mask

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3-D Computer Vision CSc83020 / Ioannis Stamos Edges & Edge Detection What are Edges? Theory of Edge Detection. Edge Operators (Convolution Masks) Edge Detection in the Brain? Edge Detection using Resolution Pyramids

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3-D Computer Vision CSc83020 / Ioannis Stamos Edges

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What are Edges? Rapid Changes of intensity in small region

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3-D Computer Vision CSc83020 / Ioannis Stamos What are Edges? Surface-Normal discontinuity Depth discontinuity Surface-Reflectance Discontinuity Illumination Discontinuity Rapid Changes of intensity in small region

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3-D Computer Vision CSc83020 / Ioannis Stamos Local Edge Detection

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3-D Computer Vision CSc83020 / Ioannis Stamos What is an Edge? Edge easy to find

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3-D Computer Vision CSc83020 / Ioannis Stamos What is an Edge? Where is edge? Single pixel wide or multiple pixels?

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3-D Computer Vision CSc83020 / Ioannis Stamos What is an Edge? Noise: have to distinguish noise from actual edge

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3-D Computer Vision CSc83020 / Ioannis Stamos What is an Edge? Is this one edge or two?

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3-D Computer Vision CSc83020 / Ioannis Stamos What is an Edge? Texture discontinuity

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3-D Computer Vision CSc83020 / Ioannis Stamos Local Edge Detection

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Edge Types Ideal Step Edges Ideal Ridge Edges Ideal Roof Edges

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3-D Computer Vision CSc83020 / Ioannis Stamos Real Edges I x Problems: Noisy Images Discrete Images

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3-D Computer Vision CSc83020 / Ioannis Stamos Real Edges We want an Edge Operator that produces: Edge Magnitude (strength) Edge direction Edge normal Edge position/center High detection rate & good localization

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3-D Computer Vision CSc83020 / Ioannis Stamos The 3 steps of Edge Detection Noise smoothing Edge Enhancement Edge Localization Nonmaximum suppression Thresholding

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3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Unit Step Function:

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3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Unit Step Function: Ideal Edge: Image Intensity (Brightness):

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3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Partial Derivatives: Directional!

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3-D Computer Vision CSc83020 / Ioannis Stamos Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Rotationally Symmetric, Non-Linear Edge Magnitude Edge Orientation Squared Gradient:

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Theory of Edge Detection x yB1,L(x,y)>0 B2,L(x,y)<0 t Laplacian: (Rotationally Symmetric & Linear) I xx Zero Crossing

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3-D Computer Vision CSc83020 / Ioannis Stamos Difference Operators Ii,j+1Ii+1,j+1 Ii,jIi+1,j ε Finite Difference Approximations

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3-D Computer Vision CSc83020 / Ioannis Stamos Squared Gradient x y

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3-D Computer Vision CSc83020 / Ioannis Stamos Squared Gradient ifthreshold then we have an edge [Roberts ’65]

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3-D Computer Vision CSc83020 / Ioannis Stamos Squared Gradient – Sobel Mean filter convolved with first derivative filter

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3-D Computer Vision CSc83020 / Ioannis Stamos Examples First derivative Sobel operator

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3-D Computer Vision CSc83020 / Ioannis Stamos Second Derivative Edge occurs at the zero-crossing of the second derivative

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3-D Computer Vision CSc83020 / Ioannis Stamos Laplacian Rotationally symmetric Linear computation (convolution)

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3-D Computer Vision CSc83020 / Ioannis Stamos Discrete Laplacian Ii,j+1Ii+1, j+1 Ii,jIi+1,j Finite Difference Approximations Ii+1,j-1Ii,j-1Ii-1,j-1 Ii-1,j Ii-1,j+1

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3-D Computer Vision CSc83020 / Ioannis Stamos Discrete Laplacian Rotationally symmetric Linear computation (convolution) More accurate

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3-D Computer Vision CSc83020 / Ioannis Stamos Discrete Laplacian Laplacian of an image

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3-D Computer Vision CSc83020 / Ioannis Stamos Discrete Laplacian Laplacian is sensitive to noise First smooth image with Gaussian

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3-D Computer Vision CSc83020 / Ioannis Stamos From Forsyth & Ponce.

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3-D Computer Vision CSc83020 / Ioannis Stamos From Shree Nayar’s notes.

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3-D Computer Vision CSc83020 / Ioannis Stamos Discrete Laplacian w/ Smoothing

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3-D Computer Vision CSc83020 / Ioannis Stamos From Shree Nayar’s notes.

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3-D Computer Vision CSc83020 / Ioannis Stamos Difference Operators – Second Derivative

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3-D Computer Vision CSc83020 / Ioannis Stamos From Forsyth & Ponce.

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3-D Computer Vision CSc83020 / Ioannis Stamos Edge Detection – Human Vision LoG convolution in the brain – biological evidence! LoGFlipped LoG

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Image Resolution Pyramids Can save computations. Consolidation: Average pixels at one level to find value at higher level. Template Matching: Find match in COARSE resolution. Then move to FINER resolution.

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3-D Computer Vision CSc83020 / Ioannis Stamos From Forsyth & Ponce.

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Lecture 8: Edges and Feature Detection

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