Computer Vision : CISC 4/689 Gradients and edges Points of sharp change in an image are interesting: –change in reflectance –change in object –change in illumination –noise Sometimes called edge points General strategy –determine image gradient –now mark points where gradient magnitude is particularly large wrt neighbours (ideally, curves of such points).

Computer Vision : CISC 4/689 The Gradient and Edges Consider image intensities as a 2-D height function I(x, y). Then the image gradient is the vector field defined by: Definition of an edge –Line segment separating regions of contrasting intensity –Location: Where gradient magnitude is high –Direction: Orthogonal to the gradient

Computer Vision : CISC 4/689 Edge Causes Depth discontinuity Surface orientation discontinuity Reflectance discontinuity (i.e., change in surface material properties) Illumination discontinuity (e.g., shadow)

Computer Vision : CISC 4/689 Edge Detection An edge point can be regarded as a point in an image where a discontinuity (in gradient) occurs across some line. A discontinuity may be classified as one of five types Searching for Edges: –Filter: Smooth image –Enhance: Apply numerical derivative approximation –Detect: Threshold to find strong edges –Localize/analyze: Reject spurious edges, include weak but justified edges Gradient Discontinuity -- where the gradient of the pixel values changes across a line. This type of discontinuity can be classed as roof edges ramp edges convex edges concave edges by noting the sign of the component of the gradient perpendicular to the edge on either side of the edge. Ramp edges have the same signs in the gradient components on either side of the discontinuity, while roof edges have opposite signs in the gradient components. A Jump or Step Discontinuity -- where pixel values themselves change suddenly across some line. A Bar Discontinuity -- where pixel values rapidly increase then decrease again (or vice versa) across some line. Source: LOCAL_COPIES/MARSHALL/node28.html

Computer Vision : CISC 4/689 Step edge detection: First Derivative Operators Method: Differentiate and find extrema Examples –Sobel operator (Matlab: edge(I, ‘sobel’) ) –Prewitt, Roberts cross –Derivative of Gaussian Sobel xSobel y Book uses this format

Computer Vision : CISC 4/689 Sobel Edge Filtering Example Rotate

Computer Vision : CISC 4/689 Step

Computer Vision : CISC 4/689 Step

Computer Vision : CISC 4/689 Step

Computer Vision : CISC 4/689 Step edge effect from zero- padding

Computer Vision : CISC 4/689 Sobel Edge Filtering Example: Result (pad with zeroes again, the boundary) and then we threshold…

Computer Vision : CISC 4/689 Sobel Edge Detection: Gradient Approximation Horizontal diff.Vertical diff Note anisotropy of edge finding

Computer Vision : CISC 4/689 Sobel These can then be combined together to find the absolute magnitude of the gradient at each point and the orientation of that gradient. The gradient magnitude is given by: an approximate magnitude is computed using: which is much faster to compute. The angle of orientation of the edge (relative to the pixel grid) giving rise to the spatial gradient is given by: In this case, orientation 0 is taken to mean that the direction of maximum contrast from black to white runs from left to right on the image, and other angles are measured anti-clockwise from this.

Computer Vision : CISC 4/689 Derivative of Gaussian

Computer Vision : CISC 4/689 Smoothing and Differentiation Issue: noise –smooth before differentiation –two convolutions: to smooth, then differentiate? –actually, no - we can use a derivative of Gaussian filter because differentiation is convolution, and convolution is associative

Computer Vision : CISC 4/689 The Laplacian of Gaussian Another way to detect an extremal first derivative is to look for a zero second derivative –the Laplacian Bad idea to apply a Laplacian without smoothing –smooth with Gaussian, apply Laplacian –this is the same as filtering with a Laplacian of Gaussian filter Now mark the zero points where there is a sufficiently large (first) derivative, and enough contrast

Computer Vision : CISC 4/689 Marr-Hildreth operator The Laplacian is linear and rotationally symmetric. Thus, we search for the zero crossings of the image that is first smoothed with a Gaussian mask and then the second derivative is calculated; or we can convolve the image with the Laplacian of the Gaussian, also known as the LoG operator; This defines the Marr-Hildreth operator. One can also get a shape similar to G'' by taking the difference of two Gaussians having different standard deviations. A ratio of standard deviations of 1:1.6 will give a close approximation to.This is known as the DoG operator (Difference of Gaussians), or the Mexican Hat Operator. Still sensitive to noise.

