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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

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Properties of Convolution

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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

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hhf g = f * h

a f1+ b f2 => a g1 + b g2where the response of f1 is g1and the response of f2 is g2

Linear:

Shift invariant: if f => g, then f(n-m) => g(n-m)

Composite Linear System

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h1h1f

h1h1f

h2h2

h2h2

h1 + h2

h1*h2

Nonlinear Filtering

Neighborhood filtering can be nonlinear

Median Filtering

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1 1 11 2 11 1 1

Mask [1 1 1 ]

1 1 11 1 11 1 1

Median Filtering in Denoising

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Add 10% pepper noiseOriginal Image

Median Filtering for Denoising

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Median filter with 3x3 square structure element

Median Filtering for Denoising

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Median filter with 5x5 square structure element

Compared with Gaussian Filtering

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Kernel size 5x5 and sigma 3 Kernel size 11x11 and sigma 5

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Image Features

Image Local Structures

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Step Ridge

Peak

Valley

Corner Junction

Image Local Structures

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Step Ridge

Peak

Valley

Corner Junction

Line Structures:“Edge”

Point Structures:“Corners”

Regions

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An Example

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edgeRegion

corners

Edge Detection in Matlab

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>> im = imread('flower.jpg');>> im = im2double(im);>> im = rgb2gray(im);>> ed = edge(im, 'canny', 0.15);

How to Find an Edge?

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A 1D edge

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f(x)

f’(x)

f’’(x)

Extend to 2D

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ab

There is a direction in whichimage f(x,y) increases thefastest. The direction is calledthe 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

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Noise Reduction

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0.01 noise

0.03 noise

Noise Reduction

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Gaussian Filtering in Edge Detection

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Gaussian Filtering in Edge Detection

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h * (g * f) = (h * g) * f

Difference Kernel Gaussian Kernel

image

Difference ofGaussian 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.

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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)

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27D. 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.

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Step LinePeak

Flat region

Find a Corner

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Compute matrix H =

in each window. If the ratio(Ix2 * Iy2 – Ixy ^2 )------------------------ > T (Ix2 + Iy2 + eps)We have a corner

Ix2 IxyIxy Iy2=

Corner Detection Programming

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