IIT Bombay Slide 75 Laplacian Operator • The Laplacian operator is based on the Laplace equation given by ∂ ∂ 2 2 f f + = 0 ∂ ∂ 2 2 x y • Laplacian operator is discretized version of the above equation and is based on second derivatives along x and y directions GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 76 Laplacian Operator • Filter coefficients • The discrete version of the second derivative operator: • [1 -2 1] and [1 -2 1]T in the horizontal and vertical directions 0 -1 0 • Superimposing the two, -1 4 -1 we get the discrete Laplace 0 -1 0 operator GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 77 Properties of Laplace Operator • Isotropic operator – cannot give orientation information • Any noise in image gets amplified • Faster since only one filter mask involved • Smoothing the image first prior to Laplace operator is often needed for reliable edges GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 78 Zero-Crossing Edge Detectors • First derivative maximum: exactly where second derivative zero crossing • In order to detect edges, we look at pixels where the intensity gradient is high, or the first derivative magnitude is maximum • First derivative maximum implies a zero when the second derivative is computed • Edges are located at those positions where there is a positive value on one side and a negative value on the other side, in other words a zero-crossing GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 79 A step edge, whose first derivative is an impulse, and whose second derivative shows a transition from a positive to a negative Edge location corresponds to the point where a sign change occurs from positive to negative (or vice versa) GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 80 Zero-Crossing Edge Detectors • Laplacian of a function I(r,c) ∂ ∂ ∂ ∂ 2 2 2 2 I I ∇ = + = + 2 I ( ) I ∂ ∂ ∂ ∂ 2 2 2 2 r c r c Two commonly used masks for Laplacian operator GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 81 Zero Crossing Edge Detector • Direct operation on the image using the Laplacian operator results in a very noisy result • Derivative operator amplifies the high frequency noise • Preprocess the input image by a smoothing operator prior to application of the Laplacian GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 82 Zero Crossing Edge Detector • The Gaussian shaped smoothing operator is found to be ideal as a preprocessing operator • Therefore the Laplacian operator is applied on Gaussian smoothed input image • ZC(image) = Laplacian [gaussian(image)] GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 83 LOG operator • Both Laplacian operator and Gaussian operator are linear, and hence can be combined into one Laplacian of Gaussian (LoG) operator • Laplacian[Gaussian(image)] = [Laplacian(Gaussian)](image) GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 84 LOG operator • Laplacian[Gaussian(image)] = [Laplacian(Gaussian)](image) 2 2 + 1 r c 2 1 r − ( ) = − − + 2 LOG r c ( , ) e 2 σ (1 ) πσ σ 4 2 2 2 2 + 1 r c 2 1 c − ( ) − − 2 σ [ e 2 (1 )] Verify! πσ σ 4 2 2 2 2 + 1 r c + 2 2 1 r c − ( ) = − − 2 (2 ) e 2 σ πσ σ 4 2 2 GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 85 LOG operator • LoG operator is a sampled version of the function 2 2 + 1 r c + 2 2 1 r c − ( ) = − − 2 LOG r c ( , ) (2 ) e 2 σ πσ σ 4 2 2 • For a given value of σ , the size of the Gaussian filter is -3 σ to +3 σ • Computationally more expensive due to convolution with large filter masks GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 86 Zero-Crossing Edge Detectors Properties • Edges depend on the value of σ • For small value of σ all edges are detected • For large value of σ only major edges are detected • Any minor difference in intensity between neighbors can be captured using LoG filter • Significant zero crossings can be identified using suitable threshold GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 87 Zero-Crossing Edge Detectors • A pixel at (m,n) is declared to have a zero crossing if f’’(m,n) > T and f’’(m+ δ m, n+ δ n) < -T OR f’’(m,n) < -T and f’’(m+ δ m, n+ δ n) > T GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 88 Edge Detection in Multispectral Images • Simple approaches: – Compute gradient by taking Euclidean distance between multispectral vectors of data at adjacent pixels instead of differences in gray levels – Find independent gradients for different bands, edges and combine edges – Find independent gradients, combine gradients, and find edge from multiband gradient GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 89 Edge Detection in Multispectral Images GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 90 Edge Detection in Multispectral Images GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 92 Image Sharpening For example, • Sharpened image = Original image + k. gradient magnitude • Scale factor k can determine whether gradient magnitude is added as it is or a fraction of it. The sum may be rescaled to 0-255 to display like an image GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 93 Original image (left), Sharpened Image (right) GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 94 Unsharp Masking • Sample convolution mask 0 0 0 0 0 0 1 1 1 0 1 0 + 0 1 0 - (1/9) 1 1 1 0 0 0 0 0 0 1 1 1 G = F + | (F – Fmean) | NR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 95 NR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 96 Line Enhancement Difference between a line and an edge Line is a physical entity Edge is a perceptual entity GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 97 Lines NR607 Lecture 28 B. Krishna Mohan
IIT Bombay Slide 98 Line Enhancement Detection of a physical line involves High to low transition Low to high transition OR Low to high transition High to low transition GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 99 Line Enhancement Masks • These masks look for positive to negative and negative to positive transitions in vertical/horizontal/diagonal directions GNR607 Lecture 13-16 B. Krishna Mohan
IIT Bombay Slide 100 Summary of Gradient Operators • Edges or boundaries convey very important information for image understanding • Gradient operators emphasize the local intensity or other property differences thereby making visible object boundaries • Gradient operations in normal course are only the first step in reliable edge extraction GNR607 Lecture 13-16 B. Krishna Mohan
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