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Computational Vision

CSCI 363, Fall 2012Lecture 6

Edge Detection

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Stages for Edge Detection

Detecting Edges:

• Smoothing--Eliminates noise. Determines spatial scale.

• Differentiation--Localizes the intensity change

Feature Extraction:

• Determine the feature that caused the intensity change.

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Smoothing

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Localization of an edge•Changes in intensity are not instantaneous, particularly in a smoothed image.•Humans can localize edges to within a few seconds of arc (a few mm for a line a distance of 1 meter from the observer).•Accurate localization is necessary for stereo vision.•Differentiation allows us to find the location of the most rapidintensity change.•The first derivative gives a peak at the location of the most rapid change.•The second derivative gives a zero at this location.• Marr and Hildreth suggested using these zero crossings to indicate edges.

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Intensity Derivative

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Zero Crossings and Edges

ImageImage after smoothing and second derivative

Black = Negative

White = Positive

Zero Crossings

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Convolution

•To smooth a function, we replace the value at each x position with the averaged values around that position.

•This process is called convolution.

•Smoothing works best if the average is weighted: nearby neighbors count more, distant neighbors count less.

•A Gaussian function works well for this weighting.

€

G(x) =1

σ

⎛ ⎝ ⎜

⎞ ⎠ ⎟e

−x2

2σ 2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Convolution of I with G is written: G(x) * I(x)

€

y(x)= I(τ )G(x − τ )dτ−∞

∞

∫For continuous functions:

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The Derivative of a Convolution

€

d

dxG(x) * I(x)( ) =

d

dxG(x)

⎛ ⎝ ⎜

⎞ ⎠ ⎟* I(x)

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d

dx

d

dxG(x) * I(x)

⎛ ⎝ ⎜

⎞ ⎠ ⎟=d 2

dx 2G(x)

⎛

⎝ ⎜

⎞

⎠ ⎟* I(x)

Gaussian and its derivativesInstead of taking the derivative of a convolution of the image and the Gaussian, we can convolve image with the derivative of the Gaussian

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2nd Derivative of Gaussian Looks Like...

Retinal Ganglion cells and bipolar cells have receptive fields that exhibit a center-surround structure.

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A 2D Gaussian

For a 2D image, I(x, y), we convolve with a 2D Gaussian:

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G(x,y) =1

2πσ

⎛ ⎝ ⎜

⎞ ⎠ ⎟e

−(x2 +y2 )

2σ 2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

To differentiate, we use the Laplacian operator:

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∇2 =∂ 2 I

∂x 2+∂ 2 I

∂y2

⎛

⎝ ⎜

⎞

⎠ ⎟

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∇2G(x,y) looks like a center surround receptive field.

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∇2G(x,y) can be approximated by G(1, x, y) - G(2, x, y)(A "Difference of Gaussians" or DOG)

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Difference of Gaussians

Note: A single retinal ganglion cell cannot distinguish between a zero crossing and uniform intensity.To find edges, one must evaluate the difference in responses of neighboring cells.

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Detecting Edges with Discrete Samples

Normally, images will be a group of discrete values:

14 7 25 9 276 5 18

15 9 2 7 25

2D Image1D Image

We use an approximation of convolution to perform the smoothing and differentiation.

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Smoothing a 1D array

Image: 1 3 6 4 5 2 3 6 6 3 1 1

Operator: 1 3 5 3 1

Position 1: 1x1 + 3x3 + 6x5 + 4x3 + 5x1 = 57

Image: 1 3 6 4 5 2 3 6 6 3 1 1

Operator: 1 3 5 3 1

Position 2: 3x1 + 6x3 + 4x5 + 5x3 + 2x1 = 58

Position: 1 2 3 4 5 6 7 8Value: 57 58 52 44 50 62 81 43

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Other Operators

Averaging: 1/5 1/5 1/5 1/5 1/5

Differentiation: 1 -1

Second Derivative: 1 -2 1

2D Convolution:

0 0 0 00 2 3 20 1 2 10 0 0 0

0 0 0 00 2 5 50 3 8 80 1 3 3

1 11 1

* =

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Matlab MatricesEntering a Matrix:>> A = [1 2 3; 4 5 6; 7 8 9]A =

1 2 34 5 67 8 9

User types

Matlab replies

>> Image = zeros(4, 5)Image =

0 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0

All zeros

>> B = ones(3, 4)B =

1 1 1 11 1 1 11 1 1 1

All ones

>> C = 5*ones(2, 3)C =

5 5 55 5 5

All fives

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Accessing Array Values

A = 1 23 4

A(1, 1) = ?A(2, 1) = ?etc.

Use a colon to specify a range of values. 4:7 indicates 4 5 6 and 7.A colon by itself indicates an entire row or column.

B = 1 2 34 5 67 8 9

B(2:3, 1:2) = ? B(1:2, :) = ?

>> A = zeros(3, 3)>> B = [5 6; 7 8]>> A(1:2, 2:3) = B

A = ?

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4 57 8

1 2 34 5 6

0 5 60 7 80 0 0

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Control FlowConditional Example:if (A(1, 1) < 5)

A(1, 1) = A(1, 1) + 1end

For loop example:for i = 1:5

A(i, 1) = 2*i;end

Stepping by 2:for i = 2:2:10

A(i, 2) = 3;end

Relational operators: < > == & | ~=

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Matrix OperationsAddition and Subtraction are pairwise by element:

1 23 4

5 67 8

6 810 12+ = ?

Matrix Multiplication (*)

1 2 3456

* = ? 32

Pairwise multiplication (.*)

1 23 4

5 67 8

5 1221 32.* = ?

Matrix Division (/)(Multiply first matrix by inverse of second)

Pairwise Division ( ./)

1 23 4

5 55 5

.2 .4

.6 .8./ = ?