Computer Vision : CISC 4/689 Step edge detection: 2 nd -Derivative Operators Method: 2 nd derivative is 0 for 1 st -derivative extrema, so find “zero-crossings” –Laplacian Isotropic (finds edges regardless of orientation. Three commonly used discrete approximations to the Laplacian filter. (Note, we have defined the Laplacian using a negative peak because this is more common, however, it is equally valid to use the opposite sign convention.) Source:

Computer Vision : CISC 4/689 Laplacian of Gaussian Matlab: fspecial(‘log’,…) Below: Discrete approximation to LoG function with Gaussian 1.4

Computer Vision : CISC 4/689 Sobel vs. LoG Edge Detection: Matlab Automatic Thresholds SobelLoG

Computer Vision : CISC 4/689 There are three major issues: 1) The gradient magnitude at different scales is different; which should we choose? 2) The gradient magnitude is large along thick trail (for 3 rd fig); how do we identify the significant points? 3) How do we link the relevant points up into curves? = 1 = 2

Computer Vision : CISC 4/689 We wish to mark points along the curve where the magnitude is biggest. We can do this by looking for a maximum along a slice normal to the curve (non-maximum suppression). These points should form a curve. There are then two algorithmic issues: at which point is the maximum, and where is the next one?

Computer Vision : CISC 4/689 Non-maximum suppression At q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.

Computer Vision : CISC 4/689 Predicting the next edge point Assume the marked point is an edge point. Then we construct the tangent (along) to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s).

Computer Vision : CISC 4/689 Remaining issues Check that maximum value of gradient value is sufficiently large –drop-outs? use hysteresis use a high threshold to start edge curves and a low threshold to continue them.

Computer Vision : CISC 4/689

fine scale high Threshold (be strict in Accepting Edge points)

Computer Vision : CISC 4/689 coarse scale, high threshold

Computer Vision : CISC 4/689 coarse scale low threshold

Computer Vision : CISC 4/689 Canny Edge Detection Steps 1.Apply derivative of Gaussian (not Laplacian!) 2.Non-maximum suppression Thin multi-pixel wide “ridges” down to single pixel 3.Thresholding Low, high edge-strength thresholds Accept all edges over low threshold that are connected to edge over high threshold (in the stage of predicting next edge point) Matlab: edge(I, ‘canny’)

Computer Vision : CISC 4/689 Edge “Smearing” from Forsyth & Ponce Sobel filter example: Yields 2-pixel wide edge “band” We want to localize the edge to within 1 pixel Input Result

Computer Vision : CISC 4/689 Non-Maximum Suppression: Steps 1.Consider 9-pixel neighborhood around each edge candidate (i.e., already over a threshold) 2.Interpolate edge strengths E at neighborhood boundaries in negative & positive gradient directions from the center pixel 3.If the pixel under consideration is not greater than these two values (i.e. not a maximum), it is suppressed Interpolating the E value: E(r) = (1 ¡ a)E(x, y) + aE(x + 1, y) a1 ¡ a (x, y)(x + 1, y) r

Computer Vision : CISC 4/689 Example: Non-Maximum Suppression courtesy of G. Loy Original imageGradient magnitude Non-maxima suppressed

Computer Vision : CISC 4/689 Edge “Streaking” Can predict next pixel in edge orthogonal to gradient to make edge chain –Can also just use 8-connectedness to define chains Streaking: Gaps in edge chain due to edge strength dipping below threshold courtesy of G. Loy Original imageStrong edges gap

Computer Vision : CISC 4/689 Edge Hysteresis Hysteresis: A lag or momentum factor Idea: Maintain two thresholds k high and k low –Use k high to find strong edges to start edge chain –Use k low to find weak edges which continue edge chain Usual ratio of thresholds is roughly k high / k low = 2 or 3

Computer Vision : CISC 4/689 Example: Canny Edge Detection courtesy of G. Loy gap is gone Original image Strong edges only Strong + connected weak edges Weak edges

Computer Vision : CISC 4/689 Example: Canny Edge Detection (Matlab automatically set thresholds)

Computer Vision : CISC 4/689 Image 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 & Ponce

Computer Vision : CISC 4/689 Gaussian Pyramid Let the base (the finest resolution) of an n -level Gaussian pyramid be defined as P 0 = I. Then the i th level is reduced from the level below it by: Upsampling S " (I) : Double size of image, interpolate missing pixels courtesy of Wolfram Gaussian pyramid

Computer Vision : CISC 4/689 Reconstruction

Computer Vision : CISC 4/689 Laplacian Pyramids The tip (the coarsest resolution) of an n -level Laplacian pyramid is the same as the Gaussian pyramid at that level: L n (I) = P n (I) The i th level is obtained from the level above according to L i (I) = P i (I) ¡ S " (P i+1 (I)) Synthesizing the original image: Get I back by summing upsampled Laplacian pyramid levels

