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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. Properties of Convolution. 1. Commutative: f * g = g * f 2. Associative

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image features i

Image Features - I

Hao Jiang

Computer Science Department

Sept. 22, 2009

outline
Outline
  • Summary of convolution and linear systems
  • Image features
    • Edges
    • Corners
  • Programming Corner Detection
properties of convolution
Properties of Convolution

1. Commutative:

f * g = g * f

2. Associative

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

3. Superposition

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

N

M

full

(N+M-1)x(N+M-1)

linear system
Linear System

f

h

g = f * h

Linear:

a f1+ b f2 => a g1 + b g2

where the response of f1 is g1

and the response of f2 is g2

Shift invariant:

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

composite linear system
Composite Linear System

f

h1

h2

h1 + h2

f

h1

h2

h1*h2

nonlinear filtering
Nonlinear Filtering
  • Neighborhood filtering can be nonlinear
  • Median Filtering
  • 1 1
  • 1 2 1
  • 1 1 1

Mask [1 1 1 ]

  • 1 1
  • 1 1 1
  • 1 1 1
median filtering in denoising
Median Filtering in Denoising

Original Image

Add 10% pepper noise

median filtering for denoising
Median Filtering for Denoising

Median filter with 3x3

square structure element

median filtering for denoising1
Median Filtering for Denoising

Median filter with 5x5

square structure element

compared with gaussian filtering
Compared with Gaussian Filtering

Kernel size 5x5 and sigma 3

Kernel size 11x11 and sigma 5

image local structures
Image Local Structures

Step

Ridge

Valley

Peak

Corner

Junction

image local structures1
Image Local Structures

Line

Structures:

“Edge”

Step

Ridge

Valley

Point

Structures:

“Corners”

Peak

Corner

Junction

an example
An Example

edge

Region

corners

edge detection in matlab
Edge Detection in Matlab

>> im = imread(\'flower.jpg\');

>> im = im2double(im);

>> im = rgb2gray(im);

>> ed = edge(im, \'canny\', 0.15);

slide18

f(x)

f’(x)

f’’(x)

extend to 2d
Extend to 2D

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)

a

finite difference
Finite Difference
  • Approximating derivatives using finite difference.
  • Finite difference and convolution
noise reduction
Noise Reduction

0.01 noise

0.03 noise

gaussian filtering in edge detection1
Gaussian Filtering in Edge Detection

image

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

Difference of

Gaussian Kernel

Difference Kernel

Gaussian Kernel

edge detection in images
Edge Detection in Images
  • Gaussian smoothed filtering in x and y directions: Ix, Iy
  • Non-maximum suppression for |Ix|+|Iy|
  • Edge Tracing – double thresholding.
edge detection using matlab
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)

slide27

Berkeley Segmentation DataSet [BSDS]

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

corner detection
Corner Detection
  • Corner is a point feature that has large changing rate in all directions.

Peak

Step

Line

Flat region

find a corner
Find a Corner

Compute matrix H =

Ix2 Ixy

Ixy Iy2

=

in each window. If the ratio

(Ix2 * Iy2 – Ixy ^2 )

------------------------ > T

(Ix2 + Iy2 + eps)

We have a corner

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