1 Image Features - I Hao Jiang Computer Science Department Sept. 22, 2009

Outline  Summary of convolution and linear systems  Image features  Edges  Corners  Programming Corner Detection 2

Properties of Convolution 3 1. Commutative: f * g = g * f 2. Associative (f * g) * h = f *(g * h) 3. Superposition (f + g) * h = f * h + g * h full (N+M-1)x(N+M-1) N M

Linear System 4 h h fg = f * h a f1+ b f2 => a g1 + b g2 where the response of f1 is g1 and the response of f2 is g2 Linear: Shift invariant: if f => g, then f(n-m) => g(n-m)

Composite Linear System 5 h1 f f h2 h1 + h2 h1*h2

Nonlinear Filtering  Neighborhood filtering can be nonlinear  Median Filtering Mask [1 1 1 ] 11 1

Median Filtering in Denoising 7 Add 10% pepper noiseOriginal Image

Median Filtering for Denoising 8 Median filter with 3x3 square structure element

Median Filtering for Denoising 9 Median filter with 5x5 square structure element

Compared with Gaussian Filtering 10 Kernel size 5x5 and sigma 3Kernel size 11x11 and sigma 5

11 Image Features

Image Local Structures 12 Step Ridge Peak Valley CornerJunction

Image Local Structures 13 Step Ridge Peak Valley CornerJunction Line Structures: “Edge” Point Structures: “Corners”

Regions 14

An Example 15 edge Region corners

Edge Detection in Matlab 16 >> im = imread('flower.jpg'); >> im = im2double(im); >> im = rgb2gray(im); >> ed = edge(im, 'canny', 0.15);

How to Find an Edge? 17 A 1D edge

18 f(x) f’(x) f’’(x)

Extend to 2D 19 a b There is a direction in which image f(x,y) increases the fastest. The direction is called the gradient direction. Gradient [df/dx df/dy] Magnitude: sqrt(fx^2 + fy^2) Direction: atan2(fy, fx)

Finite Difference  Approximating derivatives using finite difference.  Finite difference and convolution 20

Noise Reduction noise 0.03 noise

Noise Reduction 22

Gaussian Filtering in Edge Detection 23

Gaussian Filtering in Edge Detection 24 h * (g * f) = (h * g) * f Difference KernelGaussian Kernel image Difference of Gaussian Kernel

Edge Detection in Images  Gaussian smoothed filtering in x and y directions: Ix, Iy  Non-maximum suppression for |Ix|+|Iy|  Edge Tracing – double thresholding. 25

Edge Detection Using Matlab  Canny edge detector: edge(image, ‘canny’, threshold)  Sobel edge detector: edge(image, ‘sobel’, threshold)  Prewitt edge detector: edge(image, ‘prewitt’, threshold) 26

27 D. Martin, C. Fowlkes, D. Tal, J. Malik. "A Database of Human Segmented Natural Images and its Application to Evaluating Segmentation Algorithms and Measuring Ecological Statistics”, ICCV, 2001 Berkeley Segmentation DataSet [BSDS]

Corner Detection  Corner is a point feature that has large changing rate in all directions. 28 StepLine Peak Flat region

Find a Corner 29 Compute matrix H = in each window. If the ratio (Ix2 * Iy2 – Ixy ^2 ) > T (Ix2 + Iy2 + eps) We have a corner Ix2 Ixy Ixy Iy2 =

Corner Detection Programming 30