Computer Vision : CISC 4/689 Laplacian 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 coarsest level. Laplacian is largely uncorrelated, and so may be represented pixel by pixel with many fewer bits than Gaussian. courtesy of Wolfram

Computer Vision : CISC 4/689 Splining 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 image

Computer Vision : CISC 4/689 Splining (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 4/689 Example images from GTech Image-1 bit-mask image-2 Direct addition splining bad bit-mask choice

Computer Vision : CISC 4/689 Outline Corner detection RANSAC

Computer Vision : CISC 4/689 Matching with Invariant Features Darya Frolova, Denis Simakov The Weizmann Institute of Science March 2004

Computer Vision : CISC 4/689 Example: Build a Panorama M. Brown and D. G. Lowe. Recognising Panoramas. ICCV 2003

Computer Vision : CISC 4/689 How do we build panorama? We need to match (align) images

Computer Vision : CISC 4/689 Matching with Features Detect feature points in both images

Computer Vision : CISC 4/689 Matching with Features Detect feature points in both images Find corresponding pairs

Computer Vision : CISC 4/689 Matching with Features Detect feature points in both images Find corresponding pairs Use these pairs to align images

Computer Vision : CISC 4/689 Matching with Features Problem 1: –Detect the same point independently in both images no chance to match! We need a repeatable detector

Computer Vision : CISC 4/689 Matching with Features Problem 2: –For each point correctly recognize the corresponding one ? We need a reliable and distinctive descriptor

Computer Vision : CISC 4/689 More motivation… Feature points are used also for: –Image alignment (homography, fundamental matrix) –3D reconstruction –Motion tracking –Object recognition –Indexing and database retrieval –Robot navigation –… other

Computer Vision : CISC 4/689 Corner Detection Basic idea: Find points where two edges meet—i.e., high gradient in two directions “Cornerness” is undefined at a single pixel, because there’s only one gradient per point –Look at the gradient behavior over a small window Categories image windows based on gradient statistics –Constant: Little or no brightness change –Edge: Strong brightness change in single direction –Flow: Parallel stripes –Corner/spot: Strong brightness changes in orthogonal directions

Computer Vision : CISC 4/689 Corner Detection: Analyzing Gradient Covariance Intuitively, in corner windows both I x and I y should be high –Can’t just set a threshold on them directly, because we want rotational invariance Analyze distribution of gradient components over a window to differentiate between types from previous slide: The two eigenvectors and eigenvalues ¸ 1, ¸ 2 of C (Matlab: eig(C) ) encode the predominant directions and magnitudes of the gradient, respectively, within the window Corners are thus where min(¸ 1, ¸ 2 ) is over a threshold courtesy of Wolfram

Computer Vision : CISC 4/689 Contents Harris Corner Detector –Description –Analysis Detectors –Rotation invariant –Scale invariant –Affine invariant Descriptors –Rotation invariant –Scale invariant –Affine invariant

Computer Vision : CISC 4/689 Harris Detector: Mathematics Change of intensity for the shift [u,v]: Intensity Shifted intensity Window function or Window function w(x,y) = Gaussian1 in window, 0 outside Taylor series: F(x+dx,y+dy) = f(x,y) +fx(x,y)dx+fy(x,y)dy+…

Computer Vision : CISC 4/689 Harris Detector: Mathematics For small shifts [u,v ] we have a bilinear approximation: where M is a 2  2 matrix computed from image derivatives:

Computer Vision : CISC 4/689 Harris Detector: Mathematics Intensity change in shifting window: eigenvalue analysis 1, 2 – eigenvalues of M direction of the slowest change direction of the fastest change ( max ) -1/2 ( min ) -1/2 Ellipse E(u,v) = const

Computer Vision : CISC 4/689 Harris Detector: Mathematics 1 2 “Corner” 1 and 2 are large, 1 ~ 2 ; E increases in all directions 1 and 2 are small; E is almost constant in all directions “Edge” 1 >> 2 “Edge” 2 >> 1 “Flat” region Classification of image points using eigenvalues of M:

Computer Vision : CISC 4/689 Harris Detector: Mathematics Measure of corner response: (k – empirical constant, k = )

Computer Vision : CISC 4/689 Harris Detector: Mathematics 1 2 “Corner” “Edge” “Flat” R depends only on eigenvalues of M R is large for a corner R is negative with large magnitude for an edge |R| is small for a flat region R > 0 R < 0 |R| small

Computer Vision : CISC 4/689 Harris Detector The Algorithm: –Find points with large corner response function R (R > threshold) –Take the points of local maxima of R